Remarks on K(1) -local K -theory
aa r X i v : . [ m a t h . K T ] M a y REMARKS ON K (1) -LOCAL K -THEORY BHARGAV BHATT, DUSTIN CLAUSEN, AND AKHIL MATHEW
Abstract.
We prove two basic structural properties of the algebraic K -theory of rings after K (1)-localization at an implicit prime p . Our first result (also recently obtained by Land–Meier–Tamme by different methods) states that L K (1) K ( R ) is insensitive to inverting p on R ; we deducethis from recent advances in prismatic cohomology and TC. Our second result yields a K¨unnethformula in K (1)-local K -theory for adding p -power roots of unity to R . Introduction
In this note, we consider the algebraic K -theory spectrum K ( R ) of a ring R , after applyingthe operation L K (1) of K (1)-localization at a prime p which is fixed throughout. The construction R L K (1) K ( R ) featured in the work of Thomason [Tho85] connecting algebraic K -theory and ´etalecohomology, cf. [Mit97] for a survey. Here we record two basic structural features of L K (1) K ( R ).We first show that K (1)-local K -theory is insensitive to inverting p ; a stronger result (forconnective K (1)-acyclic E -rings) has been obtained recently by Land–Meier–Tamme in [LMT20,Cor. 3.28]. Theorem 1.1.
Let A be an associative ring, or even an E -algebra over Z . Then the map ofspectra K ( A ) → K ( A [1 /p ]) induces an equivalence L K (1) K ( A ) ≃ L K (1) K ( A [1 /p ]) . Example 1.2 ( p -power torsion rings) . When A is p -power torsion, we conclude that L K (1) K ( A ) =0. When A is simple p -torsion (i.e., an F p -algebra), this follows from Quillen’s calculation [Qui72]of the K -theory of finite fields, in particular that K ( F p ; Z p ) ≃ H Z p . However, for Z /p n , one knowsthe p -adic K -theory only in a certain range [Ang15, Bru01], so it seems difficult to prove the resultby direct computation.In [LMT20], Land–Meier–Tamme give a purely homotopy-theoretic proof of the result of Exam-ple 1.2, applying more generally to certain ring spectra; from this Theorem 1.1 is a consequence.Our first goal is to give an arithmetic proof of Theorem 1.1, as a K -theoretic version of the´etale comparison theorem of [BS19, Th. 9.1]. In fact, the assertion L K (1) K ( Z /p n ) = 0 is a quickconsequence of recent advances in topological cyclic homology [BMS19] and the theory of prismaticcohomology [BS19]. While we do not know the K -theory of Z /p n , the work [BMS19, CMM18, BS19]leads to a relatively explicit calculation of the K -theory of O C /p n via TC, for C the completedalgebraic closure of Q p and O C ⊂ C the ring of integers. We can calculate directly there that theBott element is p -adically nilpotent, and then we use [CMNN20] to descend.In fact, we can obtain (via [CMM18]) the following consequence, which is a K -theoretic versionof the ´etale comparison theorem: Corollary 1.3.
Let R be any commutative ring which is henselian along ( p ) . Then there is anatural equivalence L K (1) TC( R ) ≃ L K (1) K ( R [1 /p ]) . Date : May 13, 2020.
Our second result is a type of K¨unneth formula in K (1)-local K -theory. In general, K -theorydoes not satisfy a K¨unneth formula: it is only a lax symmetric monoidal, not a symmetric monoidalfunctor. Here we show that in the special case of adding p -power roots of unity, one does have aK¨unneth formula which one can make explicit.To formulate the result, we recall that Z × p naturally acts both on Z [ ζ p ∞ ] and on the p -complete E ∞ -ring KU ˆ p , by Galois automorphisms and Adams operations respectively. For a ring R , we write R [ ζ p ∞ ] = R ⊗ Z Z [ ζ p ∞ ]. Theorem 1.4.
Let R be a commutative ring. Then there are natural, Z × p -equivariant equivalencesof E ∞ -rings L K (1) K ( R [ ζ p ∞ ]) ≃ ( K ( R ) ⊗ KU ˆ p ) ˆ p . Theorem 1.4 is related to results of Dwyer–Mitchell [DM98] and Mitchell [Mit00]; our construc-tion of the comparison map is based on the description of Snaith [Sna81] of KU . Furthermore, onecan obtain an analog of this formula for any localizing invariant over Z [1 /p ]-algebras which com-mutes with filtered colimits. Using these ideas, we also give a complete description of K (1)-local K -theory as an ´etale sheaf of spectra on Z [1 /p ]-algebras (under appropriate finiteness conditions),cf. Theorem 3.9, yielding a spectrum-level version of Thomason’s spectral sequence from [Tho85]. Acknowledgments.
We thank Lars Hesselholt, Jacob Lurie, and Peter Scholze for helpful discus-sions. We are also grateful to the anonymous referee for their helpful comments on the first versionof this paper. The third author would like to thank the University of Copenhagen for its hospitalityduring which some of this work was done. The first author was supported by NSF grant
Proof of Theorem 1.1 δ -ring calculations. In this section, we prove a simple nilpotence result (Proposition 2.5). We freely use the theory of δ -rings introduced in [Joy85]. Given a δ -ring ( R, δ ), we let ϕ : R → R be the map ϕ ( x ) = x p + pδ ( x ), so that ϕ is a ring homomorphism. We recall the basic formulas δ ( ab ) = a p δ ( b ) + b p δ ( a ) + pδ ( a ) δ ( b ) = ϕ ( a ) δ ( b ) + δ ( a ) b p , (1) δ ( a + b ) = δ ( a ) + δ ( b ) − X
An element x of a p -complete δ -ring R is called weakly k -distinguished if ( x, δ ( x ) , . . . , δ k ( x ))is the unit ideal. As in Remark 2.13 below, one could replace its use below with that of the ´etale comparison theorem of [BS19]. δ -rings also arise as the natural structure on the homotopy groups of K (1)-local E ∞ -ring spectra (where theyare often called θ -algebras or Frobenius algebras), cf. [Hop14]. We will not use this fact here. EMARKS ON K (1)-LOCAL K -THEORY 3 Example 2.2.
The element p k is weakly k -distinguished in any p -complete δ -ring. It suffices tocheck this in Z p . Indeed, the formula δ ( x ) = x − x p p (valid for x ∈ Z p ) shows easily that if the p -adicvaluation v p ( x ) is positive, then v p ( δ ( x )) = v p ( x ) −
1. Inductively, we thus get that v p ( δ k ( p k )) = 0,so p k is weakly k -distinguished. Definition 2.3.
Let R be a δ -ring. Let I ⊂ R be an ideal. We define δ ( I ) as the ideal generatedby δ ( x ) , x ∈ I . Example 2.4.
Suppose I = ( x ). Then δ ( I ) ⊂ ( x, δ ( x )). More generally, if I ⊂ R is an idealgenerated by elements ( f , . . . , f n ), then(3) δ ( I ) ⊂ ( f , . . . , f n , δ ( f ) , . . . , δ ( f n )) . This follows easily from the formulas (1) and (2) above.
Proposition 2.5 (Nilpotence criterion) . Let R be a δ -ring, and let x, y ∈ R . Suppose R is ( p, x ) -adically complete and we have the equation xy = p k . Then y is weakly ( k − -distinguished and x is p -adically nilpotent.Proof. We first claim that y is weakly ( k − p k ) = ( xy ).We claim that for each i ≥
1, we have that(4) δ i ( p k ) ∈ ( ϕ i ( x ) δ i ( y ) , δ i − ( y ) , . . . , y ) . To see this, we use induction on i . For i = 1, we have δ ( xy ) = ϕ ( x ) δ ( y ) + δ ( x ) y p , as desired. If wehave proven (4) for a given i , then we can apply δ to both sides and use (3) to conclude the resultfor i + 1, together with δ ( ϕ i ( x ) δ i ( y )) = ϕ i +1 ( x ) δ i +1 ( y ) + δ ( ϕ i ( x )) δ i ( y ) p . By induction on i , thisproves (4) in general.Taking i = k in (4) and using that δ k ( p k ) is a unit, we find that ϕ k ( x ) δ k ( y ) , δ k − ( y ) , . . . , y generate the unit ideal in R . But since ϕ k ( x ) is contained in the Jacobson radical of R (as R is( p, x )-adically complete and ϕ k ( x ) ≡ x p k modulo p ), we conclude that δ k − ( y ) , . . . , y generate theunit ideal of R . Thus, y is weakly ( k − x is p -adically nilpotent. Consider the p -adic completion R ′ of R [1 /x ];this is also a p -complete δ -ring, and it suffices to show that R ′ = 0. But the image of y in R ′ isboth a unit multiple of p k and weakly ( k − y, δ ( y ) , . . . , δ k − ( y )) isboth contained in ( p ) and the unit ideal. This now shows that R ′ = 0 as desired. (cid:3) The vanishing result for L K (1) TP( O C /p n ) . In this subsection, we let C be the completion ofthe algebraic closure of Q p , let O C be its ring of integers, and let A inf denote Fontaine’s period ring,with its canonical surjective map θ : A inf → O C . The kernel of θ is generated by a nonzerodivisor,a choice of which we denote d . With respect to the unique δ -structure on A inf , d is a distinguishedelement and ( A inf , ( d )) is the perfect prism corresponding to the integral perfectoid ring O C , [BS19,Th. 3.10] and [BMS18, Sec. 3].We can fix such a d as follows. Consider a system (1 , ζ p , ζ p , . . . ) of compatible primitive p -powerroots of unity in O C and let ǫ denote the corresponding element in O ♭C = lim ←− Frob O C /p . Then wecan take d to be the element d = [ ǫ ] − ǫ /p ] − ∈ A inf = W ( O ♭C ) . It is well-known that this choice of d generates the kernel of θ . See [BMS18, Sec. 3] for a treatmentof all of these facts. BHARGAV BHATT, DUSTIN CLAUSEN, AND AKHIL MATHEW
Next we recall the calculation of topological Hochschild invariants of O C , using the notation andlanguage of [NS18]. Proposition 2.6 (Hesselholt [Hes06], Bhatt–Morrow–Scholze [BMS19, Sec. 6]) . We can chooseisomorphisms TC − ( O C ; Z p ) ≃ A inf [ u, v ] / ( uv − d ) , TP( O C ; Z p ) ≃ A inf [ σ ± ] , | u | = 2 , | v | = − , | σ | = 2 , such that the canonical map is the identity on A inf and carries v σ − , u d · σ and the cyclotomicFrobenius map is the Frobenius on A inf and carries u σ . Remark 2.7.
In degree zero, the above isomorphisms are canonical. However, in nonzero degrees,they are not canonical; for example, they are not Galois-equivariant. The canonical form of theabove proposition involves the so-called Breuil–Kisin twists as in [BMS19].
Construction 2.8 ( K (1)-localization explicitly) . Recall from [Niz98, Lemma 3.1] or [HN19, Lemma1.3.7] that the localization sequence shows K ( O C ; Z p ) ∼ → K ( C ; Z p ), and Suslin’s rigidity theorem[Sus83] shows that the latter is isomorphic to ku b p (i.e., p -complete connective topological K -theory)as a ring spectrum by choosing any ring isomorphism C ∼ = C . The K (1)-localization of ku is imple-mented by inverting the generator in degree 2 and then p -completing, as is clear from the definition.It follows that the K (1)-localization of K ( O C ; Z p ), or more generally of any p -complete K ( O C ; Z p )-module M , can be obtained in the analogous way: L K (1) M = M [ β − ] b p , where β ∈ π K ( O C ; Z p ) ∼ = Z p is any generator.Next we trace this into TP, where one can identify the image of the cyclotomic trace. Proposition 2.9 ([HN19, Th. 1.3.6]) . With respect to the above identifications, the cyclotomictrace K ∗ ( O C ; Z p ) → TP ∗ ( O C ; Z p ) carries β to a Z × p -multiple of ([ ǫ ] − σ . Let R be a quasiregular semiperfectoid O C -algebra (in the sense of [BMS19, Sec. 4]), e.g., thequotient of a perfectoid by a regular sequence. Then one can construct [BS19, Sec. 7] a ( p, d )-adically complete and d -torsion-free δ -ring ∆ R , which receives a canonical map from A inf , and amap R → ∆ R / ( d ); moreover, ∆ R is universal for this structure. The ring ∆ R is equipped withthe Nygaard filtration (also defined in loc. cit.) whose completion is denoted b ∆ R , and acquires a δ -structure itself. Our primary tool in this paper, which connects algebraic K -theory (or ratherTP) and δ -rings, is the following result. Theorem 2.10 ([BMS19] and [BS19, Sec. 13]) . For a quasiregular semiperfectoid O C -algebra R , TP ∗ ( R ; Z p ) is concentrated in even degrees, is 2-periodic, and there is a canonical isomorphism π TP( R ; Z p ) ≃ b ∆ R . Using this, we can give a direct description of the K (1)-localization of TP in terms of b ∆ . Corollary 2.11.
For a quasiregular semiperfectoid O C -algebra R , there is a canonical isomorphism π ( L K (1) TP( R )) ≃ ( b ∆ R [1 /d ]) ˆ p .Proof. The spectrum L K (1) TP( R ) is obtained by inverting (in the p -complete category) the imageof the Bott element from K ∗ ( O C ; Z p ) via the trace map. As we saw, the map K ∗ ( O C ; Z p ) → TP ∗ ( O C ; Z p ) carries the class of β to a graded unit times the class of [ ǫ ] − ∈ A inf . However, in A inf we have [ ǫ ] − ≡ ([ ǫ /p ] − p (modulo p ) and d ≡ ([ ǫ /p ] − p − (modulo p ); thus, invertingeither [ ǫ ] − d in the p -complete sense is the same operation, completing the proof. (cid:3) EMARKS ON K (1)-LOCAL K -THEORY 5 Finally, we can conclude the main vanishing result that was the goal of this section.
Corollary 2.12.
For each n , we have that L K (1) (TP( O C /p n )) = 0 .Proof. As there is a ring map ∆ O C /p n → b ∆ O C /p n , by the above it suffices to show that d is p -adicallynilpotent in ∆ O C /p n . But by definition ∆ O C /p n is a ( p, d )-adically complete δ -ring such that thereis a homomorphism O C /p n → ∆ O C /p n /d . It follows that we can solve the equation dy = p n in ∆ O C /p n , and we deduce that d is p -adically nilpotent by Proposition 2.5, as desired. (cid:3) Remark 2.13.
The main result that was shown above is that if R is a p -power torsion O C -algebrawhich is quasiregular semiperfectoid, then d is p -adically nilpotent in ∆ R . This is a special case ofthe ´etale comparison theorem [BS19, Theorem 9.1], since in this case the generic fiber of Spf( R )vanishes; in particular, the use of the ´etale comparison theorem could replace Proposition 2.5 above.2.3. The K (1) -local K -theory of Z /p n . Here we prove the following special case of our mainresult.
Proposition 2.14.
For each n , we have L K (1) K ( Z /p n ) = 0 .Proof. We first prove the weaker assertion that if C is as in the previous section, then L K (1) K ( O C /p n ) =0. Indeed, by [CMM18, Th. C], the cyclotomic trace K ( O C /p n ; Z p ) → TC( O C /p n ; Z p ) is an equiv-alence, so it suffices to show that L K (1) TC( O C /p n ; Z p ) = 0. Furthermore, according to [NS18],TC( O C /p n ; Z p ) is an equalizer of two maps,(5) TC( O C /p n ; Z p ) = eq (cid:0) TC − ( O C /p n ; Z p ) ⇒ TP( O C /p n ; Z p ) (cid:1) . The first (canonical) map has cofiber given by Σ THH( O C /p n ; Z p ) hS , which is clearly K (1)-acyclic as a homotopy colimit of Eilenberg-MacLane spectra. Thus, L K (1) TC − ( O C /p n ; Z p ) ≃ L K (1) TP( O C /p n ; Z p ), and the latter vanishes by Corollary 2.12. Using the formula (5), we get that L K (1) TC( O C /p n ; Z p ) = 0 as desired.Now we descend to prove the result for Z /p n . Let E range over the finite extensions of Q p inside Q p . For any such, we have a finite flat morphism Z /p n → O E /p n . The colimit over E yields O C /p n .Therefore, in the ∞ -category of p -complete E ∞ -rings, we havelim −→ E L K (1) K ( O E /p n ) = L K (1) K ( O C /p n ) . Since we have just shown that the target vanishes, the source does too. Now the source is a filteredcolimit in ( p -complete) ring spectra, and a ring spectrum vanishes if and only if its unit is null-homotopic. We conclude that for some finite extension E , L K (1) K ( O E /p n ) vanishes. Finally, bythe descent results of [CMNN20] (in particular, finite flat descent for L K (1) K ( − ) on commutativerings), we find that L K (1) K ( Z /p n ) ≃ Tot (cid:16) L K (1) K ( O E /p n ) ⇒ L K (1) K ( O E /p n ⊗ Z /p n O E /p n ) →→→ . . . (cid:17) . Since this is a diagram of E ∞ -rings, we conclude that all the terms in the totalization must vanish,and we get L K (1) K ( Z /p n ) = 0 as desired. (cid:3) The main result for Z -linear ∞ -categories. In this section, we explain the deduction ofTheorem 1.1. This argument also appears in [LMT20, Sec. 3.1].Let R be a commutative ring, and let C be a small R -linear stable ∞ -category (always assumedidempotent-complete). Given a nonzerodivisor (for simplicity) x ∈ R , we say that C is x -powertorsion if for each object Y ∈ C , we have that x n : Y → Y is nullhomotopic for some n ≥
0. For
BHARGAV BHATT, DUSTIN CLAUSEN, AND AKHIL MATHEW instance, the kernel of the map Perf( R ) → Perf( R [1 /x ]), i.e., those perfect complexes of R -moduleswhich are acyclic outside of ( x ), forms such an R -linear stable ∞ -category. Moreover, for each R -algebra R ′ such that R ′ is perfect as an R -module, we can define the ∞ -category of R ′ -modulesin C , which we denote Mod R ′ ( C ); this is then an R ′ -linear stable ∞ -category. Examples of objectsin Mod R ′ ( C ) include objects of the form R ′ ⊗ Y for Y ∈ C ; these are given by extension of scalarsfrom C . For simplicity, when working with these objects, we will simply write Hom R ′ instead ofHom Mod R ′ ( C ) .We use the following basic fact. Proposition 2.15.
Let C be an R -linear (idempotent-complete) stable ∞ -category which is x -powertorsion. Then we have, in R -linear stable ∞ -categories (6) lim −→ n Mod
R/x n ( C ) ≃ C , via the natural restriction of scalars maps.Proof. In the following, all tensor products of R -modules are derived. Let M, N ∈ Mod
R/x n ( C ).Then for m ≥ n , we have by adjunctionHom R/x m ( M, N ) = Hom
R/x n ( M ⊗ R/x m R/x n , N ) = Hom R/x n ( M ⊗ R/x n ( R/x n ⊗ R/x m R/x n ) , N ) , where the relative tensor products are regarded as R/x n -modules in Ind( C ). Similarly, we haveHom R ( M, N ) = Hom
R/x n ( M ⊗ R/x n ( R/x n ⊗ R R/x n ) , N ) . It therefore suffices to show that the tower in (
R/x n , R/x n )-bimodules (cid:8) R/x n ⊗ R/x m R/x n (cid:9) m ≥ n is pro-constant with value R/x n ⊗ R R/x n ; this will prove thatHom R ( M, N ) = lim −→ m ≥ n Hom
R/x m ( M, N ) , and that the functor in (6) is fully faithful. It is easy to see that any object in C is (at least up toretracts) in the essential image, since generating objects R/x ⊗ Y are in the essential image.Now the pro-constancy claim follows from the following more precise assertion: the tower ofsimplicial commutative rings (cid:8) R/x n ⊗ R/x m R/x n (cid:9) m ≥ n is pro-constant with value R/x n ⊗ R R/x n .Indeed, R/x n ⊗ R R/x n is the free simplicial commutative ring over R/x n on a class in degree 1, anda short computation shows that for m ≥ n , R/x n ⊗ R/x m R/x n is the free simplicial commutativering on classes in degree 1 and 2; moreover, the classes in degree two form a pro-zero system. (cid:3) Finally, we can prove Theorem 1.1, which we restate for arbitrary Z -linear stable ∞ -categories. Theorem 2.16.
Let C be a Z -linear stable ∞ -category. Then L K (1) K ( C ) = L K (1) K ( C [1 /p ]) .Proof. Let C tors ⊂ C be the subcategory of p -power torsion objects. Then we have a localizationsequence C tors → C → C [1 /p ], so the induced sequence in algebraic K -theory shows that it sufficesto prove that L K (1) K ( C tors ) = 0. But we have seen (Proposition 2.15) that C tors is a filtered colimitof a sequence of stable ∞ -categories, each of which is Z /p n -linear for some n . By Proposition 2.14, L K (1) K vanishes for each of these; thus, it vanishes for C tors as desired. (cid:3) EMARKS ON K (1)-LOCAL K -THEORY 7 Complements.
Combining with the main result of [CMM18], we get the following.
Theorem 2.17.
Let R be a commutative ring. Then there is a natural equivalence L K (1) TC( R ) ≃ L K (1) K ( R b p [1 /p ]) . If R is henselian along p , then these are naturally equivalent to L K (1) K ( R [1 /p ]) . Remark 2.18.
In the above statement, the p -completion R b p can be taken to be either derivedor ordinary p -completion; it doesn’t matter for the statement, as the (mod p ) K -theory of Z [1 /p ]-algebras is nil-invariant and truncating in the sense of [LT19]. Proof.
We claim that all of the natural mapsTC( R ) → TC( R b p ) ← K ( R b p ) → K ( R b p [1 /p ])are K (1)-equivalences. For the left map, this is because TC /p is invariant under (mod p ) equiv-alences. For the right map this is by Theorem 1.1. For the middle map, [CMM18] gives a fibersquare K ( R b p ; Z p ) (cid:15) (cid:15) / / TC( R b p ; Z p ) (cid:15) (cid:15) K ( R/p ; Z p ) / / TC(
R/p ; Z p ) . But K (1)-localization annihilates the bottom row since L K (1) K ( F p ) = 0. Thus we obtain thedesired equivalence L K (1) K ( R b p ) ≃ L K (1) TC( R b p ). The deduction in the p -henselian case followssimilarly from [CMM18]. (cid:3) Remark 2.19.
This result is a form of the “´etale comparison theorem” of Bhatt–Scholze in integral p -adic Hodge theory, [BS19, Th. 9.1]. Indeed, TC( R ) is closely related to the complexes Z p ( n ) of[BMS19], whereas L K (1) K ( R b p [1 /p ]) is related in a similar manner to the standard ´etale Z p ( n )’s onthe rigid analytic generic fiber [Tho85]. With respect to appropriate motivic filtrations on bothsides, we expect this result to recover the ´etale comparison theorem. Question 2.20. (1) The first statement of Theorem 2.17 also makes sense for associative rings R . It is natural to guess that the theorem holds in that greater generality, and constitutesa kind of “non-commutative p -adic Hodge theory.” We remark that the only ingredient inthe above proof which required commutativity was the rigidity result of [CMM18] for theideal ( p ) ⊂ R when R is p -complete.(2) One could also speculate about higher height analogs of Theorem 2.17, in the context ofstructured ring spectra R : is there such a thing as “ v n -adic Hodge theory”? Note that thereis a “red shift” aspect to Theorem 2.17, in that p = v is the relevant chromatic elementon the inside of the K -theory whereas v is the relevant chromatic element on the outside.This result can also be interpreted in the light of Selmer K -theory. Recall: Definition 2.21 (Selmer K -theory, [Cla17]) . Let C be a Z -linear ∞ -category. We let K Sel ( C ) =TC( C ) × L TC( C ) L K ( C ).As in [CM19], Selmer K -theory, while a noncommutative invariant (i.e., one defined for stable ∞ -categories), turns out to recover ´etale K -theory for commutative rings in degrees ≥ −
1. Thedefinition of Selmer K -theory involves a pullback square; it is built from three other noncommutativeinvariants. We observe here that the pullback, at least after p -adic completion (which we denoteby K Sel ( · ; Z p )) and for commutative rings, is exactly the arithmetic square. BHARGAV BHATT, DUSTIN CLAUSEN, AND AKHIL MATHEW
For the next result, since we need to use derived completion, we work with connective E ∞ -algebras over Z for convenience. Recall also that L K (1) : Sp → Sp is the composition of L -localization followed by p -completion. Corollary 2.22.
Let R be a connective E ∞ -algebra over Z . Then the pullback square defining K Sel ( R ; Z p ) is also the tautological pullback square (valid for any localizing invariant) K Sel ( R ˆ p ; Z p ) × K Sel ( R ˆ p [1 /p ]; Z p ) K Sel ( R [1 /p ]; Z p ) .Proof. First, TC( · ; Z p ) is invariant under passage to p -completion and agrees with K Sel ( · ; Z p ) for p -complete connective E ∞ -algebras. Second, the factor L K (1) K ( · ; Z p ) is invariant under passage toinverting p on the argument (as we showed above) for Z -algebras and agrees with K Sel ( · ; Z p ) for Z [1 /p ]-algebras. Finally, the map L K (1) K ( · ; Z p ) → L K (1) TC( · ; Z p ) is an equivalence for p -completeconnective E ∞ -algebras, thanks to Theorem 2.17 and the Dundas–Goodwillie–McCarthy theorem.These three observations imply the claim. (cid:3) Remark 2.23.
Corollary 2.22 raises the question whether there is a direct definition of Selmer K -theory (at least after p -completion), without forming the above pullback square.3. The K¨unneth formula
To begin with, we recall the K (1)-local case of the celebrated result of Goerss–Hopkins–Miller[GH04, Rez98], which describes (in this case) the E ∞ -ring KU ˆ p and its space of automorphisms.See also [Lur18, Sec. 5] for a modern account of some generalizations. Theorem 3.1 (Goerss–Hopkins–Miller) . The space of E ∞ -automorphisms of KU ˆ p is given by Z × p ,via Adams operations ψ x , x ∈ Z × p , characterized by ψ x ( t ) = x · t for all t ∈ π KU ˆ p . We can now state the main K¨unneth-style theorem in the commutative case. In fact, as theproof will show, the analogous statement also holds for non-commutative rings (minus the E ∞ -ringstructure, of course). Closely related results appear in [DM98, Mit00] (at least at the level ofhomotopy groups). Theorem 3.2.
Let R be any commutative ring. Then there exists a natural, Z × p -equivariant equiv-alence of E ∞ -rings L K (1) ( K ( R ⊗ Z Z [ ζ p ∞ ])) ≃ L K (1) ( K ( R ) ⊗ KU ˆ p ) , where Z × p acts on Z [ ζ p ∞ ] by Galois automorphisms and on KU ˆ p as in Theorem 3.1. In the above statements we are only considering Z × p as a discrete group. This is for simplicity ofexposition, but in fact we will also obtain the (appropriately formulated) analogous statements onthe level of profinite groups, essentially as a consequence of the statements on the level of discretegroups. To accomplish this we will use the following lemma. While the statement involves a non-canonical choice of g ∈ Z × p , in the end it will only be used to prove statements which are formulatedindependently of g . Lemma 3.3.
Let µ denote the torsion subgroup of Z × p (so µ = µ p − for p odd and µ = µ for p = 2 ). Further let g ∈ Z × p be an element which projects to a topological generator of the quotient Z × p /µ ( ∼ = Z p ) , and consider the homomorphism µ × Z → Z × p induced by the inclusion on the firstfactor and g on the second factor.Then the induced pullback functor π ∗ : Sh hyp ( B Z × p ) → PSh( B ( µ × Z )) EMARKS ON K (1)-LOCAL K -THEORY 9 from hypercomplete sheaves of p -complete spectra on the site of finite continuous Z × p -sets to presheavesof p -complete spectra on the one-object groupoid B ( µ × Z ) is fully faithful. Moreover, its essentialimage consists of those p -complete spectra with µ × Z -action whose (mod p ) homotopy groups havethe property that the action extends continuously to Z × p .Proof. The pullback functor is associated to a geometric morphism of topoi, and hence commuteswith (mod p ) homotopy group sheaves. Thus the pullback functor lands in the claimed full subcat-egory by the usual equivalence between abelian groups sheaves on B Z × p and abelian groups withcontinuous Z × p -action. Similarly, the pullback functor detects equivalences, as the hypercomplete-ness lets us check this on (mod p ) homotopy group objects. Thus it suffices to show that if M is a p -complete spectrum with µ × Z -action whose induced action on (mod p ) homotopy groups extendscontinuously to Z × p , then π ∗ π ∗ M ∼ → M . This can be checked on underlying p -complete spectra,where it unwinds to the claim that lim −→ H M h ( H ∩ ( µ × Z )) → M is a (mod p ) equivalence. Here H runs over all open subgroups of Z × p and the superscript standsfor homotopy fixed points, compare [CM19, Sec. 4.1]. Passing to a cofinal collection of H ’s, theabove map is equivalent to lim −→ n M h ( p n Z ) → M. Replacing M by M/p , we may as well assume that M is annihilated by a power of p , in whichcase the condition is equivalent to demanding that the action on the homotopy of M admits acontinuous extension to Z × p , or equivalently is the union of subgroups fixed by some H . As thecolimit is filtered, and the limit is uniformly finite, we can then run a d´evissage on the Postnikovtower of M and reduce to the case where M is concentrated in a single degree, which may as wellbe degree 0, and there again we can assume that M is fixed by all sufficiently small H . It followsthat the map is an equivalence in degree 0. In degree 1, analyzing the colimit on the left we findthat all the terms identify with M but the bonding maps eventually identify with multiplication by p . As M is p -torsion the colimit gives 0, as required. (cid:3) We now construct the map which will implement the equivalence of Theorem 3.2. Let µ p ∞ ⊂ Z [ ζ p ∞ ] × be the subgroup of p -power roots of unity, so µ p ∞ ≃ Q p / Z p . Consider the classifying space Bµ p ∞ as an infinite loop space; we have therefore the E ∞ -ring Σ ∞ + Bµ p ∞ . Since Z × p acts on µ p ∞ via Galois automorphisms, we obtain a Z × p -action on Σ ∞ + Bµ p ∞ . Construction 3.4.
We have a Z × p -equivariant map of E ∞ -rings ψ : (cid:0) Σ ∞ + Bµ p ∞ (cid:1) ˆ p → K ( Z [ ζ p ∞ ]) ˆ p since for any commutative ring R we have a natural map Σ ∞ + BR × → K ( R ). Moreover, the source,which is homotopy equivalent to (cid:0) Σ ∞ + BS (cid:1) ˆ p ≃ (Σ ∞ + K ( Z p , ˆ p , contains the natural Bott class β ∈ π , which is invariant under the Z × p -action up to unit multiple. Proposition 3.5. ψ carries β to an invertible element in π L K (1) ( K ( Z [ ζ p ∞ ])) .Proof. By ´etale hyperdescent for K (1)-local K -theory [Tho85], Theorem 1.1, and Gabber–Suslinrigidity [Gab92], it suffices to verify this after composing to π L K (1) ( K ( k )), where k is any separablyclosed field of characteristic = p over Z [ ζ p ∞ ]. However, this follows from Suslin’s description [Sus83]of K ( k ) ˆ p in this case. In particular, π ∗ ( L K (1) K ( k )) is a Laurent polynomial algebra on β . (cid:3) We use now the following fundamental result of Snaith [Sna81] which gives a description of KU via the above constructions (here we only use the p -complete case). See also [Lur18, Sec. 6.5] for adifferent proof. Theorem 3.6 (Snaith) . The induced map ((Σ ∞ + Bµ p ∞ ) ˆ p [ β − ]) ˆ p → KU b p is an equivalence. This furnishes a potentially different Z × p -action on KU b p from that of Theorem 3.1; but in fact itmust be the same, as it does the same thing on π . Construction 3.7.
We obtain a Z × p -equivariant map of E ∞ -rings KU ˆ p → L K (1) K ( Z [ ζ p ∞ ])obtained from the map ψ by inverting the class β (in the p -complete sense) and using Theorem 3.6.Consequently, we obtain a natural Z × p -equivariant map for any R ,(7) L K (1) ( K ( R ) ⊗ KU ˆ p ) → L K (1) K ( R ⊗ Z Z [ ζ p ∞ ]) . Let us pause and explain how to promote this to an equivariant map for the profinite Z × p ,formulated as in Lemma 3.3 in terms of hypercomplete sheaves on the topos B Z × p of finite continuous Z × p sets. Lemma 3.8.
Consider µ p ∞ as equipped with its continuous action of Z × p , hence as an abelian groupsheaf on B Z × p . Thus Σ ∞ + Bµ p ∞ promotes to a sheaf of E ∞ -ring spectra on B Z × p . Then there existsan initial p -complete hypercomplete sheaf of E ∞ -ring spectra KU ˆ p on B Z × p equipped with a map Σ ∞ + Bµ p ∞ → KU ˆ p satisfying the property that on underlying spectra (meaning,after pulling back to the basepoint ∗ → B Z × p ) it carries β to an invertible element. Moreover, on underlying spectra this KU ˆ p identifieswith the usual KU ˆ p and the map identifies with the usual one.Proof. We choose a g ∈ Z × p as in Lemma 3.3 in order to transfer this to the analogous claim forpresheaves on B ( µ × Z ). But then it is a consequence of the equivariance of the Snaith identification,explained above. (cid:3) Now we recall that L K (1) K ( Z [ ζ p ∞ ]) = L K (1) K ( Z [1 /p, ζ p ∞ ]) promotes to a hypercomplete sheafon B Z × p , by Thomason’s hyperdescent theorem applied to the p -cyclotomic tower. Moreover themap Bµ p ∞ → Ω ∞ K ( Z [1 /p, ζ p ∞ ]) used to define ψ comes from the finite level maps Bµ p n → Ω ∞ K ( Z [1 /p, ζ p n ]) and hence ψ promotes to a map of sheaves of E ∞ -ring spectra on B Z × p . Thusthe above lemma does promote our naive discrete Z × p -equivariant map KU b p → L K (1) K ( Z [ ζ p ∞ ])to an honest one. The claim that such a map (or one derived from it such as (7)) is an equivalenceis independent of whether we think of Z × p as a discrete or profinite group, since equivalences ofhypercomplete sheaves over B Z × p can be checked on pullback to the basepoint.Let us formally record this more refined construction, and its fundamental property, in thefollowing. Theorem 3.9.
Let KU ˆ p denote the hypercomplete p -complete sheaf of E ∞ -algebras on B Z × p con-structed in the previous lemma. Let also π : Spec( Z [1 /p ]) et → B Z × p be the geometric morphismof topoi encoding the p -cyclotomic extension. Then there is a natural comparison map π ∗ KU b p → L K (1) K ( − ) of sheaves of E ∞ -rings. EMARKS ON K (1)-LOCAL K -THEORY 11 Furthermore, suppose that X is an algebraic space over Z [1 /p ] of finite Krull dimension with auniform bound on the virtual (mod p ) Galois cohomological dimension of its residue fields. Then for π X : X et → B Z × p the composition of π with the natural projection X → Spec( Z [1 /p ]) , the inducedcomparison map π ∗ X KU b p → L K (1) K ( − ) identifies the target as the p -completion of the hypercompletion of the source.Proof. The first statement was proved in the discussion just before. For the second statement,by Thomason’s hyperdescent theorem in the general form proved in [CM19], it suffices to checkthis on strictly henselian local rings; by Gabber–Suslin rigidity, we can even reduce to separablyclosed fields k . Then this encodes the combination of Suslin’s identification of K ( k ) b p with Snaith’spresentation of KU b p , as already explained above. (cid:3) When R is commutative, one can use similar arguments to directly check that (7) is an equiva-lence. However, we actually prove below a more general statement for arbitrary localizing invariantsover Z [1 /p ], which we formulate next. Let R be a commutative Z [1 /p ]-algebra and let E be a lo-calizing invariant for R -linear ∞ -categories (in the sense of [BGT13]) which commutes with filteredcolimits. Since everything is linear over algebraic K -theory, we obtain as well from (7) a natural Z × p -equivariant map(8) L K (1) ( E ( R ) ⊗ KU ˆ p ) → L K (1) ( E ( R [ ζ p ∞ ])) , which we will show to be an equivalence.To do this, we will need to use a type of Galois descent for the profinite group Z × p ; recall that L K (1) S → KU ˆ p is a pro-Galois extension for the profinite group Z × p in the sense studied by Rognes[Rog08]. From this, one can obtain a type of Galois descent with respect to the profinite group Z × p ;here we formulate an equivalent primitive version using the dense discrete subgroup µ × Z ⊂ Z × p asin Lemma 3.3.First, the Z × p -action on KU ˆ p yields a functor(9) L K (1) ( · ⊗ KU ˆ p ) : L K (1) Sp → Mod L K (1) Sp ( KU ˆ p ) h ( µ × Z ) , Proposition 3.10.
The natural functor (9) is fully faithful, and the essential image is spannedby those such that on mod p homotopy groups, the stabilizer of any element under the Z -actioncontains p N Z for N ≫ .Proof. Recall that L K (1) S ≃ ( KU ˆ p ) h ( µ × Z ) . Therefore, the functor (9) is fully faithful on thecompact generator L K (1) S /p of the source. To prove that (9) is fully faithful, it suffices to provethat the image of this object in the target is compact. When p >
2, this follows because homotopyinvariants over µ × Z can be expressed as the retract of a finite limit. This requires an argument for p = 2. In this case, it suffices to show that the unit is compact in Mod( KU ) h {± } , which followsby the Galois descent equivalence Mod( KU ) h {± } ≃ Mod( KO ), cf. e.g. [Mat16, Th. 9.4] for anaccount; in particular, this shows that taking µ -invariants here can also be expressed as a retractof a finite limit.For essential surjectivity, it suffices to show that if M is a p -complete KU -module with compatible µ × Z -action satisfying the continuity property in the statement, then M = 0 if and only if M h ( µ × Z ) =0. Indeed, suppose these homotopy fixed points vanish. Then also by Galois descent up the faithful µ -Galois extension of E ∞ -rings KU hµ ˆ p → KU ˆ p , it suffices to see that M hµ = 0. Now ( M hµ ) h Z = 0.But by the homotopy fixed point spectral sequence, since Z has cohomological dimension 1, we get that M hµ = 0 as desired. Here we use that any p -adically continuous Z -action on a nonzero p -torsion abelian group has a nontrivial fixed point. (cid:3) Theorem 3.11.
Let R be a commutative Z [1 /p ] -algebra and let E be a localizing invariant on R -linear ∞ -categories which commutes with filtered colimits (or just the filtered colimit giving the p -cyclotomic extension of R ). Then the natural map (8) is an equivalence.Proof. To see that (8) is an equivalence, it suffices to prove that it becomes an equivalence aftertaking µ × Z ⊂ Z × p -homotopy fixed points thanks to Proposition 3.10. The homotopy fixed pointson the left-hand-side are given by L K (1) E ( R ). For the right-hand-side, the localizing invariant L K (1) E ( R ⊗ Z [1 /p ] − ) satisfies ´etale hyperdescent over Z [1 /p ] by [CM19, Th. 7.14]. Using the evidentcomparison between continuous cohomology on Z × p and discrete group cohomology on µ × Z (whichfollows from Lemma 3.3), we find that the natural map L K (1) E ( R ) → L K (1) ( E ( R [ ζ p ∞ ])) h ( µ × Z ) isan equivalence. Thus, (8) becomes an equivalence after taking homotopy fixed points and thus isan equivalence. (cid:3) Finally, Theorem 3.2 follows, since by Theorem 1.1 one reduces to the case where R is a Z [1 /p ]-algebra. References [Ang15] Vigleik Angeltveit,
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