Hermitian K-theory for stable ∞ -categories III: Grothendieck-Witt groups of rings
Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle
aa r X i v : . [ m a t h . K T ] S e p HERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III:GROTHENDIECK-WITT GROUPS OF RINGS BAPTISTE CALMÈS, EMANUELE DOTTO, YONATAN HARPAZ, FABIAN HEBESTREIT, MARKUS LAND,KRISTIAN MOI, DENIS NARDIN, THOMAS NIKOLAUS, AND WOLFGANG STEIMLE
To Andrew Ranicki. A BSTRACT . We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring 𝑅 to thehomotopy C -orbits of its K-theory and Ranicki’s original (non-periodic) symmetric L-theory. We use thisfibre sequence to remove the assumption that is a unit in 𝑅 from various results about Grothendieck-Wittgroups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a numberfield, calculate the various flavours of Grothendieck-Witt groups of ℤ , show that the Grothendieck-Witt groupsof rings of integers in number fields are finitely generated, and that the comparison map from quadratic tosymmetric Grothendieck-Witt theory of Noetherian rings of global dimension 𝑑 is an equivalence in degrees ≥ 𝑑 +3 . As an important tool, we establish the hermitian analogue of Quillen’s localisation-dévissage sequencefor Dedekind rings and use it to solve a conjecture of Berrick-Karoubi. C ONTENTS
Introduction Recollection 𝑚 -quadratic structures 131.3 Surgery for 𝑟 -symmetric structures 24 References
NTRODUCTION
This paper investigates the hermitian K-theory spectra of non-degenerate symmetric and quadratic formsover a ring 𝑅 and their homotopy groups: the higher Grothendieck-Witt groups of 𝑅 . Many structuraland computational features of the higher Grothendieck-Witt groups of rings 𝑅 in which 2 is a unit arewell understood, prevalently due to extensive work of Karoubi [Kar71d, Kar71b, Kar71c, Kar71a, Kar80]and Schlichting [Sch10a, Sch10b, Sch17, Sch19a]. Previously, in Paper [II] we have used the categoricalframework of Poincaré ∞ -categories to establish some fundamental properties of the higher Grothendieck-Witt groups of rings in which is not necessarily a unit, most notably a form of Karoubi periodicity [II].4.3.4and the existence of a fibre sequence relating the Grothendieck-Witt theory of any Poincaré ∞ -category withits algebraic K -theory and L-theory. This allows for the separation of K -theoretic and L -theoretic arguments,and the theme of this paper is to deduce results about Grothendieck-Witt theory from their counterparts inL-theory. Date : September 16, 2020.
Main results.
Let D be a duality on the category Proj( 𝑅 ) of finitely generated projective 𝑅 -modules, thatis, an equivalence of categories D ∶ Proj( 𝑅 ) op → Proj( 𝑅 ) . Then D is necessarily of the form D 𝑃 =hom 𝑅 ( 𝑃 , 𝑀 ) , where 𝑀 ∶= D( 𝑅 ) is an invertible module with involution (see Definition R.1). An 𝑀 -valued unimodular symmetric form on 𝑃 is then a self-dual isomorphism 𝜑 ∶ 𝑃 → D 𝑃 . Together withtheir isomorphisms these form a groupoid Unimod( 𝑅 ; 𝑀 ) , symmetric monoidal under orthogonal directsum. The classical symmetric Grothendieck-Witt theory of 𝑅 is its group-completion: GW scl ( 𝑅 ; 𝑀 ) = (Unimod( 𝑅 ; 𝑀 ) , ⊕ ) 𝑔𝑝 . By construction, GW scl ( 𝑅 ; 𝑀 ) is a grouplike E ∞ -space, which we view equivalently as a connective spec-trum. Its homotopy groups are the higher symmetric Grothendieck-Witt groups GW scl , ∗ ( 𝑅 ; 𝑀 ) of 𝑅 .After inverting , the study of the higher Grothendieck-Witt groups of 𝑅 reduces to the study of theK-groups and L-groups, as by work of Karoubi there is a natural splitting GW scl , ∗ ( 𝑅 ; 𝑀 )[ ] ≅ (K ∗ ( 𝑅 ; 𝑀 )[ ]) C ⊕ (W ∗ ( 𝑅 ; 𝑀 )[ ]) , see also [BF85]. Here K( 𝑅 ; 𝑀 ) is the K-theory spectrum of 𝑅 with C -action induced by sending 𝑃 to itsdual D 𝑃 , and the first summand is the subgroup of invariants of its homotopy groups with inverted. Thesecond summand consists of the Witt groups of symmetric forms and formations, which are -periodic bydefinition. The first main result of the present paper combines the general fibre sequence of Paper [II] withRanicki’s algebraic surgery to obtain an integral version of this result. Theorem 1.
For every ring 𝑅 and duality D = hom 𝑅 (− , 𝑀 ) on Proj( 𝑅 ) , there is a fibre sequence of spectra K( 𝑅 ; 𝑀 ) hC hyp ⟶ GW scl ( 𝑅 ; 𝑀 ) ⟶ L short ( 𝑅 ; 𝑀 ) where L short ( 𝑅 ; 𝑀 ) is a canonical connective spectrum whose homotopy groups are Ranicki’s original (non4-periodic) symmetric L-groups from [Ran80] . After inverting 2, this fibre sequence recovers Karoubi’s splitting of GW scl ( 𝑅 ; 𝑀 ) , but it also allows toefficiently treat the behaviour of Grothendieck-Witt theory at the prime , as we will explain below. Withoutinverting 2 on the outside, but when is a unit in 𝑅 , Ranicki’s L-groups L short∗ ( 𝑅 ; 𝑀 ) are still -periodic andisomorphic to the Witt groups W ∗ ( 𝑅 ; 𝑀 ) , and in this case the sequence of Theorem 1 is due to Schlichting[Sch17, §7]. However, if 2 is not invertible in 𝑅 , there are several variants of L-spectra in addition to L short ( 𝑅 ; 𝑀 ) , most notably the 4-periodic symmetric L-theory L s ( 𝑅 ; 𝑀 ) used by Ranicki in later work[Ran92]. Our insight is that it is the non-periodic classical symmetric L-theory of Ranicki [Ran80] whichmakes Theorem 1 true for all rings.Coming back to the 2-local behaviour of Grothendieck-Witt theory, we note that sending a symmetric bi-linear form to its underlying finitely generated projective module leads to a canonical map GW scl ( 𝑅 ; 𝑀 ) → K( 𝑅 ; 𝑀 ) hC . The question whether this map is a 2-adic equivalence in positive degrees is known as Thoma-son’s homotopy limit problem [Tho83], which admits a positive solution for many rings in which 2 is in-vertible, notably by work of Hu, Kriz, and Ormsby [HKO11], Bachmann and Hopkins [BH20] and Berrick,Karoubi, Schlichting and Østvær [BKSØ15]. In §3.1 we will show: Theorem 2.
Let 𝑅 be a Dedekind ring whose fraction field is a number field. Then the canonical map GW scl ( 𝑅 ; 𝑀 ) → K( 𝑅 ; 𝑀 ) hC is a 2-adic equivalence in non-negative degrees. To the best of our knowledge this is the first general result on the homotopy limit problem for a classof rings which are not fields and in which 2 is not assumed to be a unit. The strategy we adopt to proveTheorem 2 is to use Theorem 1 to reduce it to the case of 𝑅 [ ] , where it holds by [BKSØ15]. For a generalring 𝑅 , we further observe that the failure of 4-periodicity of L short ( 𝑅 ; 𝑀 ) in high degrees provides a purelyL-theoretic obstruction for the homotopy limit problem map GW scl ( 𝑅 ; 𝑀 ) → K( 𝑅 ; 𝑀 ) hC to be a 2-adicequivalence in positive degrees; see Proposition 3.1.10.Theorem 1 does not only provide a conceptual description of the Grothendieck-Witt spectrum, but itcan also be used for explicit calculations. When 𝑅 = ℤ there are two dualities on Proj( ℤ ) , leading to thesymmetric and symplectic Grothendieck-Witt groups of ℤ , respectively. In §3.2 we explicitly calculate thesegroups in a range of degrees < , and beyond that conditionally on the Kummer-Vandiver conjecture. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 3 Similar calculations have also been announced by Schlichting in [Sch19b], see the remark at the end of theintroduction.
Theorem 3.
The symmetric and symplectic Grothendieck-Witt groups of ℤ are given in the table of Theo-rem 3.2.1, and their quadratic versions in Theorems 3.2.9 and 3.2.13. Proof strategy and further results.
We approach GW scl by investigating Grothendieck-Witt theory in thegeneral context of Poincaré ∞ -categories, as defined by Lurie [Lur11] and further developed in Paper [I],Paper [II]. We briefly recall that a Poincaré ∞ -category consists of a small stable ∞ -category C equippedwith a Poincaré structure, that is a functor Ϙ ∶ C op → Sp which is quadratic and satisfies a non-degeneracycondition, which allows to extract an induced duality D ∶ C op → C . We refer to Paper [I] for a generalintroduction to Poincaré ∞ -categories, and to Paper [II] for the construction of their Grothendieck-Witt andL-spectra and their universal properties. In [II].4.4.14, we showed that for any Poincaré ∞ -category ( C , Ϙ ) there is a natural fibre sequence(1) K( C , Ϙ ) hC ⟶ GW( C , Ϙ ) ⟶ L( C , Ϙ ) . where K( C , Ϙ ) is the K-theory spectrum of C with the C -action induced by D . To connect this general fibresequence to Theorem 1, we will be concerned with studying appropriate Poincaré structures on the derived ∞ -category of perfect complexes D p ( 𝑅 ) . Some immediate examples of Poincaré structures on D p ( 𝑅 ) arethe quadratic and symmetric Poincaré structures given at a perfect complex 𝑋 by the formulae(2) Ϙ q 𝑀 ( 𝑋 ) = hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) hC and Ϙ s 𝑀 ( 𝑋 ) = hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) hC , where 𝑀 is an invertible module with involution over 𝑅 (see Definition R.1), and the C -action is givenby conjugating the flip action on 𝑋 ⊗ 𝑋 and the C -action on 𝑀 . These two Poincaré structures are thehomotopy theoretic analogues of quadratic and symmetric forms in algebra, which on a finitely generatedprojective 𝑅 -module 𝑃 are respectively the groups of coinvariants and invariants(3) Hom
𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑀 ) C and Hom
𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑀 ) C for the same C -action as above. One insight in our series of papers is that the abstract framework of Paper[I], Paper [II] allows us to work with Poincaré structures on D p ( 𝑅 ) which are more intimately related toalgebra than the naive homotopy theoretic constructions of (2). These are the non-abelian derived functorsof the algebraic constructions of (3), which we call the genuine quadratic and genuine symmetric Poincaréstructures and that we denote respectively by Ϙ gq 𝑀 and Ϙ gs 𝑀 . There are canonical comparison maps Ϙ q 𝑀 ⟶ Ϙ gq 𝑀 ⟶ Ϙ gs 𝑀 ⟶ Ϙ s 𝑀 relating these Poincaré structures. When 2 is a unit in 𝑅 , all of them are equivalences, and we showed in§[II].B that the corresponding Grothendieck-Witt spectra coincide with previous constructions of Grothendieck-Witt spectra due to Schlichting and Spitzweck [Sch17, Spi16]. In general, when 2 is not necessarily aunit, the fourth and ninth authors [HS20] related the genuine Grothendieck-Witt spectra GW gs ( 𝑅 ; 𝑀 ) ∶=GW( D p ( 𝑅 ); Ϙ gs 𝑀 ) and GW gq ( 𝑅 ; 𝑀 ) ∶= GW( D p ( 𝑅 ); Ϙ gq 𝑀 ) to the classical ones, by providing natural equiv-alences GW scl ( 𝑅 ; 𝑀 ) ≃ ⟶ 𝜏 ≥ GW gs ( 𝑅 ; 𝑀 ) and GW qcl ( 𝑅 ; 𝑀 ) ≃ ⟶ 𝜏 ≥ GW gq ( 𝑅 ; 𝑀 ) , where GW qcl ( 𝑅 ; 𝑀 ) denotes, similary to GW scl ( 𝑅 ; 𝑀 ) , the group completion of the category of unimodularquadratic forms, and 𝜏 ≥ denotes the connective cover. Writing similarly L gs ( 𝑅 ; 𝑀 ) for L( D p ( 𝑅 ); Ϙ gs 𝑀 ) ,we therefore obtain a fibre sequence K( 𝑅 ; 𝑀 ) hC → GW gs ( 𝑅 ; 𝑀 ) → L gs ( 𝑅 ; 𝑀 ) , and Theorem 1 is thenimplied by the following result, see Theorem 1.2.18. Theorem 4.
For any ring 𝑅 and non-negative integer 𝑛 , the genuine symmetric L-groups L gs 𝑛 ( 𝑅 ; 𝑀 ) arecanonically isomorphic to Ranicki’s original symmetric L-groups from [Ran80] . Thus, in the notation ofTheorem 1, we have L short ( 𝑅 ; 𝑀 ) = 𝜏 ≥ L gs ( 𝑅 ; 𝑀 ) . We recall that the original symmetric L-groups of Ranicki are defined so that elements of the 𝑛 ’th L-groupare represented by Poincaré chain complexes of length at most 𝑛 , for 𝑛 ≥ . Ranicki then defines negative CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE symmetric L-groups in an ad hoc manner, and we show that these negative L-groups are also canonicallyisomorphic to the corresponding negative genuine symmetric L-groups: Concretely they are given by L gs 𝑛 ( 𝑅 ; 𝑀 ) = { L ev 𝑛 +2 ( 𝑅 ; − 𝑀 ) if 𝑛 = −2 , −1L q 𝑛 ( 𝑅 ; 𝑀 ) if 𝑛 ≤ −3 , where L ev∗ and L q∗ are respectively the even and quadratic L-groups of [Ran80]. In particular, Theorem 4 andthe described addendum show that the classical symmetric L-groups can be realised as the homotopy groupsof the non-connective spectrum L gs ( 𝑅 ; 𝑀 ) . The general form of Karoubi periodicity of Paper [II], whichwe review in Theorem R.8 below, relates the Poincaré structures Ϙ gs 𝑀 and Ϙ gq 𝑀 and their GW and L-spectra,in particular showing that Σ L gs ( 𝑅 ; 𝑀 ) ≃ L gq ( 𝑅 ; 𝑀 ) ; see Corollary R.10. From the fibre sequence forgeneral Poincaré ∞ -categories we therefore also obtain a quadratic version of Theorem 1, given by the fibresequence K( 𝑅 ; 𝑀 ) hC hyp ⟶ GW qcl ( 𝑅 ; 𝑀 ) ⟶ 𝜏 ≥ (Σ L gs ( 𝑅 ; 𝑀 )) . We prove Theorem 4 in § 1.2 using Ranicki’s procedure of algebraic surgery, which allows us to com-pare the L-groups of various Poincaré structures in a range of degrees. We discuss this technique also forconnective ring spectra in Corollary 1.2.24, and in § 1.3, to obtain the following comparison result. We willwrite GW s ( 𝑅 ; 𝑀 ) ∶= GW s ( D p ( 𝑅 ); Ϙ s 𝑀 ) for the homotopy symmetric Grothendieck-Witt theory, and writelikewise L s ( 𝑅 ; 𝑀 ) ∶= L( D p ( 𝑅 ); Ϙ s 𝑀 ) for periodic symmetric L-theory. Theorem 5.
Suppose 𝑅 is a commutative Noetherian ring of finite global dimension 𝑑 . Then:i) the map L gs 𝑛 ( 𝑅 ; 𝑀 ) → L s 𝑛 ( 𝑅 ; 𝑀 ) is injective for 𝑛 ≥ 𝑑 − 2 and an isomorphism for 𝑛 ≥ 𝑑 − 1 ,ii) the map L gq 𝑛 ( 𝑅 ; 𝑀 ) → L gs 𝑛 ( 𝑅 ; 𝑀 ) is injective for 𝑛 ≥ 𝑑 + 2 and an isomorphism for 𝑛 ≥ 𝑑 + 3 . Part i) of Theorem 5, together with Theorem 4, improve a similar comparison result of Ranicki [Ran80,Proposition 4.5], where he proves injectivity for non-negative 𝑛 ≥ 𝑑 −3 and bijectivity for non-negative 𝑛 ≥ 𝑑 − 2 . Combining Theorem 1 and Theorem 5, we obtain the following surprising result, see Remarks 1.3.8and 1.3.13: Corollary 6.
Suppose 𝑅 is a commutative Noetherian ring of finite global dimension 𝑑 . Then the map GW qcl ,𝑛 ( 𝑅 ; 𝑀 ) → GW scl ,𝑛 ( 𝑅 ; 𝑀 ) is injective for 𝑛 ≥ 𝑑 + 2 and an isomorphism for 𝑛 ≥ 𝑑 + 3 . Moreover,for 𝑀 = 𝑅 and if 𝑅 is 2-torsion free, the map is injective for 𝑛 ≥ 𝑑 and an isomorphism for 𝑛 ≥ 𝑑 + 1 . Furthermore, part i) of Theorem 5 implies that the map GW scl ( 𝑅 ; 𝑀 ) → 𝜏 ≥ GW s ( 𝑅 ; 𝑀 ) is an equiva-lence if 𝑅 is a Dedekind domain. Thus in order to study the classical Grothendieck-Witt groups of Dedekindrings, it suffices to study the homotopy symmetric Grothendieck-Witt theory GW s . This is an interestinginvariant in its own right which enjoys pleasant properties not shared with the genuine variant GW gs . Mostnotably, we prove in Theorem 2.1.8 that 4-periodic symmetric L-theory L s , and hence also GW s , satisfiesa dévissage theorem. In particular, we obtain the hermitian analogue of Quillen’s famous localization-dévissage fibre sequence [Qui73], see Corollary 2.1.9: Theorem 7.
Let 𝑅 be a Dedekind ring, 𝑇 ⊂ 𝑅 a multiplicative subset, and 𝔽 𝔭 the residue field 𝑅 ∕ 𝔭 at amaximal ideal 𝔭 ⊆ 𝑅 . Then restriction, localisation, and a choice of uniformiser for every 𝔭 induce a fibresequence of spectra ⨁ 𝔭 ∩ 𝑇 ≠ ∅ GW s ( 𝔽 𝔭 ; ( 𝑀 ∕ 𝔭 )[−1]) ⟶ GW s ( 𝑅 ; 𝑀 ) ⟶ GW s ( 𝑅 [ 𝑇 −1 ]; 𝑅 [ 𝑇 −1 ] ⊗ 𝑅 𝑀 ) , where 𝑅 [ 𝑇 −1 ] is obtained from 𝑅 by inverting the elements of 𝑇 . We in fact construct a more general fibre sequence for localisations of 𝑅 away from a set of non-emptyprime ideals of 𝑅 , see Corollary 2.1.9. This result establishes a conjecture of Berrick and Karoubi whichasserts that the map ℤ → ℤ [ ] induces an equivalence on the positive, 2-localised Grothendieck-Wittgroups [BK05]. In fact, this result holds for general rings of integers in number fields as we observe inRemark 3.2.14. In § 3 we then combine Theorem 7 with work of Berrick, Karoubi, Schlichting, and Østvær,[BKSØ15] to deduce Theorem 2, as well as the calculations for the integers of Theorem 3.Finally, we also use Theorem 1, together with a calculation of the symmetric and quadratic L-groupsof Dedekind rings to deduce the following finiteness result; see Corollary 2.2.17. When 𝑀 = 𝑅 with the ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 5 involution given by multiplication by 𝜖 = ±1 , we write GW scl ,𝑛 ( 𝑅 ; 𝜖 ) for GW scl ,𝑛 ( 𝑅 ; 𝑀 ) , and similarly for GW qcl ,𝑛 ( 𝑅 ; 𝜖 ) . Corollary 8.
Let O be a number ring, that is, a localisation of the ring of integers in a number field awayfrom finitely many primes, and 𝜖 = ±1 . Then its classical 𝜖 -symmetric and 𝜖 -quadratic Grothendieck-Wittgroups GW scl ,𝑛 ( O ; 𝜖 ) and GW qcl ,𝑛 ( O ; 𝜖 ) are finitely generated. In the quadratic case, one can prove this result also through homological stability, but in the general-ity presented here the argument is not known to carry over to the symmetric case, as we explain in Re-mark 2.2.18.
Remark.
Some of the results presented above have also been announced in [Sch19b]: The calculations of theGrothendieck-Witt groups of the integers of Theorem 3 in the symmetric, symplectic and quadratic cases(although they deviate from ours away from the prime , see Remark 3.2.5), the localisation-dévissage se-quence of Theorem 7 in non-negative degrees, and Corollary 6 for the ring 𝑅 = ℤ with the trivial involution. Notation and Conventions.
All tensor products appearing without further explanation are derived tensorproducts over ℤ , and will be denoted by ⊗ rather than ⊗ L ℤ . We always denote by D = hom 𝑅 (− , 𝑀 ) thedualities on Proj( 𝑅 ) and D p ( 𝑅 ) determined by an invertible module with involution 𝑀 . Acknowledgements.
For useful discussions about our project, we heartily thank Tobias Barthel, LukasBrantner, Mauricio Bustamante, Denis-Charles Cisinski, Dustin Clausen, Uriya First, Søren Galatius, RuneHaugseng, André Henriques, Lars Hesselholt, Gijs Heuts, Geoffroy Horel, Marc Hoyois, Max Karoubi,Daniel Kasprowski, Ben Knudsen, Manuel Krannich, Achim Krause, Henning Krause, Sander Kupers,Wolfgang Lück, Ib Madsen, Cary Malkiewich, Mike Mandell, Akhil Matthew, Lennart Meier, Irakli Patchko-ria, Nathan Perlmutter, Andrew Ranicki, Oscar Randal-Williams, George Raptis, Marco Schlichting, PeterScholze, Stefan Schwede, Graeme Segal, Markus Spitzweck, Jan Steinebrunner, Georg Tamme, UlrikeTillmann, Maria Yakerson, Michael Weiss, and Christoph Winges.Besides these discussions, we owe a tremendous intellectual debt to Jacob Lurie.The authors would also like to thank the Hausdorff Center for Mathematics at the University of Bonn,the Newton Institute at the University of Cambridge, the University of Copenhagen and the MathematicalResearch Institute Oberwolfach for hospitality and support while parts of this project were undertaken.BC was supported by the French National Centre for Scientific Research (CNRS) through a “délégation”at LAGA, University Paris 13. ED was supported by the German Research Foundation (DFG) through thepriority program “Homotopy theory and Algebraic Geometry” (DFG grant no. SPP 1786) at the Universityof Bonn and WS by the priority program “Geometry at Infinity” (DFG grant no. SPP 2026) at the Universityof Augsburg. YH and DN were supported by the French National Research Agency (ANR) through thegrant “Chromatic Homotopy and K-theory” (ANR grant no. 16-CE40-0003) at LAGA, University of Paris13. FH is a member of the Hausdorff Center for Mathematics at the University of Bonn (DFG grant no.EXC 2047 390685813) and TN of the cluster “Mathematics Münster: Dynamics-Geometry-Structure” atthe University of Münster (DFG grant no. EXC 2044 390685587). FH, TN and WS were further supportedby the Engineering and Physical Sciences Research Council (EPSRC) through the program “Homotopyharnessing higher structures” at the Isaac Newton Institute for Mathematical Sciences (EPSRC grants no.EP/K032208/1 and EP/R014604/1). FH was also supported by the European Research Council (ERC)through the grant “Moduli spaces, Manifolds and Arithmetic” (ERC grant no. 682922) and KM by thegrant “ K -theory, L -invariants, manifolds, groups and their interactions” (ERC grant no. 662400). ML andDN were supported by the collaborative research centre “Higher Invariants” (DFG grant no. SFB 1085)at the University of Regensburg. ML was further supported by the research fellowship “New methods inalgebraic K -theory” (DFG grant no. 424239956) and by the Danish National Research Foundation (DNRF)through the Center for Symmetry and Deformation (DNRF grant no. 92) and the Copenhagen Centre forGeometry and Topology (DNRF grant GeoTop) at the University of Copenhagen. KM was also supportedby the K&A Wallenberg Foundation. CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE R ECOLLECTION
In this section we recall some of the material from Paper [I] and Paper [II] on Poincaré structures on theperfect derived ∞ -category of a ring and their Grothendieck-Witt and L spectra, as we rely on this frameworkin the rest of the paper. We will also review the general form of Karoubi’s periodicity Theorem [II].4.3.4,which does not require that is a unit in the base ring.In Paper [II], we view Grothendieck-Witt theory as an invariant of what we call a Poincaré ∞ - category .A Poincaré ∞ -category is a pair ( C , Ϙ ) consisting of a small stable ∞ -category C equipped with a Poincaréstructure Ϙ , that is a functor Ϙ ∶ C op → S 𝑝 which is reduced and -excisive in the sense of Goodwillie’s func-tor calculus, and whose symmetric cross-effect B ∶ C op × C op → S 𝑝 is of the form B( 𝑋, 𝑌 ) = hom C ( 𝑋, D 𝑌 ) for some equivalence of categories D ∶ C op → C . Poincaré ∞ -categories were introduced by Lurie as a novelframework for Ranicki’s L-theory (see [Lur11] and Paper [II]). A Poincaré structure provides a formal no-tion of “hermitian form” on the objects of C . Indeed, there is as a space of Poincaré objects Pn( C , Ϙ ) whichconsists of pairs ( 𝑋, 𝑞 ) where 𝑋 is an object of C and 𝑞 ∈ Ω ∞ Ϙ ( 𝑋 ) is such that a certain canonical map 𝑋 → D 𝑋 is an equivalence (see Definition [I].2.1.3). There are then canonical transformations Pn( C , Ϙ ) → Ω ∞ GW( C , Ϙ ) and Pn( C , Ϙ ) → Ω ∞ L( C , Ϙ ) which exhibit the Grothendieck-Witt and L-theory functors as the universal approximation of Pn by a Verdierlocalising, respectively a bordism invariant, functor (see Observation [II].4.1.2 and Theorem [II].4.4.12).These universal properties are similar to the universal property of the map from the groupoid core to K-theory Cr C → K( C ) of a small stable ∞ -category provided by [BGT13].In the present paper we will be concerned with the perfect derived ∞ -category D p ( 𝑅 ) of a ring 𝑅 ,which is the ∞ -categorical localisation of the category of bounded chain complexes of finitely generatedprojective (left) 𝑅 -modules at the quasi-isomorphisms, or equivalently the ∞ -category of compact objectsof the localisation D ( 𝑅 ) of all chain complexes at the quasi-isomorphisms. Given a Poincaré structure Ϙ ∶ D p ( 𝑅 ) op → S 𝑝 , we will denote the corresponding Grothendieck-Witt spectrum by GW( 𝑅 ; Ϙ ) ∶= GW( D p ( 𝑅 ) , Ϙ ) . We are going to consider a specific collection of Poincaré structures on D p ( 𝑅 ) associated to a module withinvolution, which we now introduce. For what follows, ⊗ denotes the underived tensor product of ringsover ℤ , but see below for a relation with the derived tensor product. Given an 𝑅 ⊗ 𝑅 -module 𝑀 , we let 𝑀 op denote the 𝑅 ⊗ 𝑅 -module defined by 𝑀 with the module action 𝑟 ⊗ 𝑠 ⋅ 𝑚 ∶= 𝑠 ⊗ 𝑟 ⋅ 𝑚 for all 𝑟, 𝑠 in 𝑅 and 𝑚 in 𝑀 .R.1. Definition. A module with involution over 𝑅 is an 𝑅 ⊗ 𝑅 -module 𝑀 together with an 𝑅 ⊗ 𝑅 -modulemap ∙ ∶ 𝑀 𝑜𝑝 → 𝑀 such that ̄̄𝑚 = 𝑚 . We say that 𝑀 is invertible if it is finitely generated projective foreither of its 𝑅 -module structures, and the map 𝑅 ⟶ hom 𝑅 ( 𝑀, 𝑀 ) which sends to ∙ is an isomorphism, where 𝑀 is regarded as an 𝑅 -module via the first 𝑅 -factor in thesource, and the second one in the target.We warn the reader that the modules with involution over 𝑅 of Definition R.1 correspond to the moduleswith involution over the Eilenberg-MacLane spectrum of 𝑅 in the sense of Definition [I].3.1.1 (definedusing the derived tensor product), which are moreover discrete. To see this, we first note that there is a mapof H ℤ -algebras 𝑅 ⊗ L ℤ 𝑅 → 𝑅 ⊗ 𝑅 so that any module over the latter is canonically one over the former.Furthermore, this map is the canonical 0-truncation map which any connective H ℤ -algebra has. If 𝑀 is adiscrete H ℤ -module, we deduce from the coconnectivity of hom ℤ ( 𝑀, 𝑀 ) that any 𝑅 ⊗ 𝑅 -module structureextends essentially uniquely to an
𝑅 ⊗ L ℤ 𝑅 -module structure.R.2. Example. i) When 𝑅 is commutative, any line bundle 𝐿 over 𝑅 gives rise to an invertible module with involutionover 𝑅 , with 𝑀 = 𝐿 and ∙ = id .ii) Let 𝜖 ∈ 𝑅 be a unit. We recall that an 𝜖 -involution on 𝑅 consists of a ring isomorphism ∙ ∶ 𝑅 → 𝑅 op such that ̄̄𝑟 = 𝜖𝑟𝜖 −1 and 𝜖 = 𝜖 −1 . In this case 𝑀 = 𝑅 equipped with the 𝑅 ⊗ 𝑅 -module structure 𝑟 ⊗ 𝑠 ⋅ 𝑥 = 𝑟𝑥𝑠 and the involution 𝜖 (∙) is an invertible module with involution over 𝑅 , that we denoteby 𝑅 ( 𝜖 ) . This is the structure commonly used by Ranicki as input for L-theory [Ran80]. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 7 iii) Given a module with involution 𝑀 over 𝑅 , we can define a new module with involution over 𝑅 denoted − 𝑀 , with the same underlying 𝑅 ⊗ 𝑅 -module 𝑀 but with involution −(∙) . In the case where 𝑀 = 𝑅 we have by definition that − 𝑅 = 𝑅 (−1) .For every pair of objects 𝑋 and 𝑌 of D p ( 𝑅 ) , we may form the mapping spectrum B( 𝑋, 𝑌 ) ∶= hom
𝑅⊗𝑅 ( 𝑋 ⊗ 𝑌 , 𝑀 ) in the stable ∞ -category D p ( 𝑅 ⊗ 𝑅 ) . Then 𝐵 is a symmetric bilinear functor, so the spectrum B( 𝑋, 𝑋 ) inherits a C -action by conjugating the flip action on 𝑋 ⊗ 𝑋 and the involution of 𝑀 ; see §[I].3.1.Given spectrum with C -action 𝑋 ∶ 𝐵 C → S 𝑝 , we denote by 𝑋 hC and 𝑋 hC its homotopy fixed pointsand homotopy orbits, respectively. Similarly, we let 𝑋 tC denote its Tate construction, defined as the cofibreof the norm map 𝑁 ∶ 𝑋 hC → 𝑋 hC as defined in [Lur17, §6.1.6], see also [NS18, I.1.11]. As in §[I].4.2we make the following definition.R.3. Definition.
Let 𝑀 be an invertible module with involution over 𝑅 . For every 𝑚 ∈ ℤ ∪ {±∞} , wedefine a functor Ϙ ≥ 𝑚𝑀 ∶ D p ( 𝑅 ) op → S 𝑝 as the pullback Ϙ ≥ 𝑚𝑀 ( 𝑋 ) hom 𝑅 ( 𝑋, 𝜏 ≥ 𝑚 𝑀 tC )hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) hC hom 𝑅 ( 𝑋, 𝑀 tC ) . Here, the right hand vertical map is induced by the 𝑚 -connective cover 𝜏 ≥ 𝑚 𝑀 tC → 𝑀 tC , and the bottomhorizontal map is induced by a canonical equivalence hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) tC ≃ hom 𝑅 ( 𝑋, 𝑀 tC ) , see Lemma [I].3.2.4. In the special cases where 𝑚 = ±∞ we will denote these functors by Ϙ q 𝑀 ∶= Ϙ ≥ ∞ 𝑀 = hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) hC and Ϙ s 𝑀 ∶= Ϙ ≥ −∞ 𝑀 = hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) hC . The functors Ϙ ≥ 𝑚𝑀 are indeed Poincaré structures by Examples [I].3.2.7. By construction, they all sharethe same underlying duality D 𝑋 = hom 𝑅 ( 𝑋, 𝑀 ) , where the mapping spectrum acquires a residual 𝑅 -module structure from the 𝑅⊗𝑅 -module structure of 𝑀 .The canonical connective cover maps 𝜏 ≥ 𝑚 +1 → 𝜏 ≥ 𝑚 define an infinite sequence of natural transformations Ϙ q 𝑀 = Ϙ ≥ ∞ 𝑀 → ⋯ → Ϙ ≥ ( 𝑚 +1) 𝑀 → Ϙ ≥ 𝑚𝑀 → Ϙ ≥ ( 𝑚 −1) 𝑀 → ⋯ → Ϙ ≥ −∞ 𝑀 = Ϙ s 𝑀 , and hence analogous sequences between the corresponding Grothendieck-Witt and L spectra.R.4. Remark.
Let ̂ H 𝑚 (C ; 𝑀 ) = 𝜋 − 𝑚 𝑀 tC denote the Tate cohomology of C with coefficients in the un-derlying ℤ [C ] -module of 𝑀 . When ̂ H − 𝑚 (C ; 𝑀 ) = 0 the map 𝜏 ≥ 𝑚 +1 𝑀 tC → 𝜏 ≥ 𝑚 𝑀 tC is an equivalence.Therefore, in this case, Ϙ ≥ 𝑚 +1 𝑀 → Ϙ ≥ 𝑚𝑀 is an equivalence, and it induces equivalences on the correspondingGrothendieck-Witt and L spectra GW( 𝑅 ; Ϙ ≥ ( 𝑚 +1) 𝑀 ) ∼ ⟶ GW( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) and L( 𝑅 ; Ϙ ≥ ( 𝑚 +1) 𝑀 ) ∼ ⟶ L( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) . Moreover, ̂ H ∗ (C ; 𝑀 ) is -periodic, so if this happens for 𝑚 it also does for all 𝑚 + 2 𝑘 . In particular, if 𝑅 is a unit all the natural transformations Ϙ ≥ 𝑚 +1 𝑀 → Ϙ ≥ 𝑚𝑀 are equivalences. If is not invertible however,the Grothendieck-Witt and L spectra for different 𝑚 are not generally equivalent, for instance this is the casefor 𝑅 = ℤ .R.5. Remark.
Among the Poincaré structures of Definition R.3, Ϙ ≥ 𝑀 , Ϙ ≥ 𝑀 and Ϙ ≥ 𝑀 are the ones which sendfinitely generated projective 𝑅 -modules 𝑃 (regarded as chain complexes concentrated in degree zero) toabelian groups (regarded as discrete spectra). The values of Ϙ ≥ 𝑀 and Ϙ ≥ 𝑀 are the abelian groups of strictcoinvariants and invariants, respectively, Ϙ ≥ 𝑀 ( 𝑃 ) = hom 𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑀 ) C and Ϙ ≥ 𝑀 ( 𝑃 ) = hom 𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑀 ) C , CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE which are canonically isomorphic to the usual abelian groups of 𝑀 -valued quadratic and symmetric formson 𝑃 , respectively, see §[I].4.2. Moreover, the group Ϙ ≥ 𝑀 ( 𝑃 ) is the image of the norm (or symmetrization)map Ϙ ≥ 𝑀 ( 𝑃 ) → Ϙ ≥ 𝑀 ( 𝑃 ) . The functors Ϙ ≥ 𝑀 , Ϙ ≥ 𝑀 and Ϙ ≥ 𝑀 are the non-abelian derived functors of these functorsof classical forms on modules, as shown in Proposition [I].4.2.15. We call them the genuine quadratic, genuine even, and genuine symmetric Poincaré structures respectively, and we denote them by Ϙ gq 𝑀 ∶= Ϙ ≥ 𝑀 , Ϙ ge 𝑀 ∶= Ϙ ≥ 𝑀 and Ϙ gs 𝑀 ∶= Ϙ ≥ 𝑀 . The connective covers of the associated Grothendieck-Witt spectra are the group-completions of the corre-sponding spaces of forms 𝜏 ≥ GW( 𝑅 ; Ϙ gq 𝑀 ) ≃ GW qcl ( 𝑅 ; 𝑀 ) , 𝜏 ≥ GW( 𝑅 ; Ϙ ge 𝑀 ) ≃ GW evcl ( 𝑅 ; 𝑀 ) , 𝜏 ≥ GW( 𝑅 ; Ϙ gs 𝑀 ) ≃ GW scl ( 𝑅 ; 𝑀 ) by the main result of [HS20]. In particular if 𝑀 = 𝑅 ( 𝜖 ) is the module with involution defined from an 𝜖 -involution on 𝑅 these are the classical Grothendieck-Witt spaces of 𝜖 -quadratic, 𝜖 -even, and 𝜖 -symmetricforms on 𝑅 .There is a periodicity phenomenon that relates the Poincaré structures Ϙ ≥ 𝑚𝑀 , that we now review. Werecall that a hermitian morphism of Poincaré ∞ -categories ( C , Ϙ ) → ( C ′ , Ϙ ′ ) consists of an exact functor 𝑓 ∶ C → C ′ and a natural transformation 𝜂 ∶ Ϙ → 𝑓 ∗ Ϙ ′ = Ϙ ′ ◦ 𝑓 . We say that a hermitian morphism ( 𝑓 , 𝜂 ) is a Poincaré morphism if a canonical induced map 𝑓 D → D 𝑓 is an equivalence; see §[I].1.2. A Poincaré morphism ( 𝑓 , 𝜂 ) is an equivalence of Poincaré ∞ -categoriesprecisely when 𝑓 is an equivalence of categories and 𝜂 is a natural equivalence. We recall the followingproposition (see Proposition [I].3.4.2 and Corollary [II].4.3.4), first observed by Lurie in the cases where 𝑚 = ±∞ .R.6. Proposition.
For every invertible module with involution 𝑀 over 𝑅 and 𝑚 ∈ ℤ ∪ {±∞} , the loopfunctor Ω ∶ D p ( 𝑅 ) → D p ( 𝑅 ) extends to an equivalence of Poincaré ∞ -categories ( D p ( 𝑅 ) , ( Ϙ ≥ 𝑚𝑀 ) [2] ) ∼ ⟶ ( D p ( 𝑅 ) , Ϙ ≥ 𝑚 +1− 𝑀 ) , where Ϙ [ 𝑘 ] ∶= Σ 𝑘 Ϙ denotes the 𝑘 -fold shift of a Poincaré structure, and − 𝑀 is the twist by a sign ofExample R.2. R.7.
Remark.
For a commutative ring 𝑅 , we may apply Proposition R.6 with 𝑀 = 𝑅 . If we set GW [ 𝑛 ] ( 𝑅 ) = 𝜏 ≥ GW( D p ( 𝑅 ); ( Ϙ ≥ 𝑅 ) [ 𝑛 ] ) , we obtain from [HS20] the equivalences GW [0] ( 𝑅 ) ≃ GW scl ( 𝑅 ) , GW [2] ( 𝑅 ) ≃ GW −evcl ( 𝑅 ) and GW [4] ( 𝑅 ) ≃ GW qcl ( 𝑅 ) . These equivalences were also announced by Schlichting, see [Sch19b, Theorem 3.1], where GW [2] ( 𝑅 ) isdescribed in terms of symplectic forms. Concretely, symplectic forms are those skew-symmetric forms 𝑏 ∶ 𝑃 ⊗ 𝑃 → 𝑅 which vanish on the diagonal, i.e. 𝑏 ( 𝑥, 𝑥 ) = 0 for all 𝑥 in 𝑃 , compare [Sch19a, Definition3.8 & Example 3.11]. This condition is in fact equivalent to admitting a (−1) -quadratic refinement, so sym-plectic forms are precisely the (−1) -even forms. To see this, we claim that ̂ H (C ; hom 𝑅⊗𝑅 ( 𝑃 ⊗𝑃 , 𝑅 (−1))) is isomorphic to hom 𝑅 ( 𝑃 , 𝑅 ) where 𝑅 denotes the 2-torsion in 𝑅 . Combining this isomorphism with thecanonical map from ordinary cohomology to Tate cohomology gives a map H (C ; hom 𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑅 (−1))) ⟶ hom 𝑅 ( 𝑃 , 𝑅 ) . Elements of the domain are skew-symmetric forms 𝑏 , and they are sent under this map to the map 𝑥 ↦ 𝑏 ( 𝑥, 𝑥 ) . Note that this is an additive map which indeed takes values in the 2-torsion of 𝑅 if 𝑏 is skew-symmetric. Hence the obstruction to lifting a skew-symmetric form 𝑏 along the norm map H (C ; hom 𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑅 (−1))) ⟶ H (C ; hom 𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑅 (−1))) , is given by the vanishing of 𝑏 on the diagonal as claimed. Of course, one can also give a direct argumentfor the existence of a quadratic refinement under the assumption 𝑏 ( 𝑥, 𝑥 ) = 0 . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 9 The shifted quadratic functor relates to that of the original Poincaré ∞ -category by means of the Bott-Genauer sequence, which we now recall. Given a Poincaré ∞ -category ( C , Ϙ ) we can functorially form an ∞ -category Met( C , Ϙ ) whose underlying ∞ -category is the ∞ -category of arrows in C , where the Poincaréstructure is defined by Ϙ met ( 𝑓 ∶ 𝐿 → 𝑋 ) = f ib ( Ϙ ( 𝑓 ) ∶ Ϙ ( 𝑋 ) → Ϙ ( 𝐿 )) , and with underlying duality D( 𝑓 ∶ 𝐿 → 𝑋 ) = (D( 𝑋 ∕ 𝐿 ) → D 𝑋 ) , see Definition [I].2.3.5. The Poincaréobjects of Met( C , Ϙ ) are given by Poincaré objects 𝑋 of ( C , Ϙ ) equipped with a Lagrangian 𝐿 (see §[I].2.3);classically, forms equipped with a Lagrangian are called metabolic forms, hence the notation Met( C , Ϙ ) .The Bott-Genauer sequence is the sequence of Poincaré ∞ -categories ( C , Ϙ [−1]) ⟶ Met( C , Ϙ ) ⟶ ( C , Ϙ ) where the underlying functors send an object 𝐿 of C to the arrow 𝐿 → , and an object 𝑓 ∶ 𝐿 → 𝑋 in thearrow category to its target 𝑋 , respectively; see Lemma [I].2.3.7. The Bott-Genauer sequence is both a fibreand a cofibre sequence of Poincaré ∞ -categories, that is a Poincaré-Verdier sequence in the terminology ofPaper [II], see Example [II].1.2.5. One of the main results of Paper [II] is that Grothendieck-Witt theory isVerdier localising, that is that it sends Poincaré-Verdier sequences to fibre sequences of spectra. There ismoreover a natural equivalence GW(Met( C , Ϙ )) ≃ K( C ) established in Corollary [II].4.3.1. By combining these ingredients with the periodicity of Proposition R.6we obtain the following general form of Karoubi’s periodicity theorem. Let hyp ∶ K( 𝑅 ) → GW( 𝑅 ; Ϙ ) and f gt ∶ GW( 𝑅 ; Ϙ ) → K( 𝑅 ) denote the hyperbolic and forgetful map, respectively, from and to the K-theoryspectrum of 𝑅 , and let U( C , Ϙ ) and V( C , Ϙ ) be their respective fibres.R.8. Theorem ([II].4.3.4) . Let 𝑅 be a ring and 𝑀 an invertible module with involution over 𝑅 . Then thereis a natural equivalence V( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) ≃ Ω U( 𝑅 ; Ϙ ≥ ( 𝑚 +1)− 𝑀 ) for every 𝑚 ∈ ℤ , where − 𝑀 is the 𝑅 ⊗ 𝑅 -module 𝑀 with the involution (∙) replaced by −(∙) . R.9.
Remark. If 𝑅 is a unit, Theorem R.8 is due to Karoubi [Kar80]. Since in this case the Poincaréstructures Ϙ ≥ 𝑚𝑀 are all equivalent, it takes the form V( 𝑅 ; Ϙ s 𝑀 ) ≃ Ω U( 𝑅 ; Ϙ s− 𝑀 ) . There is another case where this theorem simplifies, but where does not need to be invertible. Let 𝑅 be a commutative ring which is 2-torsion free, for instance the ring of integers in a number field, and let 𝑀 = 𝑅 with the trivial involution. In this case ̂ H (C ; − 𝑅 ) = 0 and ̂ H −1 (C ; 𝑅 ) = 0 , and by Remark R.4we have that Ϙ ge− 𝑅 = Ϙ gs− 𝑅 and Ϙ gq 𝑅 = Ϙ ge 𝑅 . Therefore the periodicity Theorem gives us that V( 𝑅 ; Ϙ gs 𝑅 ) ≃ Ω U( 𝑅 ; Ϙ gs− 𝑅 ) , V( 𝑅 ; Ϙ gs− 𝑅 ) ≃ Ω U( 𝑅 ; Ϙ gq 𝑅 ) and V( 𝑅 ; Ϙ gq 𝑅 ) ≃ Ω U( 𝑅 ; Ϙ gq− 𝑅 ) . Curiously, V( 𝑅 ; Ϙ gq− 𝑅 ) ≃ Ω U( 𝑅 ; Ϙ ≥ 𝑅 ) , and to the best of our knowledge Ϙ ≥ 𝑅 cannot be expressed in terms ofclassical forms.Given any invariant F of Poincaré ∞ -categories which is Verdier localising, the Bott-Genauer sequenceinduces a fibre sequence upon applying F . If in addition F (Met( C , Ϙ )) = 0 for any Poincaré ∞ -category ( C , Ϙ ) (i.e. in the terminology of Paper [II] F is in addition bordism invariant), one obtains a canonicalequivalence F ( C , Ϙ [ 𝑛 ] ) ≃ Σ 𝑛 F ( C , Ϙ ) for every 𝑛 ∈ ℤ (see [Lur11] and Proposition [II].3.5.8). Examples of bordism invariant functors areL-theory L( C , Ϙ ) and the Tate construction on K-theory K( C , Ϙ ) tC , where K( C , Ϙ ) denotes the K-theoryspectrum of C with the C -action induced by the duality underlying Ϙ . In fact, for K( C , Ϙ ) tC this is animmediate consequence of classical additivity: K(Met( C , Ϙ )) is equivalent to C ⊗ K( C ) , so that its Tateconstruction vanishes. We can again combine these results with the periodicity of Proposition R.6 to obtainthe following. R.10.
Corollary.
Let 𝑅 be a ring and 𝑀 an invertible module with involution over 𝑅 . Then there are naturalequivalences L( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) ≃ Ω L( 𝑅 ; Ϙ ≥ ( 𝑚 +1)− 𝑀 ) and K( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) tC ≃ Ω K( 𝑅 ; Ϙ ≥ ( 𝑚 +1)− 𝑀 ) tC . In particular, the spectra L( 𝑅 ; Ϙ s 𝑀 ) , L( 𝑅 ; Ϙ q 𝑀 ) and K( 𝑅 ; 𝑀 ) tC are -periodic, and periodic if 𝑅 is an 𝔽 -algebra. The last observation on the periodicity of the quadratic and symmetric L-spectra is of course due toRanicki, and it has been reworked in the present language by Lurie [Lur11].R.11.
Notation.
Let F be a functor, such as GW or L , from the category of Poincaré ∞ -categories to spectra.We introduce the following compact notation for the value of F at the perfect derived ∞ -category of 𝑅 withone of the Poincaré structures Ϙ 𝛼𝑀 discussed above: F 𝛼 ( 𝑅 ; 𝑀 ) ∶= F ( D p ( 𝑅 ) , Ϙ 𝛼𝑀 ) If 𝑀 = 𝑅 ( 𝜖 ) is the module with involution associated to an 𝜖 -involution on 𝑅 as in Example R.2, we write Ϙ 𝛼𝜖 ∶= Ϙ 𝛼𝑅 ( 𝜖 ) and F 𝛼 ( 𝑅 ; 𝜖 ) ∶= F 𝛼 ( 𝑅 ; 𝑅 ( 𝜖 )) for any of the decorations 𝛼 above. In the special cases where 𝜖 = ±1 we will further write Ϙ 𝛼 ∶= Ϙ 𝛼 = Ϙ 𝛼𝑅 Ϙ 𝛼 − ∶= Ϙ 𝛼 −1 = Ϙ 𝛼𝑅 (−1) F 𝛼 ( 𝑅 ) ∶= F 𝛼 ( 𝑅 ; 1) = F 𝛼 ( 𝑅 ; 𝑅 ) F − 𝛼 ( 𝑅 ) ∶= F 𝛼 ( 𝑅 ; −1) = F 𝛼 ( 𝑅 ; 𝑅 (−1)) . The homotopy groups of any of these spectra will be denoted by adding a subscript F 𝛼𝑛 ( 𝑅 ; 𝑀 ) ∶= 𝜋 𝑛 F 𝛼 ( 𝑅 ; 𝑀 ) for every 𝑛 ∈ ℤ . 1. L- THEORY AND ALGEBRAIC SURGERY
This section is devoted to exploring L -theory in the context of modules with involution. In §1.1 werecall the generators and relations description of the L-groups, and an important construction which allowsto manipulate representatives in such L-groups (without changing the class in L-theory) called algebraicsurgery .In §1.2, we prove a surgery result for Poincaré structures which we call 𝑚 -quadratic, for 𝑚 ∈ ℤ , anduse this to represent L -theory classes by Poincaré objects which satisfy certain connectivity bounds. Inparticular this allows us to show that the L -groups L gs 𝑛 ( 𝑅 ; 𝑀 ) coincide with Ranicki’s original definition ofsymmetric L-theory of short complexes, Theorem 4 from the introduction.Finally, in §1.3 we prove a surgery result for Poincaré structures which we call 𝑟 -symmetric, for 𝑟 ∈ ℤ ,in case the ring under consideration is Noetherian of finite global dimension. We will use this to show thatthe genuine symmetric L-groups are isomorphic to the symmetric L-groups in sufficiently high degrees,and consequently the analogous statement for the Grothendieck-Witt groups, which are Theorem 5 andCorollary 6 of the introduction.1.1. L-theoretic preliminaries.
For the whole section we let 𝑅 be a ring, 𝑀 an invertible module withinvolution over 𝑅 , and D = hom 𝑅 (− , 𝑀 ) the corresponding duality on D p ( 𝑅 ) . We recall that Ϙ q 𝑀 denotesthe quadratic Poincaré structure on D p ( 𝑅 ) , defined as the homotopy coinvariants Ϙ q 𝑀 ( 𝑋 ) = hom 𝑅⊗𝑅 ( 𝑋 ⊗𝑋, 𝑀 ) hC , and that the symmetric Poincaré structure Ϙ s 𝑀 is defined in an analogous way by taking homotopyinvariants.1.1.1. Remark.
If a Poincaré structure Ϙ ∶ D p ( 𝑅 ) op → S 𝑝 has underlying duality D , we will say that Ϙ is compatible with 𝑀 . In this case, the canonical map Ϙ q 𝑀 → Ϙ s 𝑀 factors as in Construction [I].3.2.5 into apair of natural transformations Ϙ q 𝑀 ⟶ Ϙ ⟶ Ϙ s 𝑀 , exhibiting Ϙ q 𝑀 and Ϙ s 𝑀 respectively as the initial and the final Poincaré structure compatible with 𝑀 , seeCorollary [I].1.3.6. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 11 We recall that a spectrum 𝐸 is 𝑚 -connective for some integer 𝑚 ∈ ℤ if 𝜋 𝑘 𝐸 = 0 for all 𝑘 < 𝑚 , and 𝑚 -truncated if 𝜋 𝑘 𝐸 = 0 for all 𝑘 > 𝑚 .1.1.2. Definition.
For every 𝑟 ∈ ℤ we will say that Ϙ is 𝑟 -symmetric if for every finitely generated projectivemodule 𝑃 ∈ Proj( 𝑅 ) the fibre of Ϙ ( 𝑃 [0]) → Ϙ s 𝑀 ( 𝑃 [0]) is (− 𝑟 ) -truncated. Dually, for 𝑚 ∈ ℤ we will saythat Ϙ is 𝑚 -quadratic if the cofibre of Ϙ q 𝑀 ( 𝑃 [0]) → Ϙ ( 𝑃 [0]) is 𝑚 -connective for every 𝑃 ∈ Proj( 𝑅 ) .1.1.3. Remark.
Note that the fibre of Ϙ → Ϙ s 𝑀 and the cofibre of Ϙ q 𝑀 → Ϙ are exact (contravariant) functors.It thus suffices to check the conditions in the definition for 𝑚 -quadratic and 𝑟 -symmetric Poincaré structuresonly in the case where 𝑃 = 𝑅 .It also follows that the collection of 𝑋 ∈ D p ( 𝑅 ) for which the above fibre is (− 𝑟 ) -truncated for a given 𝑟 ∈ ℤ is closed under suspensions and extensions. In particular, if Ϙ is 𝑟 -symmetric then the fibre of Ϙ ( 𝑋 ) → Ϙ s 𝑀 ( 𝑋 ) is (− 𝑟 − 𝑘 ) -truncated for every 𝑘 -connective 𝑋 .Dually, for a given 𝑚 ∈ ℤ the collection of D 𝑋 ∈ D p ( 𝑅 ) for which the above cofibre is 𝑚 -connectiveis closed under suspensions and extensions. In particular, if Ϙ is 𝑚 -quadratic then the cofibre of Ϙ q 𝑀 ( 𝑋 ) → Ϙ ( 𝑋 ) is ( 𝑚 + 𝑘 ) -connective whenever D 𝑋 is 𝑘 -connective.1.1.4. Example.
The symmetric Poincaré structure Ϙ s 𝑀 is 𝑟 -symmetric for every 𝑟 and the quadratic Poincaréstructure Ϙ q 𝑀 is 𝑚 -quadratic for every 𝑚 . More generally, from the exact sequences 𝜏 ≤ 𝑚 −2 Ω 𝑀 tC → Ϙ ≥ 𝑚𝑀 ( 𝑅 ) → Ϙ s 𝑀 ( 𝑅 ) and Ϙ q 𝑀 ( 𝑅 ) → Ϙ ≥ 𝑚𝑀 ( 𝑅 ) → 𝜏 ≥ 𝑚 𝑀 tC we find that the Poincaré structure Ϙ ≥ 𝑚𝑀 is 𝑚 -quadratic and (2 − 𝑚 ) -symmetric. In particular, Ϙ gs 𝑀 is -symmetric and -quadratic, Ϙ ge 𝑀 is -symmetric and -quadratic and Ϙ gq 𝑀 is -symmetric and -quadratic.As explained earlier, one goal of this paper is to show that the genuine symmetric L-groups coincidewith Ranicki’s classical symmetric L-groups of [Ran80], for which elements can be represented by chaincomplexes 𝑋 which are concentrated in a specific range of degrees. The following lemma shows that thiscan be equivalently phrased in terms of connectivity estimates for 𝑋 and D 𝑋 . The latter will be moreconvenient to work with for us.1.1.5. Lemma.
Let 𝑋 ∈ D p ( 𝑅 ) a perfect 𝑅 -module and 𝑘 ≤ 𝑙 integers. Then the following conditions areequivalent:i) 𝑋 can be represented by a chain complex of the form ⋯ → → 𝑃 𝑙 → 𝑃 𝑙 −1 → ⋯ → 𝑃 𝑘 → → ⋯ where each 𝑃 𝑖 is a finitely generated projective 𝑅 -module concentrated in homological degree 𝑖 .ii) 𝑋 is 𝑘 -connective and D 𝑋 is (− 𝑙 ) -connective.Proof. The implication i) ⇒ ii) is clear. For the other implication, let 𝐶 be a complex of finitely generatedprojective 𝑅 -modules of minimum length representing 𝑋 . We claim that 𝐶 is concentrated in the range [ 𝑘, 𝑙 ] . Let 𝑖 be the minimal integer such that 𝐶 𝑖 ≠ . We claim that 𝑖 ≥ 𝑘 . Indeed, suppose that 𝑖 < 𝑘 . Since 𝑋 is 𝑘 -connective we have that H 𝑖 ( 𝐶 ) = 0 and so the differential 𝐶 𝑖 +1 → 𝐶 𝑖 is a surjection of projectivemodules, hence a split surjection of projective modules, hence a surjection whose kernel 𝑁 ∶= ker( 𝐶 𝑖 +1 → 𝐶 𝑖 ) is projective. Removing 𝐶 𝑖 and replacing 𝐶 𝑖 +1 with 𝑁 thus yields a shorter complex representing 𝑋 ,contradicting the minimality of 𝐶 . We may hence conclude that 𝐶 is concentrated in degrees ≥ 𝑘 .Let now D 𝐶 be the complex given by (D 𝐶 ) 𝑖 ∶= D( 𝐶 − 𝑖 ) . Since 𝑀 is finitely generated projective D 𝐶 = hom 𝑅 ( 𝐶, 𝑀 ) ∈ Ch b ( 𝑅 ) represents D 𝑋 ∈ D p ( 𝑅 ) , and is thus also a complex of minimal lengthrepresenting D 𝑋 . Since D 𝑋 is assumed to be (− 𝑙 ) -connective, the same argument as above shows that D 𝐶 is concentrated in degrees ≥ − 𝑙 . It then follows that 𝐶 is concentrated in degrees ≤ 𝑙 , and hence in therange [ 𝑘, 𝑙 ] , as desired. (cid:3) Remark.
Lemma 1.1.5 does not really require a duality. In its absence the statement still holds if wetreat D 𝑋 = hom 𝑅 ( 𝑋, 𝑅 ) as an object of D p ( 𝑅 op ) . For later use, we also remark that our proof also showsthat 𝑋 is 𝑘 -connective if and only if it can be represented by a chain complex of finitely generated projectivemodules which are trivial below degree 𝑘 . Remark. If 𝑀 is moreover free as an 𝑅 -module, the proof of Lemma 1.1.5 works verbatim toshow that for 𝑋 ∈ D f ( 𝑅 ) , condition ii) above is equivalent to 𝑋 being representable by a complex asin Lemma 1.1.5 with each 𝑃 𝑖 a finitely generated stably free 𝑅 -module. More generally, if Free( 𝑅 ) ⊆ C ⊆ Proj( 𝑅 ) is any intermediate full subcategory closed under the duality and under direct sums, and 𝑋 can berepresented by a bounded complex valued in C , then the argument in the proof below yields that condition ii)above is equivalent to 𝑋 being representable by a complex as in i) with each 𝑃 𝑖 stably in C (that is, suchthat there exist 𝑄 𝑖 ∈ C with 𝑃 𝑖 ⊕ 𝑄 𝑖 ∈ C ). L-theory and surgery.
The purpose of this subsection is to recall some fundamental properties of L-theory.For the construction of the L-theory spectra, we refer to [Lur11] and §[II].4.4. However, a key featureof the L-spectrum L( C , Ϙ ) is that its homotopy groups have a very simple presentation: They are givenby cobordism groups of Poincaré objects. Let us explain what this means precisely, as we rely on thisconstruction throughout the section. We recall that the space Pn( C , Ϙ ) is the space of Poincaré objects, thatis of pairs ( 𝑋, 𝑞 ) of an object 𝑋 in C and a point 𝑞 ∈ Ω ∞ Ϙ ( 𝑋 ) such that a canonically associated map 𝑞 ♯ ∶ 𝑋 ⟶ D 𝑋 is an equivalence. Likewise, there is the space Pn 𝜕 ( C , Ϙ ) of Poincaré pairs, that is of triples ( 𝑓 ∶ 𝐿 → 𝑋, 𝑞, 𝜂 ) with 𝑞 ∈ Ω ∞ Ϙ ( 𝑋 ) and 𝜂 a nullhomotopy of 𝑓 ∗ ( 𝑞 ) , such that the canonically associated map 𝜂 ♯ ∶ 𝑋 ∕ 𝐿 ⟶ D 𝐿 induced on the quotient 𝑋 ∕ 𝐿 by 𝜂 is an equivalence. In this case we say that 𝐿 is a Lagrangian in 𝑋 (oralso that 𝐿 is a nullcobordism of 𝑋 ). It turns out that Pn 𝜕 ( C , Ϙ ) = Pn(Met( C , Ϙ )) where Met( C , Ϙ ) is themetabolic category associated to Ϙ as in §[I].2.3. We find that forgetting the Lagrangian provides a map Pn 𝜕 ( C , Ϙ ) → Pn( C , Ϙ ) , which is induced from the Poincaré functor Met( C , Ϙ ) → ( C , Ϙ ) sending 𝐿 → 𝑋 to 𝑋 .1.1.8. Definition.
We say that Poincaré objects ( 𝑋, 𝑞 ) and ( 𝑋 ′ , 𝑞 ′ ) are cobordant if ( 𝑋 ⊕ 𝑋 ′ , 𝑞 ⊕ (− 𝑞 ′ )) admits a Lagrangian, i.e. is nullcobordant. We define the 𝑛 ’th L-group L 𝑛 ( C , Ϙ ) as the group of cobordismclasses of Poincaré objects ( 𝑋, 𝑞 ) for the Poincaré structure Ϙ [− 𝑛 ] ∶= Ω 𝑛 Ϙ .We remark that the cobordism relation really is a congruence relation with respect to ⊕ , and that thediagonal 𝑋 → 𝑋 ⊕ 𝑋 is a canonical Lagrangian for ( 𝑋 ⊕ 𝑋, 𝑞 ⊕ (− 𝑞 )) , so that L 𝑛 ( C , Ϙ ) is indeed an abeliangroup.1.1.9. Notation.
In the case of the category C = D p ( 𝑅 ) , we will denote the L-groups and L-spectra respec-tively by L 𝑛 ( 𝑅 ; Ϙ ) ∶= L 𝑛 ( D p ( 𝑅 ) , Ϙ ) and L( 𝑅 ; Ϙ ) ∶= L( D p ( 𝑅 ) , Ϙ ) . When Ϙ = Ϙ 𝛼 is one of the genuine functors associated to an invertible module with involution 𝑀 analysedin the previous section, we use the notation L 𝛼 ( 𝑅 ; 𝑀 ) established in Notation R.11 for the correspondingL-groups.1.1.10. Remark.
It is immediate from the definition that L q ( 𝑅 ; 𝑀 ) and L s ( 𝑅 ; 𝑀 ) are respectively the usualquadratic and symmetric L -theory spectra of 𝑅 of [Lur11] which also agree with the L -spectra of Ranicki[Ran92]. The other variants are, however, more mysterious, and their study is the focus of this section.1.1.11. Remark.
It is immediate from the definition that the L-groups fit into an exact sequence of monoids 𝜋 (Pn 𝜕 ( C , Ϙ [− 𝑛 ] )) 𝜕 ⟶ 𝜋 (Pn( C , Ϙ [− 𝑛 ] )) ⟶ L 𝑛 ( C , Ϙ ) ⟶ , so that they agree with Definition [I].2.3.11. We remark that the quotient monoid of 𝜋 (Pn( C , Ϙ [− 𝑛 ] )) by thesubmonoid of metabolic objects is a priori smaller than L 𝑛 ( C , Ϙ ) , since two Poincaré objects are identifiedin the quotient if they become isomorphic after adding metabolic forms.Let us see directly that L 𝑛 ( C , Ϙ ) is indeed the quotient monoid. We need to verify that if a Poincaré object ( 𝑋, 𝑞 ) in 𝜋 (Pn( C , Ϙ [− 𝑛 ] )) is zero in the quotient, that is if there are metabolic Poincaré objects ( 𝑋 ′ , 𝑞 ′ ) and ( 𝑋 ′′ , 𝑞 ′′ ) and an isomorphism ( 𝑋 ⊕ 𝑋 ′ , 𝑞 ⊕ 𝑞 ′ ) ≅ ( 𝑋 ′′ , 𝑞 ′′ ) , then ( 𝑋, 𝑞 ) is itself metabolic. If 𝐿 ′ → 𝑋 ′ and 𝐿 ′′ → 𝑋 ′′ are Lagrangians for 𝑞 ′ and 𝑞 ′′ respectively, then it is straightforward to verify that 𝐿 ′ × 𝑋 ′ 𝐿 ′′ → 𝑋 is a Lagrangian for 𝑞 , where the pullback is formed via the second projection 𝐿 ′′ → 𝑋 ′′ ≅ 𝑋 ⊕ 𝑋 ′ → 𝑋 ′ ,and it maps to 𝑋 via the first projection 𝐿 ′′ → 𝑋 ′′ ≅ 𝑋 ⊕ 𝑋 ′ → 𝑋 . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 13 We note that given a Lagrangian for ( 𝑋, 𝑞 ) , i.e. a Poincaré object of the metabolic category, through theeyes of L-theory, we may replace ( 𝑋, 𝑞 ) by . Such a procedure in fact works more generally if we start withonly a hermitian object for the metabolic category and is the content of algebraic surgery. We recall that thehermitian objects of the metabolic category consist of triples ( 𝑓 ∶ 𝐿 → 𝑋, 𝑞, 𝜂 ) such that 𝑞 ∈ Ω ∞ Ϙ ( 𝑋 ) and 𝜂 is a nullhomotopy of 𝑓 ∗ ( 𝑞 ) . In many cases of interest, the object ( 𝑋, 𝑞 ) is Poincaré, and in this situationwe will refer to 𝐿 , or more precisely to ( 𝑓 , 𝜂 ) , as a surgery datum on ( 𝑋, 𝑞 ) . The non-degeneracy conditionfor this triple to be a Poincaré object for the metabolic category is that the map 𝜂 ♯ ∶ 𝑋 ∕ 𝐿 ⟶ D 𝐿 definedabove is an equivalence, i.e. if its fibre 𝑋 ′ is . In general, 𝑋 ′ need not vanish, but nevertheless acquires acanonical Poincaré form 𝑞 ′ induced from ( 𝑓 , 𝑞, 𝜂 ) . In fact, we have the following result; see §[II].2.3 for ageneral discussion of algebraic surgery.1.1.12. Proposition.
Let ( 𝑋, 𝑞 ) be a Poincaré object for Ϙ with surgery datum ( 𝑓 ∶ 𝐿 → 𝑋, 𝜂 ) . Then theobject 𝑋 ′ carries a canonical Poincaré form 𝑞 ′ such that ( 𝑋, 𝑞 ) and ( 𝑋 ′ , 𝑞 ′ ) are cobordant. Remark.
The underlying object of 𝑋 ′ and the of the cobordism 𝜒 ( 𝑓 ) between 𝑋 and 𝑋 ′ are sum-marised in the following surgery diagram consisting of horizontal and vertical fibre sequences.(4) 𝐿 𝐿 𝜒 ( 𝑓 ) 𝑋 D 𝐿𝑋 ′ 𝑋 ∕ 𝐿 D 𝐿 D 𝑓 ◦ 𝑞 ♯ 𝜂 ♯ We can use this to perform the following construction, which we will refer to as
Lagrangian surgery .1.1.14.
Construction.
Let ( 𝐿 → 𝑋, 𝑞, 𝜂 ) be a Lagrangian for a Poincaré object ( 𝑋, 𝑞 ) . Equivalently, wemay view ( 𝐿 → 𝑋, 𝑞, 𝜂 ) as a Poincaré object of the metabolic category Met( C , Ϙ ) . Now, given a surgerydatum for this Poincaré object, i.e. a commutative diagram 𝑍 𝑊𝐿 𝑋 and a null-homotopy of Φ ∗ ( 𝑞, 𝜂 ) in Ϙ met ( 𝑍 → 𝑊 ) , we may thus perform surgery by Proposition 1.1.12to obtain a new Poincaré object ( 𝐿 ′ → 𝑋 ′ , 𝑞 ′ , 𝜂 ′ ) of Met( C , Ϙ ) . We observe that the map 𝑊 → 𝑋 iscanonically a surgery datum on ( 𝑋, 𝑞 ) and that ( 𝑋 ′ , 𝑞 ′ ) is the result of surgery with this surgery datum. Inparticular, if 𝑊 = 0 , then ( 𝑋 ′ , 𝑞 ′ ) is canonically equivalent to ( 𝑋, 𝑞 ) . Moreover, by diagram (4) the newLagrangian 𝐿 ′ sits inside a fibre sequence 𝐿 ′ ⟶ 𝐿 ∕ 𝑍 ⟶ ΩD 𝑍. We will refer to such surgery data as
Lagrangian surgery data and refer to the surgery as a
Lagrangiansurgery . For future reference, we notice that the underlying map of a Lagrangian surgery datum is equiva-lently described by a map 𝑍 → 𝑁 = f ib( 𝐿 → 𝑋 ) . If we denote by 𝑁 ′ the fibre of the map 𝐿 ′ → 𝑋 , thenwe obtain likewise a fibre sequence 𝑁 ′ → 𝑁 ∕ 𝑍 → ΩD 𝑍 .1.2. Surgery for 𝑚 -quadratic structures. In this section we will show how to apply algebraic surgery toPoincaré structure which are sufficiently quadratic and use this to show that the genuine symmetric L-groupscoincide with Ranicki’s symmetric L-groups of short complexes; see Theorem 1.2.18. We also show thatin sufficiently small degrees, the L-groups of an 𝑚 -quadratic functor coincide with the quadratic L-groups;see Corollary 1.2.8. The surgery arguments we present below are designed to replace (shifted) Poincaréobjects and Lagrangians by cobordant counterparts which are suitably connective. The following definitionsummarises the kind of connectivity we seek:1.2.1. Definition.
Let 𝑀 be an invertible module with involution over 𝑅 , Ϙ a Poincaré structure on D p ( 𝑅 ) compatible with 𝑀 . Let 𝑛, 𝑎, 𝑏 ∈ ℤ be such that 𝑎, 𝑏 ≥ −1 , 𝑏 ≥ 𝑎 − 1 , and ( 𝑛 + 𝑎 ) is even. i) We denote by Pn 𝑎𝑛 ( 𝑅, Ϙ ) ⊆ Pn( D p ( 𝑅 ) , Ϙ [− 𝑛 ] ) the subspace spanned by those Poincaré objects ( 𝑋, 𝑞 ) such that 𝑋 is ( − 𝑛 − 𝑎 ) -connective.ii) We denote by Pn 𝜕 ( D p ( 𝑅 ) , Ϙ [− 𝑛 ] ) the subspace spanned by those Poincaré pairs ( 𝐿 → 𝑋, 𝑞, 𝜂 ) such that 𝑋 is ( − 𝑛 − 𝑎 ) -connective, 𝐿 is ⌈ − 𝑛 −1− 𝑏 ⌉ -connective and 𝑁 ∶= f ib( 𝐿 → 𝑋 ) ≃ Ω 𝑛 +1 D 𝐿 is ⌊ − 𝑛 −1− 𝑏 ⌋ -connective. We refer to such an 𝐿 as an allowed Lagrangian for ( 𝑋, 𝑞 ) .Finally, we define L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) = coker ( 𝜋 M 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) → 𝜋 Pn 𝑎𝑛 ( 𝑅, Ϙ ) ) as the cokernel in the category of monoids of the map that forgets the Lagrangian.1.2.2. Remark. If 𝑏 ≥ 𝑎 , the diagonal inside ( 𝑋, 𝑞 ) ⊕ ( 𝑋, − 𝑞 ) is an allowed Lagrangian, so that L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) is in fact a group. In Proposition 1.2.3, we will show that this is also the case for 𝑏 = 𝑎 − 1 , under additionalhypotheses on the Poincaré structure Ϙ .Furthermore, in case i), D 𝑋 ≃ 𝑋 [ 𝑛 ] is ( 𝑛 − 𝑎 ) -connective so that 𝑋 can be represented by a complex oflength 𝑎 concentrated in degrees [ − 𝑛 − 𝑎 , − 𝑛 + 𝑎 ] by Lemma 1.1.5. In particular, Pn −1 𝑛 ( 𝑅, Ϙ ) ≃∗ . In case ii), 𝑋 can be represented by a complex concentrated in degrees [ − 𝑛 − 𝑎 , − 𝑛 + 𝑎 ] , 𝐿 by a complex concentrated in de-grees [ ⌈ − 𝑛 −1− 𝑏 ⌉ , ⌈ − 𝑛 −1+ 𝑏 ⌉ ] and 𝑁 by a complex concentrated in degrees [ ⌊ − 𝑛 −1− 𝑏 ⌋ , ⌊ − 𝑛 −1+ 𝑏 ⌋ ] . Notice thatfor the conclusion that 𝐿 is concentrated in a certain range of degrees, we have used both the connectivityof 𝐿 and 𝑁 , as D 𝐿 = 𝑁 [ 𝑛 + 1] .For the remainder of the section we fix an invertible module with involution 𝑀 over 𝑅 , and we consideronly Poincaré structures Ϙ on D p ( 𝑅 ) which are compatible with 𝑀 , and we denote the underlying duality by D = hom 𝑅 (− , 𝑀 ) . To put the assumptions of the next result into context, recall that the Poincaré structure Ϙ ≥ 𝑚𝑀 is 𝑚 -quadratic.1.2.3. Proposition (Surgery for 𝑚 -quadratic Poincaré structures) . Let Ϙ be an 𝑚 -quadratic Poincaré struc-ture on D p ( 𝑅 ) . Fix an 𝑛 ∈ ℤ and let 𝑎, 𝑏 ≥ be two non-negative integers with 𝑏 ≥ 𝑎 − 1 , and suchthat ∙ ( 𝑛 + 𝑎 ) ≡ ( 𝑛 + 1 + 𝑏 ) ≡ modulo 2, and ∙ 𝑎 ≥ 𝑛 − 2 𝑚 and 𝑏 ≥ 𝑛 − 2 𝑚 + 1 .Then the map L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) → L 𝑛 ( 𝑅 ; Ϙ ) is an isomorphism. The proof of Proposition 1.2.3 will require the following connectivity estimate:1.2.4.
Lemma.
Suppose that Ϙ is an 𝑚 -quadratic Poincaré structure on D p ( 𝑅 ) . Then for every projectivemodule 𝑃 ∈ Proj( 𝑅 ) and every 𝑘 ∈ ℤ the spectrum Ϙ ( 𝑃 [ 𝑘 ]) is min(−2 𝑘, 𝑚 − 𝑘 ) -connective.Proof. The cofibre of the map Ϙ q 𝑀 ( 𝑃 [ 𝑘 ]) → Ϙ ( 𝑃 [ 𝑘 ]) is ( 𝑚 − 𝑘 ) -connective by the assumption that Ϙ is 𝑚 -quadratic, see Remark 1.1.3. Furthermore, Ϙ q 𝑀 ( 𝑃 [ 𝑘 ]) = (hom 𝑅⊗𝑅 ( 𝑃 ⊗ 𝑃 , 𝑀 )[−2 𝑘 ]) hC and is thus (−2 𝑘 ) -connective as 𝑃 is projective and homotopy orbits preserve connectivity. (cid:3) Proof of Proposition 1.2.3.
We start with the surjectivity of the map in question. For this, it suffices toshow that every Poincaré object ( 𝑋, 𝑞 ) ∈ Pn( D p ( 𝑅 ) , Ϙ [− 𝑛 ] ) is cobordant to one whose underlying object is ( − 𝑛 − 𝑎 ) -connective. If 𝑋 itself is ( − 𝑛 − 𝑎 ) -connective then we are done. Otherwise, since 𝑋 is perfect thereexists some 𝑘 < − 𝑛 − 𝑎 such that 𝑋 is 𝑘 -connective. By Lemma 1.1.5 the object 𝑋 can be represented by achain complex of projectives concentrated in degrees ≥ 𝑘 and so there exists a projective module 𝑃 and amap 𝑓 ∶ 𝑃 [ 𝑘 ] → 𝑋 which is surjective on H 𝑘 . By Lemma 1.2.4, the spectrum Ϙ ( 𝑃 [ 𝑘 ]) is min(−2 𝑘, 𝑚 − 𝑘 ) -connective and since 𝑘 < − 𝑛 − 𝑎 we have that min(−2 𝑘, 𝑚 − 𝑘 ) > min ( 𝑛 + 𝑎, 𝑚 + 𝑛 + 𝑎 ) ≥ 𝑛 by the inequalities in our assumptions. It then follows that Ω ∞+ 𝑛 Ϙ ( 𝑃 [ 𝑘 ]) is connected and hence 𝑞 restrictedto 𝑃 [ 𝑘 ] is null-homotopic, so that any nullhomotopy 𝜂 , makes ( 𝑃 [ 𝑘 ] → 𝑋, 𝑞, 𝜂 ) a hermitian form for themetabolic category. We may therefore apply Proposition 1.1.12 and perform surgery along 𝑓 ∶ 𝑃 [ 𝑘 ] → 𝑋 to obtain a cobordant Poincaré object 𝑋 ′ , given by the fibre of the induced map 𝑋 ∕ 𝑃 [ 𝑘 ] → D( 𝑃 )[− 𝑘 − 𝑛 ] .Since −2 𝑘 − 1 > 𝑛 + 𝑎 ≥ 𝑛 (here we use that 𝑛 + 𝑎 is even) we have that − 𝑘 − 𝑛 > 𝑘 + 1 and so H 𝑘 ′ ( 𝑋 ′ ) = H 𝑘 ′ ( 𝑋 ) = 0 for 𝑘 ′ < 𝑘 and H 𝑘 ( 𝑋 ′ ) ≅ coker[ 𝑃 → H 𝑘 ( 𝑋 )] = 0 , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 15 which means that 𝑋 ′ is ( 𝑘 + 1) -connective. Proceeding inductively we may thus obtain a Poincaré object ( 𝑋 ′′ , 𝑞 ′′ ) which is cobordant to ( 𝑋, 𝑞 ) and which is ( − 𝑛 − 𝑎 ) -connective. It then follows that the class [ 𝑋, 𝑞 ] is in the image of L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) → L 𝑛 ( 𝑅 ; Ϙ ) , so we have established surjectivity.To prove injectivity, we represent an element of L 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) by a Poincaré complex ( 𝑋, 𝑞 ) such that 𝑋 is ( − 𝑛 − 𝑎 ) -connective, and we suppose that it represents zero in L 𝑛 ( 𝑅 ; Ϙ ) . We need to verify that in this case ( 𝑋, 𝑞 ) already represents the zero element in L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) . By Remark 1.1.11 it admits a Lagrangian ( 𝐿 → 𝑋, 𝑞, 𝜂 ) . Let 𝑁 be the fibre of the map 𝐿 → 𝑋 . If 𝑁 is − 𝑛 −1− 𝑏 -connective, since 𝑏 ≥ 𝑎 − 1 then so is 𝐿 andwe are done. Otherwise, let 𝑙 < − 𝑛 −1− 𝑏 be such that 𝑁 is 𝑙 -connective. We can then find a projective module 𝑃 and a map 𝑃 [ 𝑙 ] → 𝑁 which is surjective on H 𝑙 . We may view the map 𝑃 [ 𝑙 ] → 𝑁 equivalently as a map ( 𝑃 [ 𝑙 ] → → ( 𝐿 → 𝑋 ) in the metabolic category. We claim that this map extends to a Lagrangian surgerydatum in the sense of Construction 1.1.14, for which it suffices to see that Ϙ met ( 𝑃 [ 𝑙 ] →
0) ≃ Ω Ϙ ( 𝑃 [ 𝑙 ]) is ( 𝑛 + 1) -connective. By Lemma 1.2.4, the spectrum Ϙ ( 𝑃 [ 𝑙 ]) is then min(−2 𝑙, 𝑚 − 𝑙 ) -connective and since 𝑙 < − 𝑛 −1− 𝑏 we have that min(−2 𝑙, 𝑚 − 𝑙 ) > min ( 𝑛 + 1 + 𝑏, 𝑚 + 𝑛 + 1 + 𝑏 ) ≥ 𝑛 + 1 by the inequalities in our assumptions. We may therefore perform Lagrangian surgery along 𝑃 [ 𝑙 ] → 𝐿 , seeConstruction 1.1.14, to obtain a new Lagrangian 𝐿 ′ → 𝑋 such that the fibre 𝑁 ′ of the map 𝐿 ′ → 𝑋 fits ina fibre sequence 𝑁 ′ ⟶ 𝑁 ∕ 𝑃 [ 𝑙 ] ⟶ D( 𝑃 )[− 𝑙 − 𝑛 − 1] . Since 𝑙 < − 𝑛 − 1 − 𝑏 and 𝑛 + 1 + 𝑏 is even, we have that −2 𝑙 − 1 > 𝑛 + 1 + 𝑏 ≥ 𝑛 + 1 . Thus − 𝑙 − 𝑛 − 1 > 𝑙 + 1 ,and so H 𝑙 ′ ( 𝑁 ′ ) = H 𝑙 ′ ( 𝑁 ) = 0 for 𝑙 ′ < 𝑙 and H 𝑙 ( 𝑁 ′ ) ≅ coker[ 𝑃 → H 𝑙 ( 𝑁 )] = 0 , which means that 𝑁 ′ is ( 𝑙 + 1) -connective. Proceeding inductively we may thus obtain a Lagrangian 𝐿 ′′ → 𝑋 for which 𝑁 ′′ , and thus 𝐿 ′′ , is ( − 𝑛 −1− 𝑏 ) -connective. This shows thati) the kernel of the map L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) → L 𝑛 ( 𝑅 ; Ϙ ) is trivial, andii) L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) is a group.Indeed, to see ii), we note that ( 𝑋 ⊕ 𝑋, 𝑞 ⊕ (− 𝑞 )) admits a Lagrangian, and hence by the above argumentalso a Lagrangian with suitable connectivity properties, which shows that ( 𝑋, − 𝑞 ) is an inverse of ( 𝑋, 𝑞 ) inthe monoid L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) . This shows the proposition. (cid:3) Remark.
The definition of L 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) as a cokernel in the category of commutative monoids meansin particular that a class [ 𝑋, 𝑞 ] ∈ 𝜋 Pn 𝑎𝑛 ( 𝑅, Ϙ ) maps to zero in L 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) if and only if there exists metabolicclasses [ 𝐿 ′ → 𝑋 ′ , 𝑞 ′ , 𝜂 ′ ] , [ 𝐿 ′′ → 𝑋 ′′ , 𝑞 ′′ , 𝜂 ′′ ] ∈ 𝜋 M 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) such that [ 𝑋, 𝑞 ] + [ 𝑋 ′ , 𝑞 ′ ] = [ 𝑋 ′′ , 𝑞 ′′ ] in 𝜋 Pn 𝑎𝑛 ( 𝑅, Ϙ ) . When Ϙ is 𝑚 -quadratic for some 𝑚 ∈ ℤ and 𝑎, 𝑏, 𝑛 satisfy the assumptions of Proposition 1.2.3then the surgery argument in the proof of that proposition allows us to slightly refine this statement: if ( 𝑋, 𝑞 ) is a ( − 𝑛 − 𝑎 ) -connective Poincaré object in ( D p ( 𝑅 ) , Ϙ [− 𝑛 ] ) which represents zero in L 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) then ( 𝑋, 𝑞 ) itselfadmits a Lagrangian 𝐿 → 𝑋 such that 𝐿 and f ib( 𝐿 → 𝑋 ) are ( − 𝑛 −1− 𝑏 ) -connective. In other words, thesequence 𝜋 M 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) ⟶ 𝜋 Pn 𝑎𝑛 ( 𝑅, Ϙ ) ⟶ L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) is exact in the middle, just as in the case of ordinary L-groups; see Remark 1.1.11.1.2.6. Example.
The quadratic Poincaré structure Ϙ q 𝑀 is 𝑚 -quadratic for every 𝑚 . Given 𝑛 = 2 𝑘 ∈ ℤ , wemay apply Proposition 1.2.3 to Ϙ q 𝑀 with ( 𝑎, 𝑏 ) = (0 , and get that every class in L q 𝑛 ( 𝑅 ; 𝑀 ) can be repre-sented by a Poincaré object which is concentrated in degree − 𝑘 , and that such a Poincaré object representszero in L q 𝑛 ( 𝑅 ; 𝑀 ) if and only if it admits a Lagrangian which is concentrated in degrees [− 𝑘 −1 , − 𝑘 ] . On theother hand, if 𝑛 = 2 𝑘 + 1 is odd we may apply Proposition 1.2.3 to Ϙ q 𝑀 with ( 𝑎, 𝑏 ) = (1 , and get that everyclass in L q 𝑛 ( 𝑅 ; 𝑀 ) can be represented by a Poincaré object which is concentrated in degree [− 𝑘 − 1 , − 𝑘 ] ,and that such a Poincaré object represents zero in L q 𝑛 ( 𝑅 ; 𝑀 ) if and only if it admits a Lagrangian whichis concentrated in degree − 𝑘 − 1 . This is often referred to in the literature as surgery below the middledimension . In fact, Proposition 1.2.3 gives this statement for any 𝑚 -quadratic Poincaré structure, as long aswe take 𝑛 ≤ 𝑚 . Corollary.
For any 𝑛 ≥ , the canonical map L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) → L 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) = L gs 𝑛 ( 𝑅 ; 𝑀 ) is anisomorphism.Proof. The genuine symmetric Poincaré structure Ϙ gs 𝑀 is -quadratic. Given 𝑛 ≥ ℤ we may thus applyProposition 1.2.3 to Ϙ gs 𝑀 with ( 𝑎, 𝑏 ) = ( 𝑛, 𝑛 + 1) so that L( 𝑅 ; Ϙ gs 𝑀 ) ≅ L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) . (cid:3) That is, every class in L gs 𝑛 ( 𝑅 ; 𝑀 ) can be represented by a Poincaré object which is concentrated in degrees [− 𝑛, , and such a Poincaré object represents zero in L gs 𝑛 ( 𝑅 ; 𝑀 ) if and only if it admits a Lagrangian whichis concentrated in degrees [− 𝑛 − 1 , .1.2.8. Corollary. If Ϙ is 𝑚 -quadratic (e.g., Ϙ = Ϙ ≥ 𝑚𝑀 ) then the natural map L q 𝑛 ( 𝑅 ; 𝑀 ) = L 𝑛 ( 𝑅 ; Ϙ q 𝑀 ) ⟶ L 𝑛 ( 𝑅 ; Ϙ ) is surjective for 𝑛 ≤ 𝑚 − 2 and bijective for 𝑛 ≤ 𝑚 − 3 .Proof. Let 𝑎, 𝑏 ∈ {0 , be such that 𝑛 + 𝑎 and 𝑛 + 1 + 𝑏 are even. Notice that 𝑛 + 𝑎 ≤ 𝑚 − 2 . ByProposition 1.2.3, we have L 𝑛 ( 𝑅, Ϙ q 𝑀 ) ≅ L 𝑎,𝑏𝑛 ( 𝑅, Ϙ q 𝑀 ) and L 𝑛 ( 𝑅, Ϙ ) ≅ L 𝑎,𝑏𝑛 ( 𝑅, Ϙ ) whenever 𝑛 ≤ 𝑚 , and soto prove the surjectivity statement it will suffice to show that the monoid map 𝜋 Pn 𝑎𝑛 ( 𝑅, Ϙ q 𝑀 ) ⟶ 𝜋 Pn 𝑎𝑛 ( 𝑅, Ϙ ) is surjective when 𝑛 ≤ 𝑚 − 2 . So let 𝑋 be ( − 𝑛 − 𝑎 ) -connective and equipped with a Poincaré form 𝑞 for Ϙ [− 𝑛 ] . It will suffice to prove that the map 𝜋 Ω 𝑛 Ϙ q 𝑀 ( 𝑋 ) → 𝜋 Ω 𝑛 Ϙ ( 𝑋 ) is surjective. As Ϙ is 𝑚 - quadratic, D 𝑋 is ( 𝑛 − 𝑎 ) -connective by Remark 1.2.2, so that the cofibre of the map Ω 𝑛 Ϙ q 𝑀 ( 𝑋 ) → Ω 𝑛 Ϙ ( 𝑋 ) is 𝑚 − 𝑛 + ( 𝑛 − 𝑎 ) -connective; see Remark 1.1.3. Since 𝑚 − 𝑛 + ( 𝑛 − 𝑎 ) = 𝑚 −( 𝑛 + 𝑎 )2 ≥ the claim follows.To prove injectivity, let us now assume that 𝑛 ≤ 𝑚 − 3 . In light of Remark 1.2.5, it will suffice to showthat if ( 𝑋, 𝑞 ) is a ( − 𝑛 − 𝑎 ) -connective Poincaré object in ( D p ( 𝑅 ) , ( Ϙ q 𝑀 ) [− 𝑛 ] ) whose associated Poincaré objectin ( D p ( 𝑅 ) , Ϙ [− 𝑛 ] ) admits a Lagrangian 𝐿 → 𝑋 such that 𝐿 is ( − 𝑛 −1− 𝑏 ) -connective and 𝑁 = f ib( 𝐿 → 𝑋 ) isalso ( − 𝑛 −1− 𝑏 ) -connective, then 𝐿 can be refined to a Lagrangian of ( 𝑋, 𝑞 ) with respect to Ϙ q 𝑀 . For this, itwill suffice to show that in this situation, the map Ω 𝑛 Ϙ q 𝑀 ( 𝐿 ) ⟶ Ω 𝑛 Ϙ ( 𝐿 ) is injective on 𝜋 and surjective on 𝜋 , which follows if the cofibre of this map has trivial 𝜋 . Now, D 𝐿 ≃Σ 𝑛 +1 𝑁 is ( 𝑛 +1− 𝑏 ) -connective. Hence by Remark 1.1.3 and the assumption that Ϙ is 𝑚 -quadratic we deducethat Ω 𝑛 Ϙ ( 𝐿 ) is 𝑚 − 𝑛 +1− 𝑏 -connective. Since 𝑚 ≥ 𝑛 + 3 , we can then estimate 𝑚 − 𝑛 +1− 𝑏 ≥ 𝑏 > , whichfinishes the proof. (cid:3) Remark.
The range in which the map of Corollary 1.2.8 is an isomorphism is essentially optimal.For example for 𝑅 = ℤ and 𝑚 = 0 , the map ℤ ∕2 ≅ L q−2 ( ℤ ) ⟶ L gs−2 ( ℤ ) = 0 is not an isomorphism, see Example 2.2.12 for the calculations of the groups.A particular case of -quadratic Poincaré structures Ϙ on D p ( 𝑅 ) which is of special interest is the onewhere Ϙ ( 𝑃 [0]) is discrete (i.e. is the Eilenberg-MacLane spectrum of an abelian group) for every 𝑃 ∈Proj( 𝑅 ) . In this case the restriction of Ϙ to Proj( 𝑅 ) can be regarded as a functor Ϙ ∶ Proj( 𝑅 ) ⟶ A 𝑏 which is quadratic in the sense of Eilenberg-MacLane [EM54, §9], that is, its second cross effect is additivein each variable separately, and Ϙ can be identified with the non-abelian derived functor of Ϙ as discussedin Propositions [I].4.2.12 and [I].4.2.15. The data of a Poincaré object in D p ( 𝑅 ) which is concentrated indegree can then be defined purely in terms of Proj( 𝑅 ) and Ϙ , namely as pairs ( 𝑃 , 𝑞 ) where 𝑃 ∈ Proj( 𝑅 ) and 𝑞 ∈ Ϙ ( 𝑃 ) is such that the induced map 𝑞 ♯ ∶ 𝑃 → D 𝑃 is an isomorphism (where we note that theassociation 𝑞 ↦ 𝑞 ♯ depends only on Ϙ ). Let us then write Pn(Proj( 𝑅 ) , Ϙ ) for the groupoid of such Poincaréobjects. We will say that a Poincaré object ( 𝑃 , 𝑞 ) ∈ Pn(Proj( 𝑅 ) , Ϙ ) is strictly metabolic if it admits aLagrangian 𝐿 → 𝑃 such that 𝐿 is concentrated in degree . The question of whether a given ( 𝑃 , 𝑞 ) as aboveis strictly metabolic again depends only on Proj( 𝑅 ) and Ϙ . We may then associate to the pair (Proj( 𝑅 ) , Ϙ ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 17 its corresponding Witt group
W(Proj( 𝑅 ); Ϙ ) , defined as the quotient of the monoid 𝜋 Pn(Proj( 𝑅 ) , Ϙ ) by thesubmonoid of strictly metabolic objects. We note that there is an evident map W(Proj( 𝑅 ); Ϙ ) → L ( 𝑅 ; Ϙ ) induced by the inclusion Proj( 𝑅 ) → D p ( 𝑅 ) .1.2.10. Proposition.
Let Ϙ be a Poincaré structure on D p ( 𝑅 ) which is compatible with the invertible modulewith involution 𝑀 , and such that Ϙ ∶= Ϙ | Proj( 𝑅 ) takes values in A 𝑏 ⊆ S 𝑝 . Then the natural map W(Proj( 𝑅 ); Ϙ ) ⟶ L ( 𝑅 ; Ϙ ) is an isomorphism.Proof. As observed above, Ϙ is -quadratic. Applying Proposition 1.2.3 to Ϙ , we conclude that the map L , ( 𝑅 ; Ϙ ) → L ( 𝑅 ; Ϙ ) is an isomorphism. On the other hand, by construction, the map under considerationfactors through a surjective map W(Proj( 𝑅 ); Ϙ ) ⟶ L , ( 𝑅 ; Ϙ ) . It will hence suffice to show that this map is also injective. By Remark 1.2.5 it will suffice to show thatif ( 𝑃 [0] , 𝑞 ) is a Poincaré object concentrated in degree , and it admits a Lagrangian ( 𝑓 ∶ 𝐿 → 𝑃 [0] , 𝑞, 𝜂 ) such that 𝐿 can be represented by a complex [ 𝑄 𝑑 → 𝑁 ] concentrated in degrees [−1 , , then ( 𝑃 [0] , 𝑞 ) represents zero in W(Proj( 𝑅 ); Ϙ ) . To see this, we note that the object 𝐿 sits in a fibre sequence of the form 𝑁 [−1] → 𝐿 → 𝑄 [0] . Consider the commutative square 𝑁 [−1] 𝑁 [−1] 𝐿 𝑃 [0] 𝑓 as a morphism in Met( D p ( 𝑅 ) , Ϙ ) from the top row to the bottom row. In particular, the object correspondingto the bottom horizontal arrow carries a Poincaré structure in Met( D p ( 𝑅 ) , Ϙ ) which exhibits 𝐿 → 𝑃 [0] asa Lagrangian. The top row on the other hand admits no non-trivial hermitian structure in Met( D p ( 𝑅 ) , Ϙ ) since Ϙ met (id 𝑁 [−1] ) = 0 . In particular, the Poincaré structure on the bottom row becomes uniquely null-homotopic when restricted to the top row. We may thus consider it as providing a surgery datum on thebottom row (note that this does not constitute a Lagrangian surgery datum in the sense of Construction 1.1.14since 𝑁 [−1] ≠ ). Performing surgery, i.e. applying Proposition 1.1.12, we obtain a new Poincaré object ( 𝐿 ′ → 𝑃 ′ , 𝑞 ′ , 𝜂 ′ ) in Met( D p ( 𝑅 ) , Ϙ ) in which ( 𝑃 ′ , 𝑞 ′ ) is the Poincaré object obtained from performing surgeryalong 𝑁 [−1] → 𝑃 [0] and where 𝐿 ′ = 𝑄 [0] is concentrated in degree 0. Let us now identify ( 𝑃 ′ , 𝑞 ′ ) . Thesurgery datum on 𝑃 [0] is given by restricting the null homotopy of 𝑞 | 𝐿 to 𝑁 [−1] along the map 𝑁 [−1] → 𝐿 .However, the map 𝑁 [−1] → 𝑃 [0] is null homotopic for degree reasons as 𝑁 is projective, so that we mayidentify the restriction of 𝑞 to 𝑁 [−1] with using a null homotopy. The surgery datum on 𝑃 [0] is henceequivalently given by a loop in Ϙ ( 𝑁 [−1]) . It follows that ( 𝑃 ′ , 𝑞 ′ ) is the orthogonal sum of ( 𝑃 , 𝑞 ) and theoutput of surgery on 𝑁 [−1] → , with surgery datum given by said loop in Ϙ ( 𝑁 [−1]) . By construction,a surgery on along 𝑁 [−1] is a metabolic form on 𝑁 [0] : it is given by (D 𝑁 [0] ⊕ 𝑁 [0] , 𝑞 ′′ ) for which D 𝑁 [0] is a Lagrangian. Hence we obtain ( 𝑃 ′ , 𝑞 ′ ) ≃ ( 𝑃 [0] , 𝑞 ) ⊕ (D 𝑁 [0] ⊕ 𝑁 [0] , 𝑞 ′′ ) . Thus, since ( 𝑃 ′ , 𝑞 ′ ) and (D 𝑁 [0] ⊕ 𝑁 [0] , 𝑞 ′′ ) admit Lagrangians concentrated in degree , namely 𝑄 [0] and D 𝑁 [0] respectively, we find that ( 𝑃 [0] , 𝑞 ) vanishes in W(Proj( 𝑅 ) , Ϙ ) as claimed. (cid:3) Remark.
Suppose that Ϙ is 1-quadratic and that ( 𝑃 , 𝑞 ) is a Poincaré object with 𝑃 finitely generatedprojective concentrated in degree 0. Suppose ( 𝑃 , 𝑞 ) admits a Lagrangian 𝐿 which is itself concentrated indegree . Since 𝐿 is a Lagrangian, we find that the fibre of the map 𝐿 → 𝑃 is equivalent to (D 𝐿 )[−1] and that 𝑃 ≅ 𝐿 ⊕ D 𝐿 . By the algebraic Thom isomorphism [I].2.3.20, the space of Poincaré structureson the object 𝐿 → 𝑃 of the metabolic category is equivalently described by the space of shifted forms Ω Ϙ (D 𝐿 [−1]) , which is connected as Ϙ is 1-quadratic. It follows that ( 𝑃 , 𝑞 ) is equivalent to hyp( 𝐿 ) , thehyperbolic form on 𝐿 . This recovers the well-known classical fact that a strictly metabolic quadratic formon a finitely generated projective module is hyperbolic.1.2.12. Corollary. i) L gs0 ( 𝑅 ; 𝑀 ) is naturally isomorphic to the Witt group of 𝑀 -valued symmetric forms over 𝑅 .ii) L ge0 ( 𝑅 ; 𝑀 ) is naturally isomorphic to the Witt group of 𝑀 -valued even forms over 𝑅 .iii) L ( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) is naturally isomorphic to the Witt group of 𝑀 -valued quadratic forms over 𝑅 for every 𝑚 ≥ .Proof. For L ( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) with 𝑚 = 0 , , we simply apply Proposition 1.2.10 and invoke the explicit descrip-tion of Ϙ ≥ 𝑚𝑀 ( 𝑃 [0]) for 𝑚 = 0 , , in terms of symmetric, even and quadratic forms, respectively. The caseof L ( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) for 𝑚 > reduces to that of 𝑚 = 2 since the natural map L q0 ( 𝑅 ; 𝑀 ) ⟶ L ( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) is an isomorphism for 𝑚 ≥ by Corollary 1.2.8. (cid:3) Remark.
Combining Corollary 1.2.12 and the equivalences L gs−2 𝑘 ( 𝑅 ; 𝑀 ((−1) 𝑘 )) ≅ L ( 𝑅 ; Ϙ ≥ 𝑘𝑀 ) given by Corollary R.10 we obtain a description of all the even non-positive genuine symmetric L -groups of 𝑅 in terms of Witt groups of symmetric, even or quadratic forms, see Theorem 1.2.18. A similar descriptioncan be obtained for the corresponding odd L -groups of degrees ≤ in terms of symmetric (in degree ),even (in degree −1 ) and quadratic (in odd degrees ≤ −3 ) formations . We leave the details to the motivatedreader.1.2.14. Remark.
When 𝑀 is free as an 𝑅 -module, the results of this section apply equally well if we restrictattention to the full subcategory D f ( 𝑅 ) ⊆ D p ( 𝑅 ) of finitely generated complexes. Indeed, in the proof ofProposition 1.2.3 we may simply choose 𝑃 to be free, in which case the algebraic surgery procedure stayswithin D f ( 𝑅 ) . In addition, the connectivity bounds obtained by that proposition translate into the same typeof representation by complexes concentrated in certain intervals, only that now these complexes consist ofstably free modules, see Remark 1.1.7. For example, if Ϙ is a -quadratic Poincaré structure then any elementof L 𝑛 ( D f ( 𝑅 ) , Ϙ ) with 𝑛 ≥ can be represented by a Poincaré form on a complex of stable free 𝑅 -modulesconcentrated in degrees [− 𝑛, , and such a Poincaré complex represents zero if and only if it admits aLagrangian represented by a complex of stable free modules concentrated in degrees [− 𝑛 − 1 , . Similarly,the proof of Proposition 1.2.10 can be run verbatim with stably free modules instead of projective modules,and so we get that in the situation of that proposition, L ( D f ( 𝑅 ) , Ϙ ) is isomorphic to the correspondingWitt groups of stably free Poincaré objects. More generally, one can take any intermediate subcategory D f ( 𝑅 ) ⊆ C ⊆ D p ( 𝑅 ) which is closed under the duality. A typical such C is the full subcategory of objectswhose class in K ( 𝑅 ) lies in a given involution-closed subgroup of K ( 𝑅 ) . We will consider this frameworkagain in §2 when we will discuss control on GW and L spectra. Genuine symmetric L-theory.
In this subsection, we use the previous surgery results to identify the genuinesymmetric L-groups L gs 𝑛 ( 𝑅 ; 𝑀 ) ∶= L 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) with Ranicki’s original definition of symmetric L-groups, which we recall now.We let Ch b (Proj( 𝑅 )) be the category of bounded chain complexes of finitely generated projective 𝑅 -modules. We will say that 𝐶 ∈ Ch b (Proj( 𝑅 )) is 𝑛 -dimensional if it is concentrated in the range [0 , 𝑛 ] , thatis, if 𝐶 𝑖 = 0 whenever 𝑖 < or 𝑖 > 𝑛 . Recall that the ∞ -category D p ( 𝑅 ) of perfect left 𝑅 -modules canbe identified with the ∞ -categorical localisation Ch b (Proj( 𝑅 ))[ 𝑊 −1 ] of Ch b (Proj( 𝑅 )) with respect to thecollection 𝑊 of quasi-isomorphisms.1.2.15. Definition.
We let 𝑛 ≥ be a non-negative integer. An 𝑛 -dimensional Poincaré complex in thesense of [Ran80, §1] is a pair ( 𝐶, 𝑞 ) where 𝐶 ∈ Ch b (Proj( 𝑅 )) is an 𝑛 -dimensional complex and 𝑞 is anelement of H 𝑛 ( H 𝑜𝑚 𝑅 (D 𝐶, 𝐶 ) hC ) whose image in H 𝑛 ( H 𝑜𝑚 𝑅 (D 𝐶, 𝐶 )) = [D( 𝐶 )[ 𝑛 ] , 𝐶 ] is an isomorphism D( 𝐶 )[ 𝑛 ] → 𝐶 in the homotopy category of Ch b (Proj( 𝑅 )) . Here, H 𝑜𝑚 𝑅 (− , −) denotes the internal Homcomplex and (−) hC is the homotopy fixed point construction (described explicitly in [Ran80] using thestandard projective resolution of ℤ as a trivial 𝐶 -module). An ( 𝑛 + 1) -dimensional Poincaré pair is a pair ( 𝑓 , 𝜂 ) where 𝑓 ∶ 𝐶 → 𝐶 ′ is a map in Ch b (Proj( 𝑅 )) from an 𝑛 -dimensional complex to an ( 𝑛 +1) -dimensionalcomplex and 𝜂 is an element of H 𝑛 (f ib[ H 𝑜𝑚 𝑅 (D 𝐶, 𝐶 ) → H 𝑜𝑚 𝑅 (D 𝐶 ′ , 𝐶 ′ )] hC ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 19 whose respective images in [D( 𝐶 )[ 𝑛 ] , 𝐶 ] and [cof(D 𝐶 ′ → D 𝐶 )[ 𝑛 ] , 𝐶 ′ ] are isomorphisms in the homotopycategory. Every Poincaré pair ( 𝐶 → 𝐶 ′ , 𝜂 ) determines, in particular, a Poincaré complex ( 𝐶, 𝜂 | 𝐶 ) , and wesay that a Poincaré complex is null-cobordant if it is obtained in this way. Similarly, two 𝑛 -dimensionalPoincaré complexes ( 𝐶, 𝑞 ) , ( 𝐶 ′ , 𝑞 ′ ) are said to be cobordant if ( 𝐶 ⊕ 𝐶 ′ , 𝑞 ⊕ − 𝑞 ′ ) is null-cobordant. The setof equivalence classes of 𝑛 -dimensional Poincaré complexes modulo the cobordism relation above formsan abelian group L short 𝑛 ( 𝑅 ; 𝑀 ) under direct sum, with the inverse of ( 𝐶, 𝑞 ) given by ( 𝐶, − 𝑞 ) . We will referto these groups as the short symmetric L -groups of 𝑅 .1.2.16. Remark. i) The groups L short 𝑛 ( 𝑅 ; 𝑀 ) are formally defined in [Ran80] only when 𝑀 is of the form 𝑅 ( 𝜖 ) for some 𝜖 -involution on 𝑅 . However, the definition only makes use of the induced duality on chain complexesand it therefore makes sense for any invertible module 𝑀 with involution.ii) In [Ran80] Ranicki extends the definition of the classical symmetric L-groups to negative integers asfollows: L short 𝑛 ( 𝑅 ; 𝑀 ) = { L short , ev 𝑛 +2 ( 𝑅 ; − 𝑀 ) for 𝑛 = −2 , −1L q 𝑛 ( 𝑅 ; 𝑀 ) for 𝑛 ≤ −3 Here L short , ev 𝑛 ( 𝑅 ; 𝑀 ) are the even L-groups from [Ran80, §3]. We recall that an 𝑛 -dimensional Poincarécomplex ( 𝐶, 𝜑 ) is called even in [Ran80] if a certain Wu class 𝑣 ( 𝜙 ) ∶ H 𝑛 ( 𝐶 ) → ̂ H (C ; 𝑀 ) vanishes.Likewise, an ( 𝑛 + 1) -dimensional Poincaré pair 𝑓 ∶ 𝐶 → 𝐶 ′ is called even if its relative Wu class H 𝑛 +1 ( 𝑓 ) → ̂ H (C ; 𝑀 ) vanishes. The short even L -groups L short , ev 𝑛 ( 𝑅 ; 𝑀 ) are then the cobordismgroups of 𝑛 -dimensional even complexes. For 𝑛 ≥ we will also show that they are equivalent to ourgenuine even L-groups, see the proof of Theorem 1.2.18.1.2.17. Remark.
Two Poincaré complexes ( 𝐶, 𝑞 ) , ( 𝐶 ′ , 𝑞 ′ ) are said to be quasi-isomorphic if there exists aquasi-isomorphism 𝑓 ∶ 𝐶 → 𝐶 ′ such that 𝑓 ∗ 𝑞 = 𝑞 ′ . This yields an equivalence relation which is finerthan cobordism: Given a quasi-isomorphism 𝑓 ∶ ( 𝐶, 𝑞 ) → ( 𝐶 ′ , 𝑞 ′ ) one can construct a Poincaré pair ofthe form (id , 𝑓 ) ∶ 𝐶 → 𝐶 ⊕ 𝐶 ′ witnessing ( 𝐶, 𝑞 ) and ( 𝐶 ′ , 𝑞 ′ ) as cobordant. In particular, if Pn short 𝑛 ( 𝑅 ) denotes the monoid of quasi-isomorphism classes of 𝑛 -dimensional Poincaré complexes and M short 𝑛 ( 𝑅 ) the monoid of 𝑛 -dimensional Poincaré pairs, then L short 𝑛 ( 𝑅 ; 𝑀 ) is naturally isomorphic to the cokernel of M short 𝑛 ( 𝑅 ) → Pn short 𝑛 ( 𝑅 ) in the category of commutative monoids.The remainder of this subsection is devoted to a proof of the following theorem.1.2.18. Theorem.
Let 𝑅 be a ring and 𝑀 an invertible module with involution over 𝑅 . Then for all integers 𝑛 , there is a natural isomorphism L short 𝑛 ( 𝑅 ; 𝑀 ) ≅ L gs 𝑛 ( 𝑅 ; 𝑀 ) between Ranicki’s classical symmetric L-groups and the genuine symmetric L-groups. The proof will proceed in several steps. We first compare, for 𝑛 ≥ , Ranicki’s classical L-group L short 𝑛 ( 𝑅 ; 𝑀 ) to the group L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) of Definition 1.2.1 as follows. Let I 𝑛 ⊆ Ch b (Proj( 𝑅 )) be thesubcategory consisting of the 𝑛 -dimensional complexes and quasi-isomorphisms between them, and let J [0 ,𝑛 ] ⊆ D p ( 𝑅 ) ≃ be the full sub- ∞ -groupoid spanned by those perfect 𝑅 -modules which can be representedby a complex in Proj( 𝑅 ) concentrated in degrees [0 , 𝑛 ] . The canonical localisation map then restricts to afunctor 𝜋 ∶ I 𝑛 → J [0 ,𝑛 ] , and it induces isomorphisms 𝜌 ∶ H 𝑛 ( H 𝑜𝑚 𝑅 (D( 𝐶 ) , 𝐶 ) hC ) ≅ ⟶ 𝜋 𝑛 (hom 𝑅 (D( 𝜋𝐶 ) , 𝜋𝐶 ) hC ) ≅ 𝜋 Ω 𝑛 Ϙ s 𝑀 ( 𝜋 D 𝐶 ) natural in the object 𝐶 of I 𝑛 . We can then define a map of sets Pn short 𝑛 ( 𝑅 ) ⟶ 𝜋 Pn 𝑛𝑛 ( 𝑅 ; Ϙ s 𝑀 ) by sending an 𝑛 -dimensional Poincaré complex ( 𝐶, 𝑞 ) to the component determined by ( 𝜋 (D 𝐶 ) , 𝜌 ( 𝑞 )) . Thismap is in fact an isomorphism, since the localisation 𝜋 ∶ I 𝑛 → J [0 ,𝑛 ] is an equivalence on homotopy cat-egories, and 𝜌 is an isomorphism. Since the localisation functor Ch b ( 𝑅 ) → D p ( 𝑅 ) preserves direct sumsthis is moreover an isomorphism of monoids. Proposition.
For every 𝑛 ≥ , the previously defined map induces a group isomorphism L short 𝑛 ( 𝑅 ; 𝑀 ) ≅ L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) . Proof.
By replacing Ch b ( 𝑅 ) and D p ( 𝑅 ) by their arrow categories a similar construction provides a mor-phism of monoids M short 𝑛 ( 𝑅 ) ⟶ 𝜋 M 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) which is compatible with the morphism Pn short 𝑛 ( 𝑅 ) → 𝜋 Pn 𝑛𝑛 ( 𝑅 ; Ϙ s 𝑀 ) . Thus we obtain a well-defined grouphomomorphism on L-groups. Since every arrow in D p ( 𝑅 ) can be lifted to a map of chain complexes,an argument similar to the one above shows that this map is also surjective. This suffices to induce anisomorphism on quotients. (cid:3) The next step for the proof of Theorem 1.2.18 is to see that on 𝑛 -dimensional complexes, the datum of asymmetric form is the same as the datum of a genuine symmetric form. We record here the correspondingstatement for L-groups, For the following result, we keep in mind that Ϙ gs 𝑀 is 2-symmetric:1.2.20. Lemma.
Let Ϙ be 𝑟 -symmetric for 𝑟 ∈ ℤ . Let 𝑎, 𝑏, 𝑛 ∈ ℤ be as in Definition 1.2.1, and supposeadditionally that 𝑎 ≤ 𝑛 + 2 𝑟 − 4 and 𝑏 ≤ 𝑛 + 2 𝑟 − 3 . Then the map L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) → L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ s 𝑀 ) is anisomorphism. In particular, the map L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) ⟶ L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) is an isomorphism for every 𝑛 ≥ .Proof. It will suffice to show that the monoid homomorphisms 𝜋 Pn 𝑎𝑛 ( 𝑅 ; Ϙ ) → 𝜋 Pn 𝑎𝑛 ( 𝑅 ; Ϙ s 𝑀 ) and 𝜋 M 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) → 𝜋 M 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ s 𝑀 ) are isomorphisms. Now the left homomorphism is an isomorphism since Ϙ is 𝑟 -symmetric and so the map Ω ∞+ 𝑛 Ϙ ( 𝑋 ) → Ω ∞+ 𝑛 Ϙ s 𝑀 ( 𝑋 ) is an equivalence by Remark 1.1.3 whenever 𝑋 is ( − 𝑛 − 𝑎 ) -connective, takinginto account that − 𝑟 − − 𝑛 − 𝑎 = −2 𝑟 + 𝑛 + 𝑎 ≤ 𝑛 − 2 by our assumption. Concerning the right map, it will sufficeto show that whenever 𝐿 → 𝑋 is such that 𝐿 is ⌈ − 𝑛 −1− 𝑏 ⌉ -connective and 𝑋 is ( − 𝑛 − 𝑎 ) -connective the map Ϙ Met ( 𝐿 → 𝑋 ) = f ib[ Ϙ ( 𝑋 ) → Ϙ ( 𝐿 )] → f ib[ Ϙ s 𝑀 ( 𝑋 ) → Ϙ s 𝑀 ( 𝐿 )] = Ϙ sMet ( 𝐿 → 𝑋 ) has an ( 𝑛 − 2) -truncated fibre. Equivalently, this is the same as saying that the square Ϙ ( 𝑋 ) Ϙ s 𝑀 ( 𝑋 ) Ϙ ( 𝐿 ) Ϙ s 𝑀 ( 𝐿 ) has ( 𝑛 − 2) -truncated total fibre. As in the first part of the proof, we find that the top horizontal map is ( 𝑛 − 2) -truncated and the bottom horizontal map is ( 𝑛 − 1) -truncated since 𝐿 is ⌈ − 𝑛 −1− 𝑏 ⌉ -connective and − 𝑟 − ⌈ − 𝑛 −1− 𝑏 ⌉ ≤ −2 𝑟 + 𝑛 +1+ 𝑏 ≤ 𝑛 − 1 . (cid:3) Proof of Theorem 1.2.18.
First, we consider the case 𝑛 ≥ where we simply combine the isomorphisms L short 𝑛 ( 𝑅 ; 𝑀 ) ≅ L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) ≅ L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) ≅ L 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) of Proposition 1.2.19, Lemma 1.2.20 and Corollary 1.2.7, respectively.The case 𝑛 ≤ −3 is covered by Corollary 1.2.8. It then suffices to treat the case 𝑛 = −2 , −1 . Here, weuse the “periodicity” L gs 𝑛 ( 𝑅 ; 𝑀 ) ≅ L ge 𝑛 +2 ( 𝑅 ; − 𝑀 ) of Corollary R.10 and will now argue more generally that for 𝑛 ≥ , a canonical map L short , ev 𝑛 ( 𝑅 ; 𝑀 ) → L ge 𝑛 ( 𝑅 ; 𝑀 ) is an isomorphism. To construct the map, we consider an 𝑛 -dimensional even complex ( 𝐶, 𝜑 ) and obtain from the construction preceding Proposition 1.2.19 a canonical element ( 𝑋, 𝑞 ) of L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ gs 𝑀 ) ,represented by D 𝐶 . We want to argue that ( 𝑋, 𝑞 ) refines to an element of L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ ge 𝑀 ) . Let us considerthe fibre sequence Ω 𝑛 Ϙ ge ( 𝑋 ) ⟶ Ω 𝑛 Ϙ gs ( 𝑋 ) ⟶ hom 𝑅 ( 𝑋 ; ̂ H (C ; 𝑀 )[− 𝑛 ]) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 21 We note the equivalence Ω ∞ hom 𝑅 ( 𝑋 ; ̂ H (C ; 𝑀 )[− 𝑛 ]) ≃ Hom 𝑅 (H − 𝑛 ( 𝑋 ) , ̂ H (C ; 𝑀 )) . Tracing throughthe definitions, the symmetric structure 𝑞 is sent to the Wu class 𝑣 ( 𝜑 ) which is zero by assumption. Wededuce that ( 𝑋, 𝑞 ) canonically refines to an element of L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ ge 𝑀 ) . Likewise, an ( 𝑛 + 1) -dimensionaleven pair gives rise to a genuine even structure on the associated cobordism. Reversing the above argument,we deduce that the map L short , ev 𝑛 ( 𝑅 ; 𝑀 ) ⟶ L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ ge 𝑀 ) is an isomorphism. Combining this with the isomorphism L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ ge 𝑀 ) ⟶ L 𝑛 ( 𝑅 ; Ϙ ge 𝑀 ) obtained from Proposition 1.2.3 just as Corollary 1.2.7, the theorem follows. (cid:3) Remark.
In [Ran80] Ranicki also defines, for 𝑛 ≥ , the 𝑛 ’th quadratic L -group , using quadraticPoincaré complexes of dimension 𝑛 . The same argument as above shows that this group coincides with L 𝑛,𝑛 +1 𝑛 ( 𝑅 ; Ϙ q 𝑀 ) , and hence with L 𝑛 ( 𝑅 ; Ϙ q 𝑀 ) by Proposition 1.2.3. As shown in [Ran80], these are also thesame as the quadratic L -groups of Wall [Wal99], which arise in manifold theory as the natural recipientof surgery obstructions. We warn the reader that these groups do not agree with L 𝑛 ( 𝑅 ; Ϙ gq 𝑀 ) , which byperiodicity are isomorphic to L 𝑛 −4 ( 𝑅 ; Ϙ gs 𝑀 ) . Surgery for connective ring spectra.
In this section we apply the surgery arguments previously developedto the case where 𝑅 is a connective ring spectrum (that is a connective E -algebra in the monoidal ∞ -category of spectra). The perfect derived category is then replaced with the ∞ -category Mod 𝜔𝑅 of compact 𝑅 -module spectra, and finitely generated projective modules with the full subcategory Proj( 𝑅 ) of Mod 𝜔𝑅 consisting of the retracts of finitely generated free modules, i.e. those equivalent to 𝑅 ⊕𝑛 for some 𝑛 . Theduality underlying a Poincaré structure Ϙ on Mod 𝜔𝑅 is then induced by an invertible module with involution 𝑀 as defined in Definition [I].3.1.1. We refer to §[I].3 for a complete treatment of Poincaré structures oncategories of module spectra.As in Definition 1.1.2, we say that Ϙ is 𝑚 -quadratic if the cofibre of the map Ϙ q 𝑀 ( 𝑋 ) = hom 𝑅⊗𝑅 ( 𝑋 ⊗ 𝑋, 𝑀 ) hC ⟶ Ϙ ( 𝑋 ) sends 𝑅 , or equivalently the category Proj( 𝑅 ) , to 𝑚 -connective spectra. Here and in the rest of the section ⊗ denotes the tensor product over the sphere spectrum 𝕊 , or in other words the smash product of spectra.When 𝑅 and 𝑀 are discrete Ϙ q 𝑀 on Mod 𝜔 H 𝑅 corresponds to the quadratic Poincaré structure on D p ( 𝑅 ) underthe equivalence D p ( 𝑅 ) ≃ Mod 𝜔 H 𝑅 .We notice that the definition of the groups L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) carry over to this more general situation, as theyonly make use of the notion of connectivity for objects of Mod 𝜔𝑅 .1.2.22. Remark.
Since a connective module over a connective ring spectrum is projective if and only ifits dual is also connective, the proof of Proposition 1.2.3 applies equally well to the case where 𝑅 is aconnective ring spectrum. As a result, one obtains for instance that L 𝑎,𝑏𝑛 ( 𝑅 ; Ϙ ) ⟶ L 𝑛 ( 𝑅 ; Ϙ ) is an isomorphism for 𝑛 ≤ 𝑚 if 𝑎, 𝑏 ∈ {0 , are such that 𝑛 + 𝑎 and 𝑛 + 1 + 𝑏 are even, and hence that for 𝑛 ≤ 𝑚 − 3 the canonical map L 𝑛 ( 𝑅 ; Ϙ q 𝑀 ) ⟶ L 𝑛 ( 𝑅 ; Ϙ ) is an isomorphism, as in Corollary 1.2.8.One may for instance apply this for the universal Poincaré structure Ϙ u on Mod 𝜔 𝕊 from Example [I].1.2.15,to obtain that L q 𝑛 ( 𝕊 ) → L u 𝑛 ( 𝕊 ) is an equivalence in degrees ≤ −3 . Interestingly, since 𝕊 tC ≃ 𝕊 ∧ is connectiveby Lin’s Theorem [Lin80], the symmetric Poincaré structure Ϙ s on Mod 𝜔 𝕊 is also -quadratic. Thus, the map L q 𝑛 ( 𝕊 ) → L s 𝑛 ( 𝕊 ) is also an isomorphism for 𝑛 ≤ −3 . Both of these observations also follow from work ofWeiss-Williams [WW14] who give an explicit formula for the cofibre of the maps in question.Using such surgery methods, we obtain the following results, see also [Lur11, Lecture 14] for a proofof the algebraic 𝜋 - 𝜋 -theorem, Corollary 1.2.24i) below. We say that a map of spectra is 𝑘 -connective forsome 𝑘 ∈ ℤ if its fibre is. We also write L Ϙ for the linear approximation of a quadratic functor Ϙ . Proposition.
Let 𝑓 ∶ 𝑅 → 𝑆 be a 1-connective map of connective ring spectra and Ϙ 𝑅 and Ϙ 𝑆 be 𝑚 -quadratic Poincaré structures on Mod 𝜔𝑅 and Mod 𝜔𝑆 respectively. Suppose that the extension of scalars func-tor 𝑓 ! is enhanced to a Poincaré functor (Mod 𝜔𝑅 , Ϙ 𝑅 ) → (Mod 𝜔𝑆 , Ϙ 𝑆 ) such that the induced map L Ϙ 𝑅 ( 𝑅 ) → L Ϙ 𝑆 ( 𝑆 ) is ( 𝑚 + 1) -connective. Then the induced map (5) L 𝑛 ( 𝑅 ; Ϙ 𝑅 ) ⟶ L 𝑛 ( 𝑆 ; Ϙ 𝑆 ) is an isomorphism for 𝑛 ≤ 𝑚 and surjective for 𝑛 = 2 𝑚 + 1 .Proof. First, we observe that for every 𝑟 ≤ 𝑚 and every projective 𝑅 -module 𝑃 , the map Ϙ 𝑅 ( 𝑃 [− 𝑟 ]) → Ϙ 𝑆 ( 𝑓 ! 𝑃 [− 𝑟 ]) is (2 𝑟 + 1) -connective: Consider the diagram of horizontal cofibre sequences Ϙ q 𝑅 ( 𝑃 [− 𝑟 ]) Ϙ 𝑅 ( 𝑃 [− 𝑟 ]) L Ϙ 𝑅 ( 𝑃 )[ 𝑟 ] Ϙ q 𝑆 ( 𝑓 ! 𝑃 [− 𝑟 ]) Ϙ 𝑆 ( 𝑓 ! 𝑃 [− 𝑟 ]) L Ϙ 𝑆 ( 𝑓 ! 𝑃 )[ 𝑟 ] in which the left vertical map is (2 𝑟 +1) -connective by inspection: By writing 𝑃 as a retract of a free module 𝑅 ⊕𝑛 , we may assume that 𝑃 is itself free, in which case it follows immediately from the assumption that 𝑓 is 1-connective. Furthermore, the right vertical map is ( 𝑟 + 𝑚 + 1) -connective by assumption. Since 𝑟 ≤ 𝑚 we have 𝑟 + 1 ≤ 𝑟 + 𝑚 + 1 and the claim follows.Let us now discuss the bijectivity of the map (5). Suppose first that 𝑛 = 2 𝑘 is even, and let ( 𝑋, 𝑞 ) representan element of L 𝑛 ( 𝑆 ; Ϙ 𝑆 ) . Since 𝑛 ≤ 𝑚 the surgery step above allows us to assume that 𝑋 = 𝑃 ′ [− 𝑘 ] forsome projective 𝑆 -module 𝑃 ′ . We can find a projective 𝑅 -module 𝑃 such that 𝑓 ! ( 𝑃 ) = 𝑃 ′ by the assumptionthat 𝑓 is a 𝜋 -isomorphism. Also, 𝑛 = 2 𝑘 ≤ 𝑚 , so that 𝑘 ≤ 𝑚 and we find that Ϙ 𝑅 ( 𝑃 [− 𝑘 ]) ⟶ Ϙ 𝑆 ( 𝑃 ′ [− 𝑘 ]) is (2 𝑘 + 1) -connective, and hence surjective on 𝜋 𝑘 (in fact an isomorphism) by our first observation. Wemay thus lift the form 𝑞 to a form on 𝑃 which is automatically Poincaré (as this may be tested after basechange to 𝜋 𝑅 ≅ 𝜋 𝑆 ), and we deduce surjectivity. Next, we show that it is also injective. So let ( 𝑋, 𝑞 ) be aPoincaré object for Ϙ 𝑅 [−2 𝑘 ] which is sent to zero in L 𝑘 ( 𝑆 ; Ϙ 𝑆 ) . We may assume that 𝑋 = 𝑃 [− 𝑘 ] for someprojective 𝑅 -module 𝑃 , and set 𝑃 ′ [− 𝑘 ] ∶= 𝑓 ! 𝑋 . Using an argument similar to that of Corollary 1.2.10,we may assume that its image ( 𝑃 ′ [− 𝑘 ] , 𝑞 ′ ) admits a Lagrangian 𝐿 ′ [− 𝑘 ] → 𝑃 ′ [− 𝑘 ] which is itself of theform 𝐿 ′ = 𝑓 ! 𝐿 for a projective 𝑆 -module 𝐿 . We can then lift the map 𝐿 ′ [− 𝑘 ] → 𝑃 ′ [− 𝑘 ] to a map 𝐿 [− 𝑘 ] → 𝑃 [− 𝑘 ] as 𝑓 is a 𝜋 -isomorphism. Since the map Ϙ 𝑅 ( 𝐿 [− 𝑘 ]) → Ϙ 𝑆 ( 𝐿 ′ [− 𝑘 ]) is injective on 𝜋 𝑘 ,we deduce that the form 𝑞 restricted to 𝐿 [− 𝑘 ] is . We obtain an induced null homotopy of the composite 𝐿 [− 𝑘 ] → 𝑃 [− 𝑘 ] → (D 𝐿 )[− 𝑘 ] which becomes a fibre sequence after applying 𝑓 ! . It follows that it isin fact a fibre sequence before applying 𝑓 ! , so that 𝐿 [− 𝑘 ] is indeed a Lagrangian. We conclude that for 𝑛 = 2 𝑘 ≤ 𝑚 , the map (5) is an isomorphism as claimed.We now turn to the case 𝑛 = 2 𝑘 + 1 . By the above surgery arguments, we find that for the maps L , 𝑘 +1 ( 𝑅 ; Ϙ 𝑅 ) ⟶ L , 𝑘 +1 ( 𝑅 ; Ϙ 𝑅 ) ⟶ L 𝑘 +1 ( 𝑅 ; Ϙ 𝑅 ) the composite is an isomorphism provided 𝑘 + 1 ≤ 𝑚 and the latter map is an isomorphism for 𝑘 + 1 =2 𝑚 + 1 , and likewise for 𝑆 . In order to prove surjectivity of the map (5) we may thus represent an elementof L 𝑘 +1 ( 𝑆 ; Ϙ 𝑆 ) by a Poincaré object ( 𝑋 ′ , 𝑞 ′ ) where 𝑋 ′ is the cofibre of a map 𝑃 ′ [− 𝑘 − 1] → 𝑄 ′ [− 𝑘 − 1] ,for some 𝑃 ′ , 𝑄 ′ in Proj( 𝑆 ) . We can lift this map to a map 𝑃 [− 𝑘 − 1] → 𝑄 [− 𝑘 − 1] , and set 𝑋 to be itscofibre. We wish to argue that the map Ϙ 𝑅 ( 𝑋 ) → Ϙ 𝑆 ( 𝑋 ′ ) is surjective on 𝜋 𝑘 +1 . We consider the diagram Ϙ q 𝑅 ( 𝑋 ) Ϙ 𝑅 ( 𝑋 ) L Ϙ 𝑅 ( 𝑋 ) Ϙ q 𝑆 ( 𝑋 ′ ) Ϙ 𝑆 ( 𝑋 ′ ) L Ϙ 𝑆 ( 𝑋 ′ ) where the left vertical map is (2 𝑘 + 1) -connective. The right vertical map is the fibre of two maps whichare each ( 𝑚 + 𝑘 + 2) -connective, and is consequently ( 𝑚 + 𝑘 + 1) -connective. Again, 𝑘 + 1 ≤ 𝑚 + 𝑘 + 1 as we have assumed 𝑘 + 1 ≤ 𝑚 + 1 . Thus Ϙ 𝑅 ( 𝑋 ) → Ϙ 𝑆 ( 𝑋 ′ ) is (2 𝑘 + 1) -connective, and we can again liftthe form on 𝑋 ′ to a form on 𝑋 which is automatically Poincaré. The surjectivity claim is hence proven. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 23 It remains to prove that the map (5) is also injective. So let ( 𝑋, 𝑞 ) be a Poincaré object over 𝑅 , withoutloss of generality assume that 𝑋 = cof( 𝑃 [− 𝑘 − 1] → 𝑄 [− 𝑘 − 1]) and that 𝑋 ′ = 𝑓 ! ( 𝑋 ) admits a Lagrangian 𝐿 ′ [− 𝑘 − 1] → 𝑋 ′ ; it is here where we use that 𝑘 + 1 ≤ 𝑚 , in the case 𝑘 + 1 = 2 𝑚 + 1 we can in generalnot push the Lagrangian into a single degree. Now we recall from the algebraic Thom construction that theLagrangian 𝐿 ′ [− 𝑘 − 1] → 𝑋 ′ is determined by an induced (possibly degenerate) form on f ib( 𝐿 ′ [− 𝑘 − 1] → 𝑋 ′ ) with respect to the Poincaré structure Ϙ 𝑆 [−2 𝑘 − 2] . We find that this fibre is given by 𝑁 ′ [− 𝑘 − 1] for aprojective 𝑆 -module 𝑁 ′ (namely D 𝐿 ′ ). We may then lift 𝑁 ′ to a projective 𝑅 -module 𝑁 shifted in degree − 𝑘 − 1 . Moreover, since 𝑘 + 1 ≤ 𝑚 , we find that the map Ϙ 𝑅 ( 𝑁 [− 𝑘 − 1]) ⟶ Ϙ 𝑆 ( 𝑁 ′ [− 𝑘 − 1]) is (2( 𝑘 + 1) + 1) -connective and hence surjective on 𝜋 𝑘 +2 as needed. The proposition follows. (cid:3) We can now apply Proposition 1.2.23 to compare the L-spectra of some canonical Poincaré structuresover sphere spectrum and over the integers. We compare the universal Poincaré structure Ϙ u 𝕊 on Mod 𝜔 𝕊 from §[I].4.1, the Tate Poincaré structure Ϙ t ℤ on D p ( ℤ ) from Example [I].3.2.11, and the Burnside Poincaréstructure Ϙ b ℤ on D p ( ℤ ) . These are defined respectively by the pullback squares Ϙ u 𝕊 ( 𝑋 ) hom 𝕊 ( 𝑋, 𝕊 ) Ϙ s 𝕊 ( 𝑋 ) hom 𝕊 ( 𝑋, 𝕊 tC ) Ϙ t ℤ ( 𝑌 ) hom ℤ ( 𝑌 , ℤ ) Ϙ s ℤ ( 𝑌 ) hom ℤ ( 𝑌 , ℤ tC ) Ϙ b ℤ ( 𝑌 ) hom ℤ ( 𝑌 , 𝜏 ≥ ℤ tC ) Ϙ s ℤ ( 𝑌 ) hom ℤ ( 𝑌 , ℤ tC ) where the right-hand vertical maps in the first two diagrams are induced by the Tate-valued Frobenii of the E ∞ -rings 𝕊 and 𝐻 ℤ . In the third diagram we have denoted 𝜏 ≥ ℤ tC ∶= ( 𝜏 ≥ ℤ tC ) × H ℤ ∕2 H ℤ , and the name Burnside indicates the fact that Ϙ b ( ℤ ) is the Burnside ring of C . There are canonical Poincaréfunctors (Mod 𝜔 𝕊 , Ϙ u 𝕊 ) ⟶ ( D p ( ℤ ) , Ϙ t ℤ ) ⟶ ( D p ( ℤ ) , Ϙ b ℤ ) ⟶ ( D p ( ℤ ) , Ϙ gs ℤ ) where the first functor is the base change along 𝕊 → H ℤ , and the second Poincaré functor is induced bythe Tate valued Frobenius of H ℤ . We denote the corresponding L-spectra by L u ( 𝕊 ) , L t ( ℤ ) and L b ( ℤ ) ,respectively. The connective cover of L b ( ℤ ) is equivalent to the spectrum L g ( 𝔸 ) considered in [DO19].1.2.24. Corollary.
For any connective ring spectrum 𝑅 we have that:i) The map L q ( 𝑅 ) → L q ( 𝜋 𝑅 ) is an equivalence.ii) The map L u 𝑛 ( 𝕊 ) → L t 𝑛 ( ℤ ) is an isomorphism for 𝑛 ≤ and surjective for 𝑛 = 1 .iii) The map L u 𝑛 ( 𝕊 ) → L b 𝑛 ( ℤ ) is an isomorphism for 𝑛 ≤ and surjective for 𝑛 = 1 .iv) The map L u 𝑛 ( 𝕊 ) → L gs 𝑛 ( ℤ ) is an isomorphism for 𝑛 ≤ −1 and surjective for 𝑛 = 0 .Proof. All claims follow from Proposition 1.2.23. For i) we observe that Ϙ q 𝑅 and Ϙ q 𝜋 𝑅 are 𝑚 -quadratic forevery 𝑚 . Furthermore the linear terms vanish, and therefore they are equivalent. For ii) notice that both Ϙ u 𝕊 and Ϙ t ℤ are -quadratic, and that the map on linear terms evaluated at the sphere spectrum is the map 𝕊 → H ℤ which is 1-connective. For iii) we observe that Ϙ t ℤ is also -quadratic, and that the map on linearterms is 𝕊 → ( 𝜏 ≥ ℤ tC )× H ℤ ∕2 H ℤ , which is -connective (notice that the target has trivial 𝜋 ). For iv), recallthat also Ϙ gs is -quadratic and that the map on linear terms on the sphere spectrum is the map 𝕊 → 𝜏 ≥ ℤ tC ,which is -connective. We can then apply Proposition 1.2.23 for 𝑚 = −1 which shows the claim for 𝑛 ≤ −2 .We conclude that L u−2 ( 𝕊 ) = 0 since L gs−2 ( ℤ ) is well-known to vanish, but see also Example 2.2.12 for a proof.The vanishing of L u−2 ( 𝕊 ) was also shown in Proposition [II].4.6.4 by explicit means. In loc. cit. it is alsoargued how one can use the fibre sequence of Weiss and Williams [WW14, Theorem 4.5] L q ( 𝕊 ) ⟶ L u ( 𝕊 ) ⟶ 𝕊 ⊕ MTO(1) and i) to deduce that also L u−1 ( 𝕊 ) = 0 , so the map L u 𝑛 ( 𝕊 ) → L gs 𝑛 ( ℤ ) is a bijection up to degree −1 . Here, MTO(1) denotes the Thom spectrum of − 𝛾 , where 𝛾 is the tautological bundle over BO(1) . Namely, oneuses that
MTO(1) is (−1) -connective and that 𝜋 −1 (MTO(1)) = ℤ ∕2 . As 𝜋 (MTO(1)) = 0 , we also deduce from the fibre sequence that L u0 ( 𝕊 ) ≅ ℤ ⊕ ℤ , and that the map in 𝜋 has target ℤ and hits . The lowdimensional homotopy groups of MTO(1) are calculated by describing
MTO(1) as the fibre of the transferfor the universal cover of
BO(1) , see again Proposition [II].4.6.4 for the details. (cid:3)
Remark.
We end this section with the observation that L u2 ( 𝕊 ) → L t2 ( ℤ ) is not an isomorphism.Using algebraic surgery, one can show that the map L q2 ( 𝕊 ) → L t2 ( ℤ ) is surjective. However, using the exactsequence L q2 ( 𝕊 ) ⟶ L u2 ( 𝕊 ) ⟶ 𝜋 ( 𝕊 ⊕ MTO(1)) ⟶ we deduce that the map L q2 ( 𝕊 ) → L u2 ( 𝕊 ) is not surjective.1.3. Surgery for 𝑟 -symmetric structures. In this section, we prove a comparison result between genuinesymmetric and symmetric L-theory. Algebraic surgery for symmetric Poincaré structures is not as straight-forward as for the quadratic ones, and we will need to further assume that the base ring is Noetherian offinite global dimension.Let 𝑅 be a ring and 𝑀 an invertible module with involution over 𝑅 . For an integer 𝑑 ≥ , we recall that 𝑅 has global dimension ≤ 𝑑 if every 𝑅 -module has a projective resolution of length at most 𝑑 . When 𝑅 isin addition Noetherian, one can find such a resolution where the modules are moreover finitely generated.In this case the connective cover functor 𝜏 ≥ and the truncation functor 𝜏 ≤ preserve perfect 𝑅 -modules,and so D p ( 𝑅 ) inherits from D ( 𝑅 ) its Postnikov 𝑡 -structure, so that D p ( 𝑅 ) ≥ consists of the -connectiveperfect 𝑅 -modules and D p ( 𝑅 ) ≤ of the -truncated perfect 𝑅 -modules. The duality D ∶ D p ( 𝑅 ) op → D p ( 𝑅 ) induced by 𝑀 interacts with the 𝑡 -structure as follows: D( D p ( 𝑅 ) ≥ ) ⊆ D p ( 𝑅 ) ≤ and D( D p ( 𝑅 ) ≤ ) ⊆ D p ( 𝑅 ) ≥ − 𝑑 . The first inclusion is immediate from Remark 1.1.6, and the second one follows from the Universal Coef-ficient spectral sequence computing H ∗ hom 𝑅 ( 𝑋, 𝑀 ) , since Ext 𝑖𝑅 = 0 for every 𝑖 ≥ 𝑑 + 1 as 𝑅 has globaldimension ≤ 𝑑 .We will cast the algebraic surgery argument for symmetric Poincaré structures in the setting of a gen-eral Poincaré ∞ -category ( C , Ϙ ) equipped with a 𝑡 -structure which interacts with the underlying duality D ∶ C op → C in the way described above. Similarly to Definition 1.1.2 (see also Remark 1.1.3), we willsay that the Poincaré structure Ϙ is 𝑟 -symmetric if the fibre of Ϙ ( 𝑋 ) → Ϙ sD ( 𝑋 ) is (− 𝑟 ) -truncated for every 𝑋 ∈ C ≥ , where Ϙ sD ( 𝑋 ) = hom C ( 𝑋, D 𝑋 ) hC is the symmetric Poincaré structure associated to the duality D . Given integers 𝑎, 𝑏 ≥ −1 we define Pn 𝑎𝑛 ( C , Ϙ ) , M 𝑎,𝑏𝑛 ( C , Ϙ ) and L 𝑎,𝑏𝑛 ( C , Ϙ ) as in Definition 1.2.1, where we interpret the connectivity requirementon Poincaré objects and Lagrangians as pertaining to the given 𝑡 -structure on C .1.3.1. Proposition (Surgery for 𝑟 -symmetric Poincaré structures) . Let C be a stable ∞ -category with abounded 𝑡 -structure C ≥ , C ≤ . Let Ϙ an 𝑟 -symmetric Poincaré structure on C with duality D ∶ C → C op suchthat D( C ≤ ) ⊆ C ≥ − 𝑑 for some integer 𝑑 ≥ . Fix an 𝑛 ∈ ℤ and let 𝑎 ≥ 𝑑 − 1 , 𝑏 ≥ 𝑑 be integers with 𝑏 ≥ 𝑎 ,and such that ∙ 𝑛 + 𝑎 is even, and ∙ 𝑎 ≥ − 𝑛 + 2 𝑑 − 2 𝑟 .Then the canonical map L 𝑎,𝑏𝑛 ( C , Ϙ ) → L 𝑛 ( C , Ϙ ) is an isomorphism.Proof. We start with the surjectivity of the map in question. For this, it suffices to show that every Poincaréobject ( 𝑋, 𝑞 ) ∈ Pn( C , Ϙ [− 𝑛 ] ) is cobordant to one whose underlying object is ( − 𝑛 − 𝑎 ) -connective. Let 𝑘 = − 𝑛 − 𝑎 −22 , define 𝑊 ∶= Ω 𝑛 D 𝜏 ≤ 𝑘 𝑋 and let 𝑓 ∶ 𝑊 ⟶ Ω 𝑛 D 𝑋 𝑞 ≃ 𝑋 be the map dual to the truncation map 𝑋 → 𝜏 ≤ 𝑘 𝑋 . Since D( C ≤ ) ⊆ C ≥ − 𝑑 we have that 𝑊 is (− 𝑛 − 𝑘 − 𝑑 ) -connective. Since D 𝑊 ≃ Σ 𝑛 𝜏 ≤ 𝑘 𝑋 is ( 𝑛 + 𝑘 ) -truncated we conclude that Ω 𝑛 hom C ( 𝑊 , D 𝑊 ) hC is ( 𝑛 + 2 𝑘 + 𝑑 ) -truncated. On the other hand, since Ϙ is 𝑟 -symmetric, the fibre of the map Ω 𝑛 Ϙ ( 𝑊 ) ⟶ Ω 𝑛 Ϙ sD ( 𝑊 ) = Ω 𝑛 hom C ( 𝑊 , D 𝑊 ) hC ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 25 is ( 𝑘 + 𝑑 − 𝑟 ) -truncated, so that Ω 𝑛 Ϙ ( 𝑊 ) is max( 𝑛 + 2 𝑘 + 𝑑, 𝑘 + 𝑑 − 𝑟 ) -truncated. Spelling out the definitionof 𝑘 and using the estimates in the assumptions, we find that max( 𝑛 + 2 𝑘 + 𝑑, 𝑘 + 𝑑 − 𝑟 ) < . We hence get that Ω 𝑛 Ϙ ( 𝑊 ) is (−1) -truncated and so Ω ∞+ 𝑛 Ϙ ( 𝑊 ) ≃∗ . The restriction of 𝑞 to 𝑊 is con-sequently null-homotopic, and we may therefore perform surgery along 𝑓 ∶ 𝑊 → 𝑋 to obtain a newPoincaré object ( 𝑋 ′ , 𝑞 ′ ) , given by the cofibre of the resulting map 𝑊 → 𝜏 ≥ 𝑘 +1 𝑋 (see diagram (4)). Since 𝑊 is (− 𝑛 − 𝑘 − 𝑑 ) -connective it is in particular ( 𝑘 + 1) -connective (since −2 𝑘 ≥ 𝑛 + 𝑎 + 2 ≥ 𝑛 + 𝑑 + 1 ),and so 𝑋 ′ is ( 𝑘 + 1) -connective. Since 𝑘 + 1 = − 𝑛 − 𝑎 , surjectivity is shown.To prove injectivity, we may represent an element of L 𝑎,𝑏𝑛 ( C , Ϙ ) by a Poincaré complex ( 𝑋, 𝑞 ) such that 𝑋 is ( − 𝑛 − 𝑎 ) -connective. Such an element maps to zero in L( C , Ϙ ) if and only if it admits a Lagrangian ( 𝐿 → 𝑋, 𝑞, 𝜂 ) . We need to verify that in this case ( 𝑋, 𝑞 ) already represents the zero element in L 𝑎,𝑏𝑛 ( C , Ϙ ) .Let 𝑁 ∶= f ib( 𝐿 → 𝑋 ) , so that 𝐿 ≃ Ω 𝑛 +1 D 𝑁 . If 𝐿 is ⌈ − 𝑛 −1− 𝑏 ⌉ -connective then 𝑁 is ⌊ − 𝑛 −1− 𝑏 ⌋ -connective(since 𝑋 is ( − 𝑛 − 𝑎 ) -connective and 𝑏 ≥ 𝑎 ) and we are done. Otherwise, let 𝑙 = ⌈ − 𝑛 −1− 𝑏 ⌉ − 1 , define 𝑁 ′ ∶= Ω 𝑛 +1 D 𝜏 ≤ 𝑙 𝐿 and let 𝑓 ∶ 𝑁 ′ ⟶ 𝑁 be the map dual to the truncation map 𝐿 → 𝜏 ≤ 𝑙 𝐿 . We may view this map as a map ( 𝑁 ′ → → ( 𝐿 → 𝑋 ) in the metabolic category, and we claim that it extends to a Lagrangian surgery datum for which itsuffices to show that Ϙ met ( 𝑁 ′ →
0) ≃ Ω Ϙ ( 𝑁 ′ ) is ( 𝑛 − 1) -truncated. Since D( C ≤ ) ⊆ C ≥ − 𝑑 we havethat 𝑁 ′ is (− 𝑛 − 1 − 𝑙 − 𝑑 ) - connective. Since D 𝑁 ′ ≃ Σ 𝑛 +1 𝜏 ≤ 𝑙 𝐿 is ( 𝑛 + 1 + 𝑙 ) -truncated we have that Ω 𝑛 +1 hom C ( 𝑁 ′ , D 𝑁 ′ ) hC is ( 𝑛 + 1 + 2 𝑙 + 𝑑 ) -truncated. On the other hand since Ϙ is 𝑟 -symmetric the fibreof the map Ω 𝑛 +1 Ϙ ( 𝑁 ′ ) ⟶ Ω 𝑛 +1 hom C ( 𝑁 ′ , D 𝑁 ′ ) hC is ( 𝑙 + 𝑑 − 𝑟 ) -truncated, so that Ω 𝑛 +1 Ϙ ( 𝑁 ′ ) is max( 𝑛 + 1 + 2 𝑙 + 𝑑, 𝑙 + 𝑑 − 𝑟 ) -truncated. Now by definitionof 𝑙 and the estimates in the assumptions we have that max( 𝑛 + 1 + 2 𝑙 + 𝑑, 𝑙 + 𝑑 − 𝑟 ) < We hence get that Ω 𝑛 +1 Ϙ ( 𝑁 ′ ) is (−1) -truncated as needed. We may therefore perform Lagrangian surgeryalong 𝑁 ′ → 𝑁 to obtain a new Lagrangian 𝐿 ′ → 𝑋 , such that 𝐿 ′ is given by the cofibre of the resultingmap 𝑁 ′ → 𝜏 ≥ 𝑙 +1 𝐿 . Since 𝑁 ′ is (− 𝑛 − 1 − 𝑙 − 𝑑 ) -connective it is in particular ( 𝑙 + 1) -connective (since −2 𝑙 ≥ 𝑛 + 𝑏 + 2 ≥ 𝑛 + 𝑑 + 2 ), and so 𝐿 ′ is ( 𝑙 + 1) -connective. Since 𝑙 + 1 = ⌈ − 𝑛 −1− 𝑏 ⌉ , it follows that theclass ( 𝑋, 𝑞 ) already represents zero in L 𝑎,𝑏𝑛 ( C , Ϙ ) , and so we have established injectivity. The proposition isshown. (cid:3) Remark.
Similarly to Remark 1.2.5, the surgery argument in the proof of Proposition 1.3.1 allows usto conclude that in the situation of that proposition, the sequence 𝜋 M 𝑎,𝑏𝑛 ( C , Ϙ ) ⟶ 𝜋 Pn 𝑎𝑛 ( C , Ϙ ) ⟶ L 𝑎,𝑏𝑛 ( C , Ϙ ) is exact in the middle, when 𝑎, 𝑏 and 𝑛 satisfy the assumptions of Proposition 1.3.1.Unwinding the definitions, the case 𝑑 = 0 of Proposition 1.3.1 gives the following result. Let C be astable ∞ -category with a 𝑡 -structure ( C ≥ , C ≤ ) and a duality D ∶ C → C op such that D sends C ≤ to C ≥ and vice versa. In particular, D induces a duality D ♡ ∶ C ♡ → C ♡ on the heart of C , and we may consider C ♡ an abelian category with duality. As in the previous section, we let Ϙ gsD ♡ ∶ C ♡ → A 𝑏 be the quadraticfunctor that takes 𝐴 ∈ C ♡ to the abelian subgroup of strict invariants Ϙ gsD ♡ ( 𝐴 ) ∶= hom C ♡ ( 𝐴, D ♡ 𝐴 ) C . We let W( C ♡ , Ϙ gsD ♡ ) be the corresponding symmetric Witt group, defined as the quotient of the monoid 𝜋 Pn( C ♡ , Ϙ gsD ♡ ) by the submonoid of strictly metabolic objects. Thus two elements of 𝜋 Pn( C ♡ , Ϙ gsD ♡ ) areidentified in the Witt group if they are isomorphic after adding strictly metabolic objects. We also write −D ♡ for the duality on C ♡ defined by the functor D ♡ but where we replace the isomorphism 𝜂 ∶ id → (D ♡ ) op D ♡ with − 𝜂 . Corollary.
In the above situation, let Ϙ sD ∶ C → S 𝑝 be the symmetric Poincaré structure associatedto D . Then there are canonical isomorphisms L 𝑛 ( C ; Ϙ sD ) ≅ ⎧⎪⎨⎪⎩ W( C ♡ , Ϙ gsD ♡ ) for 𝑛 ≡ mod , W( C ♡ , Ϙ gs−D ♡ ) for 𝑛 ≡ mod , else.In particular, every element of 𝜋 Pn( C ♡ , Ϙ gsD ♡ ) which is zero in W( C ♡ , Ϙ gsD ♡ ) is metabolic.Proof. Apply Proposition 1.3.1 in the case of 𝑟 = ∞ , 𝑑 = 0 and take ( 𝑎, 𝑏 ) to be (0 , when 𝑛 is even and (−1 , when 𝑛 is odd. (cid:3) We now specialize Proposition 1.3.1 to the main case of interest, where C = D p ( 𝑅 ) :1.3.4. Corollary.
Let 𝑀 be an invertible module with involution over 𝑅 and suppose that 𝑅 is Noetherianof finite global dimension 𝑑 . Let Ϙ be an 𝑟 -symmetric compatible Poincaré structure on D p ( 𝑅 ) , for 𝑟 ∈ ℤ ,e.g. Ϙ = Ϙ ≥ 𝑟𝑀 . Then for 𝑛 ≥ 𝑑 − 2 𝑟 the following holds:i) If 𝑛 + 𝑑 is even, the canonical map L 𝑑,𝑑𝑛 ( 𝑅 ; Ϙ ) ⟶ L 𝑛 ( 𝑅 ; Ϙ ) is an isomorphism.ii) If 𝑛 + 𝑑 is odd, the canonical map L 𝑑 −1 ,𝑑𝑛 ( 𝑅 ; Ϙ ) ⟶ L 𝑛 ( 𝑅 ; Ϙ ) is an isomorphism. Corollary.
Let 𝑀 be an invertible module with involution over 𝑅 and suppose that 𝑅 is Noetherianof finite global dimension . Then the following holds:i) L 𝑘 −1 ( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) = 0 whenever 𝑘 ≥ 𝑚 − 1 ;ii) L 𝑘 ( 𝑅 ; Ϙ ) ≅ W(Proj( 𝑅 ) , 𝜋 Ϙ ≥ 𝑚 (−1) 𝑘 𝑀 ) whenever 𝑘 ≥ 𝑚 − 2 ;iii) every symmetric, even or quadratic 𝑀 -valued Poincaré object in Proj( 𝑅 ) which is zero in the Wittgroup is strictly metabolic. Remark.
Notice that a ring 𝑅 is Noetherian of global dimension 0 if and only if it is semisimple.Part i) above hence recovers Ranicki’s result that the odd-dimensional symmetric and quadratic L-groups ofsemisimple rings vanish [Ran92, Proposition 22.7]: Indeed, the symmetric case follows from the above with 𝑚 = −∞ . For the quadratic case, by Corollary R.10 applied to Ϙ q 𝑀 , it suffices to show that L q−3 ( 𝑅 ; 𝑀 ) = 0 .But by Corollary 1.2.8 we have that L q−3 ( 𝑅 ; 𝑀 ) ≅ L gs−3 ( 𝑅 ; 𝑀 ) = L −3 ( 𝑅 ; Ϙ gs 𝑀 ) , so i) applies for 𝑘 = −1 .For completeness, we note that if 𝐾 is a field of characteristic different from , also L q4 𝑘 +2 ( 𝐾 ) ≅ L s4 𝑘 +2 ( 𝐾 ) vanishes: By Corollary 1.3.5 it is given by the Witt group of anti-symmetric forms over 𝐾 , but any suchform admits a symplectic basis and hence a Lagrangian.Our next goal is to use the surgery results above in order to identify the L -groups of an 𝑟 -symmetricstructure with the corresponding symmetric L -groups in a suitable range. The following corollary shouldbe compared with Corollary 1.2.8 above:1.3.7. Corollary.
Let 𝑀 be an invertible module with involution over 𝑅 and suppose that 𝑅 is Noetherianof finite global dimension 𝑑 . Let Ϙ be an 𝑟 -symmetric Poincaré structure on D p ( 𝑅 ) compatible with 𝑀 , for 𝑟 ∈ ℤ . Then the canonical map L 𝑛 ( 𝑅 ; Ϙ ) ⟶ L 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) = L s 𝑛 ( 𝑅 ; 𝑀 ) is injective for 𝑛 ≥ 𝑑 − 2 𝑟 + 2 and bijective for 𝑛 ≥ 𝑑 − 2 𝑟 + 3 .Proof. We consider the commutative diagram L 𝑑,𝑑𝑛 ( 𝑅 ; Ϙ ) L 𝑛 ( 𝑅 ; Ϙ ) L 𝑑 −1 ,𝑑𝑛 ( 𝑅 ; Ϙ )L 𝑑,𝑑𝑛 ( 𝑅 ; Ϙ s 𝑀 ) L 𝑛 ( 𝑅 ; Ϙ s 𝑀 ) L 𝑑 −1 ,𝑑𝑛 ( 𝑅 ; Ϙ s 𝑀 ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 27 and use Corollary 1.3.4 and Lemma 1.2.20 to conclude the bijectivity claim of the corollary. To see injec-tivity for 𝑛 = 𝑑 − 2 𝑟 + 2 , again using Corollary 1.3.4, it will suffice to show that the left vertical map in theabove diagram is injective. In light of Remark 1.3.2 it will suffice to show that if ( 𝑋, 𝑞 ) is a (− 𝑑 + 𝑟 − 1) -connective Poincaré object in ( D p ( 𝑅 ) , Ϙ [− 𝑛 ] ) whose associated Poincaré object in ( D p ( 𝑅 ) , ( Ϙ s 𝑀 ) [− 𝑛 ] ) admitsa Lagrangian 𝐿 → 𝑋 such that 𝐿 is (− 𝑑 + 𝑟 − 1) -connective then 𝐿 can be refined to a Lagrangian of ( 𝑋, 𝑞 ) with respect to Ϙ . For this, it will suffice to show that for an 𝐿 with this connectivity bound, the map Ω 𝑛 Ϙ ( 𝐿 ) ⟶ Ω 𝑛 Ϙ s 𝑀 ( 𝐿 ) is surjective on 𝜋 and injective on 𝜋 . Indeed, this map is (−1) -truncated by Remark 1.1.3 since Ϙ is 𝑟 -symmetric. (cid:3) Remark.
By Theorem 1, the canonical squares GW gq ( 𝑅 ; 𝑀 ) GW gs ( 𝑅 ; 𝑀 ) GW s ( 𝑅 ; 𝑀 )L gq ( 𝑅 ; 𝑀 ) L gs ( 𝑅 ; 𝑀 ) L s ( 𝑅 ; 𝑀 ) are pullbacks. Corollary 1.3.7 implies that for 𝑅 Noetherian of finite global dimension 𝑑 , the map L gs 𝑛 ( 𝑅 ; 𝑀 ) ⟶ L s 𝑛 ( 𝑅 ; 𝑀 ) is injective for 𝑛 ≥ 𝑑 − 2 and bijective for 𝑛 ≥ 𝑑 − 1 , since Ϙ gs 𝑀 is 2-symmetric. Likewise the map L gq 𝑛 ( 𝑅 ; 𝑀 ) ⟶ L s 𝑛 ( 𝑅 ; 𝑀 ) is injective for 𝑛 ≥ 𝑑 + 2 and bijective for 𝑛 ≥ 𝑑 + 3 , since Ϙ gq 𝑀 is 0-symmetric. Combined with the abovepullbacks, these statements prove Theorem 5 and the first part of Corollary 6 from the introduction.1.3.9. Remark.
In Theorem 1.2.18 we have shown that the non-negative genuine symmetric L -groups coin-cide with Ranicki’s L -groups of short complexes. The comparison range above then improves on Ranicki’sclassical theorem that established injectivity of the map L gs 𝑛 ( 𝑅 ) → L s ( 𝑅 ) for non-negative 𝑛 ≥ 𝑑 − 3 andbijectivity for non-negative 𝑛 ≥ 𝑑 − 2 .1.3.10. Remark.
The range of Corollary 1.3.7 is essentially optimal. For example for 𝑅 = ℤ and 𝑟 = 0 , themap −gq3 ( ℤ ) ⟶ L −s3 ( ℤ ) ≅ ℤ ∕2 is not an isomorphism. See Example 2.2.12 and Remark 1.3.13 for the calculations of the groups.Since Dedekind rings have global dimension ≤ , and by applying the fibre sequence of Theorem 1 weimmediately find:1.3.11. Corollary.
Let 𝑅 be a Dedekind ring, e.g. the ring of integers in an algebraic number field. Thenthe canonical maps L gs 𝑛 ( 𝑅 ; 𝑀 ) ⟶ L s 𝑛 ( 𝑅 ; 𝑀 ) and GW gs 𝑛 ( 𝑅 ; 𝑀 ) ⟶ GW s 𝑛 ( 𝑅 ; 𝑀 ) are injective for 𝑛 = −1 and bijective for 𝑛 ≥ . In particular, the non-negative homotopy groups of L gs 𝑛 ( 𝑅 ; 𝑀 ) are 4-periodic. Similarly, the maps L gq 𝑛 ( 𝑅 ; 𝑀 ) ⟶ L s 𝑛 ( 𝑅 ; 𝑀 ) and GW gq 𝑛 ( 𝑅 ; 𝑀 ) ⟶ GW s 𝑛 ( 𝑅 ; 𝑀 ) are injective for 𝑛 = 3 and bijective for 𝑛 ≥ . We recall that by the main Theorem of [HS20] the connective covers of GW gs 𝑛 ( 𝑅 ; 𝑀 ) and GW gq 𝑛 ( 𝑅 ; 𝑀 ) are equivalent to the classical Grothendieck-Witt groups respectively of symmetric and quadratic forms, andwe further find that:1.3.12. Corollary.
Let 𝑅 be a Dedekind ring and 𝑀 an invertible module with involution over 𝑅 . Then thecanonical maps GW scl ( 𝑅 ; 𝑀 ) ⟶ 𝜏 ≥ GW s ( 𝑅 ; 𝑀 ) and 𝜏 ≥ GW qcl ( 𝑅 ; 𝑀 ) ⟶ 𝜏 ≥ GW s ( 𝑅 ; 𝑀 ) are equivalences. Remark.
We now prove the second part of Corollary 6 from the introduction. So let 𝑅 be a com-mutative Noetherian ring of finite global dimension d. We have equivalences Σ L gs ( 𝑅 ) ≃ L −ge ( 𝑅 ) , and Σ L ge ( 𝑅 ) ≃ L −gq ( 𝑅 ) . If 𝑅 is in addition 2-torsion free, for instance a Dedekind domain whose fraction fieldis of characteristic different from 2, then the canonical maps Ϙ −ge → Ϙ −gs and Ϙ gq → Ϙ ge are equivalencesby Remark R.4. We deduce that for such rings, there are in fact canonical equivalences Σ L gs ( 𝑅 ) ≃ L −gs ( 𝑅 ) and Σ L gq ( 𝑅 ) ≃ L −gq ( 𝑅 ) . The comparison map is compatible with these equivalences, so we deduce that the map L gq ( 𝑅 ) → L gs ( 𝑅 ) is a 2-fold loop of the map L −gq ( 𝑅 ) → L −gs ( 𝑅 ) . Remark 1.3.8 then implies that the map L gq 𝑛 ( 𝑅 ) → L gs 𝑛 ( 𝑅 ) ,and consequently the map GW qcl ,𝑛 ( 𝑅 ) → GW scl ,𝑛 ( 𝑅 ) is injective for 𝑛 = 𝑑 and an isomorphism for 𝑛 ≥ 𝑑 +1 .1.3.14. Remark.
The surgery results of this section can now be used to almost fully determine the L-groupsof L b ( ℤ ) appearing in Corollary 1.2.24, or equivalently L g ( 𝔸 ) from [DO19]. Recall that in Corollary 1.2.24we have determined the non-positive homotopy groups of L b ( ℤ ) . Making use of the fact that Ϙ b ℤ is also -symmetric, we obtain from Corollary 1.3.7 that the map L b ( ℤ ) → L s ( ℤ ) is an isomorphism on homotopygroups of degree at least ; here we have in addition used that L s3 ( ℤ ) = 0 . Finally, from Corollary 1.2.24, wealso deduce that the map L u1 ( 𝕊 ) → L b1 ( ℤ ) is surjective. By the results of Weiss-Williams [WW14, Theorem4.5] mentioned in the proof of Corollary 1.2.24, we find that L u1 ( 𝕊 ) ≅ ( ℤ ∕2) , with generators havingunderlying object 𝕊 ⊕ Ω 𝕊 . We find that both generators are mapped to an Ω Ϙ b -form on ℤ ⊕ ℤ [−1] . But since Ω Ϙ b ( ℤ ) is connected, we find that both these forms admit a Lagrangian. Hence the map L u1 ( 𝕊 ) → L b1 ( ℤ ) isboth zero and surjective from which we conclude that L b1 ( ℤ ) = 0 . This leaves open only the group L b2 ( ℤ ) .2. L- THEORY OF D EDEKIND RINGS
The goal of this section is to extend Quillen’s localisation-dévissage sequence [Qui73, Corollary of The-orem 5] for Dedekind rings to hermitian K-theory, thereby proving Theorem 7 of the introduction. We recallthat Dedekind rings are commutative regular Noetherian domains of global dimension , and the main ex-ample of interest to us are those whose fraction field is a number field. Other notable examples are discretevaluation rings and rings of functions of smooth affine curves over fields.To prove the theorem we first construct a Poincaré-Verdier sequence induced by the map from 𝑅 to itslocalisation away from a set 𝑆 of non-zero prime ideals . The functor GW takes such sequences to cofibresequences of spectra. Using a dévissage result for symmetric GW -theory we then identify the fibre term asthe sum of GW -spectra of the residue fields 𝑅 ∕ 𝔭 , where 𝔭 ranges through the set 𝑆 .Results of this type have appeared in the literature from early on. Three- or four-term localisationsequences for Witt groups appear in the work of Knebusch [Kne70], Milnor-Husemoller [MH73] andthese are extended to long exact sequences of L-groups by Ranicki [Ran81, §4.2] and Balmer-Witt groupsin [Bal05, §1.5.2]. For Grothendieck-Witt groups there are exact sequences due to Karoubi [Kar74, Kar75]as well as Hornbostel-Schlichting [Hor02, HS04], both under the assumption that 2 is invertible in the ring.Our results hold with no assumption on invertibility of , for the homotopy theoretic symmetric Grothendieck-Witt spectrum GW s . By the results of the previous section (see Corollary 1.3.11 and 1.3.12) the non-negative homotopy groups of GW s ( 𝑅 ) agree with the classical higher Grothendieck-Witt groups, but it isonly GW s ( 𝑅 ) which is well-behaved in all degrees, see Remark 2.1.11.2.1. The localisation-dévissage sequence.
Let 𝑅 be a Dedekind ring and 𝑆 a set of non-zero prime idealsof 𝑅 . We let 𝑅 𝑆 = O ( 𝑈 ) where O is the structure sheaf of spec( 𝑅 ) and 𝑈 = spec( 𝑅 ) ⧵ 𝑆 is the complementof the set 𝑆 (in case 𝑆 is infinite 𝑈 is not open in spec( 𝑅 ) , and O ( 𝑈 ) is defined as the colimit of O onthe open subsets of spec( 𝑅 ) containing 𝑈 ). Concretely, one can describe the ring 𝑅 𝑆 as follows: For anynon-zero prime ideal 𝔭 , the localisation 𝑅 ( 𝔭 ) at 𝔭 is a discrete valuation ring, and the fraction field 𝐾 of 𝑅 hence acquires a 𝔭 -adic valuation 𝜈 𝔭 . Then 𝑅 𝑆 identifies with the subring of 𝐾 given by all elements 𝑥 ∈ 𝐾 such that 𝜈 𝔭 ( 𝑥 ) ≥ for all 𝔭 not contained in 𝑆 . The ring 𝑅 𝑆 can be thought of as the localisationof 𝑅 away from the set of primes 𝑆 .2.1.1. Example.
Let 𝑅 be a Dedekind ring.i) If 𝑆 consists of all the non-zero prime ideals of 𝑅 , then 𝑅 𝑆 is the fraction field 𝐾 , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 29 ii) Given a multiplicative subset 𝑇 ⊂ 𝑅 we may consider the set of primes ideals 𝑆 = { 𝔭 | 𝔭 ∩ 𝑇 ≠ ∅} .Then 𝑅 𝑆 = 𝑅 [ 𝑇 −1 ] is obtained from 𝑅 by inverting the elements of 𝑇 . In particular:iii) If 𝑆 = { 𝔭 , … , 𝔭 𝑛 } is such that the ideal 𝔭 𝑟 ⋅ ⋯ ⋅ 𝔭 𝑟 𝑛 𝑛 = ( 𝑥 ) is a principal ideal, then 𝑅 𝑆 = 𝑅 [ 𝑥 ] .These are the cases that we will use most. We warn the reader that, in general, 𝑅 𝑆 is not obtained from 𝑅 by inverting a multiplicative subset.2.1.2. Lemma.
Let 𝑅 be a Dedekind ring and 𝑆 a set of non-zero prime ideals of 𝑅 . Then the map 𝑅 → 𝑅 𝑆 is a derived localisation, 𝑅 𝑆 is a flat 𝑅 -module, and the extension of scalars functor D ( 𝑅 ) → D ( 𝑅 𝑆 ) hasperfectly generated fibre.Proof. We recall from Definition [II].A.4.2 that a map of rings 𝐴 → 𝐵 is called a derived localisation ifthe map H 𝐵 ⊗ H 𝐴 H 𝐵 → H 𝐵 of ring spectra is an equivalence. We note that 𝑅 𝑆 is described as the filteredcolimit 𝑅 𝑆 ≅ colim 𝑆 ′ ⊆𝑆 O ( 𝑈 ′ ) where 𝑈 ′ = spec( 𝑅 ) ⧵ 𝑆 ′ and 𝑆 ′ ranges through the finite subsets of 𝑆 . As 𝑅 is a Dedekind ring, a finiteset of primes 𝑆 ′ is a closed subset of spec( 𝑅 ) , so the map 𝑅 → 𝑅 𝑆 arises as a filtered colimit of ring maps,each of which is given by restricting the structure sheaf to an affine open subset. Any such map is a derivedlocalisation and has perfectly generated fibre by [Rou10, Theorem 3.6]. By passing to filtered colimits,we deduce that also the map D ( 𝑅 ) → D ( 𝑅 𝑆 ) is a derived localisation and has perfectly generated fibre.Finally 𝑅 𝑆 , as a submodule of the fraction field 𝐾 , is a torsion free and hence flat 𝑅 -module [SP18, Lemma0AUW]. (cid:3) Remark.
From the description of 𝑅 𝑆 as a subring of the fraction field, it also follows immediatelythat 𝑅 → 𝑅 𝑆 is a derived localisation: Since 𝑅 𝑆 is a flat module, it suffices to note that the underived tensorproduct 𝑅 𝑆 ⊗ 𝑅 𝑅 𝑆 is, via the multiplication map, isomorphic to 𝑅 𝑆 . Likewise, we remark that the kernel D ( 𝑅 ) 𝑆 of D ( 𝑅 ) → D ( 𝑅 𝑆 ) is in fact generated by the perfect 𝑅 -modules 𝑅 ∕ 𝔭 where 𝔭 ranges through theelements of 𝑆 . Note that 𝔭 is a finitely generated projective 𝑅 -module (again, because it is a torsion freemodule, hence flat, and thus also projective by finite generation), so that 𝑅 ∕ 𝔭 is equivalent to the perfectcomplex ( 𝔭 → 𝑅 ) .First, we observe that for each non-zero prime 𝔭 , the module 𝑅 ∕ 𝔭 is indeed in D ( 𝑅 ) 𝑆 . To see that theobjects 𝑅 ∕ 𝔭 indeed generate the kernel it suffices to consider the case where 𝑆 = { 𝔭 } for a prime ideal 𝔭 of 𝑅 . Let us write 𝑅 [ 𝔭 ] for the ring 𝑅 𝑆 . The fibre sequence, 𝑅 ⟶ 𝑅 [ 𝔭 ] ⟶ 𝑅 [ 𝔭 ]∕ 𝑅 together with the fact that 𝑅 generates D ( 𝑅 ) , implies that it suffices to prove that 𝑅 [ 𝔭 ]∕ 𝑅 is in the subcat-egory generated by 𝑅 ∕ 𝔭 . For this, one constructs a filtration on 𝑅 [ 𝔭 ] using a lower bound on the 𝔭 -adicvaluation of elements of 𝑅 𝑆 ⊆ 𝐾 . This induces a filtration on the quotient 𝑅 [ 𝔭 ]∕ 𝑅 and the successivefiltration quotients are then equivalent to 𝑅 ∕ 𝔭 as an 𝑅 -module.We denote by D p ( 𝑅 ) 𝑆 ⊆ D p ( 𝑅 ) the fibre of the map D p ( 𝑅 ) → D p ( 𝑅 𝑆 ) . By the above, it coincides withthe full subcategory spanned by those perfect 𝑅 -modules whose homotopy groups are 𝑆 -primary torsionmodules. Now we let 𝑀 be a line bundle over 𝑅 with an 𝑅 -linear involution (which in fact can only bemultiplication with ±1 ), which we regard as a module with involution over 𝑅 as in Definition R.1. We recallthat a line bundle over a commutative ring is a finitely generated projective module of rank 1.We then endow D p ( 𝑅 ) with the symmetric Poincaré structure Ϙ s 𝑀 and D p ( 𝑅 𝑆 ) with the symmetricPoincaré structure Ϙ s 𝑀 𝑆 associated to the localised line bundle 𝑀 𝑆 ∶= 𝑅 𝑆 ⊗ 𝑅 𝑀. The extension of scalars is then a Poincaré functor, see Lemma [I].3.3.3, so that D p ( 𝑅 ) 𝑆 is closed under theduality of D p ( 𝑅 ) induced by 𝑀 and becomes a Poincaré subcategory, with the restricted Poincaré structure.We will denote this restricted Poincaré structure again by Ϙ s 𝑀 ∶ D p ( 𝑅 ) 𝑆 op → S 𝑝 . Proposition.
Let 𝑅 be a Dedekind ring and 𝑀 a line-bundle over 𝑅 with 𝑅 -linear involution. Thenthe sequence of Poincaré ∞ -categories ( D p ( 𝑅 ) 𝑆 , Ϙ s 𝑀 ) ⟶ ( D p ( 𝑅 ) , Ϙ s 𝑀 ) ⟶ ( D p ( 𝑅 𝑆 ) , Ϙ s 𝑀 𝑆 ) is a Poincaré-Verdier sequence. In particular, it induces a fibre sequence of GW and L -spectra.Proof. The last statement says that GW and L are Verdier-localising functors, which was proven in Corol-lary [II].4.4.15 and Corollary [II].4.4.6. We now wish to apply Proposition [II].1.4.8. For this we needto check that 𝑀 is compatible with the localisation 𝑅 → 𝑅 𝑆 in the sense of Definition [II].1.4.3, whichfollows from the fact that 𝑅 and 𝑅 𝑆 are commutative and 𝑀 is an 𝑅⊗𝑅 -module through the multiplicationmap of 𝑅 , see Example [II].1.4.4. Furthermore, we have observed earlier that 𝑅 𝑆 is a flat 𝑅 -module. Inaddition, the map K ( 𝑅 ) → K ( 𝑅 𝑆 ) is surjective: As filtered colimits along surjections are surjections, itsuffices to argue this in the case where 𝑆 is finite. In this case, 𝑅 𝑆 is itself a Dedekind ring, so it sufficesto argue that the map Pic( 𝑅 ) → Pic( 𝑅 𝑆 ) is surjective. This follows from the observation that Pic( 𝑅 ) and Pic( 𝑅 𝑆 ) are respectively the quotients of the free abelian groups generated by the prime ideals of 𝑅 and 𝑅 𝑆 . The proposition then follows from Lemma 2.1.2. (cid:3) The objective of dévissage is then to identify
GW( D p ( 𝑅 ) 𝑆 , Ϙ s 𝑀 ) in terms of the GW -spectra of the residuefields 𝔽 𝔭 ∶= 𝑅 ∕ 𝔭 for 𝔭 ∈ 𝑆 . To establish this, we begin by refining the restriction of scalars functor to aPoincaré functor.We recall that for a ring homomorphism 𝑓 ∶ 𝐴 → 𝐵 , the extension of scalars functor 𝑓 ! ∶ D ( 𝐴 ) → D ( 𝐵 ) is left adjoint to the restriction of scalars functor 𝑓 ∗ ∶ D ( 𝐵 ) → D ( 𝐴 ) , which admits a further right adjoint 𝑓 ∗ ∶ D ( 𝐴 ) → D ( 𝐵 ) . If 𝐵 is moreover perfect as an 𝐴 -module (that is, it admits a finite resolution by finitelygenerated projective 𝐴 -modules), then 𝑓 ∗ restricts to a functor 𝑓 ∗ ∶ D p ( 𝐵 ) → D p ( 𝐴 ) on perfect objects.2.1.5. Lemma.
Let 𝑓 ∶ 𝐴 → 𝐵 be a ring homomorphism such that 𝐵 is a perfect 𝐴 -module, and let 𝑀 and 𝑁 be invertible modules with involution, respectively over 𝐴 and 𝐵 . Then any map Ψ ∶ ( 𝑓 ⊗ 𝑓 ) ∗ ( 𝑁 ) → 𝑀 induces a hermitian structure on the restriction functor 𝑓 ∗ ∶ ( D p ( 𝐵 ) , Ϙ s 𝑁 ) → ( D p ( 𝐴 ) , Ϙ s 𝑀 ) . This hermitianstructure is Poincaré if and only if the map 𝑁 → 𝑓 ∗ ( 𝑀 ) induced by Ψ is an equivalence in D ( 𝐵 ) .Proof. For such a Ψ , the hermitian structure on 𝑓 ∗ ∶ D p ( 𝐵 ) → D p ( 𝐴 ) is given by the natural transformation Ϙ s 𝑁 ( 𝑋 ) = hom 𝐵⊗𝐵 ( 𝑋 ⊗ 𝑋, 𝑁 ) hC ⟶ hom 𝐴⊗𝐴 (( 𝑓 ⊗ 𝑓 ) ∗ ( 𝑋 ⊗ 𝑋 ) , ( 𝑓 ⊗ 𝑓 ) ∗ 𝑁 ) hC ⟶ hom 𝐴⊗𝐴 ( 𝑓 ∗ ( 𝑋 ) ⊗ 𝑓 ∗ ( 𝑋 ) , ( 𝑓 ⊗ 𝑓 ) ∗ 𝑁 ) hC Ψ ∗ ⟶ hom 𝐴⊗𝐴 ( 𝑓 ∗ ( 𝑋 ) ⊗ 𝑓 ∗ ( 𝑋 ) , 𝑀 ) hC = Ϙ s 𝑀 ( 𝑓 ∗ ( 𝑋 )) . This hermitian structure is Poincaré if and only if the map 𝛾 ∶ 𝑁 → 𝑓 ∗ 𝑀 , adjoint to the map 𝑓 ∗ 𝑀 → 𝑁 obtained from Ψ by restricting the 𝐵 ⊗ 𝐵 -module structure to one factor, is an equivalence: To see this,observe that since D p ( 𝐵 ) is generated by 𝐵 via finite colimits and retracts it will suffice to show that theassociated natural transformation 𝑓 ∗ D( 𝑋 ) ⟶ D 𝑓 ∗ ( 𝑋 ) evaluates to an equivalence on 𝑋 = 𝐵 . Unwinding the definitions, the above map for 𝑋 = 𝐵 identifies withthe map 𝑓 ∗ 𝑁 → hom 𝐴 ( 𝑓 ∗ 𝐵, 𝑀 ) = 𝑓 ∗ 𝑓 ∗ 𝑀 which is the image under 𝑓 ∗ of 𝛾 ∶ 𝑁 → 𝑓 ∗ 𝑀 . Since 𝑓 ∗ isconservative this image is an equivalence if and only if 𝛾 is an equivalence. (cid:3) Now suppose that 𝔭 ⊆ 𝑅 is a non-zero prime ideal, and let 𝑝 ∶ 𝑅 → 𝔽 𝔭 ∶= 𝑅 ∕ 𝔭 be the quotient map. Wenote that 𝔭 is a rank 1 projective module and hence is ⊗ -invertible, where the inverse is given by the dual 𝔭 −1 ≃ hom 𝑅 ( 𝔭 , 𝑅 ) , see [SP18, Tag 0AUW]. We find that 𝔽 𝔭 is perfect as an 𝑅 -module, as it is representedby the chain complex ( 𝔭 → 𝑅 ) with 𝔭 in degree 1 and 𝑅 in degree . Let us then consider the adjunction 𝑝 ∗ ∶ D ( 𝔽 𝔭 ) ⟂ D ( 𝑅 ) ∶ 𝑝 ∗ and note that 𝑝 ∗ preserves compact objects since 𝑝 ∗ ( 𝔽 𝔭 ) is compact. It follows that 𝑝 ∗ preserves filteredcolimits, and since it is exact it must in fact preserve all colimits. We recall that 𝑝 ∗ is given by the formula 𝑝 ∗ ( 𝑋 ) = hom 𝑅 ( 𝑝 ∗ 𝔽 𝔭 , 𝑋 ) regarded as an 𝔽 𝔭 -module via the functoriality in the first variable. Moreover 𝑝 ∗ ( 𝑅 ) is an 𝔽 𝔭 ⊗ 𝑅 -modulevia the functoriality of 𝑝 ∗ , and there is an equivalence of functors 𝑝 ∗ ( 𝑋 ) ≃ 𝑋 ⊗ 𝑅 𝑝 ∗ ( 𝑅 ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 31 as both preserve colimits and agree on 𝑋 = 𝑅 . To determine the functor 𝑝 ∗ it hence suffices to calculate 𝑝 ∗ ( 𝑅 ) which we do in the following lemma, see also [SP18, Tag 0BZH].2.1.6. Lemma.
Let 𝑅 be a Dedekind ring and 𝔭 ⊆ 𝑅 a non-zero prime ideal. Then there is a canonicalequivalence 𝑝 ∗ ( 𝑅 )[1] ≃ 𝑝 ! ( 𝔭 −1 ) . A choice of uniformiser 𝜋 for 𝔭 thus induces an equivalence 𝑝 ∗ ( 𝑅 )[1] ≃ 𝔽 𝔭 .Proof. By applying the functor hom 𝑅 (− , 𝑅 ) to the fibre sequence 𝔭 → 𝑅 → 𝔽 𝔭 , and using that hom 𝑅 ( 𝔭 , 𝑅 ) = 𝔭 −1 , we find a fibre sequence of 𝑅 -modules 𝑝 ∗ 𝑝 ∗ ( 𝑅 ) ⟶ 𝑅 ⟶ 𝔭 −1 . From this, we find a canonical equivalence 𝑝 ∗ 𝑝 ∗ ( 𝑅 )[1] ≃ 𝔽 𝔭 ⊗ 𝑅 𝔭 −1 ≃ 𝑝 ∗ 𝑝 ! ( 𝔭 −1 ) . In fact, one obtains a canonical equivalence 𝑝 ∗ ( 𝑅 )[1] ≃ 𝑝 ! ( 𝔭 −1 ) since on discrete modules 𝑝 ∗ is fully faithful.Finally, we note that the map 𝑝 ∶ 𝑅 → 𝔽 𝔭 factors through the localisation 𝑅 ( 𝔭 ) of 𝑅 at 𝔭 . The base change 𝔭 ⊗ 𝑅 𝑅 ( 𝔭 ) is the maximal ideal in the local Dedekind ring 𝑅 ( 𝔭 ) . As any local Dedekind ring is a principalideal domain, one can choose a generator 𝜋 ∈ 𝑅 ( 𝔭 ) for this maximal ideal, called a uniformiser. Hence,the choice of a uniformiser determines an equivalence 𝑝 ! ( 𝔭 ) ≃ 𝔽 𝔭 and hence an equivalence 𝑝 ! ( 𝔭 −1 ) ≃ 𝑝 ! ( 𝔭 ) −1 ≃ 𝔽 𝔭 . (cid:3) Remark.
For a choice of uniformiser 𝜋 for 𝔭 , the induced equivalence 𝑝 ∗ ( 𝑅 )[1] ≃ 𝔽 𝔭 is concretelygiven as follows. As before, it suffices to describe the equivalence after applying 𝑝 ∗ . In this case we findthat 𝑝 ∗ 𝑝 ∗ ( 𝑅 )[1] = hom 𝑅 ( 𝔽 𝔭 , 𝑅 )[1] = cof ( 𝑅 𝜋 → 𝑅 ) = 𝑅 ∕ 𝜋 since 𝔽 𝔭 is the cofibre of the map 𝜋 ∶ 𝑅 → 𝑅 . In addition, the counit map 𝑝 ∗ 𝑝 ∗ ( 𝑅 ) → 𝑅 is the 𝑅 -lineardual of the map 𝑅 → 𝑅 ∕ 𝜋 , which is the Bockstein map 𝑅 ∕ 𝜋 [−1] → 𝑅 . We will use this observation inLemma 2.2.1.For what follows, we fix a uniformiser 𝜋 for 𝔭 , and remark that the corresponding isomorphism 𝑝 ∗ ( 𝑅 )[1] ≃ 𝔽 𝔭 only depends on the class of 𝜋 in 𝔭 ∕ 𝔭 . Suppose as earlier that 𝑀 is a line-bundle over 𝑅 with 𝑅 -linearinvolution. From the discussion preceding Lemma 2.1.6, we obtain an induced equivalence 𝑝 ! 𝑀 ∶= 𝔽 𝔭 ⊗ 𝑅 𝑀 ≃ ( 𝑝 ∗ 𝑅 [1]) ⊗ 𝑅 𝑀 = 𝑝 ∗ 𝑀 [1] where the involution on 𝑝 ! 𝑀 has the same sign as the one on 𝑀 . The adjoint of the equivalence abovedefines a map of modules with involution 𝑝 ∗ 𝑝 ! 𝑀 → 𝑀 [1] , and we may apply Lemma 2.1.5 to promote therestriction functor 𝑝 ∗ ∶ D p ( 𝔽 𝔭 ) → D p ( 𝑅 ) to a Poincaré functor ( D p ( 𝔽 𝔭 ) , Ϙ s 𝑝 ! 𝑀 ) ⟶ ( D p ( 𝑅 ) , Ϙ s 𝑀 [1] ) ≃ ( D p ( 𝑅 ) , ( Ϙ s 𝑀 ) [1] ) . The image of this functor lands in D p ( 𝑅 ) 𝑆 , yielding in particular a Poincaré functor 𝜓 𝔭 ∶ ( D p ( 𝔽 𝔭 ) , Ϙ s 𝑝 ! 𝑀 ) ⟶ ( D p ( 𝑅 ) 𝑆 , ( Ϙ s 𝑀 ) [1] ) . Theorem (Dévissage) . Let 𝑅 be a Dedekind ring, 𝑆 a set of non-zero prime ideals of 𝑅 with chosenuniformisers, and 𝑀 a line bundle over 𝑅 with 𝑅 -linear involution. Then for every 𝑚 ∈ ℤ the direct sumPoincaré functor (6) 𝜓 𝑆 ∶ ⊕ 𝔭 ∈ 𝑆 ( D p ( 𝔽 𝔭 ) , ( Ϙ s 𝑝 ! 𝑀 ) [ 𝑚 ] ) ⟶ ( D p ( 𝑅 ) 𝑆 , ( Ϙ s 𝑀 ) [ 𝑚 +1] ) induces equivalences on algebraic K -theory, GW -theory and L -theory spectra.Proof. First, note that by the fibre sequence of Corollary [II].4.4.14, it will be enough to prove the theoremfor algebraic K -theory and L -theory. Second, both sides of 𝜓 𝑆 depend on 𝑆 in a manner that preservesfiltered colimits. More specifically, if we write 𝑆 as a filtered colimit 𝑆 = colim 𝑆 ′ ⊆𝑆, | 𝑆 | < ∞ 𝑆 ′ of its finitesubsets then the direct sum on the left hand side of (6) is the colimit of the corresponding finite direct sums,while on the right hand side the full subcategory D p ( 𝑅 ) 𝑆 ⊆ D p ( 𝑅 ) is the union of all the full subcategories D p ( 𝑅 ) 𝑆 ′ for finite 𝑆 ′ ⊆ 𝑆 . Since both algebraic K -theory and L -theory commute with filtered colimits wemay reduce to the case where 𝑆 is finite. In this case, the left hand side of (6) can also be written as the product of D p ( 𝔽 𝔭 ) for varying 𝔭 ∈ 𝑆 (recall that Cat p∞ is semi-additive, see Proposition [I].6.1.7). In addition, each D p ( 𝔽 𝔭 ) , being the perfectderived category of a field, supports a 𝑡 -structure inherited from D ( 𝔽 𝔭 ) , and so we can endow the left handside of (6) with the product of the corresponding 𝑡 -structures. The heart of this product 𝑡 -structure is thenthe direct sum ⊕ 𝔭 ∈ 𝑆 Vect( 𝔽 𝔭 ) where Vect( 𝔽 𝔭 ) is the abelian category of finite dimensional 𝔽 𝔭 -vector spaces.We can also identify it with the category of finitely generated modules over the product ring ∏ 𝔭 ∈ 𝑆 𝔽 𝔭 .Concerning the right hand side, since 𝑅 and 𝑅 𝑆 have global dimension ≤ , the perfect derived categories D p ( 𝑅 ) and D p ( 𝑅 𝑆 ) inherit 𝑡 -structures from the respective unbounded derived categories. In addition, since 𝑅 𝑆 is flat over 𝑅 the localisation functor D p ( 𝑅 ) → D p ( 𝑅 𝑆 ) preserves both connective and truncated objects,and hence commutes with truncations and connective covers. As a result, its kernel D p ( 𝑅 ) 𝑆 ⊆ D p ( 𝑅 ) isclosed under truncation and connective covers, and so inherits a 𝑡 -structure from D p ( 𝑅 ) , so that the inclusion D p ( 𝑅 ) 𝑆 ⊆ D p ( 𝑅 ) commutes with truncations and connective covers. The heart of this 𝑡 -structure is then theabelian category Mod fin ( 𝑅 ) 𝑆 of finitely generated 𝑆 -primary torsion 𝑅 -modules. Now since (6) is inducedby the various restriction functors 𝑝 ∗ ∶ D p ( 𝔽 𝔭 ) → D p ( 𝑅 ) it preserves connective and truncated objects withrespect to the 𝑡 -structures just discussed. The functor(7) ⊕ 𝔭 ∈ 𝑆 Vect( 𝔽 𝔭 ) ⟶ Mod fin ( 𝑅 ) 𝑆 induced by (6) on the respective hearts is then fully-faithful (even though (6) itself is not fully-faithful) andcan be identified with the inclusion of the full subcategory of finitely generated 𝑆 -torsion modules insideall finitely generated 𝑆 -primary torsion modules (i.e. the full subcategory of semi-simple objects inside theabelian category Mod fin ( 𝑅 ) 𝑆 ). By the main result of Barwick [Bar15] the inclusion of hearts induces anequivalence on algebraic K -theory spectra on both the domain and codomain of (6). The desired claimfor algebraic K -theory is hence equivalent to saying that the inclusion of abelian categories (7) induces anequivalence on algebraic K -theory, which in turn follows from Quillen’s classical dévissage theorem foralgebraic K -theory [Qui73, Theorem §5.4].We will now show that (6) induces an equivalence on L -theory. Recall that by Corollary R.10, L-theorysupports natural equivalences L( C , Ϙ [1] ) ≃ Σ L( C , Ϙ ) . It will thus suffice to prove the claim for 𝑚 = 0 . Nowsince each 𝔽 𝔭 has global dimension the duality D 𝑝 ! 𝑀 on D p ( 𝔽 𝔭 ) maps -connective objects to -truncatedobjects and vice versa . The same hence holds for the product duality on ∏ 𝔭 ∈ 𝑆 D p ( 𝔽 𝔭 ) with respect to theproduct 𝑡 -structure. We now claim that this also holds for the Poincaré ∞ -category ( D p ( 𝑅 ) 𝑆 , ( Ϙ s 𝑀 ) [1] ) . Tosee this, note first that since 𝑅 has global dimension the shifted duality ΣD 𝑀 on D p ( 𝑅 ) sends D p ( 𝑅 ) ≥ to D p ( 𝑅 ) ≤ and D p ( 𝑅 ) ≤ to D p ( 𝑅 ) ≥ . On the other hand, if 𝑋 ∈ D p ( 𝑅 ) ≥ is 𝑆 ∞ -torsion then 𝜋 ΣD 𝑀 ( 𝑋 ) ≅ 𝜋 hom 𝑅 ( 𝑋, 𝑀 [0]) ≅ 𝜋 hom 𝑅 ( 𝜏 ≤ 𝑋, 𝑀 [0]) = Hom 𝑅 ( 𝜋 ( 𝑋 ) , 𝑀 ) = 0 since 𝜋 ( 𝑋 ) is 𝑆 -primary torsion and 𝑀 is torsion free. We then get that the duality on D p ( 𝑅 ) 𝑆 restrictedfrom D 𝑀 [1] sends ( D p ( 𝑅 ) 𝑆 ) ≥ to ( D p ( 𝑅 ) 𝑆 ) ≤ and vice versa. Hence for both sides of (6) we are in the situ-ation of Corollary 1.3.3, and so to finish the proof it will suffice to show that (7) induces an isomorphism onsymmetric and anti-symmetric Witt groups. But this follows from the dévissage result of [QSS79, Corollary6.9, Theorem 6.10]. (cid:3) The combination of the classical dévissage and localisation theorems of Quillen give rise to the fibresequence of K-theory spectra ⊕ 𝔭 ∈ 𝑆 K( 𝔽 𝔭 ) ⟶ K( 𝑅 ) ⟶ K( 𝑅 𝑆 ) . From Theorem 2.1.8 we obtain the corresponding sequences for the symmetric L and GW-spectra.2.1.9.
Corollary (Localisation-dévissage) . Under the assumptions of Theorem 2.1.8, the restriction andlocalisation functors yield fibre sequences of spectra ⊕ 𝔭 ∈ 𝑆 GW( 𝔽 𝔭 ; ( Ϙ s 𝑝 ! 𝑀 ) [ 𝑚 −1] ) GW( 𝑅 ; ( Ϙ s 𝑀 ) [ 𝑚 ] ) GW( 𝑅 𝑆 ; ( Ϙ s 𝑀 𝑆 ) [ 𝑚 ] ) ⊕ 𝔭 ∈ 𝑆 L( 𝔽 𝔭 ; ( Ϙ s 𝑝 ! 𝑀 ) [ 𝑚 −1] ) L( 𝑅 ; ( Ϙ s 𝑀 ) [ 𝑚 ] ) L( 𝑅 𝑆 ; ( Ϙ s 𝑀 𝑆 ) [ 𝑚 ] ) for every 𝑚 ∈ ℤ . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 33 Remark.
A variant of Corollary 2.1.9 for L -theory of short complexes in non-negative degrees(which, for Dedekind rings, coincides with symmetric L -theory in non-negative degrees by Theorem 1.2.18and Corollary 1.3.11), was proven by Ranicki in [Ran81, §4.2]. For Grothendieck-Witt theory, Hornbosteland Schlichting prove a dévissage statement and obtain a localisation sequence of the type of Corollary 2.1.9under the assumption that is a unit in 𝑅 , see [Hor02], [HS04]. Apart from the announcement [Sch19b, The-orem 3.2] which provides the above fibre sequence for GW after passing to connective covers, we arenot aware of any previous results in the literature for Grothendieck-Witt spaces, along the lines of Corol-lary 2.1.9, for rings in which is not invertible.2.1.11. Remark.
The dévissage result above is indeed a special feature of the symmetric Poincaré structure:It is the only among the genuine structures for which this results holds on the spectrum level (not just in arange of degrees). Indeed, to see this it suffices, by Corollary 1.2.8, to argue that dévissage fails for quadraticL-theory. For an explicit example, one can note that the maps
Ω L q ( 𝔽 ) ⟶ L q ( ℤ ) ⟶ L q ( ℤ [ ]) cannot be part of a fibre sequence, for instance because it would imply that L q2 ( ℤ ) = 0 , which is not the case.Here, we use that L q2 ( ℤ [ ]) ≅ L s2 ( ℤ [ ]) = 0 , which is of course well-known, but see also Corollary 2.2.4below, as well as the vanishing of L q3 ( 𝔽 ) , see Remark 1.3.6.In the next subsection, we will use the localisation-dévissage sequence to calculate the symmetric L-theory of Dedekind rings. By Remark 2.1.11, this strategy, however, will not allow us to calculate thequadratic L-groups of Dedekind rings. Instead we will make use of a general localisation-completion prop-erty, Proposition 2.1.12 below, and a rigidity property of quadratic L-theory, Proposition 2.1.13. Following§[II].A.4, for a subgroup c ⊂ K ( 𝑅 ) fixed by the involution, we let D c ( 𝑅 ) denote the full subcategory of D p ( 𝑅 ) spanned by the complexes whose K -class lies in c .2.1.12. Proposition.
Let 𝑅 be a ring, 𝑀 an invertible module with involution over 𝑅 , and 𝑆 the multi-plicatively closed subset generated by an integer 𝓁 ∈ 𝑅 . Assume that the 𝓁 ∞ -torsion in 𝑅 is bounded, forinstance that 𝓁 is a non-zero divisor. Then the square ( D p ( 𝑅 ) , Ϙ ≥ 𝑚𝑀 ) ( D p ( 𝑅 ∧ 𝓁 ) , Ϙ ≥ 𝑚𝑀 ∧ 𝓁 )( D c ( 𝑅 [ 𝓁 ]) , Ϙ ≥ 𝑚𝑆 −1 𝑀 ) ( D c ′ ( 𝑅 ∧ 𝓁 [ 𝓁 ]) , Ϙ ≥ 𝑚𝑆 −1 ( 𝑀 ∧ 𝓁 ) ) is a Poincaré-Verdier square for all 𝑚 ∈ ℤ ∪ {±∞} , where 𝑐 = im(K ( 𝑅 ) → K ( 𝑅 [ 𝓁 ])) , and 𝑐 ′ =im(K ( 𝑅 ∧ 𝓁 ) → K ( 𝑅 ∧ 𝓁 [ 𝓁 ])) . In particular it becomes a pullback after applying GW or L .Proof. We show that the canonical maps 𝑓 ∶ 𝑅 → 𝑅 ∧ 𝓁 and 𝛼 ∶ 𝑀 → ( 𝑓 ⊗ 𝑓 ) ∗ ( 𝑀 ∧ 𝓁 ) satisfy the conditionsof Proposition [II].4.4.21. The morphism 𝑅 ∧ 𝓁 ⊗ 𝑅 𝑀 → ( 𝑅 ∧ 𝓁 ⊗ 𝑅 ∧ 𝓁 ) ⊗ 𝑅⊗𝑅 𝑀 → 𝑀 ∧ 𝓁 is indeed an equivalence: This is clear for 𝑀 = 𝑅 , which implies the general case since 𝑀 is a finitelygenerated projective 𝑅 -module. Moreover the square 𝑅 𝑅 ∧ 𝓁 𝑅 [ 𝓁 ] 𝑅 ∧ 𝓁 [ 𝓁 ] is a derived pullback, see for instance [DG02, §4], as the assumption on 𝓁 ∞ -torsion implies that 𝑅 ∧ 𝓁 is alsoa derived completion. The final thing to check is that the map 𝑀 ∧ 𝓁 [ 𝓁 ] → 𝑀 [1] induces the zero map in C -Tate cohomology. This follows from the fact that the domain is a ℚ -vector space, and so has trivial C -Tate cohomology. (cid:3) To make efficient use of the localisation-completion square, we shall also need the following result dueto Wall [Wal73, Lemma 5]. We include a guide through the proof merely for convenience of the reader,as to avoid confusion about different definitions (and versions) of L-theory. We warn the reader that whatis denoted by 𝐿 𝐾𝑖 ( 𝑅 ) in [Wal73] is what we would denote L( D f ( 𝑅 ) , Ϙ q ) , i.e. quadratic L-theory based oncomplexes of (stably) free modules.2.1.13. Proposition.
Let 𝑅 be ring, complete in the 𝐼 -adic topology for an ideal 𝐼 of 𝑅 . Then the canonicalmap L q ( 𝑅 ) → L q ( 𝑅 ∕ 𝐼 ) is an equivalence.Proof. First, we claim that the functor
Unimod q ( 𝑅 ; 𝜖 ) → Unimod q ( 𝑅 ∕ 𝐼 ; 𝜖 ) induces a bijection on isomor-phism classes, for 𝜖 = ±1 . To see this, we first observe that the functor Proj( 𝑅 ) → Proj( 𝑅 ∕ 𝐼 ) is full andessentially surjective. Moreover, for any finitely generated projective 𝑅 module 𝑃 , the map Ϙ q 𝜖 ( 𝑃 ) → Ϙ q 𝜖 ( 𝑃 ⊗ 𝑅 𝑅 ∕ 𝐼 ) is surjective on 𝜋 , and furthermore an 𝜖 -quadratic form is unimodular if and only if its image over 𝑅 ∕ 𝐼 is; this is as a consequence of Nakayama’s lemma: 𝐼 is contained in the Jacobsen radical as we have as-sumed that 𝑅 is 𝐼 -complete. We deduce that the above map is surjective. To see injectivity, we apply[Wal70, Theorem 2]: amongst other things, it says that given forms ( 𝑃 , 𝑞 ) and ( 𝑃 ′ , 𝑞 ′ ) over 𝑅 , then anyisometry between their induced forms over 𝑅 ∕ 𝐼 can be lifted to an isometry over 𝑅 . In particular, themap Unimod q ( 𝑅 ; 𝜖 ) → Unimod q ( 𝑅 ∕ 𝐼 ; 𝜖 ) is also injective on isomorphism classes. We deduce that themap GW q0 ( 𝑅 ; 𝜖 ) → GW q0 ( 𝑅 ∕ 𝐼 ; 𝜖 ) is an isomorphism. Since likewise the map K ( 𝑅 ) → K ( 𝑅 ∕ 𝐼 ) is anisomorphism, we deduce that L q0 ( 𝑅 ; 𝜖 ) → L q0 ( 𝑅 ∕ 𝐼 ; 𝜖 ) is an isomorphism as well. We then consider thediagram 𝜋 (K( 𝑅 ; 𝜖 ) hC ) GW q1 ( 𝑅 ; 𝜖 ) L q1 ( 𝑅 ; 𝜖 ) K ( 𝑅 ; 𝜖 ) C GW q0 ( 𝑅 ; 𝜖 ) 𝜋 (K( 𝑅 ∕ 𝐼 ; 𝜖 ) hC ) GW q1 ( 𝑅 ∕ 𝐼 ; 𝜖 ) L q1 ( 𝑅 ∕ 𝐼 ; 𝜖 ) K ( 𝑅 ∕ 𝐼 ; 𝜖 ) C GW q0 ( 𝑅 ∕ 𝐼 ; 𝜖 ) ≅ ≅ where [Wal73, Corollary 1 & Lemma 1] give that the two left most vertical maps are surjective, and[Wal73, Proposition 4] that the induced map on vertical kernels is surjective. This implies that the map L q1 ( 𝑅 ; 𝜖 ) → L q1 ( 𝑅 ∕ 𝐼 ; 𝜖 ) is an isomorphism. From the general periodicity L q 𝑛 ( 𝑅 ; 𝜖 ) ≅ 𝐿 q 𝑛 +2 ( 𝑅 ; − 𝜖 ) we de-duce the proposition. (cid:3) Symmetric and quadratic L-groups of Dedekind rings.
In this section we show that the classicalsymmetric and quadratic Grothendieck-Witt groups of certain Dedekind rings are finitely generated. ByTheorem 1 it will suffice to prove the finite generation of the corresponding L-groups, provided the finitegeneration of the K-groups is known. On the L-theory side we in fact do much more: We give a fullcalculation of the quadratic and symmetric L-groups of Dedekind rings whose field of fractions is notof characteristic 2. We first treat the symmetric case, where we need to make the boundary map of thelocalisation-dévissage sequence explicit: Shifting the L-theory fibre sequence of Corollary 2.1.9 once tothe right, we obtain a fibre sequence L s ( 𝑅 ) ⟶ L s ( 𝐾 ) 𝜕 ⟶ ⊕ 𝔭 L s ( 𝔽 𝔭 ) where we recall that 𝑅 is a Dedekind ring with field of fractions 𝐾 , and 𝔽 𝔭 is the residue field at a primeideal 𝔭 of 𝑅 . As we shall use it momentarily, let us make the effect of the map 𝜕 on 𝜋 explicit. Clearly, itsuffices to describe the composite of 𝜕 with the projection to L s ( 𝔽 𝔭 ) for each prime 𝔭 of 𝑅 . By naturalityof the dévissage theorem and the localisation sequence, to describe this composition we may replace 𝑅 by its localisation 𝑅 ( 𝔭 ) which is a local Dedekind ring and hence a discretely valued ring, as the choice ofuniformiser for 𝔭 is (by definition) also a uniformiser for 𝔭 , viewed as prime ideal in 𝑅 ( 𝔭 ) . Without loss ofgenerality, we may hence assume that 𝑅 was a discretely valued ring to begin with. Let 𝜋 be a uniformiserof the maximal ideal of 𝑅 , so that every non-zero element in 𝐾 is uniquely of the form 𝜋 𝑖 𝑢 for some unit 𝑢 in 𝑅 . Clearly, it suffices to describe the map 𝜕 ∶ L s0 ( 𝐾 ) → L s0 ( 𝔽 𝔭 ) on generators of the L-group, whichare given by the forms ⟨ 𝑥 ⟩ = ( 𝐾, 𝑥 ) for units 𝑥 of 𝐾 , where we have identified canonically 𝜋 ( Ϙ s ( 𝐾 )) with 𝐾 . Indeed, if the characteristic of 𝐾 is not 2, then every form itself is isomorphic to a diagonal form. If the ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 35 characteristic is 2, then any unimodular form is the sum of a diagonalisable form and one which admits aLagrangian, see [MH73, I §3]. By a change of basis, one finds the relation ⟨ 𝑥 ⟩ = ⟨ 𝑥𝑦 ⟩ for any other unit 𝑦 . We may thus suppose without loss of generality that 𝑥 is either of the form 𝜋𝑢 or of the form 𝑢 , againfor 𝑢 a unit in 𝑅 . By exactness of the localisation-dévissage sequence, we have 𝜕 ⟨ 𝑢 ⟩ = 0 , so it remains todescribe 𝜕 ⟨ 𝜋𝑢 ⟩ .2.2.1. Lemma.
In the notation just established, we have 𝜕 ( ⟨ 𝜋𝑢 ⟩ ) = ⟨ 𝑢 ⟩ in L s0 ( 𝔽 𝔭 ) . Here we view 𝑢 also asa unit of 𝔽 𝔭 via the projection 𝑅 → 𝔽 𝔭 . In particular, the map L s0 ( 𝐾 ) → L s0 ( 𝔽 𝔭 ) is surjective.Proof. We note that, by construction, the composite L s0 ( 𝐾 ) 𝜕 ⟶ L s0 ( 𝔽 𝑝 ) ≅ ⟶ L ( D p ( 𝑅 ) ( 𝜋 ) , ( Ϙ s ) [1] ) , where the second map is the dévissage isomorphism of Theorem 2.1.8, is the boundary map associated tothe Poincaré-Verdier sequence ( D p ( 𝑅 ) ( 𝜋 ) , Ϙ s ) ⟶ ( D p ( 𝑅 ) , Ϙ s ) ⟶ ( D p ( 𝐾 ) , Ϙ s ) . It hence suffices to prove that the image of ⟨ 𝑢 ⟩ under the dévissage isomorphism is mapped to the imageof ⟨ 𝜋𝑢 ⟩ under this boundary map. The boundary map for this localisation sequence is explicitly describedas follows, see Proposition [II].4.4.8. Starting with the Poincaré object ( 𝐾, 𝜋𝑢 ) , we may view the her-mitian object ( 𝑅, 𝜋𝑢 ) as a surgery datum on for ( Ϙ s ) [1] . The output of surgery is a Poincaré object for ( D p ( 𝑅 ) ( 𝜋 ) , ( Ϙ s ) [1] ) which is the value of the boundary map at ( 𝐾, 𝜋𝑢 ) . It is immediate from the diagram ofRemark 1.1.13 that the underlying object of the surgery output is the complex 𝑅 ∕ 𝜋𝑢 . In order to describe itsPoincaré form we interpret the surgery as an instance of the algebraic Thom isomorphism, which induces amap from (possibly degenerate) forms for Ϙ to Poincaré objects for Ϙ [1] . In general, this map is implementedby the following construction. Let 𝛼 be a form in 𝜋 ( Ϙ ( 𝐿 )) regarded as a surgery datum ( 𝐿, 𝛼 ) on for Ϙ [1] .The output of surgery is a Poincaré form on the cofibre 𝐶 of the map 𝛼 ♯ ∶ 𝐿 → D 𝐿 . The form is constructedfrom the cofibre sequence(8) Ϙ (D 𝐿 ) ⟶ Ϙ ( 𝐿 ) × hom( 𝐿, D 𝐿 ) hom(D 𝐿, D 𝐿 ) 𝜕 ⟶ Σ Ϙ ( 𝐶 ) , from Example [I].1.1.21, by noticing that the pair ( 𝛼, id D 𝐿 ) canonically refines to a point in the pullbackabove. Its image under 𝜕 is the form on 𝐶 we seek. We wish to make the boundary map 𝜕 explicit, as theform on 𝐶 is the image of an explicit point in the pullback. We again take a step back and consider a generalcofibre sequence 𝑋 → 𝑌 → 𝐶 instead of 𝐿 → D 𝐿 → 𝐶 . The above cofibre sequence (8) is a reformulationof the fact that Ϙ ( 𝐶 ) is the total fibre of the left square in the diagram of horizontal cofibre sequences(9) Ϙ ( 𝑌 ) Ϙ ( 𝑋 ) 𝑀 B( 𝑌 , 𝑋 ) B(
𝑋, 𝑋 ) 𝑀 ′ where the left vertical map is the composite Ϙ ( 𝑌 ) → B( 𝑌 , 𝑌 ) → B( 𝑌 , 𝑋 ) where the latter map is inducedby the map 𝑋 → 𝑌 . We note that Σ Ϙ ( 𝐶 ) , being the suspension of the total fibre, is equivalently given by thefibre of the map of horizontal cofibres, i.e. the fibre of the map 𝑀 → 𝑀 ′ displayed above. The boundarymap above is then given as follows: a map to the fibre of 𝑀 → 𝑀 ′ consists of a map to 𝑀 , together with anull homotopy of the composite to 𝑀 ′ . Now the pullback of the left upper square canonically maps to Ϙ ( 𝐴 ) which in turn maps to 𝑀 . The composite of this map to 𝑀 ′ factors through the lower horizontal cofibresequence and is thus canonically trivialised. These two pieces of data together determine the map from thepullback to Σ Ϙ ( 𝐶 ) .Spelling this diagram out for the fibre sequence 𝑅 𝑥 → 𝑅 → 𝑅 ∕ 𝑥 , for an element 𝑥 in 𝑅 , we obtain thediagram Ϙ s ( 𝑅 ) Ϙ s ( 𝑅 ) ( 𝑅 ∕ 𝑥 ) hC hom 𝑅 ( 𝑅, 𝑅 ) hom 𝑅 ( 𝑅, 𝑅 ) 𝑅 ∕ 𝑥 ( ⋅ 𝑥 ) ∗ ( ⋅ 𝑥 ) ∗ where we have used the identification Ϙ s ( 𝑅 ) = 𝑅 hC and that the map induced by multiplication with 𝑥 on 𝑅 becomes the map induced by multiplication with 𝑥 . We notice that the left vertical map identifieswith the composite of the forgetful map 𝑅 hC → 𝑅 and the multiplication by 𝑥 on 𝑅 . Now the point in thepullback is given by the point in 𝑅 , the point 𝑥 in 𝑅 hC and the canonical identification of their imagesin 𝑅 along the two maps. Since 𝑅 has no nonzero zero divisors, 𝑅 ∕ 𝑥 is discrete, and hence the first map inthe sequence 𝜋 (Σ Ϙ s 𝑅 ( 𝑅 ∕ 𝑥 )) ⟶ 𝜋 (( 𝑅 ∕ 𝑥 ) hC ) ⟶ 𝜋 ( 𝑅 ∕ 𝑥 ) is injective. We hence obtain an isomorphism 𝛼 𝑥 ∶ ( 𝑥 )∕( 𝑥 ) → 𝜋 (Σ Ϙ s 𝑅 ( 𝑅 ∕ 𝑥 )) , as the latter of the above twomaps canonically identifies with the map 𝑅 ∕ 𝑥 → 𝑅 ∕ 𝑥 . Now, the image of the point ( 𝑥, in 𝜋 (( 𝑅 ∕ 𝑥 ) hC ) ≅ 𝑅 ∕ 𝑥 is the image of 𝑥 in the quotient 𝑅 ∕ 𝑥 . Summarising, we find that for the element 𝜋𝑢 of 𝑅 , we havethat the output of surgery is given by the pair ( 𝑅 ∕( 𝜋𝑢 ) , 𝛼 𝜋𝑢 ( 𝜋𝑢 ) ∈ 𝜋 (Σ Ϙ s 𝑅 ( 𝑅 ∕( 𝜋𝑢 ))) ) . Next, we claim that the diagram Ϙ s 𝔽 𝔭 ( 𝑅 ∕ 𝜋 ) ≃ ( 𝑅 ∕ 𝜋 ) hC hom 𝑅 ( 𝑅 ∕ 𝜋 ⊗ 𝑅 𝑅 ∕ 𝜋, 𝑅 ∕ 𝜋 ) hC hom 𝑅 ( 𝑅 ∕ 𝜋 ⊗ 𝑅 𝑅 ∕ 𝜋, 𝑅 [1]) hC 𝑝 ∗ dev 𝜋 𝛽 𝜋 commutes, where dev 𝜋 denotes the map induced from the Poincaré functor 𝑝 ∗ as constructed after Lemma 2.1.6.This follows from the observation that the counit 𝑝 ∗ 𝑝 ∗ ( 𝑅 )[1] → 𝑅 [1] identifies with the Bockstein map 𝑅 ∕ 𝜋 → 𝑅 [1] , see Remark 2.1.7. It is easy to see that both upper terms have 𝜋 canonically isomorphic to 𝑅 ∕ 𝜋 , and that the horizontal map identifies with the identity. To identify the effect of the Bockstein mapon 𝜋 of the above diagram, we note that the functor Φ( 𝑋 ) = hom 𝑅 ( 𝑋 ⊗ 𝑅 𝑋, 𝑅 ∕ 𝜋 ) hC is a quadratic functor (though not non-degenerate), so we may apply the same method as for Σ Ϙ s 𝑅 ( 𝑅 ∕ 𝑥 ) , in-dicated in diagram (9), to calculate Φ( 𝑅 ∕ 𝑥 ) . The Bockstein map induces a comparison map of the diagrams.Passing to 𝜋 in the above triangle gives the diagram 𝑅 ∕ 𝜋 𝜋 (hom 𝑅 ( 𝑅 ∕ 𝜋 ⊗ 𝑅 𝑅 ∕ 𝜋, 𝑅 ∕ 𝜋 ) hC ) 𝑅 ∕ 𝜋𝜋 (hom 𝑅 ( 𝑅 ∕ 𝜋 ⊗ 𝑅 𝑅 ∕ 𝜋, 𝑅 [1]) hC ) ( 𝜋 )∕( 𝜋 ) 𝛼 𝜋 and from this it is an explicit calculation to see that the dashed vertical map in the above diagram is the mapgiven by multiplication by 𝜋 . We deduce that the map dev 𝜋 takes the form 𝑢 on 𝑅 ∕ 𝜋 to the form 𝛼 𝜋 ( 𝜋𝑢 ) on 𝑅 ∕ 𝜋 . We are thus left to compare the elements ( 𝑅 ∕ 𝜋, 𝛼 𝜋 ( 𝜋𝑢 )) and ( 𝑅 ∕( 𝜋𝑢 ) , 𝛼 𝜋𝑢 ( 𝜋𝑢 )) of L ( D p ( 𝑅 ) 𝔭 , Σ Ϙ s ) . For this, we need to work out a formula for the isomorphism 𝛼 𝜋𝑢 in terms of 𝛼 𝜋 . Thiswe do as follows. We consider the diagram 𝑅 𝑅 𝑅 ∕ 𝜋𝑅 𝑅 𝑅 ∕ 𝜋𝑢 𝜋 𝑢 𝑢𝜋𝑢 which induces a map 𝑢 ∗ ∶ Σ Ϙ s ( 𝑅 ∕ 𝜋𝑢 ) → Σ Ϙ s ( 𝑅 ∕ 𝜋 ) . One checks that the diagram ( 𝜋𝑢 )∕( 𝜋𝑢 ) 𝜋 (Σ Ϙ s ( 𝑅 ∕ 𝜋𝑢 ))( 𝜋 )∕( 𝜋 ) 𝜋 (Σ Ϙ s ( 𝑅 ∕ 𝜋 )) 𝛼 𝜋𝑢 𝑢 −2 𝑢 ∗ 𝛼 𝜋 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 37 commutes. In formulas, we find that 𝑢 ∗ ( 𝛼 𝜋𝑢 ( 𝜋𝑢 )) = 𝛼 𝜋 ( 𝜋𝑢 −1 ) so that we obtain that the output of surgery on ( 𝑅, 𝜋𝑢 ) is given by the pair ( 𝑅 ∕ 𝜋, 𝛼 𝜋 ( 𝜋𝑢 −1 )) which equalsthe image of ( 𝑅 ∕ 𝜋, 𝑢 −1 ) under the dévissage map. Since the two forms 𝑢 −1 and 𝑢 differ by a square, theyrepresent the same class in L-theory, so we finally deduce the lemma. (cid:3) Remark.
Lemma 2.2.1 identifies the map 𝜕 ∶ L s0 ( 𝐾 ) → ⊕ 𝔭 L s0 ( 𝔽 𝔭 ) with the map induced by themaps 𝜓 ∶ W s ( 𝐾 ) → W s ( 𝔽 𝔭 ) constructed in [MH73, Chapter IV §1].We obtain the following calculation. Recall that a prime is called dyadic if it contains the ideal (2) .2.2.3. Corollary.
Let 𝑅 be a Dedekind ring whose field of fractions 𝐾 is not of characteristic 2, and let I be the (finite) set of dyadic primes of 𝑅 . Then we have L s 𝑛 ( 𝑅 ) ≅ ⎧⎪⎪⎨⎪⎪⎩ W s ( 𝑅 ) for 𝑛 ≡ ⊕ 𝔭 ∈ I W s ( 𝔽 𝔭 ) for 𝑛 ≡ for 𝑛 ≡ 𝜕 ) for 𝑛 ≡ Proof.
The case 𝑛 ≡ follows from combining Corollary 1.2.12 and Corollary 1.3.11. Since the sym-metric L -groups of 𝐾 and each residue field 𝔽 𝔭 vanish in odd degrees by Corollary 1.3.5, the long exactsequence in L -groups furnished by Corollary 2.1.9 yields for every 𝑘 an exact sequence ⟶ L s2 𝑘 ( 𝑅 ) ⟶ L s2 𝑘 ( 𝐾 ) 𝜕 𝑘 ⟶ ⊕ 𝔭 L s2 𝑘 ( 𝔽 𝔭 ) ⟶ L s2 𝑘 −1 ( 𝑅 ) ⟶ . This shows the case 𝑛 ≡ , as for odd numbers 𝑘 , the group L s2 𝑘 ( 𝐾 ) is isomorphic to the anti-symmetricWitt group of 𝐾 by Corollary 1.3.5, which vanishes as the characteristic of 𝐾 is not 2. The remaining casesare obvious from the above exact sequence, making use of the fact that the symmetric L-theory of fields ofcharacteristic 2, like 𝔽 𝔭 for dyadic primes, is 2-periodic, whereas the symmetric L-theory of fields of oddcharacteristic vanishes in degrees different from ≡ , see Remark 1.3.6. (cid:3) Corollary.
Under the assumptions of Corollary 2.2.3, assume in addition that 𝐾 is a global field andlet 𝑑 = | I | be the (finite) number of dyadic primes of 𝑅 . Then we have L s 𝑛 ( 𝑅 ) = ⎧⎪⎪⎨⎪⎪⎩ W s ( 𝑅 ) for 𝑛 ≡ ℤ ∕2) 𝑑 for 𝑛 ≡ for 𝑛 ≡ 𝑅 )∕2 for 𝑛 ≡ Proof.
The case 𝑛 ≡ follows since the assumption that 𝐾 is global says that the residue fields 𝔽 𝔭 atnon-zero primes are finite fields. The claim then follows from the fact that the symmetric Witt group ofa finite field of characteristic 2 is given by ℤ ∕2 . For the other non-trivial case, Lemma 2.2.1 gives thefollowing commutative diagram W s ( 𝐾 ) ⊕ 𝔭 W s ( 𝔽 𝔭 )L s0 ( 𝐾 ) ⊕ 𝔭 L s0 ( 𝔽 𝔭 ) 𝜓 ≅ ≅ 𝜕 It is then shown in [MH73, Chapter IV §4] that the cokernel of the upper horizontal map is given by
Pic( 𝑅 )∕2 , provided 𝐾 is a number field. In [Sch12, Chapter 6, §6, Theorem 6.11] this is extended tohold for a general global field 𝐾 . (cid:3) Remark.
We recall that there is a canonical equivalence L −s ( 𝑅 ) ≃ Σ L s ( 𝑅 ) , so that Corollaries 2.2.3and 2.2.4 also determine the (−1) -symmetric L-groups.When the fraction field of 𝑅 is a global field of characteristic we have a similar result: Corollary.
Let 𝑅 be a Dedekind ring whose field of fractions 𝐾 is a global field of characteristic .Then L s 𝑛 ( 𝑅 ) = { W s ( 𝑅 ) for 𝑛 ≡ 𝑅 )∕2 for 𝑛 ≡ Proof.
As in the proof of Corollary 2.2.3, we have an exact sequence ⟶ L s2 𝑘 ( 𝑅 ) ⟶ L s2 𝑘 ( 𝐾 ) 𝜕 𝑘 ⟶ ⊕ 𝔭 L s2 𝑘 ( 𝔽 𝔭 ) ⟶ L s2 𝑘 −1 ( 𝑅 ) ⟶ . Since 𝑅 is an 𝔽 -algebra, the L-groups L s 𝑛 ( 𝑅 ) are 2-periodic and since 𝐾 is a global field, all residue fields 𝔽 𝔭 are (finite) fields of characteristic 2. We recall that there is an exact sequence of abelian groups 𝐾 × div ⟶ Div( 𝑅 ) ⟶ Pic( 𝑅 ) ⟶ where Div( 𝑅 ) is the free abelian group generated by the prime ideals of 𝑅 , and the group homomorphism div is determined by the following: For a non-zero element 𝑥 of 𝑅 , write ( 𝑥 ) = 𝔭 𝑟 ⋅ ⋯ ⋅ 𝔭 𝑟 𝑛 𝑛 with naturalnumbers 𝑟 𝑖 . Then div( 𝑥 ) = ∑ 𝑛𝑖 =1 𝑟 𝑖 ⋅ 𝔭 𝑖 . Now consider the diagram ℤ ∕2[ 𝐾 × ] Div( 𝑅 )∕2 Pic( 𝑅 )∕2 0W s ( 𝐾 ) ⊕ 𝔭 W s ( 𝔽 𝔭 ) L s1 ( 𝑅 ) 0 ⟨ − ⟩ ≅ 𝜕 consisting of exact horizontal sequences and the left most top vertical map is induced by the map div above.Here, the middle vertical isomorphism is induced from the isomorphism Div( 𝑅 )∕2 ≅ ⊕ 𝔭 ℤ ∕2 and theisomorphisms W s ( 𝔽 𝔭 ) ≅ ℤ ∕2 . The left square commutes by an explicit check, so that there exists a dashedarrow as indicated. By construction, the dashed map is a surjection, and an injection by the observation thatthe left most vertical map is surjective, as the Witt group W s ( 𝐾 ) is generated by the forms ⟨ 𝑥 ⟩ for 𝑥 ∈ 𝐾 × by [MH73, I §3]. (cid:3) Remark.
Let 𝑅 be a local Dedekind ring with fraction field 𝐾 and residue field 𝑘 . Then the map 𝜕 ∶ L s0 ( 𝐾 ) → L s0 ( 𝑘 ) is surjective by Lemma 2.2.1, and we have seen earlier that its kernel is L s0 ( 𝑅 ) .Assuming that the characteristic of 𝐾 is not , we deduce from Corollary 2.2.3 that L s 𝑛 ( 𝑅 ) vanishes for 𝑛 ≡ , . Furthermore, for 𝑛 ≡ , we find that L s 𝑛 ( 𝑅 ) is either isomorphic to W s ( 𝑘 ) , ifthe characteristic of 𝑘 is 2, or is trivial otherwise. If the characteristic of 𝐾 is 2, we deduce from the proofof Corollary 2.2.6 that L s1 ( 𝑅 ) = 0 .We now want to give a formula for the quadratic L-groups of Dedekind rings, similar to Corollary 2.2.3and Corollary 2.2.4. We set out to prove the following result:2.2.8. Proposition.
Let 𝑅 be a Dedekind ring whose field of fractions 𝐾 is not of characteristic 2, and let I be the (finite) set of dyadic primes of 𝑅 . Then we have L q 𝑛 ( 𝑅 ) ≅ ⎧⎪⎨⎪⎩ W q ( 𝑅 ) for 𝑛 ≡ for 𝑛 ≡ ⊕ 𝔭 ∈ I W q ( 𝔽 𝔭 ) for 𝑛 ≡ The isomorphism in degrees 𝑛 ≡ is induced by the canonical maps 𝑅 → 𝔽 𝔭 for each dyadic prime 𝔭 .For 𝑛 ≡ there is a short exact sequence ⟶ 𝐴 ⟶ L q 𝑛 ( 𝑅 ) ⟶ L s 𝑛 ( 𝑅 ) ⟶ where 𝐴 is the total cokernel, that is the cokernel of the map induced on cokernels, of the commutativesquare L q0 ( 𝑅 ) L s0 ( 𝑅 )L q0 ( 𝑅 ∧ ) L s0 ( 𝑅 ∧ ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 39 Proof.
The canonical map W q ( 𝑅 ) → L q0 ( 𝑅 ) is an isomorphism by Corollary 1.2.12. To see the other cases,we consider the cube L s ( 𝑅 ) L s ( 𝑅 ∧ )L q ( 𝑅 ) L q ( 𝑅 ∧ )L s ( 𝑅 [ ]) L s ( 𝑅 ∧ [ ])L q ( 𝑅 [ ]) L q ( 𝑅 ∧ [ ]) which is obtained by mapping the quadratic localisation-completion square appearing in Proposition 2.1.12to the symmetric one. We note that no control terms are needed since localisations of Dedekind ringsinduce surjections on K . In this cube, the front and back squares are pullbacks by Proposition 2.1.12, andthe bottom square is a pullback since in all rings that appear 2 is invertible. We deduce that the diagram(10) L q ( 𝑅 ) L q ( 𝑅 ∧ )L s ( 𝑅 ) L s ( 𝑅 ∧ ) is also a pullback.Now all remaining statements to be proven follow from the long exact Mayer-Vietoris sequence associ-ated to this pullback, using the following:i) L q 𝑛 ( 𝑅 ∧ ) = 0 for odd 𝑛 , by Proposition 2.1.13,ii) L s 𝑛 ( 𝑅 ∧ ) = 0 for 𝑛 ≡ , because 𝑅 ∧ is a product of local Dedekind rings; see Remark 2.2.7,iii) L s 𝑛 ( 𝑅 ) = L s 𝑛 ( 𝑅 ∧ ) = 0 for 𝑛 ≡ by Corollary 2.2.3, andiv) the map L s1 ( 𝑅 ) → L s1 ( 𝑅 ∧ ) is an isomorphism. This can be seen from the localisation-completionsquare for symmetric L-theory and iii). (cid:3) Corollary.
Under the assumptions of Proposition 2.2.8, assume in addition that 𝐾 is a number fieldand let 𝑑 = | I | be the (finite) number of dyadic primes of 𝑅 . Then we have L q 𝑛 ( 𝑅 ) ≅ ⎧⎪⎨⎪⎩ W q ( 𝑅 ) for 𝑛 ≡ for 𝑛 ≡ ℤ ∕2) 𝑑 for 𝑛 ≡ The invariants in the case 𝑛 ≡ are given by the Arf invariants of the images in the L-theory of 𝔽 𝔭 foreach dyadic prime 𝔭 . Moreover, there is an exact sequence ⟶ 𝐴 ⟶ L q−1 ( 𝑅 ) ⟶ Pic( 𝑅 )∕2 ⟶ where 𝐴 is as in Proposition 2.2.8 and is a finite 2-group.Proof. First we recall from Corollary 2.2.4 that for 𝑛 ≡ , we have L s ( 𝑅 ) ≅ Pic( 𝑅 )∕2 , and that 𝐴 is aquotient of L s0 ( 𝑅 ∧ ) . We have (2) = ( 𝔭 𝑒 ⋅ ⋯ ⋅ 𝔭 𝑒 𝑘 𝑘 ) for some numbers 𝑒 𝑖 , where the 𝔭 𝑖 are the dyadic primes.It follows that there is an isomorphism L s0 ( 𝑅 ∧ ) ≅ 𝑘 ∏ 𝑖 =1 L s0 ( 𝑅 ∧ 𝔭 𝑖 ) . It thus suffices to recall thati) the map L s0 ( 𝑅 ∧ 𝔭 𝑖 ) → L s0 ( 𝑅 ∧ 𝔭 𝑖 [ ]) is injective; see the proof of Corollary 2.2.3, and thatii) L s0 ( 𝑅 ∧ 𝔭 𝑖 [ ]) is a finite 2-group: The fraction field 𝑅 ∧ 𝔭 𝑖 [ ] of 𝑅 ∧ 𝔭 𝑖 is a finite extension of ℚ , so we mayappeal to [Lam05, Theorem VI 2.29].Finally, we note that the residue fields 𝔽 𝔭 are finite fields of characteristic 2, so that the Arf invariant providesan isomorphism W q ( 𝔽 𝔭 ) ≅ ℤ ∕2 . (cid:3) Remark.
As in the symmetric case, we recall that there is a canonical equivalence L −q ( 𝑅 ) ≃Σ L q ( 𝑅 ) , so that Proposition 2.2.8 and Corollary 2.2.9 also determine the (−1) -quadratic L-groups.2.2.11. Remark.
If the number 𝑑 of dyadic primes of 𝑅 is at least 2, then 𝐴 is not trivial: Taking the rankmod 2 induces the right horizontal surjections in the following diagram. L q0 ( 𝑅 ) L s0 ( 𝑅 ) ℤ ∕2L q0 ( 𝑅 ∧ ) L s0 ( 𝑅 ∧ ) ( ℤ ∕2) 𝑑 Both horizontal composites are zero, therefore we obtain a commutative diagram coker(L q0 ( 𝑅 ) → L s0 ( 𝑅 )) ℤ ∕2coker(L q0 ( 𝑅 ∧ ) → L 𝑠 ( 𝑅 ∧ )) ( ℤ ∕2) 𝑑 whose horizontal arrows are surjective. The induced map on vertical cokernels is a map 𝐴 → ( ℤ ∕2) 𝑑 −1 which is therefore again surjective.2.2.12. Example.
Let us consider the case 𝑅 = ℤ . From the pullback diagram (10), we obtain an exactsequence ⟶ L q0 ( ℤ ) (0 , ⟶ L q0 ( ℤ ∧ ) ⊕ L s0 ( ℤ ) ⟶ L s0 ( ℤ ∧ ) ⟶ L q−1 ( ℤ ) ⟶ , where the map L q0 ( ℤ ) → L q0 ( ℤ ∧ ) is the zero map: By Proposition 2.1.13, it suffices to know that the map L q0 ( ℤ ) → L q0 ( 𝔽 ) is the zero map. For this, one calculates that the Arf invariant of the 𝐸 -form (viewed asa form over 𝔽 ) is zero. Furthermore, the map L q0 ( ℤ ) → L s0 ( ℤ ) is isomorphic to multiplication by , as the 𝐸 form generates L q0 ( ℤ ) . We therefore obtain a short exact sequence(11) ⟶ ℤ ∕2 ⊕ ℤ ∕8 ⟶ L s0 ( ℤ ∧ ) ⟶ L q−1 ( ℤ ) ⟶ . Furthermore, by localisation-dévissage, there is a short exact sequence ⟶ L s0 ( ℤ ∧ ) ⟶ L s0 ( ℚ ) ⟶ ℤ ∕2 ⟶ and from [Lam05, Theorem 2.29 & Corollary 2.23], we know that L s0 ( ℚ ) has 32 elements. We deducethat L s0 ( ℤ ∧ ) has 16 elements, and hence the above injection ℤ ∕2 ⊕ ℤ ∕8 ⊆ L s0 ( ℤ ∧ ) is an isomorphism. Forcompleteness, we observe that the exact sequence involving L s0 ( ℚ ) splits, so one obtains the well knownisomorphism L s0 ( ℚ ) ≅ ( ℤ ∕2) ⊕ ℤ ∕8 [Lam05, Theorem 2.29]. A concrete splitting is given by the element ⟨ −1 , ⟩ . The only thing that needs checking is that this element has order 2.From the above and the exact sequence (11), we find that L q−1 ( ℤ ) = 0 . In particular, we obtain the wellknown calculations of the symmetric and quadratic L-groups of ℤ : L s 𝑛 ( ℤ ) ≅ ⎧⎪⎪⎨⎪⎪⎩ ℤ for 𝑛 ≡ ℤ ∕2 for 𝑛 ≡ for 𝑛 ≡ for 𝑛 ≡ q 𝑛 ( ℤ ) ≅ ⎧⎪⎪⎨⎪⎪⎩ ℤ for 𝑛 ≡ for 𝑛 ≡ ℤ ∕2 for 𝑛 ≡ for 𝑛 ≡ Together with Theorem 1.2.18, Corollary 1.3.11, and Remark 1.3.13 this determines L gs 𝑛 ( ℤ ) . In addition,we find that the map L gs ( ℤ )[ ] → L s ( ℤ )[ ] is an equivalence. We will make use of this fact in Proposi-tion 3.1.12.2.2.13. Example.
Consider the quadratic extension 𝐾 = ℚ [ √ −3] of ℚ and let 𝑅 be its ring of integers.Concretely, 𝑅 is the ring of Eisenstein integers 𝑅 = ℤ [ √ −32 ] , which is a euclidean domain and hence aprincipal ideal domain. The discriminant of 𝐾 is (3) , and as (2) does not divide (3) , we deduce that (2) is a prime ideal in 𝑅 [Neu99, Corollary III.2.12], and hence is the single dyadic prime. We deduce that ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 41 L s2 ( 𝑅 ) = L s3 ( 𝑅 ) = 0 , as the Picard group of a principal ideal domain vanishes. Furthermore L s1 ( 𝑅 ) ≅ ℤ ∕2 and L s0 ( 𝑅 ) ≅ W s0 ( 𝑅 ) ≅ ℤ ∕4 [MH73, Corollary 4.2]. To calculate the quadratic L-groups we consider thediagram of exact sequences q0 ( ℤ ) ℤ ⊕ ℤ ∕2 L s0 ( ℤ ∧ ) 0 00 L q0 ( 𝑅 ) ℤ ∕4 ⊕ ℤ ∕2 L s0 ( 𝑅 ∧ ) 𝐴 (8 , , id) 𝜃 and deduce that 𝐴 ≅ coker( 𝜃 ) and that there is an exact sequence ⟶ ℤ ∕2 ⟶ ker( 𝜃 ) ⟶ L q0 ( 𝑅 ) ⟶ . Now, from the commutative diagram of localisation-dévissage sequences (note that 2 is a uniformiser inboth cases) L s ( ℤ ∧ ) L s ( ℚ ) L s ( ℤ ∕(2))L s ( 𝑅 ∧ ) L s ( 𝐾 ∧ ) L s ( 𝑅 ∕(2)) we deduce that the kernel and the cokernel of 𝜃 are respectively isomorphic to the kernel and the cokernelof the map 𝜃 ′ ∶ L s0 ( ℚ ) ⟶ L s0 ( 𝐾 ∧ ) . It is a general theorem about quadratic extensions of fields that the kernel of 𝜃 ′ is, as an ideal, generatedby the element ⟨ , ⟩ , [Lam05, VII Theorem 3.5]. Since −5∕3 is a square in ℚ , we deduce that ⟨ , ⟩ = ⟨ , −5 ⟩ . From [Lam05, VI Remark 2.31], we then deduce that the kernel of 𝜃 ′ is spanned by ⟨ ⟩ and ⟨ , ⟩ and thus isomorphic to ( ℤ ∕2) . We deduce that L q0 ( 𝑅 ) ≅ ℤ ∕2 . From [Lam05, VII Theorem 3.5], we alsofind that coker( 𝜃 ′ ) ≅ ker ( L s0 ( ℚ ) ⋅ ⟨ , ⟩ ⟶ L s0 ( ℚ ) ) and again from [Lam05, Remark 2.31], we find that the kernel of ⋅ ⟨ , ⟩ is additively generated by ⟨ ⟩ and ⟨ , −2 ⟩ , and deduce an isomorphism ker ( L s0 ( ℚ ) ⋅ ⟨ , ⟩ ⟶ L s0 ( ℚ ) ) ≅ ℤ ∕4 ⊕ ℤ ∕2 . In summary, we obtain the following L-groups for 𝑅 : L s 𝑛 ( 𝑅 ) ≅ ⎧⎪⎪⎨⎪⎪⎩ ℤ ∕4 for 𝑛 ≡ ℤ ∕2 for 𝑛 ≡ for 𝑛 ≡ for 𝑛 ≡ q 𝑛 ( 𝑅 ) ≅ ⎧⎪⎪⎨⎪⎪⎩ ℤ ∕2 for 𝑛 ≡ for 𝑛 ≡ ℤ ∕2 for 𝑛 ≡ ℤ ∕4 ⊕ ℤ ∕2 for 𝑛 ≡ Corollary.
Let O be a number ring, that is, a localisation of the rings of integers in a number fieldaway from finitely many primes, and 𝜖 = ±1 . Then the 𝜖 -symmetric L-groups L s 𝑛 ( O ; 𝜖 ) and the 𝜖 -quadraticL-groups L q 𝑛 ( O ; 𝜖 ) are finitely generated.Proof. It follows from Corollaries 2.2.3 and 2.2.9 and Remarks 2.2.5 and 2.2.10 that it suffices to showthat the symmetric Witt group W s ( O ) , the quadratic Witt group W q ( O ) , and the Picard group Pic( O ) arefinitely generated. The statement for the symmetric Witt group is proven in [MH73, §4, Theorem 4.1], andin fact W s ( O ) is an extension of a finite group by a free abelian group of rank given by the number of realembeddings of the number field 𝐹 . Now we claim that generally for a Dedekind ring 𝑅 whose fraction field 𝐾 is of characteristic different from 2, the canonical map W q ( 𝑅 ) → W s ( 𝑅 ) is injective, so that W q ( 𝑅 ) isfinitely generated if W s ( 𝑅 ) is. This follows from the fact that the map W q ( 𝑅 ) → W q ( 𝐾 ) is injective, see[KS71]. As the argument in loc. cit. is not explicitly written out, let us sketch a direct argument that themap W q ( 𝑅 ) → W s ( 𝑅 ) is injective: First, assume that a symmetric form ( 𝑃 , 𝜑 ) vanishes in W s ( 𝑅 ) . Thenthe same is true for its image in W s ( 𝐾 ) . By Corollary 1.3.5 we deduce that ( 𝑃 ⊗ 𝑅 𝐾, 𝜑 ⊗ 𝑅 𝐾 ) admits astrict Lagrangian. The argument written in the proof of [KS71, Lemma 1.4] then shows that ( 𝑃 , 𝜑 ) indeed itself admits a strict Lagrangian. Now, let ( 𝑃 , 𝑞 ) be a quadratic form whose image in W s ( 𝑅 ) vanishes. Wededuce that the underlying symmetric bilinear form of ( 𝑃 , 𝑞 ) admits a strict Lagrangian 𝐿 . We then observethat for each 𝑥 in 𝐿 , we have 𝑞 ( 𝑥 ) = 𝑏 ( 𝑥, 𝑥 ) = 0 , so that 𝑞 | 𝐿 = 0 as 𝑅 is 2-torsion free. It follows that 𝐿 is a Lagrangian for the quadratic form ( 𝑃 , 𝑞 ) as needed. Finally, the Picard group of the ring of integersin a number field is finite (in other words, the class number of a ring of integers is finite), and hence thePicard group of a localisation of such a ring receives a surjection from a finite group and is thus itself finite,compare to the proof of Proposition 2.1.4. (cid:3) Remark.
In the above proof, we have again restricted our attention to Dedekind rings whose fieldof fractions 𝐾 has characteristic different from 2. If the characteristic of 𝐾 is 2 we find that:i) The map W s ( 𝑅 ) → W s ( 𝐾 ) is injective, butii) the map W q ( 𝑅 ) → W s ( 𝑅 ) is zero.Indeed i) follows from the same argument given above, since also for fields 𝐾 of characteristic 2 a form ( 𝑃 , 𝑞 ) is zero in W s ( 𝐾 ) if and only if it admits a strict Lagrangian, see Corollary 1.3.5. To see ii), it sufficesto show that the composite W q ( 𝑅 ) → W s ( 𝑅 ) → W s ( 𝐾 ) vanishes, as the latter map is injective, see theproof of Corollary 2.2.6. This composite factors through the map W q ( 𝐾 ) → W s ( 𝐾 ) which is zero as theunderlying bilinear form of any quadratic form over a field of characteristic 2 has a symplectic basis andhence admits a Lagrangian.2.2.16. Corollary.
Let O be a number ring and 𝜖 = ±1 . Then for all 𝑚, 𝑛 ∈ ℤ , the groups L 𝑛 ( O ; Ϙ ≥ 𝑚𝜖 ) , andconsequently the groups GW 𝑛 ( O ; Ϙ ≥ 𝑚𝜖 ) , are finitely generated.Proof. We saw in Example 1.1.4 that the functor Ϙ ≥ 𝑚𝜖 is 𝑚 -quadratic and (2 − 𝑚 ) -symmetric. Hence, on theone hand, it follows from Corollary 1.2.8 that for 𝑛 ≤ 𝑚 − 2 the map L q 𝑛 ( O ; 𝜖 ) → L 𝑛 ( O ; Ϙ ≥ 𝑚𝜖 ) is surjective.By Corollary 2.2.14 the left hand group is finitely generated, so the same is true for L 𝑛 ( O ; Ϙ ≥ 𝑚𝜖 ) .On the other hand, Corollary 1.3.7 implies that the map L 𝑛 ( O ; Ϙ ≥ 𝑚𝜖 ) → L s 𝑛 ( O ; 𝜖 ) is injective for 𝑛 ≥ 𝑚 −1 .Again, by Corollary 2.2.14 the target group is finitely generated, it follows that L 𝑛 ( O ; Ϙ ≥ 𝑚𝜖 ) is so as well. Toobtain the consequences for Grothendieck-Witt groups, we recall from Quillen’s results that the algebraicK-groups of number rings are finitely generated. From the homotopy orbits spectral sequence, it followsthat also the homotopy groups of K( O ; 𝜖 ) hC are finitely generated, so that the desired result follows fromthe fibre sequence K( O ; 𝜖 ) hC ⟶ GW( O ; Ϙ ≥ 𝑚𝜖 ) ⟶ L( O ; Ϙ ≥ 𝑚𝜖 ) . (cid:3) Combining the above with the comparison theorem of [HS20] gives the following corollary.2.2.17.
Corollary.
Let O be a number ring, and 𝜖 = ±1 . Then the classical 𝜖 -symmetric and 𝜖 -quadraticGrothendieck-Witt groups GW scl ,𝑛 ( O ; 𝜖 ) and GW qcl ,𝑛 ( O ; 𝜖 ) are finitely generated for all 𝑛 ≥ . Remark.
The finite generation of the groups GW q ( O ; 𝜖 ) can also be deduced by a homologicalstability argument similar to the one of Quillen for algebraic K -theory of number rings: By Serre classtheory, it suffices to show that the ordinary homology groups of the components of Ω ∞ GW q ( O ; 𝜖 ) arefinitely generated. Since every 𝜖 -quadratic form is a direct summand in an 𝜖 -hyperbolic form, the groupcompletion theorem identifies any such component with the space { BO ∞ , ∞ ( O ) + for 𝜖 = 1 , BSp q∞ ( O ) + for 𝜖 = −1 where O ∞ , ∞ ( O ) and Sp q∞ ( O ) denote the colimit of the automorphism group of an 𝑛 -fold sum of the (1) - and (−1) -quadratic hyperbolic form, respectively. Charney [Cha87] has proved a homological stability resultfor those groups, so that it suffices to show that the groups O 𝑛,𝑛 ( O ) and Sp q 𝑛 ( O ) have finitely generatedhomology. Let us briefly explain why that is: First we note that both groups are arithmetic. Second, everyarithmetic group has a torsion free finite index subgroup [Ser79, 1.3 (4)], and hence also a normal torsionfree finite index subgroup. By the Serre spectral sequence for the quotient by this normal subgroup, we findthat it suffices to know that torsion free arithmetic groups have finitely generated homology, which followsfrom the fact they they admit a finite classifying space [Ser79, 1.3 (5)]. We wish to thank Manuel Krannichfor a helpful discussion about this and for making us aware of Serre’s survey. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 43 Finally, we note that in the symmetric case, it is not generally true that every form embeds into a hy-perbolic form (as any such form admits a quadratic refinement), so in order to run a similar argument onefirst needs to find a symmetric bilinear form 𝑏 such that every other form embeds into a suitable number oforthogonal copies of 𝑏 , and one needs to prove homological stability for the family of automorphism groupsof such orthogonal copies of 𝑏 . To our knowledge, this is not known to hold in the generality of numberrings, though it does hold for the integers.3. G ROTHENDIECK -W ITT GROUPS OF D EDEKIND RINGS
In this final section we consider the homotopy limit problem for Dedekind domains and finite fieldsof characteristic . In the latter case, we extend the solution of the homotopy limit problem from theGrothendieck-Witt space (where it is known to hold by the work of Friedlander) to its Grothendieck-Wittspectrum. We then combine this with the dévissage results of §3.1 to solve the homotopy limit problemfor Dedekind rings whose fraction field is a global field of characteristic , i.e. a number field, provingTheorem 2 from the introduction. Finally, we apply these ideas to the particular case of ℤ and calculate its ±1 -symmetric and genuine ±1 -quadratic Grothendieck-Witt groups conditionally on Vandiver’s conjecture,and in the range 𝑛 ≤ unconditionally.3.1. The homotopy limit problem.
A prominent question in the hermitian K -theory of rings and schemesis when the map from the Grothendieck-Witt space/spectrum to the homotopy fixed points of the associatedalgebraic K -theory space/spectrum is an equivalence. This question, first raised by Thomason in [Tho83],is commonly known as the homotopy limit problem . In the case of fields, the following theorem representsthe current state of the art; see [HKO11, BKSØ15, BH20]. We recall that the virtual mod 2 cohomologicaldimension vcd of a field 𝑘 can be defined as the ordinary mod 2 cohomological dimension cd of (theabsolute Galois group of) 𝑘 [ √ −1] . In particular, we have vcd ( 𝑘 ) ≤ cd ( 𝑘 ) . Given 𝜖 = ±1 we let K( 𝑘 ; 𝜖 ) denote the K-theory spectrum of 𝑘 with C -action induced by the duality D = hom 𝑘 (− , 𝑘 ( 𝜖 )) .3.1.1. Theorem.
Let 𝑘 be a field of characteristic different from and such that vcd ( 𝑘 ) < ∞ . Then themap of spectra GW s ( 𝑘 ; 𝜖 ) ⟶ K( 𝑘 ; 𝜖 ) hC is an equivalence after -completion. Remark.
The cited Theorem 3.1.1 was stated in [BKSØ15] using Schlichting’s model for Grothedieck-Witt spectra. Since 𝑘 is assumed to have characteristic ≠ we may invoke the comparison statement ofProposition [II].B.2.2 and identify Schlichting’s construction with ours.The characteristic case of Theorem 3.1.1 was proven in [HKO11], while the positive odd characteristiccase is established in [BKSØ15]. An alternative proof of this theorem is also provided in recent work ofBachmann and Hopkins [BH20]. Special cases of the above theorem were already known before: the caseof the field ℂ of complex numbers, for example, can be reduced to the classical equivalence BO ≃ BU hC see, e.g., [BK05, Lemma 7.3]. In fact, in loc. cit. the authors prove this also for the (−1) -symmetric variant.The equivalence for ℂ can in turn be used to deduce the same for finite fields 𝔽 𝑞 . One can express theGrothendieck-Witt spaces of 𝔽 𝑞 in terms of the Adams operations on BO and BSp in a way analogous tothe main results of Quillen’s famous paper [Qui72] on the algebraic K -theory of 𝔽 𝑞 . These results werefirst established by Friedlander in [Fri76], and later expanded and refined in see [FP06] (where also a smallmistake was corrected in the case of 𝑞 even). Combined with the positive solution of the homotopy limitproblem for ℂ they imply the following.3.1.3. Theorem.
For 𝜖 = ±1 and every prime power 𝑞 the natural map GW scl ( 𝔽 𝑞 ; 𝜖 ) ⟶ K( 𝔽 𝑞 ; 𝜖 ) hC is an equivalence on connective covers. The results of [Fri76] and [FP06] on which this approach relies use lengthy computations in the co-homology of various finite matrix groups. We shall now present an alternative and significantly shorterproof of the Theorem 3.1.3 in the case of 𝑞 even, using Theorem 1 from the introduction. We recall that GW scl ( 𝔽 𝑞 ; 𝜖 ) → GW 𝑠 ( 𝔽 𝑞 ; 𝜖 ) is an equivalence on connective covers, Corollary 1.3.12, so it suffices to prove the following proposition, covering not only the Grothendieck-Witt space, but also the corresponding spec-trum, and which applies to arbitrary shifts of the symmetric Poincaré structure:3.1.4. Proposition.
Let 𝑞 = 2 𝑟 for some positive integer 𝑟 . Then the map of spectra GW( 𝔽 𝑞 ; ( Ϙ s ) [ 𝑚 ] ) ⟶ K( 𝔽 𝑞 ; ( Ϙ s ) [ 𝑚 ] ) hC is an equivalence for every 𝑚 ∈ ℤ .Proof. Corollary [II].4.4.14 provides, for every ring 𝑅 and Poincaré structure Ϙ on D p ( 𝑅 ) , a pullback square GW( 𝑅 ; Ϙ ) L( 𝑅 ; Ϙ )K( 𝑅 ; Ϙ ) hC K( 𝑅 ; Ϙ ) tC , It therefore suffices to show that the canonical map(12) L( 𝔽 𝑞 ; ( Ϙ s ) [ 𝑚 ] ) ⟶ K( 𝔽 𝑞 ; ( Ϙ s ) [ 𝑚 ] ) tC is an equivalence for every 𝑚 . Applying the transformation L(−) → K(−) tC to the Bott-Genauer sequenceof Example [II].1.2.5 gives a commutative diagram L( 𝔽 𝑞 ; ( Ϙ s ) [ 𝑚 ] ) Σ 𝑚 L( 𝔽 𝑞 ; Ϙ s )K( 𝔽 𝑞 ; ( Ϙ s ) [ 𝑚 ] ) tC Σ 𝑚 K( 𝔽 𝑞 ; Ϙ s ) tC ≃≃ whose horizontal arrows are equivalences; see the discussion before Corollary R.10. It will thus suffice totreat the case 𝑚 = 0 . Furthermore, L and Tate of K-theory are 2-periodic, see Corollary R.10, and it sufficesto check that (12) induces an isomorphism on 𝜋 and 𝜋 . By Corollary 1.3.4 we have that L ( 𝔽 𝑞 ; Ϙ s ) ≅W s ( 𝔽 𝑞 ) ≅ ℤ ∕2 is the Witt group of symmetric bilinear forms over 𝔽 𝑞 , which is isomorphic to ℤ ∕2 generatedby the class of the symmetric bilinear form ( 𝔽 𝑞 , 𝑏 ) with 𝑏 (1 ,
1) = 1 . On the other hand, the same corollaryalso gives that L ( 𝔽 𝑞 ; Ϙ s ) = 0 . To finish the proof it will hence suffice to show that 𝜋 K( 𝔽 𝑞 ; Ϙ s ) tC = 0 , that 𝜋 K( 𝔽 𝑞 ; Ϙ s ) tC = ℤ ∕2 , and that the map L ( 𝔽 𝑞 ; Ϙ s ) → 𝜋 K( 𝔽 𝑞 ; Ϙ s ) tC is non-zero.Now, by Quillen’s calculation of the K -theory of finite fields [Qui72], the K-groups K ∗ ( 𝔽 𝑞 ) are oddtorsion groups in positive degrees, so that the map K( 𝔽 𝑞 ) → H ℤ is a 2-adic equivalence. It follows thatthe induced map K( 𝔽 𝑞 ; Ϙ s ) tC → ℤ tC is an equivalence as well, which shows that the Tate-K-groups are asclaimed.To finish the proof it will hence suffice to show that the map L ( 𝔽 𝑞 ; Ϙ s ) → 𝜋 (K( 𝔽 𝑞 ; Ϙ s ) tC ) sends thegenerator to the generator. Indeed, in light of the commutative diagram 𝜋 Pn( D p ( 𝔽 𝑞 ) , Ϙ s ) GW ( 𝔽 𝑞 ; Ϙ s ) L ( 𝔽 𝑞 ; Ϙ s ) 𝜋 Cr( D p ( 𝔽 𝑞 ) , Ϙ s ) C K ( 𝔽 𝑞 ; Ϙ s ) C ̂ H (C , K ( 𝔽 𝑞 ; Ϙ s )) this simply follows from the fact that the composed forgetful functor 𝜋 Pn( D p ( 𝔽 𝑞 ) , Ϙ s ) ⟶ 𝜋 Cr( D p ( 𝔽 𝑞 ) , Ϙ s ) C ⟶ K ( 𝔽 𝑞 ; Ϙ s ) C ⟶ K ( 𝔽 𝑞 ) ≅ ℤ sends ( 𝔽 𝑞 , 𝑏 ) to the generator ( 𝔽 𝑞 ) . (cid:3) Remark.
An alternative argument can be given making use of multiplicative structures: In Paper[IV], we prove that the map L( 𝑅 ; Ϙ s ) → K( 𝑅 ; Ϙ s ) tC is a map of E ∞ -rings if 𝑅 is a commutative ring. For 𝑅 = 𝔽 𝑞 with 𝑞 even, we then know that both homotopy rings are isomorphic to 𝔽 [ 𝑥 ±1 ] , for | 𝑥 | = 2 . As anyring endomorphism of this ring is an isomorphism, the map we investigate is an equivalence. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 45 The results of Berrick at. al. [BKSØ15] on the homotopy limit problem extend significantly beyond therealm of fields. It is shown, for example, that for any Noetherian scheme 𝑋 of finite Krull dimension over ℤ [ ] , if vcd ( 𝑘 ( 𝑥 )[ √ −1]) is uniformally bounded across all points 𝑥 ∈ 𝑋 , then the map GW( 𝑋 ) ⟶ K( 𝑋 ) hC is an equivalence after -completion. Using the results of the previous sections we can now relax theassumption that is invertible from the above result. Recall that for a Dedekind ring 𝑅 with line bundle 𝑀 with involution ±1 , the canonical map GW scl ( 𝑅 ; 𝑀 ) ⟶ GW( 𝑅 ; Ϙ s 𝑀 ) is an equivalence in non-negative degrees, by Corollary 1.3.12. Combining this with the following resultgives Theorem 2 from the introduction.3.1.6. Theorem (The homotopy limit problem) . Let 𝑅 be a Dedekind ring whose fraction field is a numberfield. Then for every 𝑚 ∈ ℤ and every line bundle 𝑀 over 𝑅 with involution ±1 , the map GW( 𝑅 ; ( Ϙ s 𝑀 ) [ 𝑚 ] ) ⟶ K( 𝑅 ; ( Ϙ s 𝑀 ) [ 𝑚 ] ) hC is a 2-adic equivalence.Proof. Let 𝑆 be the (finite) set of all prime ideals in 𝑅 lying over . We observe that then 𝑅 𝑆 = 𝑅 [ ] andsimilarly that 𝑀 𝑆 = 𝑀 [ ] and consider the commutative diagram ⊕ 𝔭 ∈ 𝑆 GW( 𝔽 𝔭 ; ( Ϙ s 𝑀 𝔭 ) [ 𝑚 −1] ) GW( 𝑅 ; ( Ϙ s 𝑀 ) [ 𝑚 ] ) GW( 𝑅 [ ]; ( Ϙ s 𝑀 𝑆 ) [ 𝑚 ] ) ⊕ 𝔭 ∈ 𝑆 K( 𝔽 𝔭 ; ( Ϙ s 𝑀 𝔭 ) [ 𝑚 −1] ) hC K( 𝑅 ; ( Ϙ s 𝑀 ) [ 𝑚 ] ) hC K( 𝑅 [ ]; ( Ϙ s 𝑀 𝑆 ) [ 𝑚 ] ) hC obtained via the localisation-dévissage sequences of Corollary 2.1.9. Note that we have commuted thehomotopy fixed points with the finite direct sum in the lower left corner. The left most vertical map is anequivalence by Proposition 3.1.4, and the right most vertical map is a 2-adic equivalence by [BKSØ15,Theorem 2.2]: We need to argue that all residue fields of 𝑅 [ ] have finite mod 2 virtual cohomologicaldimension. Indeed, the residue fields at non-zero prime ideals are finite fields and hence have cohomologicaldimension one (the Galois group is ̂ ℤ ), and the residue field at is the fraction field which is number fieldand hence also has finite vcd ; [Ser02, §II.4.4]. It then follows that the middle vertical map is a 2-adicequivalence, as desired. (cid:3) Remark.
The conclusion of Theorem 3.1.6 thus holds for all Dedekind rings whose field of fractionsis a global field of characteristic different from : In the odd characteristic case [BKSØ15] applies, and thecase of characteristic zero is the content of Theorem 3.1.6.3.1.8. Remark.
Suppose again that 𝑅 is a Dedekind ring with global fraction field 𝐾 . Suppose that 𝐾 has characteristic different from 2 and is not formally real, that is, that −1 is a sum of squares. In otherwords, suppose that 𝐾 has positive odd characteristic or is a totally imaginary number field. Then the Wittgroup W s ( 𝐾 ) is a 2-primary torsion group of bounded exponent by [Sch12, Theorem 2.7.9]. As W s ( 𝑅 ) is a subgroup of W s ( 𝐾 ) , see the proof of Corollary 2.2.3, Corollary 2.2.4 implies that L s ( 𝑅 ) is (derived)2-complete. As K( 𝑅 ) tC is also 2-complete, the pullback GW( 𝑅 ; Ϙ s ) L( 𝑅 ; Ϙ s )K( 𝑅 ; Ϙ s ) hC K( 𝑅 ; Ϙ s ) tC together with Theorem 3.1.6 implies that the map of Theorem 3.1.6 is in fact an equivalence before 2-completion. Conversely, if 𝐾 admits a real embedding, then L s ( 𝑅 ) is not 2-complete: We have seen inCorollary 2.2.14 that all homotopy groups are finitely generated, so L s ( 𝑅 ) is 2-complete if and only if allsymmetric L-groups of 𝑅 are 2-complete. However, as observed in the proof of Corollary 2.2.14, W s ( 𝑅 ) has rank equal to the number of real embeddings of 𝐾 , and is thus not 2-complete. It hence follows thatthe map under investigation in Theorem 3.1.6 is not an integral equivalence if 𝐾 admits a real embedding.See also [BKSØ15, Theorem 2.4 & Proposition 4.7]. In fact, in our situation, the same result is true for W s ( 𝑅 ; 𝑀 ) for any line bundle 𝑀 on 𝑅 : The map W s ( 𝑅 ; 𝑀 ) → W s ( 𝐾 ) is an isomorphism after inverting2, and the map W s ( 𝐾 ) → W s ( ℝ ) induced from a real embedding of 𝐾 is surjective. Hence the compositeis non-zero and consequently ℤ is a direct summand inside W s ( 𝑅 ; 𝑀 ) . Hence L s ( 𝑅 ; 𝑀 ) is not 2-complete.As a side remark, we note that in the case where 𝐾 admits a real embedding, L s ( 𝑅 ) contains L 𝑠 ( ℝ ) as aretract, and L s ( ℝ ) is not 2-complete. To see that L s ( ℝ ) is indeed a retract, consider the following composite L s ( ℤ ) ⟶ L s ( 𝑅 ) ⟶ L s ( ℝ ) where the two maps are induced by the canonical map ℤ → 𝑅 and the map 𝑅 → 𝐾 ⊆ ℝ induced by a realembedding of 𝐾 . This composite admits a splitting, as was observed in [HLN20, Theorem A].3.1.9. Remark.
The proof of Theorem 3.1.6 reveals that the assumptions are not optimal. Assume forinstance that 𝑅 is the ring of integers in a non-archimedean local field 𝐾 of mixed characteristic (0 , andlet 𝑘 be the residue field of the local ring 𝑅 . For instance, assume that 𝑅 is a dyadic completion of the ringof integers in a number field. Since 𝑅 is local the line bundle 𝑀 is trivial. We again consider the diagramconsisting of horizontal fibre sequences GW( 𝑘 ; ( Ϙ s ) [ 𝑚 −1] ) GW( 𝑅 ; ( Ϙ s ) [ 𝑚 ] ) GW( 𝐾 ; ( Ϙ s ) [ 𝑚 ] )K( 𝑘 ; ( Ϙ s ) [ 𝑚 −1] ) hC K( 𝑅 ; ( Ϙ s ) [ 𝑚 ] ) hC K( 𝐾 ; ( Ϙ s ) [ 𝑚 ] ) hC First, we note that cd ( 𝐾 ) = 2 [Ser02, §4.3], so the right vertical map is a 2-adic equivalence. We deducethat the middle vertical map is a 2-adic equivalence if and only if the left vertical map is a 2-adic equivalence.Thus if we assume that 𝑘 is a finite field, the middle vertical map is a 2-adic equivalence. In fact, in thiscase, 𝐾 is a finite extension of ℚ ∧ , and as observed earlier, L s ( 𝐾 ) is 2-complete, in fact 2-power torsion[Lam05, Theorem 2.29]. It follows that the middle vertical map is in fact an equivalence.We finish this subsection by noting the following obstruction to a positive solution of the homotopy limitproblem for classical Grothendieck-Witt-theory of a discrete ring 𝑅 , see also [BKSØ15, Remark 4.9]. Theresult implies that the map GW scl ( 𝑅 )∕2 → K( 𝑅 ) hC ∕2 cannot be an equivalence in non-negative degreesunless the comparison map L gs ( 𝑅 ) → L s ( 𝑅 ) is so as well.3.1.10. Proposition.
Suppose that the fibre of the map GW scl ( 𝑅 )∕2 → K( 𝑅 ) hC ∕2 is 𝑛 -truncated for someinteger 𝑛 , where we view GW scl ( 𝑅 )∕2 as a (connective) spectrum. Then the map L gs ( 𝑅 ) → L s ( 𝑅 ) is anequivalence on ( 𝑛 + 2) -connective covers.Proof. Let 𝐹 be the fibre of the map 𝜏 ≥ 𝑛 +2 L gs ( 𝑅 ) → 𝜏 ≥ 𝑛 +2 L s ( 𝑅 ) . By Proposition 3.1.12 below, the mapis an equivalence after inverting , so it follows that 𝐹 [ ] vanishes. We will show that also 𝐹 ∕2 vanishes,which implies that 𝐹 is trivial. We recall that there are canonical shift maps ⋯ ⟶ Σ L gs ( 𝑅 ) 𝜎 ⟶ L gs ( 𝑅 ) 𝜎 ⟶ Σ −4 L gs ( 𝑅 ) ⟶ ⋯ whose filtered colimit is given by L s ( 𝑅 ) . Now, we need to show that the map 𝜋 𝑘 (L gs ( 𝑅 )∕2) → 𝜋 𝑘 (L s ( 𝑅 )∕2) is an isomorphism for 𝑘 ≥ 𝑛 + 2 . It hence suffices to argue that the top horizontal map in the diagram 𝜋 𝑘 (L gs ( 𝑅 )∕2) 𝜋 𝑘 +4 (L gs ( 𝑅 )∕2) 𝜋 𝑘 (K( 𝑅 ) tC ∕2) 𝜋 𝑘 +4 (K( 𝑅 ) tC ∕2) 𝜎𝜎 ′ is an isomorphism for all 𝑘 ≥ 𝑛 + 2 . Here, 𝜎 ′ denotes the corresponding shift map on the Tate constructionof algebraic K-theory, which is an equivalence by Corollary R.10. We claim that the vertical maps areisomorphisms, concluding the proof of the proposition. To see the claim, we first observe that the map ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 47 GW gs ( 𝑅 )∕2 → K( 𝑅 ) hC ∕2 is also 𝑛 -truncated, because the map GW scl ( 𝑅 )∕2 → GW gs ( 𝑅 )∕2 is -truncated.We then consider the pullback diagram GW gs ( 𝑅 )∕2 L gs ( 𝑅 )∕2K( 𝑅 ) hC ∕2 K( 𝑅 ) tC ∕2 and conclude that the map L gs ( 𝑅 )∕2 → K( 𝑅 ) tC ∕2 is 𝑛 -truncated as well. Thus the vertical maps induceisomorphisms for 𝑘 > 𝑛 + 1 as needed. (cid:3) Remark.
There are examples of rings for which the map L gs ( 𝑅 ; 𝑀 ) → L s ( 𝑅 ; 𝑀 ) is not an equiva-lence in non-negative degrees. Notice that all of the following rings have global dimension 2.i) Consider ℤ [ ℤ ] as ring with trivial involution and let 𝜖 = −1 . Then the map L gs0 ( ℤ [ ℤ ]; 𝜖 ) → L s0 ( ℤ [ ℤ ]; 𝜖 ) is not surjective; see [Ran80, Prop. 10.3].ii) In a similar vein, we can consider ℤ [ ℤ ] as a ring with group-ring involution 𝑡 ↦ 𝑡 −1 , for 𝑡 a generatorof ℤ . It was conjectured in [Ran80, §10] and proven in [MR90] that L gs ( ℤ [ ℤ ]; 𝜖 ) , satisfies Shane-son’s splitting. The same is true for symmetric L-theory, so the map in question, evaluated on 𝜋 isisomorphic to gs−2 ( ℤ ) ⊕ L gs−3 ( ℤ ) ⟶ L s−2 ( ℤ ) ⊕ L s−3 ( ℤ ) = ℤ ∕2 and is thus not surjective.iii) Similarly, we can consider 𝔽 [ ℤ ] as ring with group-ring involution. Again, one can use Shaneson’ssplitting to show that the map L gs0 ( 𝔽 [ ℤ ]) → L s0 ( 𝔽 [ ℤ ]) is not surjective: In this case, it is becausethe map gs−2 ( 𝔽 ) ⟶ L s−2 ( 𝔽 ) = ℤ ∕2 is not surjective.We finish this section with the promised calculation of 2-inverted genuine L-theory. At this point, wewill invoke multiplicative structures on L-theory which we develop in detail in Paper [IV].3.1.12. Proposition.
Let 𝑅 be a ring with invertible module with involution 𝑀 , and let 𝑚 ∈ ℤ ∪ {±∞} .Then the natural map L( 𝑅 ; Ϙ ≥ 𝑚𝑀 )[ ] ⟶ L( 𝑅 ; Ϙ s 𝑀 )[ ] is an equivalence.Proof. We first observe that the canonical map L gs ( ℤ ) → L s ( ℤ ) is an equivalence after inverting , seeExample 2.2.12. Moreover, the shift maps appearing in the proof of Proposition 3.1.10 are in fact given bymultiplication with an element 𝑥 ∈ L gs4 ( ℤ ) , namely the Poincaré object ℤ [−2] with its standard genuinesymmetric Poincaré structure of signature 1. Thus we find L( 𝑅 ; Ϙ s 𝑀 ) ≃ L( 𝑅 ; Ϙ ≥ 𝑚𝑀 )[ 𝑥 −1 ] ≃ L( 𝑅 ; Ϙ ≥ 𝑚𝑀 ) ⊗ L gs ( ℤ ) L s ( ℤ ) , and the result follows. (cid:3) Grothendieck-Witt groups of the integers.
In this section, we will specialise the results establishedearlier in the paper to the ring of integers ℤ , and calculate its classical 𝜖 -symmetric and 𝜖 -quadratic Grothendieck-Witt groups. We will exploit Corollary 1.3.11 and instead calculate the non-negative Grothendieck-Wittgroups GW s ( ℤ ; 𝜖 ) = GW( ℤ ; Ϙ s 𝜖 ) for the non-genuine symmetric Poincaré structure. Our calculation cru-cially relies on the knowledge of the algebraic K -groups [Wei13] of ℤ , and on the calculations of Berrick-Karoubi [BK05] of the Grothendieck-Witt groups of ℤ [ ] . Before we start the computation, we give a briefaccount of the types of classical Grothendieck-Witt groups that we are considering. (1) -Symmetric : We recall that GW scl ( ℤ ) denotes the homotopy theoretic group completion of the maximalsubgroupoid of the category of non-degenerate symmetric bilinear forms over ℤ . By [Ser61, Théorème1], there is an isomorphism 𝜋 GW scl ( ℤ ) ≅ ℤ ⊕ ℤ where the summands are generated by the classes theforms ⟨ ⟩ and ⟨ −1 ⟩ on a free module of rank 1 ℤ ; they send ( 𝑥, 𝑦 ) to 𝑥𝑦 and − 𝑥𝑦 , respectively. We write O ⟨ 𝑛,𝑛 ⟩ ( ℤ ) = Aut(( ⟨ ⟩ ⟂ ⟨ −1 ⟩ ) ⟂ 𝑛 ) ⊆ GL 𝑛 ( ℤ ) and O ⟨ ∞ , ∞ ⟩ ( ℤ ) = colim 𝑛 O ⟨ 𝑛,𝑛 ⟩ ( ℤ ) . Then the commutator subgroup of O ⟨ ∞ , ∞ ⟩ ( ℤ ) is perfect by e.g. [RW13, Proposition 3.1] and since any non-degenerate symmetricbilinear form over ℤ is an orthogonal summand in ( ⟨ ⟩ ⟂ ⟨ −1 ⟩ ) ⟂ 𝑛 for some 𝑛 ≥ , the group completiontheorem yields a homotopy equivalence of spaces 𝜏 > GW scl ( ℤ ) ≃ BO ⟨ ∞ , ∞ ⟩ ( ℤ ) + , see [MS76], or [RW13, Corollary 1.2]. (−1) -Symmetric: Similarly GW −scl ( ℤ ) is the homotopy theoretic group completion of the maximal sub-groupoid of the category of non-degenerate symplectic bilinear forms over ℤ . We let H −s be the standardsymplectic bilinear form on ℤ . As every symplectic form over ℤ is isomorphic to a finite orthogonal sum ofcopies of H −s , we find 𝜋 GW −scl ( ℤ ) ≅ ℤ , generated by H −s . We write Sp 𝑛 ( ℤ ) = Aut((H −s ) ⟂ 𝑛 ) ⊆ GL 𝑛 ( ℤ ) and Sp ∞ ( ℤ ) = colim 𝑛 Sp 𝑛 ( ℤ ) . The group Sp ∞ ( ℤ ) is again perfect, see e.g. [RW13, Proposition 3.1], andthe group completion theorem yields a homotopy equivalence of spaces 𝜏 > GW −scl ( ℤ ) ≃ BSp ∞ ( ℤ ) + . (1) -Quadratic: Now GW qcl ( ℤ ) is the homotopy theoretic group completion of the maximal subgroupoid ofthe category of non-degenerate quadratic forms over ℤ . Let H q be the standard hyperbolic quadratic formand 𝐸 the classical -dimensional quadratic form associated to the Dynkin diagram of the same name.By [Ser61, Théorème 5], every quadratic form ( 𝑃 , 𝑞 ) satisfies 𝑃 ⟂ H q ≅ H 𝑛 q ⊕ 𝐸 𝑚 for some 𝑛 and 𝑚 and 𝜋 GW qcl ( ℤ ) ≅ ℤ ⊕ ℤ with generators H q and 𝐸 . We write O 𝑛,𝑛 ( ℤ ) = Aut((H q ) ⟂ 𝑛 ) ⊆ GL 𝑛 ( ℤ ) and O ∞ , ∞ ( ℤ ) = colim 𝑛 O 𝑛,𝑛 ( ℤ ) . As above the group O ∞ , ∞ ( ℤ ) has perfect commutator subgroup and since anyquadratic form over ℤ is a direct summand of (H q ) ⟂ 𝑛 for some 𝑛 ≥ there is a homotopy equivalence ofspaces 𝜏 > GW qcl ( ℤ ) ≃ BO ∞ , ∞ ( ℤ ) + . (−1) -Quadratic : Finally, GW −qcl ( ℤ ) is similarly built from (−1) -quadratic forms over ℤ . Such a form isdetermined by its rank (which is an even number) and its Arf invariant, see [Bro12, §III.1]. Let H = ( ℤ , ( ) , 𝑥𝑦 ) and H = ( ℤ , ( ) , 𝑥 + 𝑥𝑦 + 𝑦 ) , be the standard hyperbolic (−1) -quadratic forms with Arf invariant and , respectively. Then every (−1) -quadratic form with Arf invariant is isomorphic to a direct sum of copies of H , and every (−1) -quadraticform with Arf invariant is isomorphic to a direct sum of copies of H plus one copy of H . Thus, 𝜋 GW −qcl ≅ ℤ ⊕ ℤ ∕2 . We define Sp q2 𝑛 ( ℤ ) = Aut((H ) ⟂ 𝑛 ) ⊆ Sp 𝑛 ( ℤ ) to be the group of matrices preservingboth the bilinear form and its quadratic refinement and set Sp q∞ ( ℤ ) = colim 𝑛 Sp q2 𝑛 ( ℤ ) . As above, the groupcompletion theorem yields a homotopy equivalence of spaces 𝜏 > GW −qcl ( ℤ ) ≃ BSp q∞ ( ℤ ) + . The Grothendieck-Witt groups of ℤ . We now proceed to calculate the 𝜖 -symmetric Grothendieck-Wittgroups of ℤ . Recall that the Bernoulli numbers { 𝐵 𝑛 } 𝑛 ≥ are rational numbers determined by the equa-tion 𝑥𝑒 𝑥 − 1 = ∞ ∑ 𝑛 =0 𝐵 𝑛 𝑛 ! 𝑥 𝑛 . We write 𝑐 𝑛 for the numerator of | 𝐵 𝑛 𝑛 | which is an odd number. For each 𝑘 ≥ we have the equations | K 𝑘 +2 ( ℤ ) | = 2 ⋅ 𝑐 𝑘 +1 and | K 𝑘 +6 ( ℤ ) | = 𝑐 𝑘 +2 , by [Wei13, Theorem 10.1], and if the Vandiver conjecture holds the groups in question are cyclic. Thedenominator of | 𝐵 𝑛 𝑛 | will be denoted by 𝑤 𝑛 . There are isomorphisms K 𝑘 +3 ( ℤ ) ≅ ℤ ∕2 𝑤 𝑘 +2 and K 𝑘 +7 ( ℤ ) ≅ ℤ ∕ 𝑤 𝑘 +4 , for all 𝑘 ≥ by [Wei13, Theorem 10.1]. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 49 We now arrive at the main computation of the 𝜖 -symmetric Grothendieck-Witt groups of ℤ in degree 𝑛 ≥ . We determine these groups completely up to the precise group structure in degrees which are mod , which depend on Vandiver’s conjecture. Thanks to the work of Weibel [Wei13] on the algebraic K -theory of ℤ the last uncertainty can be removed in the range 𝑛 ≤ , see Remark 3.2.2 below. For anabelian group 𝐴 , we write 𝐴 odd for the odd torsion subgroup of 𝐴 .3.2.1. Theorem.
The classical 𝜖 -symmetric Grothendieck-Witt groups ℤ are given in degrees 𝑛 ≥ by thefollowing table: 𝑛 = GW scl ,𝑛 ( ℤ ) GW −scl ,𝑛 ( ℤ )8 𝑘 ℤ ⊕ ℤ ∕2 08 𝑘 + 1 ( ℤ ∕2) 𝑘 + 2 ( ℤ ∕2) ⊕ K 𝑘 +2 ( ℤ ) 𝑜𝑑𝑑 ℤ ⊕ K 𝑘 +2 ( ℤ ) odd 𝑘 + 3 ℤ ∕ 𝑤 𝑘 +2 ℤ ∕2 𝑤 𝑘 +2 𝑘 + 4 ℤ ℤ ∕28 𝑘 + 5 0 ℤ ∕28 𝑘 + 6 K 𝑘 +6 ( ℤ ) odd ℤ ⊕ K 𝑘 +6 ( ℤ ) odd 𝑘 + 7 ℤ ∕ 𝑤 𝑘 +4 ℤ ∕ 𝑤 𝑘 +4 Remark.
For 𝑚 ≤ the group (K 𝑚 −2 ) odd is known to be cyclic of order 𝑐 𝑚 , see [Wei13, Example10.3.2]. This holds for all 𝑚 if Vandiver’s conjecture is true [Wei13, Theorem 10.2].3.2.3. Remark.
The number 𝑤 𝑛 is equal to the cardinality of the image of the 𝐽 -homomorphism 𝜋 𝑛 −1 ( 𝑂 ) → 𝜋 𝑛 −1 ( 𝕊 ) in the stable stem. By [Qui76, pg. 186] the unit map 𝜋 𝑛 −1 ( 𝕊 ) → K 𝑛 −1 ( ℤ ) is injective on thisimage. Since the unit map for K( ℤ ) factors through the unit map for GW scl ( ℤ ) , it follows that the groups GW scl , 𝑘 +3 ( ℤ ) and GW scl , 𝑘 +7 ( ℤ ) consist precisely of image of 𝐽 -classes. Proof of Theorem 3.2.1.
Since the groups in question are finitely generated, it suffices to prove that thetheorem holds after localisation at 2 and after inverting 2. First, we argue 2-locally. We claim that, for 𝜖 = ±1 , the canonical map GW scl ( ℤ ; 𝜖 ) ⟶ GW scl ( ℤ [ ]; 𝜖 ) is a 2-local equivalence in degrees ≥ . Once this is shown, one can compare with [BK05, Theorem B] ,where the 2-local GW -groups of ℤ [ ] are determined as displayed. In order to compare their values for 𝑘 + 3 and 𝑘 + 7 with ours, note that by work of von Staudt the largest power of which divides 𝑤 𝑛 is thesame as the largest power of which divides 𝑛 . To see that the map is a 2-local equivalence, we considerthe localisation-dévissage fibre sequence GW( 𝔽 ; ( Ϙ s 𝜖 ) [−1] ) ⟶ GW s ( ℤ ; 𝜖 ) ⟶ GW s ( ℤ [ ]; 𝜖 ) from Corollary 2.1.9 and note that GW( 𝔽 ; ( Ϙ s 𝜖 ) [−1] ) ≃ K( 𝔽 ; ( Ϙ s 𝜖 ) [−1] ) hC by Proposition 3.1.4. Let us denoteby ℤ (−1) the complex ℤ in degree 0 with the sign action of C . We now note that the map K( 𝔽 ; ( Ϙ s 𝜖 ) [−1] ) → ℤ (−1) is a C -equivariant map whose fibre has finite and odd torsion homotopy groups. It follows that thismap induces a 2-local equivalence after applying (−) hC , which shows the claim.For the 2-inverted case, we observe that the fibre sequence K( ℤ ; 𝜖 ) hC ⟶ GW s ( ℤ ; 𝜖 ) ⟶ L s ( ℤ ; 𝜖 ) from Theorem 1 splits after inverting 2, see e.g. Corollary [II].4.4.17, so that there is an equivalence ofspectra GW s ( ℤ ; 𝜖 )[ ] ≃ (K( ℤ ; 𝜖 ) hC )[ ] ⊕ L s ( ℤ ; 𝜖 )[ ] Furthermore, we note that there is an isomorphism 𝜋 𝑛 (K( ℤ ; 𝜖 ) hC [ ]) ≅ (K 𝑛 ( ℤ ; 𝜖 )[ ]) C . It then followsfrom Lemma 3.2.4 below that GW s 𝑛 ( ℤ ; 𝜖 )[ ] ≅ { L s 𝑛 ( ℤ ; 𝜖 )[ ] for 𝑛 ≡ , mod 𝑛 ( ℤ ; 𝜖 )[ ] ⊕ L s 𝑛 ( ℤ ; 𝜖 )[ ] for 𝑛 ≡ , mod Note that what we denote GW is denoted L in loc. cit. and that the homotopy groups of L ( 𝑅 ) are denoted by 𝐿 𝑖 ( 𝑅 ) . This matches with the values in the above table after tensoring with ℤ [ ] and so the desired result follows. (cid:3) Lemma.
The C -actions induced by the Poincaré structures Ϙ 𝑠 and Ϙ 𝑠 − on D p ( ℤ ) induce multiplicationby (−1) 𝑛 on the groups K 𝑛 −1 ( ℤ )[ ] and K 𝑛 −2 ( ℤ )[ ] for each 𝑛 ≥ .Proof. This follows from [FGV20, §2], and we briefly collect the arguments. We first note that the dualitiesassociated to Ϙ s and Ϙ s− have the same underlying equivalences D p ( ℤ ) → D p ( ℤ ) op so that the induced C -action on homotopy groups is the same in both cases. Hence it will suffice to prove the claim for Ϙ s . Sincethe K-groups of ℤ are finitely generated, it suffices to prove the claim on the 𝓁 -completed K-groups K 𝑛 ( ℤ ) ∧ 𝓁 for all odd primes 𝓁 . We then have the following:i) The map K( ℤ ) ∧ 𝓁 → 𝐿 𝐾 (1) K( ℤ ) induces an isomorphism on 𝜋 𝑖 for 𝑖 ≥ ; see [FGV20, Proposition2.9] and use that 𝐿 𝐾 (1) K( ℤ )∕ 𝓁 ≃ (K( ℤ )∕ 𝓁 )[ 𝛽 −1 ] where 𝛽 is the mod 𝓁 Bott element; compare[FGV20, Remark 2.8].ii) The map 𝐿 𝐾 (1) K( ℤ ) → 𝐿 𝐾 (1) K( ℤ [ 𝓁 ]) is an equivalence; this follows from the fibre sequence K( 𝔽 𝓁 ) → K( ℤ ) → K( ℤ [ 𝓁 ]) ; see also [BCM20, LMT20] for a generalisation of this equivalence.iii) The resulting map K( ℤ ) ∧ 𝓁 → 𝐿 𝐾 (1) K( ℤ [ 𝓁 ]) is equivariant with respect to the duality action on bothsides; this action is usually denoted by 𝜓 −1 , see also [FGV20, 2.3.1].iv) On the odd homotopy groups 𝜋 𝑛 −1 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])∕ 𝓁 𝑘 ) the action of 𝜓 −1 is multiplication by (−1) 𝑛 (independently of 𝑘 ) [FGV20, Lemma 2.14], hence the same is true for the inverse limit over 𝑘 tendingto ∞ . This inverse limit is 𝜋 𝑛 −1 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) , as this group is finitely generated.v) For the action on the even homotopy groups, one shows that the (−1) -eigenspace 𝜋 𝑛 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) (−) of 𝜓 −1 is trivial for 𝑛 odd, and that the (+1) -eigenspace 𝜋 𝑛 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) (+) of 𝜓 −1 is trivial foreven 𝑛 : Indeed, one has { 𝜋 𝑛 −2 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])∕ 𝓁 𝑘 ) (−) ≅ H ét (spec( ℤ [ 𝓁 ]); 𝜇 ⊗ (2 𝑛 −1) 𝓁 𝑘 ) 𝜋 𝑛 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])∕ 𝓁 𝑘 ) (+) ≅ H ét (spec( ℤ [ 𝓁 ]); 𝜇 ⊗ (2 𝑛 ) 𝓁 𝑘 ) Since ℤ [ 𝓁 ] does not have non-trivial 𝓁 -power roots of unity, one finds that the étale cohomology termsvanish upon passing to the inverse limit over 𝑘 tending to ∞ .vi) We deduce that 𝜋 𝑛 −2 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) (+) = 𝜋 𝑛 −2 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) and that 𝜋 𝑛 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) (−) = 𝜋 𝑛 ( 𝐿 𝐾 (1) K( ℤ [ 𝓁 ])) . (cid:3) Remark.
A calculation of the Grothendieck-Witt groups of the integers has also been announced in[Sch19b], but with a different odd torsion: there it is claimed that the C -action on K ∗ ( ℤ )[ ] is multipli-cation by (−1) 𝑛 +1 on K 𝑛 −2 ( ℤ )[ ] and K 𝑛 −1 ( ℤ )[ ] , but we believe this comes from an error in equation(3.3) of [Sch19b, Proof of Lemma 3.1].In low degrees the groups can be worked out explicitly.3.2.6. Proposition.
The first 24 non-negative Grothendieck-Witt groups of ℤ are given by the table 3.2.6below.Proof. The only information not already present in the table of Theorem 3.2.1 is the structure of the oddtorsion in K 𝑛 ( ℤ ) for 𝑛 = 2 , mod . This can be read off from the list of K -groups [Wei13, Example 10.3]- the only non-trivial one in this range is K ( ℤ ) = ℤ ∕691 . (cid:3) We now turn to the computation of the classical 𝜖 -quadratic Grothendieck-Witt groups of ℤ . Recallthat for 𝜖 = ±1 there is a Poincaré functor ( D p ( ℤ ) , Ϙ gq 𝜖 ) → ( D p ( ℤ ) , Ϙ gs 𝜖 ) , which by the fibre sequence of ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 51 T ABLE
1. The first 24 Grothendieck-Witt groups of ℤ 𝑘 GW s 𝑘 ( ℤ ) 𝑘 GW 𝑠𝑘 ( ℤ ) 𝑘 GW 𝑠𝑘 ( ℤ )0 ℤ ⊕ ℤ ℤ ⊕ ℤ ∕2 16 ℤ ⊕ ℤ ∕21 ( ℤ ∕2) ℤ ∕2)
17 ( ℤ ∕2) ℤ ∕2)
10 ( ℤ ∕2)
18 ( ℤ ∕2) ℤ ∕24 11 ℤ ∕504 19 ℤ ∕2644 ℤ ℤ ℤ ℤ ∕6917 ℤ ∕240 15 ℤ ∕480 23 ℤ ∕65520 𝑘 GW −s 𝑘 ( ℤ ) 𝑘 GW −s 𝑘 ( ℤ ) 𝑘 GW −s 𝑘 ( ℤ )0 ℤ ℤ ℤ ℤ ℤ ∕48 11 ℤ ∕1008 19 ℤ ∕5284 ℤ ∕2 12 ℤ ∕2 20 ℤ ∕25 ℤ ∕2 13 ℤ ∕2 21 ℤ ∕26 ℤ ℤ ℤ ⊕ ℤ ∕6917 ℤ ∕240 15 ℤ ∕480 23 ℤ ∕65520 Corollary [II].4.4.14 induces a cartesian square of spectra GW gq ( ℤ ; 𝜖 ) L gq ( ℤ ; 𝜖 )GW gs ( ℤ ; 𝜖 ) L gs ( ℤ ; 𝜖 ) . The non-negative homotopy groups of the bottom left hand spectrum were computed in Theorem 3.2.1above. To understand the spectrum GW gq ( ℤ ; 𝜖 ) we will calculate the homotopy groups of the cofibre of theright hand vertical map, which is equivalent to the cofibre of the left hand vertical map. We begin with thecase 𝜖 = 1 . Write 𝐶 for the cofibre of the map L gq ( ℤ ) → L gs ( ℤ ) and 𝐶 𝑖 for the homotopy group 𝜋 𝑖 ( 𝐶 ) .3.2.7. Lemma.
The groups 𝐶 𝑖 are given byi) 𝐶 ≅ ℤ ∕2 ,ii) 𝐶 ≅ ℤ ∕8 iii) 𝐶 −1 ≅ ℤ ∕2 ,iv) 𝐶 𝑖 = 0 for all other values of 𝑖 .Proof. Let us consider the commutative diagram L gq ( ℤ )L q ( ℤ ) L s ( ℤ )L gs ( ℤ ) , ≃ ≥ ≃ ≤ ≃ ≤ −3 ≃ ≥ −2 where the subscript on the symbol ≃ indicates the range of dimensions 𝑖 in which the map induces anisomorphism on 𝜋 𝑖 . These ranges are obtained from Corollaries 1.2.8 and 1.3.7, using that by Example 1.1.4the Poincaré structures Ϙ gq = Ϙ ge and Ϙ gs = Ϙ ≥ (−1) are respectively -symmetric and (−1) -quadratic. Usingin addition that L gs−2 ( ℤ ) = 0 , it follows that 𝐶 𝑖 is at most non-trivial in the range −1 ≤ 𝑖 ≤ as claimed. Wethen find that 𝐶 −1 ≅ L gq−2 ( ℤ ) ≅ L q−2 ( ℤ ) ≅ ℤ ∕2 . The remaining two groups sit in the exact sequence ⟶ L gs1 ( ℤ ) ⟶ 𝐶 ⟶ L gq0 ( ℤ ) ⟶ L gs0 ( ℤ ) ⟶ 𝐶 → . Since the map L gq0 ( ℤ ) → L gs0 ( ℤ ) identifies with the multiplication by map on ℤ it follows that 𝐶 ≅ ℤ ∕8 and 𝐶 ≅ L gs1 ( ℤ ) ≅ ℤ ∕2 ; see [Ran81, Prop 4.3.1]. (cid:3) Remark.
Let us denote by L 𝑛 ( 𝑅 ) the cofibre of the symmetrisation map L q ( 𝑅 ) → L s ( 𝑅 ) , called normal or hyperquadratic L-theory in Ranicki’s work [Ran79, Ran92]. We then have 𝐶 ≃ 𝜏 [−1 , L 𝑛 ( ℤ ) .3.2.9. Theorem.
The classical quadratic Grothendieck-Witt groups of ℤ are given byi) GW gq0 ( ℤ ) ≅ ℤ ⊕ ℤ ,ii) GW gq1 ( ℤ ) ≅ ℤ ∕2 ⊕ ℤ ∕2 ,iii) GW gq 𝑛 ( ℤ ) ≅ GW gs 𝑛 ( ℤ ) for 𝑛 ≥ .Proof. The group GW gq0 ( ℤ ) is well known to be freely generated by the standard hyperbolic form and thepositive definite even form 𝐸 (see the discussion at the beginning of the section). For ii) consider the exactsequence 𝐶 → GW gq1 ( ℤ ) → GW gs1 ( ℤ ) → 𝐶 → GW gq0 ( ℤ ) → GW gs0 ( ℤ ) . The map GW gq0 ( ℤ ) → GW gs0 ( ℤ ) is injective and the image has index 8. It follows that GW gs1 ( ℤ ) ≅ ( ℤ ∕2) maps surjectively onto 𝐶 ≅ ℤ ∕2 and since 𝐶 = 0 by Lemma 3.2.7 we get that GW gq1 ( ℤ ) ≅ ( ℤ ∕2) .Finally, iii) is implied by Lemma 3.2.7iii). (cid:3) We now turn to the case 𝜖 = −1 .3.2.10. Lemma.
Let 𝐷 be the cofibre of the map L −gq ( ℤ ) → L −gs ( ℤ ) . Then 𝐷 ≃ Σ 𝐶 .Proof. By Theorem R.6 and Remark R.4, we have canonical equivalences L −gq ( ℤ ) ≃ Σ L gq ( ℤ ) and L −gs ( ℤ ) ≃ Σ L gs ( ℤ ) . Under these equivalences, the symmetrisation map in the definition of 𝐷 corre-sponds to the one of the definition of 𝐶 . (cid:3) Lemma.
There are group isomorphismsi) 𝜋 K( ℤ ; Ϙ gq− ) hC ≅ ℤ ∕4 ,ii) 𝜋 K( ℤ ; Ϙ gq− ) hC = 0 . Proof.
Since the involution on K( ℤ ; Ϙ gq− ) only depends on the underlying duality the canonical map 𝜋 𝑛 K( ℤ ; Ϙ gq− ) hC → 𝜋 𝑛 K( ℤ ; Ϙ s− ) hC is an isomorphism, and we shall henceforth replace Ϙ gq− with Ϙ s− .We first compute 𝜋 K( ℤ ; Ϙ s− ) hC . Consider the homotopy orbit spectral sequence 𝐸 𝑠,𝑡 = H 𝑠 ( 𝐶 ; 𝜋 𝑡 K( ℤ ; Ϙ s− )) ⟹ 𝜋 𝑠 + 𝑡 K( ℤ ; Ϙ s− ) hC . Since H ( 𝐶 ; 𝜋 K( ℤ ; Ϙ s− )) = 0 the generator of H ( ℤ ∕2; 𝜋 K( ℤ ; Ϙ s− )) ≅ ℤ ∕2 is a permanent cycle. Thegroup 𝜋 K( ℤ ; Ϙ s− ) hC also gets a contribution from H ( ℤ ∕2; 𝜋 K( ℤ ; Ϙ s− )) which has order , so in total itmust have order . The former group sits in an exact sequence L −s2 ( ℤ ) → 𝜋 K( ℤ ; Ϙ s− ) hC → GW −s1 ( ℤ ) , where the left hand group is isomorphic to L s0 ( ℤ ) ≅ ℤ and the right hand group is trivial by Table 3.2.6. Itfollows that the middle group is cyclic and is hence isomorphic to ℤ ∕4 .We will now compute 𝜋 K( ℤ ; Ϙ s− ) hC . For this, it will be useful to embed ℤ in the field ℝ of realnumbers, and consider the topological variants of K -theory and GW -theory for ℝ , equipped with its usualtopology. For this we follow the approach of [Sch17, §10] and define these in terms of the simplicialring ℝ Δ ∙ ∈ Fun(Δ op , Ring) , whose 𝑛 -simplices are the set ℝ Δ 𝑛 of continuous maps of topological spaces | Δ 𝑛 | → ℝ , considered as a ring via pointwise operations. One then defines the topological variants of K -theory, GW -theory and L -theory by K top ( ℝ ) ∶= | K( ℝ Δ ∙ ) | = colim 𝑛 ∈Δ op K( ℝ Δ 𝑛 ) ∈ S 𝑝 GW top ( ℝ ; Ϙ s 𝜖 ) ∶= | GW( ℝ Δ ∙ ; Ϙ s 𝜖 ) | = colim 𝑛 ∈Δ op GW( ℝ Δ 𝑛 ; Ϙ s 𝜖 ) ∈ S 𝑝 and L top ( ℝ ; Ϙ s 𝜖 ) ∶= | L( ℝ Δ ∙ ; Ϙ s 𝜖 ) | = colim 𝑛 ∈Δ op L( ℝ Δ 𝑛 ; Ϙ s 𝜖 ) ∈ S 𝑝. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES III: GROTHENDIECK-WITT GROUPS OF RINGS 53 The construction above furnishes a natural map of spectra K top ( ℝ ) → ko = ℤ × BGL top∞ ( ℝ ) which is anequivalence by [Sch17, Proposition 10.2]. Similarly, by the same proposition GW top0 ( ℝ ; Ϙ s 𝜖 ) ≅ GW ( ℝ ; Ϙ s 𝜖 ) and 𝜏 ≥ GW top ( ℝ ; Ϙ s 𝜖 ) is naturally equivalent to BO top∞ , ∞ ( ℝ ) when 𝜖 = 1 and to BSp top∞ ( ℝ ) when 𝜖 = −1 .The superscript top indicates that we topologise the groups as sequential colimits of Lie groups. In addition,by [Sch17, Remark 10.4] the natural map L( ℝ ; Ϙ s 𝜖 ) → L top ( ℝ ; Ϙ s 𝜖 ) is an equivalence.We now claim that the map K( ℤ ) → K top ( ℝ ) ≃ ko induces isomorphisms on 𝜋 𝑖 for 𝑖 ≤ . The groupsin question are in fact isomorphic, furthermore the composite 𝕊 → K( ℤ ) → ko is an isomorphism in theclaimed range, so the result follows.Let us write K top ( ℝ ; Ϙ s− ) for the spectrum K top ( ℝ ) considered together with the 𝐶 -action induced by theduality associated to Ϙ s− . Since taking homotopy orbits preserves connectivity we get from the above thatthe map 𝜋 𝑖 K( ℤ ; Ϙ s− ) hC → 𝜋 𝑖 K top ( ℝ ; Ϙ s− ) hC is an isomorphism for 𝑖 ≤ . To finish the proof it will hencesuffice to show that 𝜋 K top ( ℝ ; Ϙ s− ) hC vanishes. Since geometric realisations preserve fibre sequences ofspectra, the latter group sits in an exact sequence L top3 ( ℝ ; Ϙ s− ) ⟶ 𝜋 K top ( ℝ ; Ϙ s− ) hC ⟶ GW top2 ( ℝ ; Ϙ s− ) . Since ℝ is a field we have that L top3 ( ℝ ; Ϙ s− ) ≅ L ( ℝ ; Ϙ s− ) ≅ 0 and since GW top2 ( ℝ ; Ϙ s− ) ≅ 𝜋 Sp top∞ ( ℝ ) ≅ ℤ it follows that the group 𝜋 K top ( ℝ ; Ϙ s− ) hC is free. But from the homotopy orbit spectral sequence we seethat it has order at most 4, and so we conclude that it is trivial. (cid:3) Remark.
By Karoubi periodicity (as formulated, e.g. in Corollary [II].4.5.4), we know that K( ℤ ; Ϙ 𝑠 − ) ≃ 𝕊 𝜎 −2 ⊗ K( ℤ ; Ϙ s ) as spectrum with C -action. Furthermore, the map K( ℤ ) → ko is C -equivariant withrespect to the C -action induced by Ϙ s on K( ℤ ) and the trivial action on ko . The above lemma is then astatement about low dimensional homotopy groups of ( 𝕊 𝜎 −2 ⊗ ko) hC . These can also be computed usingthe cofibre sequence C → 𝑆 → 𝑆 𝜎 and some elaborations thereof.3.2.13. Theorem.
There are isomorphismsi) GW −gq0 ( ℤ ) ≅ ℤ ⊕ ℤ ∕2 ,ii) GW −gq1 ( ℤ ) ≅ ℤ ∕4 ,iii) GW −gq2 ( ℤ ) ≅ ℤ iv) GW −gq3 ( ℤ ) ≅ ℤ ∕24 v) GW −gq 𝑖 ( ℤ ) ≅ GW −gs 𝑖 ( ℤ ) for 𝑖 ≥ . We remark that statements ii) and iii) have been shown previously by Krannich and Kupers using geo-metric methods, see [KK20].
Proof.
Part v) follows immediately from Lemma 3.2.10 and Lemma 3.2.7iii). Part i) is well known, seethe discussion at the beginning of the section. For Part iii) it suffices to note that by Lemma 3.2.11 the map GW −gq2 ( ℤ ) → L −gq2 ( ℤ ) ≅ L gq0 ( ℤ ) ≅ ℤ is injective with finite cokernel.Now to show ii) consider the following commutative diagram with exact rows: L −gq2 ( ℤ ) 𝜋 K( ℤ ; Ϙ gq− ) hC GW −gq1 ( ℤ ) L −gq1 ( ℤ )L −gs2 ( ℤ ) 𝜋 K( ℤ ; Ϙ gs− ) hC GW −gs1 ( ℤ ) L −gs1 ( ℤ ) ≅ Since GW −gs1 ( ℤ ) = 0 by Table (3.2.6) the bottom left hand map must be surjective. As in the proof ofLemma 3.2.10, the map L −gq2 ( ℤ ) → L −gs2 ( ℤ ) identifies with the map L gq0 ( ℤ ) → L gs0 ( ℤ ) and hence with theinclusion ℤ ↪ ℤ . Since 𝜋 K( ℤ , Ϙ gq− ) hC ≅ ℤ ∕4 the upper left hand map must be . In addition L −gq1 ( ℤ ) ≅L −q1 ( ℤ ) ≅ 0 and so the upper middle map gives an isomorphism GW −gq1 ( ℤ ) ≅ ℤ ∕4 by Lemma 3.2.11i). Finally, to prove iv) consider the commutative diagram L −gq4 ( ℤ ) 𝜋 K( ℤ ; Ϙ gq− ) hC GW −gq3 ( ℤ ) L −gq3 ( ℤ )L −gs4 ( ℤ ) 𝜋 K( ℤ ; Ϙ gs− ) hC GW −gs3 ( ℤ ) L −gs3 ( ℤ ) ≅ where the bottom right map is surjective by Lemma 3.2.11ii). Then L −gq3 ( ℤ ) ≅ L gq1 ( ℤ ) = 0 and L −gq4 ( ℤ ) ≅L −gs4 ( ℤ ) ≅ L gs2 ( ℤ ) ≅ 0 , which implies that the top middle horizontal map in the above diagram is anisomorphism and the bottom middle horizontal map is injective with cokernel L −gs3 ( ℤ ) ≅ L −s3 ( ℤ ) ≅ ℤ ∕2 ;see Corollary 2.2.3. Since GW −gs3 ( ℤ ) ≅ ℤ ∕48 by Table (3.2.6) this implies that GW −gq3 ( ℤ ) ≅ ℤ ∕24 , asclaimed. (cid:3) Remark.
We note that for the ring of integers O in a number field, the canonical map GW scl ( O ; 𝜖 ) ⟶ GW scl ( O [ ]; 𝜖 ) is a 2-local equivalence in degrees ≥ , as the fibre is given by a finite sum of spectra of the form GW( 𝔽 𝔭 ; ( Ϙ s 𝜖 ) [−1] ) ,with 𝔽 𝔭 a finite field of characteristic 2 by Corollary 2.1.9. The same argument as in the proof of Theo-rem 3.2.1 then implies the claim. In principle, one can then use the results of [KRØ18] to calculate the2-local Grothendieck-Witt groups of O . As before, the odd torsion is controlled by the isomorphisms GW scl ,𝑛 ( O ; 𝜖 )[ ] ≅ K 𝑛 ( O ; 𝜖 )[ ] C ⊕ L s 𝑛 ( O ; 𝜖 )[ ] . To make efficient use of this, one needs a version of Lemma 3.2.4, determining the C -action on the -inverted K -groups of O , which can again be described in terms of étale cohomology of O [ ] .R EFERENCES
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