Algebraic K-theory and Grothendieck-Witt theory of monoid schemes
aa r X i v : . [ m a t h . K T ] S e p ALGEBRAIC K -THEORY AND GROTHENDIECK–WITTTHEORY OF MONOID SCHEMES JENS NIKLAS EBERHARDT, OLIVER LORSCHEID, AND MATTHEW B. YOUNG
Abstract.
We study the algebraic K -theory and Grothendieck–Witt theory ofproto-exact categories of vector bundles over monoid schemes. Our main resultsare the complete description of the algebraic K -theory space of an integral monoidscheme X in terms of its Picard group Pic( X ) and pointed monoid of regularfunctions Γ( X, O X ) and a description of the Grothendieck–Witt space of X interms of an additional involution on Pic( X ). We also prove space-level projectivebundle formulae in both settings. Contents
Introduction 11. Background material 42. K -theory and Grothendieck–Witt theory of pointed monoids 93. Algebraic K -theory of monoid schemes 164. Grothendieck–Witt theory of monoid schemes 29Appendix A. 35References 36 Introduction
In this paper we study the algebraic K -theory and Grothendieck–Witt theoryof monoid schemes. Our main results state that, under mild assumptions on thescheme X , these spaces are determined by simple algebraic invariants of X . Forthe K -theory space, these invariants are the Picard group Pic( X ) and the groupof invertible regular functions Γ( X, O X ) × . For the Grothendieck–Witt space, theadditional data of a set-theoretic involution of Pic( X ) is required.Monoid schemes form the core of algebraic geometry over the elusive field F withone element [43], [38], in the sense that every other approach to F -schemes containsmonoid schemes as a full subcategory. In the other direction, monoid schemes canbe seen as a direct generalization of toric geometry and Kato fans of logarithmicschemes; see [26], [8], [5], [2], [7] among others. The central position of monoidschemes within F -geometry is confirmed by the multiple links to other disciplines,such as Weyl groups as algebraic groups over F [28], computational methods fortoric geometry [6], [7], [13], a framework for tropical scheme theory [14], applicationsto representation theory [41], [19] and, last but not least, stable homotopy theoryas K -theory over F [9], [2], the theme on which we dwell in this paper. Date : September 29, 2020.2010
Mathematics Subject Classification.
Primary: 19D10; Secondary 19G38.
Key words and phrases.
Monoid schemes. Algebraic K -theory. Grothendieck–Witt theory. Pro-jective bundle formula. For the purpose of this introduction, we provide the reader with the followingsuggestive description: a monoid scheme is a topological space together with asheaf of commutative pointed monoids which is locally isomorphic to the spectrumof a commutative pointed monoid. Here, a pointed monoid is a monoid A with anabsorbing element 0, that is, an element satisfying 0 · a = 0 for all a ∈ A .Just as algebraic K -theory is an algebraic analogue of complex topological K -theory, Grothendieck–Witt theory is an algebraic analogue of Atiyah’s topological KR -theory [1]. A key feature of KR -theory is that it generalizes complex, realand quaternionic topological K -theory. Grothendieck–Witt theory enjoys a similarstatus in the algebraic setting. For example, whereas the algebraic K -theory ofa scheme X studies algebraic vector bundles on X , the Grothendieck–Witt the-ory of X studies algebraic vector bundles with non-degenerate bilinear form on X or, equivalently, orthogonal or symplectic vector bundles on X , depending on thesymmetry of the pairing. Grothendieck–Witt theory plays a fundamental role inKaroubi’s formulation and proof of topological and algebraic Bott periodicity andstudy of the homology of orthogonal and symplectic groups [20], [22], [21]. Re-cently, much effort has been devoted to developing the Grothendieck–Witt theoryof schemes; see, for example, [12], [33], [37], [23], [25], [24]. This paper provides thefirst results in the development of these ideas for monoid schemes.In Section 1 we recall relevant categorical and K -theoretic background. We workin the setting of proto-exact categories, a non-additive generalization of Quillen’sexact categories introduced by Dyckerhoff and Kapranov [10]. This is a convenientsetting for both K -theory and Grothendieck–Witt theory and was developed by theauthors in [11]. The first main result of [11] is a Group Completion Theorem for the K -theory of uniquely split proto-exact categories. The second main result of [11]is a description of the Grothendieck–Witt space GW Q ( A ) of a uniquely split proto-exact category with duality A satisfying additional mild assumptions, defined usingthe hermitian Q -construction, in terms of the group completion of the groupoid ofhyperbolic forms and the monoidal groupoid of isotropically simple symmetric formsin A . The second result can be seen as playing the role of the Group CompletionTheorem in the Grothendieck–Witt theory of uniquely split proto-exact categorieswith duality.In Section 2 we study the K -theory and Grothendieck–Witt theory of proto-exactcategories of modules over pointed monoids or, geometrically, vector bundles overnon-commutative affine monoid schemes. Let A be a (not necessarily commutative)pointed monoid. The category A - proj of finitely generated projective left A -moduleshas a uniquely split proto-exact structure (Lemma 2.4), and so fits into the frame-work of [11]. The category A - proj is particularly simple when A is integral, inwhich case all projective A -modules are free. This, together with the Group Com-pletion Theorem, allows for an explicit description of the K -theory space K ( A - proj )in terms of the group of units A × , thereby giving a ‘ Q = +’ theorem in this setting.In this way, we extend earlier results of Deitmar [9] and establish some unprovenclaims of Chu–Morava [3]. Our result is as follows. More generally, A need only be right partially cancellative. We work at this level of generalityin the body of the paper. ( X / F ) AND GW ( X / F ) 3 Theorem A (Theorem 2.5) . Let A be an integral pointed monoid. Then there is ahomotopy equivalence K ( A - proj ) ≃ Z × B ( A × ≀ Σ ∞ ) + , where A × ≀ Σ ∞ is the infinite wreath product lim −→ n (( A × ) n ⋊ Σ n ) . Next, we study the Grothendieck–Witt theory of projective A -modules, whereour results are new. Unlike the case of rings, the category A - proj does not admitan exact duality structure. In particular, the functor Hom A - proj ( − , A ) is poorly be-haved. For this reason, we restrict attention to the non-full proto-exact subcategory A - proj n ⊂ A - proj of normal morphisms, that is, A -module homomorphisms whosenon-empty fibres over non-basepoints are singletons. We prove in Lemma 2.2 that,if A is integral, then A - proj n admits a duality structure. From this point of view,normal morphisms are essential to Grothendieck–Witt theory. We remark thatthe K -theory of A - proj n and A - proj coincides. For earlier appearances of normalmorphisms in F -geometry, see [2], [40], [49]. The main results of this section deter-mine the Grothendieck–Witt spaces GW ⊕ ( A - proj n ) and GW Q ( A - proj n ). In partic-ular, the former space can be described in terms of “infinite orthogonal/symplecticgroups over F ”. In this way, we obtain an F -analogue of Karoubi’s results on thehermitian K -theory of rings [20]. A simplified version of this result is as follows. Theorem B (Theorem 2.11) . Let A be an integral pointed monoid with A × = { } .Then there is a homotopy equivalence GW ⊕ ( A - proj n ) ≃ Z × B ( Z / ≀ Σ ∞ ) + . By applying the results of [11], we can use Theorem B to describe the weakhomotopy type of GW Q ( A - proj n ).Having treated the local theory, we turn in Sections 3 and 4 to the global theoryof monoid schemes. Following earlier approaches [18], [9], a general definition of the K -theory of a monoid scheme X was given in [2], where a proto-exact category ofvector bundles Vect ( X ) and their normal O X -module homomorphisms was defined.We point out that this is not the only approach to the K -theory of monoid schemes;see [16] for a recent alternative. In Section 3 we bring the approach of [2] to its nat-ural conclusion by explicitly describing the K -theory space K ( X ) := K ( Vect ( X )).The key structural result is Proposition 3.12, which exhibits an extremely simplenon-full proto-exact subcategory hhO X ii [Pic( X )] of Vect ( X ) whose K -theory spaceis homotopy equivalent to K ( X ). Here, hhO X ii is the category whose objects areisomorphic to O ⊕ nX , n ∈ Z ≥ , together with all normal O X -module homomorphismsbetween them. The category hhO X ii [Pic( X )] can then be seen as the group alge-bra of Pic( X ) with coefficients in hhO X ii ; see Section 3.3 for a precise definition.Proposition 3.12 fails for the exact category of vector bundles over a field and is thesource of the relative strength of the following result. Theorem C (Theorem 3.14) . Let X be an integral monoid scheme. Then there isa homotopy equivalence K ( X ) ≃ Y ′M∈ Pic( X ) Z × B (Γ( X, O X ) × ≀ Σ ∞ ) + , where Q ′ is the restricted product of pointed topological spaces. J. N. EBERHARDT, O. LORSCHEID, AND M. B. YOUNG
Turning to Grothendieck–Witt theory, let L be a line bundle on an integralmonoid scheme X . The integrality assumption on X ensures the existence of aduality structure ( P L , Θ L ) on Vect ( X ). Write GW ( X ; L ) := GW ( Vect ( X ) , P L , Θ L )for the associated Grothendieck–Witt space, defined either via the hermitian Q -construction or group completion. The duality structure ( P L , Θ L ) is compatiblewith the subcategory hhO X ii [Pic( X )], which leads to a complete description of GW ( X ; L ) in terms of Pic( X ), together with its set-theoretic Z / L , and the pointed monoid Γ( X, O X ). Theorem D (Theorem 4.8) . Let L be a line bundle on an integral monoid scheme X . Then there is a natural homotopy equivalence GW ( X ; L ) ≃ Y ′M∈ Pic( X ) P L GW (Γ( X, O X ) - proj n ) × Y ′M∈ Pic( X ) ∗ /P L K (Γ( X, O X ) - proj n ) , where Pic( X ) P L denotes the fixed point set of Pic( X ) under the Z / -action deter-mined by L and Pic( X ) ∗ /P L is the quotient of the complement Pic( X ) \ Pic( X ) P L . In particular, this result, together with Theorems A and B, leads to an explicitdescription of GW ⊕ ( X ; L ). The space GW Q ( X ; L ) can then be described using theresults of [11].As an application of our results, we prove space-level projective bundle formulaefor K -theory and Grothendieck–Witt theory, giving analogues of well-known resultsfor schemes over fields [29], [45], [35], [31]. In our setting, the key background resultsare Theorem 3.23 and Lemma 4.11, which give a ( Z / F -setting. Instead, we use particular properties of monoidsschemes whose analogues for schemes over fields fail to hold. The projective bundleformula for Grothendieck–Witt theory is as follows; for K -theory, see Theorem 3.24. Theorem E (Theorem 4.12) . Let E be a vector bundle on an integral monoidscheme X with associated projective bundle π : P E → X and L a line bundle on X .Then there is a homotopy equivalence GW ( P E ; π ∗ L ) ≃ GW ( X ; L ) × Y ′ ( M ,i ) ∈ (Pic( X ) × Z ∗ ) / h ( P L , − i K (Γ( X, O X ) - proj n ) . Acknowledgements.
The authors thank Marco Schlichting for helpful correspon-dence. All three authors thank the Max Planck Institute for Mathematics in Bonnfor its hospitality and financial support.1.
Background material
In this section, we recall necessary background material on proto-exact categoriesand their K -theory and Grothendieck–Witt theory.1.1. Proto-exact categories.
Let A be a proto-exact category, as defined in [10, § A has a zero object 0 ∈ A and two distinguishedclasses of morphisms, called inflations (or admissible monics) and deflations (or ( X / F ) AND GW ( X / F ) 5 admissible epics) and denoted by and ։ , respectively. An admissible square in A is a bicartesian square of the form U VW X
Conflations (or admissible short exact sequences) in A are admissible squares asabove with W = 0 which, for ease of notation, we denote by U V ։ X .A functor between proto-exact categories is called proto-exact if it sends ad-missible squares to admissible squares. In particular, proto-exact functors sendconflations to conflations.The proto-exact categories of interest in this paper have a weak analogue of anadditive structure, axiomatized as follows. Definition ([11, § . An exact direct sum on a proto-exact category A is a sym-metric monoidal structure ⊕ on A such that is the monoidal unit and ⊕ is aproto-exact functor. Moreover, the following additional axioms are required to hold,where we set i U : U id U ⊕ V U ⊕ V and π U : U ⊕ V id U ⊕ V ։ U for objects U, V ∈ A .(i) The map
Hom A ( U ⊕ V, W ) → Hom A ( U, W ) × Hom A ( V, W ) , f ( f ◦ i U , f ◦ i V ) is an injection for all U, V, W ∈ A .(ii) Let U i X π V be a conflation. For each section s of π , there exists aunique isomorphism φ which makes the following diagram commute: XU U ⊕ V V. i U i φ i V s Moreover, the obvious axioms dual to (i) and (ii), with the maps π ( − ) appearing inplace of i ( − ) , are required to hold. A functor between proto-exact categories with exact direct sum is called exact ifit is proto-exact and ⊕ -monoidal.Let A be a proto-exact category with exact direct sum. A commutative diagram U X VU U ⊕ V V i πi U φ π V with φ an isomorphism is called a splitting of the conflation U i X π V . Definition ([11, § . A proto-exact category with exact direct sum is called(i) uniquely split if every conflation admits a unique splitting, and
J. N. EBERHARDT, O. LORSCHEID, AND M. B. YOUNG (ii) combinatorial if, for each inflation i : U X ⊕ X , there exist inflations i k : U k X k , k = 1 , , and an isomorphism f : U → U ⊕ U such that i = ( i ⊕ i ) ◦ f .Moreover, the obvious dual axiom involving maps π k , k = 1 , , is required to hold. Algebraic K -theory of proto-exact categories. Let A be a proto-exactcategory. The Q -construction of A can be defined as for exact categories [29, § Q ( A ). See also [11, § K -theory space of A is then K ( A ) = Ω BQ ( A ), where BQ ( A ) is pointed by 0 ∈ Q ( A ), and the K -theory groupsare K i ( A ) = π i K ( A ) , i ≥ . Lemma 1.1.
Let A and B be proto-exact categories and F : A → B an essentiallysurjective proto-exact functor which is bijective on inflations and deflations. Thenthe induced map K ( F ) : K ( A ) → K ( B ) is a homotopy equivalence.Proof. To begin, note that F is conservative. Indeed, a morphism in a proto-exactcategory is an isomorphism if and only if it is an inflation and a deflation.Since F is proto-exact, there is an induced functor Q ( F ) : Q ( A ) → Q ( B ). Essen-tial surjectivity of F implies that of Q ( F ). Moreover, Q ( F ) is full (resp. faithful)because F is surjective on inflations and deflations (resp. conservative and injectiveon inflations and deflations). Hence, Q ( F ) is an equivalence and the associated map K ( F ) is a homotopy equivalence. (cid:3) Let now ( A , ⊕ ) be a symmetric monoidal category. The maximal groupoid S ⊂ A inherits a symmetric monoidal structure. Following [15, Page 222], the direct sum K -theory space of A is the group completion of B S : K ⊕ ( A ) = B ( S − S ) . We have the following proto-exact analogue of Quillen’s Group Completion The-orem [15].
Theorem 1.2 ([11, Theorem 2.2]) . Let A be a uniquely split proto-exact category.Then there is a homotopy equivalence K ( A ) ≃ K ⊕ ( A ) . Remark 1.3.
The construction of the space K ( A ) can be refined to produce aconnective spectrum K ( A ); see [47, Remark IV.6.5.1, § IV.8.5.5]. While K ( A ) and K ( A ) have the same homotopy groups, the space K ( A ) has many technical advan-tages. For example, a functor ⊗ : A × A → A which is biexact in the sense of[47, Definition IV.6.6] induces a pairing of spectra K ( A ) ∧ K ( A ) → K ( A ). Thisgives K • ( A ) = L i ≥ K i ( A ) the structure of commutative Z ≥ -graded ring if ⊗ issymmetric monoidal.1.3. Proto-exact categories with duality.
For a detailed introduction to proto-exact categories with duality, the reader is referred to [33, § § A , P, Θ) (often simply A ) consisting of acategory A , a functor P : A op → A and a natural isomorphism Θ : id A ⇒ P ◦ P op which satisfies P (Θ U ) ◦ Θ P ( U ) = id P ( U ) , U ∈ A . (1)If A is proto-exact and P is proto-exact, then A is a proto-exact category withduality. We henceforth restrict attention to this case.A symmetric form in A is an isomorphism ψ M : M → P ( M ) which satisfies P ( ψ M ) ◦ Θ M = ψ M . An isometry φ : ( M, ψ M ) → ( N, ψ N ) is an isomorphism ( X / F ) AND GW ( X / F ) 7 φ : M → N which satisfies ψ M = P ( φ ) ◦ ψ N ◦ φ . The groupoid of symmetric formsand their isometries is A h .Let ( M, ψ M ) be a symmetric form. An inflation i : U M is called isotropic if P ( i ) ◦ ψ M ◦ i is zero and U → U ⊥ := ker( P ( i ) ◦ ψ M ) is an inflation. In this case,the reduction M//U := U ⊥ /U inherits a symmetric morphism ψ M//U : M//U → P ( M//U ), which we assume to be an isomorphism; this is the Reduction Assumptionof [48, § M, ψ M ) is called metabolic if it has a Lagrangian,that is, an isotropic subobject U M with U = U ⊥ , and is called isotropicallysimple if it has no non-zero isotropic subobjects.If A has an exact direct sum, then we require that P be exact and Θ be ⊕ -monoidal. In this case, A h is a symmetric monoidal groupoid. Given an object U ∈ A , the pair (cid:16) H ( U ) = U ⊕ P ( U ) , ψ H ( U ) = (cid:16) P ( U ) Θ U (cid:17)(cid:17) is a symmetric form in A , called the hyperbolic form on U . The assignment U H ( U ) extends to a functor H : S → A h where S is the maximal grupoid in A .A symmetric form which is isometric to ( H ( U ) , ψ H ( U ) ) for some U ∈ A is calledhyperbolic. Lemma 1.4 ([11, Lemma 1.7]) . A metabolic form in a uniquely split proto-exactcategory with duality is hyperbolic.
Example.
Let A be a category. The triple ( H ( A ) , P, id id H ( A ) ), where H ( A ) = A × A op and P ( U, V ) = (
V, U ), is called the hyperbolic category with duality on A . If A is proto-exact, then so too is H ( A ) and an exact direct sum on A inducesone on H ( A ). ⊳ A form functor (
T, η ) : ( A , P, Θ) → ( B , Q, Ξ) between categories with duality is afunctor T : A → B and a natural transformation η : T ◦ P ⇒ Q ◦ T op which makesthe diagram T ( U ) Q T ( U ) T P ( U ) QT P ( U ) Ξ T ( U ) T (Θ U ) Q ( η U ) η P ( U ) commute for each U ∈ A . The form functor is called non-singular if η is a naturalisomorphism and is called an equivalence if, moreover, T is an equivalence.1.4. Grothendieck–Witt theory of proto-exact categories.
Let A be aproto-exact category with duality. The hermitian Q -construction of A can be de-fined as for exact categories with duality [33, § Q h ( A ). Seealso [11, § F : Q h ( A ) → Q ( A ).The Grothendieck–Witt space GW Q ( A ) is the homotopy fibre of BF : BQ h ( A ) → BQ ( A ) over 0 and the Grothendieck–Witt groups are GW Qi ( A ) = π i GW Q ( A ) , i ≥ . For legibility, we have written P in place of P ◦ P op , and so on. J. N. EBERHARDT, O. LORSCHEID, AND M. B. YOUNG
Despite the name, without further assumptions, GW Q ( A ) is in fact only a pointedset. If, however, A has an exact direct sum, as will always be the case, then GW Q ( A ) is a commutative monoid. The Witt groups are defined by W Qi ( A ) = coker (cid:16) K i ( A ) H ∗ −→ GW Qi ( A ) (cid:17) , i ≥ , where H ∗ is induced by the map K ( A ) → GW Q ( A ). As for GW Q ( A ), in general W Q ( A ) is only a commutative monoid. Proposition 1.5.
Let A be a proto-exact category with associated hyperbolic cate-gory H ( A ) . Then there is a homotopy equivalence GW Q ( H ( A )) ∼ −→ K ( A ) .Proof. The proof of the corresponding result in the exact setting [33, Proposition4.7] carries over. (cid:3)
Proposition 1.6.
A non-singular proto-exact form functor ( T, η ) : ( A , P, Θ) → ( B , Q, Ξ) induces a continuous map GW Q ( T, η ) : GW Q ( A ) → GW Q ( B ) . Moreover,if ( T, η ) and ( T ′ , η ′ ) are naturally isomorphic, then GW Q ( T, η ) and GW Q ( T ′ , η ′ ) arehomotopic.Proof. The proofs of the corresponding results in the exact setting [34, § (cid:3) An obvious modification of Lemma 1.1 (and its proof) is as follows.
Lemma 1.7.
Let A and B be proto-exact categories with duality and ( F, η ) :
A → B an essentially surjective proto-exact form functor which is bijective on inflations anddeflations. Then the induced map GW Q ( F, η ) : GW Q ( A ) → GW Q ( B ) is a homotopyequivalence. Suppose now that ( A , ⊕ ) is a symmetric monoidal category with duality. Or-thogonal direct sum gives A h the structure of a symmetric monoidal groupoid. Asin [17, § GW ⊕ ( A ) = B ( A − h A h ) , with associated Grothendieck–Witt and Witt groups GW ⊕ i ( A ) = π i GW ⊕ ( A ) and W ⊕ i ( A ) = coker( K ⊕ i ( A ) H −→ GW ⊕ i ( A )), i ≥
0, respectively. Note that these areindeed groups. We remark that the obvious analogues of Propositions 1.5 and 1.6and Lemma 1.7 hold for direct sum Grothendieck–Witt theory.Let Q H ( A ) ⊂ Q h ( A ) be the full subcategory on hyperbolic objects and GW QH ( A )the homotopy fibre of BQ H ( A ) → BQ ( A ) over 0. Let also GW ⊕ H ( A ) = B ( A − H A H ) , where S H is the symmetric monoidal groupoid of hyperbolic symmetric forms.The following result plays the role of the Group Completion Theorem for theGrothendieck–Witt theory of uniquely split proto-exact categories. Compare with[32, Theorem 4.2], [35, Theorem A.1] and [36, Theorem 6.6] in the split exact set-ting. Theorem 1.8 ([11, Theorems 3.2 and 3.11]) . Let A be a uniquely split proto-exactcategory with duality(i) There is a weak homotopy equivalence GW QH ( A ) ≃ GW ⊕ H ( A ) . ( X / F ) AND GW ( X / F ) 9 (ii) If, moreover, A is combinatorial and noetherian, then there is a weak homotopyequivalence GW Q ( A ) ≃ G w ∈ W Q ( A ) BG S w × GW QH ( A ) , where S w is an isotropically simple representative of the Witt class w ∈ W Q ( A ) with self-isometry group G S w . K -theory and Grothendieck–Witt theory of pointed monoids In this section we study the K -theory and Grothendieck–Witt theory of proto-exact categories of projective modules over pointed monoids. The specializationof this section to commutative pointed monoids is the local model for the schemetheoretic considerations of Sections 3 and 4.2.1. Pointed monoids and their module categories.
We record basic mate-rial about pointed monoids and their module categories. A detailed reference forcommutative pointed monoids is [2, § A with a zero (or absorbing element) 0 andan identity 1, so that a · · a and a · a = 1 · a for all a ∈ A . Ahomomorphism of pointed monoids is a semigroup homomorphism which preservesthe zero and identity. Pointed monoids and their homomorphisms form a category f M . The full subcategory of commutative pointed monoids is M .Let I be an ideal of a commutative pointed monoid A , that is, 0 ∈ I and IA = I .The quotient pointed monoid A/I is the set ( A \ I ) ∪ { } with the multiplication a · b = ab if a, b, ab ∈ A \ I and a · b = 0 otherwise.A left A -module (also called an A -set) is a pointed set M , with basepoint denotedagain by 0, together with a left A -action under which 0 ∈ A and 1 ∈ A act by thezero and identity map of M , respectively. Right A -modules are defined similarly.Unless mentioned otherwise, by an A -module we mean a left A -module. An A -module homomorphism is a pointed A -equivariant map. Let A - Mod be the categoryof left A -modules and their homomorphisms and A - mod its full subcategory offinitely generated A -modules. An A -module P is called projective if, for every A -module homomorphism f : P → M and surjective A -module homomorphism g : N → M , there exists an A -module homomorphism h : P → N satisfying g ◦ h = f . Let A - proj ⊂ A - Mod be the full subcategory of finitely generatedprojective A -modules. Lemma 2.1 ([2, Proposition 2.27]) . Every projective A -module is of the form L i ∈ J Ae i where e i = e i are idempotents in A . An A -module homomorphism f : M → N is called normal if f − ( n ) is empty ora singleton for each n ∈ N \ { } . This definition of normality is compatible withthe categorical definition in the case of monomorphisms and epimorphisms; cf. [2,Proposition 2.15]. The zero and identity morphisms are normal, as are compositionsof normal morphisms. Denote by A - Mod n ⊂ A - Mod the subcategory of normal A -module homomorphisms, and similarly for A - mod n and A - proj n . An element a ∈ A is called right cancellative (resp. right partially cancellative,or rpc) if right multiplication · a : A → A is an injective A -module homomorphism(resp. normal A -module homomorphism). Explicitly, a ∈ A is rpc if xa = ya implies x = y or xa = ya = 0 for all x, y ∈ A . Let A rpc ⊂ A be the subset of right partiallycancellative elements and A rc ⊂ A be the subset of right cancellative elementstogether with 0 ∈ A . Both A rc and A rpc are pointed submonoids of A . We call apointed monoid A right cancellative if A rc = A and rpc if A rpc = A . Replacing rightwith left multiplication leads to the notion of a left (partially) cancellative pointedmonoid. A pointed monoid is called cancellative (resp. partially cancellative, orsimply pc) if it is both left and right cancellative (resp. partially cancellative).A pointed monoid A is called right reversible if Aa ∩ Ab = { } for any two non-zero elements a, b ∈ A . For example, a commutative cancellative pointed monoid is(both left and right) reversible, since 0 = ab ∈ Aa ∩ Ab .A pointed monoid is called right noetherian if it satisfies the ascending chaincondition for right congruences.For a family of A -modules { M i } i ∈ J , the direct sum A -module is M i ∈ J M i = (cid:16) G i ∈ J M i (cid:17) / h M i ∼ M j | i, j ∈ J i with the obvious A -action. For an A - B -bimodule M and a B - C -bimodule N , thetensor product A - C -bimodule is M ⊗ B N = ( M × N ) / { ( mb, n ) ∼ ( m, bn ) | b ∈ B } with the obvious actions of A and C . Example. (i) The initial object of f M is F := { , } . There is an equivalence of F - mod with the category set ∗ of finite pointed sets. The subcategory F - mod n = F - proj n is often denoted by Vect F in the literature.(ii) The terminal object of f M is { } , the unique monoid with 0 = 1.(iii) Let G be a group. Then F [ G ] := G ⊔ { } is a cancellative pointed monoid. Apointed monoid is cancellative and right reversible if and only if it can be embeddedin F [ G ] for some group G [4, Theorem 1.23].(iv) The subset A × ⊂ A of multiplicative units is a group and F [ A × ] ⊂ A is acancellative pointed submonoid.(v) The pointed monoid F [ t ] = { t i } i ≥ ⊔ { } is cancellative.(vi) Let n ≥
2. The pointed monoid F [ t ] / h t n = 0 i = { , , t, . . . , t n − } is notcancellative, since t · t · t n − , but is pc and reversible.(vii) The pointed monoid A = F [ t, s ] / h ts = 0 i is pc but not reversible, since At ∩ As = { } .(viii) Let n > d ≥
2. The pointed monoid F [ t ] / h t n = t d i = { , , t, . . . , t d , . . . , t n − } is not pc, since t · t d − = t · t n − . ⊳ For a left A -module M , the set Hom A - Mod ( M, A ) becomes a right A -module via( f · a )( m ) := f ( m ) a, f ∈ Hom A - Mod ( M, A ) , a ∈ A, m ∈ M. Unlike in the case of rings, the module Hom A - Mod ( M, A ) does not define a goodnotion of a module dual to M . For this reason, we instead consider the subsetHom A - Mod n ( M, A ) ⊂ Hom A - Mod ( M, A ) of normal homomorphisms. As the following ( X / F ) AND GW ( X / F ) 11 result shows, this subset is not an A -submodule without additional assumptions.Denote by A op the monoid opposite to A . Lemma 2.2.
Let A be a pointed monoid.(i) For any M ∈ A - Mod , the right A -module structure on Hom A - Mod ( M, A ) inducesa right A rpc -module structure on Hom A - Mod n ( M, A ) .(ii) If A is right reversible and rpc and M is a finitely generated free A -module,then the right A -module Hom A - mod n ( M, A ) is finitely generated and free.(iii) If A is right reversible, rpc and right noetherian, then Hom A - mod n ( − , A ) definesa ⊕ -monoidal functor P : ( A - mod n ) op → A op - mod n . Proof.
The first statement is a direct verification. For the second statement, let M = M i ∈ J As i be a finitely generated free A -module. Denote by s ∨ i : M → A the map sending as i to a and as j to 0 if j = i . We claim that the induced map M i ∈ J s ∨ i : M i ∈ J t i A → Hom A - mod n ( M, A ) , t i x i s ∨ i · x i is a right A -module isomorphism. The map is well-defined and injective since A isrpc. If | J | = 1, then the map is clearly an isomorphism. Suppose then that | J | ≥ f ∈ Hom A - Mod n ( M, A ). Set f i = f ( s i ). We claim that there is at most one i ∈ J such that f i = 0 and hence f = s ∨ i · f i . Assume that there exist distinct i, j ∈ J such that f i = 0 = f j . Since A is right reversible, there exist a, b ∈ A such that af i = bf j = 0. Hence, f ( as i ) = f ( bs j ) = 0, a contradiction. The secondstatement follows.Turning to the third statement, let M ∈ A - mod . Fix a surjection F → M with F a finitely generated free A -module. A direct check shows that P ( M ) is naturally asubmodule of P ( F ). By the first two parts of the lemma, P ( F ) is finitely generatedand free. Since A is right noetherian, P ( F ) is noetherian [2, Proposition 2.31],from which it follows that P ( M ) is finitely generated. The definition of P onmorphisms is via pre-composition and is well-defined because the composition ofnormal morphisms is normal. To prove that P is ⊕ -monoidal, let M, N ∈ A - mod .An element f ∈ P ( M ⊕ N ) determines by restriction f M ∈ P ( M ) and f N ∈ P ( N ).Suppose that neither f M nor f N is zero. Since A is right reversible, im f M ∩ im f N = { } , contradicting the assumption that f is normal. It follows that at most one of f N and f M is non-zero and there is a well-defined A -module homomorphism P ( M ⊕ N ) → P ( M ) ⊕ P ( N ) . It is straightforward to verify that this is an isomorphism. We omit the verificationthat P respects ⊕ on morphisms. (cid:3) Remark 2.3. (i) There is a right A -module isomorphismHom A - mod ( M i ∈ J As i , A ) ≃ Y i ∈ J A. In particular, the standard A -linear dual of a free A -module is in general not free.In fact, Q i ∈ J A need not even be finitely generated. For example, the F [ t ]-module F [ t ] × F [ t ] is not finitely generated. From this point of view, the normal dualHom A - mod n ( − , A ) has better properties than Hom A - mod ( − , A ).(ii) The functor Hom A - mod n ( − , A ) of Lemma 2.2(iii) does not extend to ( A - mod ) op → A op - mod , since for a non-normal morphism f : M → N , the image of Hom A - mod n ( f, A )is not contained in Hom A - mod n ( M, A ) ⊂ Hom A - mod ( M, A ).2.2. K -theory of pointed monoids. The K -theory of pointed monoids has beenstudied by a number of authors [9], [2], [3], [16]. In this section we describe thoseresults which are relevant to this paper.Let A be a pointed monoid. The category A - Mod admits a proto-exact structurewith inflations and deflations being the normal A -module homomorphisms whichare injective and surjective, respectively [2, § ⊕ is not a coproductfor A - Mod n . Indeed, for a non-zero A -module M , there is no dashed arrow in A - Mod n which makes the diagram MM M ⊕ M M i M id M i M id M commute. In particular, A - Mod n is not a quasi-exact category.Since A - proj ⊂ A - Mod is an extension closed full subcategory, it inherits a proto-exact structure from A - Mod . Lemma 2.4 (See also [2, Proposition 2.29].) . Let A be a pointed monoid. Theproto-exact category A - proj is uniquely split and combinatorial.Proof. Let U i V π W be a conflation in A - proj . Since W is projective, thereexists a section s : W → V of π . Since π is a deflation, it is normal by definitionwhich implies that that the section s is unique. Define an A -module homomorphism φ : U ⊕ W → V by φ ( u ) = i ( u ) , φ ( w ) = s ( w ) . To see that φ is injective, suppose, for example, that φ ( u ) = φ ( w ). Applying π gives 0 = π ( i ( u )) = π ( s ( w )) = w, implying u = w = 0. We claim that φ is also surjective and hence an isomorphismby [2, Lemma 2.2]. It is immediate that im i ⊂ im φ . Let v ∈ V \ im i . Then π ( v ) = 0 and hence also s ( π ( v )) = 0. Because π ◦ s = id W , the map s ◦ π isidempotent. It follows that s ( π ( v )) and v have the same (non-zero) image under s ◦ π . Since s ◦ π is normal, s ( π ( v )) = v . We conclude that φ is a splitting of theoriginal conflation.That the combinatorial property holds follows from the fact that ⊕ is definedusing disjoint union of the underlying sets. (cid:3) Note that Lemma 2.4 also implies that A - proj n is a uniquely split proto-exactcategory. The following ‘ Q = +’ theorem is the main results of this section. ( X / F ) AND GW ( X / F ) 13 Theorem 2.5.
Let A be an rpc pointed monoid. Then there is a homotopy equiva-lence K ( A - proj ) ≃ Z × B ( A × ≀ Σ ∞ ) + . In particular, if A × is finite, then K ( A - proj ) ≃ Z and K i ( A - proj ) ≃ π si ( BA × + ) , i ≥ . Proof.
Since A is rpc, projective A -modules are free. Indeed, this follows fromLemma 2.1 and the fact that a (non-trivial) rpc pointed monoid has a single non-zero idempotent, namely 1 ∈ A . By Theorem 1.2 (see also [3, Theorem 4.2]), thereis a homotopy equivalence K ( A - proj ) ≃ K ⊕ ( A - proj ). To compute K ⊕ ( A - proj ), weapply [46, Proposition 3] to the cofinal family { A ⊕ n } n ∈ Z ≥ of A - proj . We then haveAut( A - proj ) := lim −→ n Aut A - proj ( A ⊕ n ) = lim −→ n ( A × ) n ⋊ Σ n = A × ≀ Σ ∞ , giving the claimed result. (cid:3) Note that, by Lemma 1.1, the embedding A - proj n ֒ → A - proj induces a homotopyequivalence K ( A - proj n ) ≃ K ( A - proj ).2.3. Grothendieck–Witt theory of pointed monoids.
In this section, westudy the Grothendieck–Witt theory of pointed monoids. This leads to a non-additive analogue of Karoubi’s Grothendieck–Witt theory of rings [20].Let A be an rpc pointed monoid. Fix a pointed monoid involution σ : A → A op and a central element ǫ ∈ A which satisfies ǫσ ( ǫ ) = 1. For example, when A iscommutative, σ = id A and ǫ = 1 is an admissible choice. For a non-trivial case, seethe examples below.Given M ∈ A - Mod , consider P σ ( M ) := Hom A - Mod n ( M, A ) as a left A -module via( a · f )( m ) := f ( m ) σ ( a ) , f ∈ P σ ( M ) , a ∈ A, m ∈ M. Compare with Lemma 2.2.
Proposition 2.6.
Let A be a right reversible rpc pointed monoid. The naturaltransformation Θ σ,ǫ : id A - proj n ⇒ P σ ◦ ( P σ ) op with components Θ σ,ǫM ( m )( f ) = ǫσ ( f ( m )) , f ∈ P σ ( M ) , m ∈ M makes ( A - proj n , P σ , Θ σ,ǫ ) into a uniquely split combinatorial proto-exact categorywith duality.Proof. Lemma 2.4 shows that A - proj , and hence A - proj n , is uniquely split andcombinatorial. Given φ : M → N in A - proj n , the morphism P σ ( φ ) : P σ ( N ) → P σ ( M ) is defined to be ( − ) ◦ φ . This is well-defined since the composition ofnormal morphisms is normal. Since A is rpc, projective A -modules are free. That P σ is ⊕ -monoidal on A - proj n follows from Lemma 2.2. Exactness of P σ then followsfrom the splitness of A - proj n . Hence, P σ satisfies the desired properties.A direct calculation shows that Θ σ,ǫM ( m ) : P σ ( M ) → A is an A -module homomor-phism. To see that Θ σ,ǫM ( m ) is normal, fix an A -module basis M ≃ L i ∈ J As i andwrite m = xs i . When x = 0, the map Θ σ,ǫM ( m ) is zero, which is normal. Supposethen that x = 0 and let 0 = a ∈ A . We haveΘ σ,ǫM ( m ) − ( a ) = { f ∈ P σ ( M ) | ǫ ( σ ( f ( xs i )) = a } = { y ∈ A | ǫ ( σ ( ys ∨ i ( xs i )) = a } = { y ∈ A | σ ( y ) σ ( x ) = σ ( ǫ ) a } . Since A is rpc and σ is an isomorphism, the final set is empty or a singleton, asrequired. The assumption ǫσ ( ǫ ) = 1 ensures that the equalities P (Θ U ) ◦ Θ P ( U ) =id P ( U ) hold. (cid:3) Example.
Let A = F . The only possibilities are σ = id F and ǫ = 1. For each M ∈ Vect F = F - proj n , there is a canonical isomorphism δ M : M ∼ −→ P ( M ) , δ M ( m )( m ′ ) = ( m = m ′ , m = m ′ . We emphasize that such an isomorphism does not exist for a general pointed monoid.Under this identification, P squares to the identity. The triple ( Vect F , P, id id Vect F )is therefore a proto-exact category with strict duality. ⊳ Next, we turn to the classification of symmetric forms in A - proj n . Let M be afree A -module of rank one. Fix a basis M ≃ A . A symmetric form ψ M on M thentakes the form ψ M ( a )( x ) = xξσ ( a ) , a, x ∈ A for some ξ ∈ A × which satisfies ξ = ǫσ ( ξ ). Write ψ ξ for this symmetric form. Anisomorphism M → M , which is necessarily determined by an element u ∈ A × ,defines an isometry ψ uξσ ( u ) → ψ ξ . Motivated by these observations, define an A × -action on the set A × σ,ǫ = { ξ ∈ A × | ξ = ǫσ ( ξ ) } by u · ξ = uξσ ( u ). Then the set of isomorphism classes of rank one symmetric formsin A - proj n is Pic sym ( A ) := A × σ,ǫ /A × . The isometry group of ψ ξ is the stabilizer I ( ξ ) = { u ∈ A × | ξ = uξσ ( u ) } . Example.
The pointed monoid A = F := F [ Z /
3] has a unique non-trivial monoidautomorphism σ , which is an involution. Either non-identity element ǫ ∈ Z / σ . We have A × σ,ǫ = { ǫ } and I ( ǫ ) ≃ Z /
3. In particular, Pic sym ( A )is a singleton. ⊳ Given ( h, { m ξ } ) ∈ Z ≥ × Y ′ ξ ∈ Pic sym ( A ) Z ≥ in the restricted product (see Appen-dix A.2), define a symmetric form ψ h, { m ξ } = ψ ⊕ hH ( A ) ⊕ M ξ ∈ Pic sym ( A ) ψ ⊕ m ξ ξ . Proposition 2.7.
Let A be a right reversible rpc pointed monoid.(i) The assignment ( h, { m ξ } ) ψ h, { m ξ } induces a monoid isomorphism between Z ≥ × Y ′ ξ ∈ Pic sym ( A ) Z ≥ and the monoid π ( A - proj n h ) of isometry classes of symmetric forms in A - proj n .(ii) There is a group isomorphism Aut A - proj n h ( ψ h, { m ξ } ) ≃ (cid:0) ( Z / ⋉ σ A × ) ≀ Σ h (cid:1) × Y ′ ξ ∈ Pic sym ( A ) (cid:0) I ( ξ ) ≀ Σ m ξ (cid:1) , where Z / acts on A × by u σ ( u − ) . ( X / F ) AND GW ( X / F ) 15 Proof.
After using Lemma 1.4, the first statement is straightforward. The secondstatement is a direct calculation. (cid:3)
Remark 2.8.
Let M ∈ A - proj be free of rank n . Fixing a basis of M , and hence alsoof P σ ( M ), identifies a symmetric form on M with an A × -valued permutation matrix ψ = ( ψ ij ) ∈ A × ≀ Σ n which satisfies ψ ij = ǫσ ( ψ ji ), 1 ≤ i, j ≤ n . In this formulation,Proposition 2.7 becomes the classification of such matrices up to congruence.Recall the Reduction Assumption from Section 1.3. Proposition 2.9.
The Reduction Assumption holds for ( A - proj n , P σ , Θ σ,ǫ ) .Proof. An isotropic subobject U ψ h, { m ξ } necessarily factors through the sum-mand H ( A ) ⊕ h . It therefore suffices to consider only hyperbolic symmetric forms.By the combinatorial property of A - proj n , we can write U = U ⊕ P ( U ) for some U i ∈ A - proj , in which case the isotropic condition is P ( U ) P ( X/U ). The re-duction of H ( A ) ⊕ h is then canonically isometric to H (coker( P ( U ) P ( X/U )).See also [49, Lemma 1.1]. (cid:3) Remark 2.10.
In view of Proposition 2.9, we conclude, using [48, Theorem 3.10],that the forgetful morphism R • ( A - proj n ) → S • ( A - proj n ) from the R • -constructionto the Waldhausen S -construction is a relative 2-Segal space. We can thereforeapply the construction of [48, §
4] to produce a module over of the Hall algebra H ( A - proj n ). The algebra H ( A - proj n ), and its variations, have been studied bySzczesny [39], [40], [41]. In the setting of the representation theory of quivers over F , which is combinatorial but not split, modules arising from the R • -constructionhave been studied in [49] where, in particular, a version of Green’s theorem isproved.We can now state the main result of this section. Theorem 2.11.
Let A be a right reversible rpc pointed monoid with Pic sym ( A ) countable. Then GW ⊕ ( A - proj n , σ, ǫ ) is homotopy equivalent to Z × (cid:16) Y ′ ξ ∈ Pic sym ( A ) Z (cid:17) × B (cid:0) ( Z / ⋉ σ A × ) ≀ Σ ∞ (cid:1) × Y ′ ξ ∈ Pic sym ( A ) (cid:0) I ( ξ ) ≀ Σ ∞ (cid:1) + . Proof.
Index the set Pic sym ( A ) as { ξ j } j ∈ J for some subset J ⊆ Z ≥ . For each n ≥ n = M j ∈ Jj ≤ n ψ ξ j and set s n = H ( A ) ⊕ n ⊕ Ψ ⊕ nn . Then { s n } n ∈ Z ≥ is a cofinal family in A - proj n h . UsingProposition 2.7, we findAut( A - proj n h ) := lim −→ Aut A - proj n h ( s n ) ≃ (cid:0) ( Z / ⋉ σ A × ) ≀ Σ ∞ (cid:1) × Y ′ ξ ∈ Pic sym ( A ) I ( ξ ) ≀ Σ ∞ . We are therefore in the setting of [46, Proposition 3], allowing us to conclude thatthere is a homotopy equivalence GW ⊕ ( A - proj n , σ, ǫ ) ≃ K ⊕ ( A - proj n h ) × B Aut( A - proj n h ) + . Finally, use Proposition 2.7 to identify K ⊕ ( A - proj n h ) and Z × Y ′ ξ ∈ Pic sym ( A ) Z . (cid:3) Corollary 2.12.
In the setting of Theorem 2.11, there is an isomorphism W ⊕ ( A - proj n , σ, ǫ ) ≃ Y ′ ξ ∈ Pic sym ( A ) Z . Example.
Suppose that A has no non-trivial units. This is the case, for example,for A = F [ T , . . . , T n ]. Then there is an isomorphism GW ⊕ ( A - proj n , σ ) ≃ Z × B (cid:0)(cid:0) Z / ≀ Σ ∞ (cid:1) × Σ ∞ (cid:1) + . ⊳ Example.
For later use, we record the homotopy equivalence GW ⊕ ( Vect F ) ≃ Z × B (cid:0)(cid:0) Z / ≀ Σ ∞ (cid:1) × Σ ∞ (cid:1) + . To describe GW Q ( Vect F ), we use Theorem 1.8. Since { H ( F ⊕ n ) } n ≥ is a cofinalfamily of A H , arguing as in the proof of Theorem 2.11, we obtain a homotopyequivalence GW H ( Vect F ) ≃ Z × B ( Z / ≀ Σ ∞ ) + so that there is a weak homotopy equivalence GW Q ( Vect F ) ≃ G n ∈ Z ≥ B Σ n × Z × B ( Z / ≀ Σ ∞ ) + . Computations for more general monoids are similar. ⊳ Algebraic K -theory of monoid schemes In the remainder of the paper, all pointed monoids are assumed to be commuta-tive.3.1.
Monoid schemes.
We present some background on monoid schemes. Thereader is referred to [2], [5], [7], [8] for further details.A prime ideal of a pointed monoid A is an ideal p ⊂ A whose complement S = A − p is a multiplicative subset, that is, S contains 1 and is multiplicativelyclosed.Let S be a multiplicative subset of A . The localization of A at S is S − A =( S × A ) / ∼ , where ( s, a ) ∼ ( a ′ , s ′ ) if there exists t ∈ S such that tsa ′ = ts ′ a . Write as for the class of ( s, a ) in S − A . The product as · bt = abst endows S − A with thestructure of a pointed monoid and the map ι S : A → S − A , a a , is a monoidmorphism, which we call the localization map.Given h ∈ A , we write A [ h − ] for the localization of A at S = { h i } i ∈ Z ≥ . Givena prime ideal p ⊂ A , we write A p for the localization of A at S = A − p . If A is cancellative, then S = A − { } is a multiplicative subset and we define thefraction field of A as Frac A = S − A , which is a pointed group, that is, (Frac A ) × =Frac A − { } . More generally, if A is cancellative and 0 / ∈ S , then ι S : A → S − A is injective and S − A is cancellative. In this situation, we often identify A with itsimage in S − A and write a for a ∈ S − A .A monoidal space is a pair ( X, O X ) consisting of a topological space X and asheaf of pointed monoids O X . We often surpress O X from the notation. A primaryexample of a monoidal space is the spectrum X = Spec A of a pointed monoid A whose points are the prime ideals of A , whose topology is generated by the ( X / F ) AND GW ( X / F ) 17 principal open subsets U h = { p | h / ∈ p } for h ∈ A , and whose structure sheaf O X ischaracterized by the values O X ( U h ) = A [ h − ] and by its stalks O X, p = A p . We usethe short hand notation Γ X = O X ( X ) for the pointed monoid of global sections of O X .An affine monoid scheme is a monoidal space which is isomorphic to Spec( A ) forsome pointed monoid A . A monoid scheme is a monoidal space which admits anopen cover by affine monoid schemes.A morphism between monoid schemes X and Y is a continuous map ϕ : X → Y together with a morphism ϕ : O Y → ϕ ∗ ( O X ) of sheaves of pointed monoids suchthat, for every x ∈ X , the induced pointed monoid morphism ϕ x : O Y,ϕ ( x ) → O X,x is local, that is, maps non-units to non-units.A monoid scheme is of finite type if it is quasi-compact and has an affine open cov-ering by spectra of finitely generated pointed monoids. We say that X has enoughclosed points if every point y ∈ X specializes to a closed point x or, equivalently, x is contained in the topological closure of { y } . Remark 3.1.
We list some well-known properties of monoid schemes.(i) A monoid scheme is a spectral space (cf. [30]), which means, in particular, thatevery irreducible closed subset has a unique generic point, which we typically denoteby η .(ii) A pointed monoid A has a unique maximal (prime) ideal, namely, the comple-ment m = A − A × of the unit group A × . Therefore, every affine monoid scheme X = Spec A has a unique closed point x = m , and X is the only open neighbour-hood of x . As a consequence, every affine open subset U of a monoid scheme X hasa unique closed point x and Γ U = O X,x .(iii) If X has enough closed points, then it is covered by the minimal open neigh-bourhoods U x = Spec O X,x of the closed points x . This covering is the minimalopen covering of X , in the sense that every other open covering of X refines to { U x | x ∈ X closed } .(iv) If X is of finite type, then it has only finitely many points. In particular, X has enough closed points and U x is open in X for every x ∈ X . Definition.
A monoid scheme X is called(i) cancellative (resp. pc) if the pointed monoid O X,x is cancellative (resp. pc) foreach x ∈ X ,(ii) integral (resp. reversible) if the pointed monoid O X ( U ) is cancellative (resp.reversible) for each open set U ⊂ X , and(iii) torsion free if for every x ∈ X , the unit group of O X,x is torsion free, that is,if a n = 1 for n > , then a = 1 . Remark 3.2.
Note that we digress from [2] in the meaning of integrality of monoidschemes. To wit, we require that O X ( U ) is cancellative for all opens U , and notmerely for an open covering, as required in [2]. Lemma 3.3. (i) Let X be a cancellative (resp. pc) monoid scheme. Then thepointed monoid O X ( U ) is cancellative (resp. pc) for each affine open subset U ⊂ X .(ii) An integral monoid scheme is cancellative.(iii) An integral monoid scheme is reversible. Proof.
Let U ⊂ X be an open affine subset of a cancellative (resp. pc) monoidscheme. Then U = Spec O X,x where x is the unique closed point of U . It followsthat O X ( U ) is cancellative (resp. pc).The second statement follows from the fact that the stalks of an integral monoidscheme are submonoids of the (cancellative) generic stalk.The final statement follows from the first statement and the fact that cancellativepointed monoid is reversible. (cid:3) Let X be an irreducible cancellative monoid scheme with generic point η . Definethe function field of X as O X,η , which is a pointed group.
Lemma 3.4.
Let X be an irreducible cancellative monoid scheme with generic point η . For every open U ⊂ X , there is an equality O X ( U ) = T x ∈ U O X,x of pointedsubmonoids of O X,η .Proof.
Since O X,x is cancellative and O X,η is a localization of O X,x , the localizationmap O X,x → O
X,η is injective for every x ∈ X . (cid:3) Proposition 3.5.
A monoid scheme X is integral if and only if it is irreducibleand pc.Proof. Assume that X is integral. If X is not irreducible, then there exist disjointnon-empty open subsets U and U . By the sheaf axiom, we have O X ( U ∪ U ) =Γ U × Γ U , which is not cancellative, contradicting the integrality of X . Thus, X is irreducible. Let x ∈ X with affine open neighbourhood U = Spec A , so that x corresponds to a prime ideal p ⊂ A . Because X is integral, A is cancellative, as isits localization O X,x = A p . Thus, X is cancellative and, in particular, pc.Suppose instead that X is irreducible and pc. We first prove that X is cancella-tive. Let x ∈ X with affine open neighbourhood U = Spec A . Since X is irreducible, A has a unique minimal ideal p and since U is pc, A is pc by Lemma 3.3(i). Then A fails to be cancellative only if it has nontrivial zero divisor, say ab = 0 for non-zero a, b ∈ A . In this case, the inverse image p b = ι − b ( m ) of the maximal ideal m = A [ b − ] − A [ b − ] × of A [ b − ] under the localization map ι b : A → A [ b − ] is aprime ideal that contains a but not b . Similarly, there is a prime ideal p b thatcontains b but not a . Since ab = 0, the intersection p a ∩ p b cannot contain a primeideal. This contradicts the fact that A has a unique minimal ideal. Hence, A andall of its localizations are cancellative. This proves that X is cancellative.To complete the proof that X is integral, denote by η the generic point of X .Lemma 3.4 implies that O X ( U ) is a submonoid of the cancellative stalk O X,η forevery open subset U of X , and therefore is itself cancellative. Thus X is integral. (cid:3) Definition. (i) A valuation monoid is a cancellative pointed monoid A such that Frac A = { a, a − | a ∈ A } .(ii) A morphism ϕ : Y → X of monoid schemes is proper if for all valuationmonoids A with inclusion ι : A → Frac A and all morphisms µ : Spec Frac A → Y and ν : Spec A → X with ϕ ◦ µ = ν ◦ ι ∗ , there exists a unique morphism ˆ ν : Spec A → Y such that the diagram Spec Frac
A Y
Spec
A X µι ∗ ϕν ˆ ν ( X / F ) AND GW ( X / F ) 19 commutes.(iii) A monoid scheme X of finite type is proper if the terminal morphism X → Spec F is proper. Proposition 3.6.
Let X be a proper, integral and torsion free monoid scheme offinite type. Then Γ X = F .Proof. Let η be the generic point of X . Since X is integral, all stalks O X,x , x ∈ X ,are submonoids of the generic stalk O X,η ; see Lemma 3.4. ThusΓ X = \ x ∈ X O X,x , the intersection being taken in O X,η .In order to prove that Γ X = F , we consider an element f ∈ O X,η and assumethat f / ∈ { , } . Being a proper scheme, X is of finite type, and since it is torsionfree, we have O X,η ≃ F [ T ± , . . . , T ± n ] for some n ≥
0. Under this isomorphism, f corresponds to a Laurent monomial Q ni =1 T e i i for some tuple ( e , . . . , e n ) = (0 , . . . , Z n . Therefore, we find a pointed monoid morphism v : O X,η ≃ F [ T ± , . . . , T ± n ] −→ F [ T ± ]that maps f to T − i for some i >
0. Since X is of finite type, U η = Spec O X,η is anopen subscheme of X .Let ι : F [ T ] → F [ T ± ] be the canonical inclusion and consider the diagramSpec F [ T ± ] U η X Spec F [ T ] Spec F v ∗ ι ∗ ϕν ˆ ν whose outer square commutes as Spec F is terminal and where ˆ ν is the uniquemorphism given by the defining property of the proper scheme X . Let z = h T i bethe closed point of Spec F [ T ] and x = ˆ ν ( z ). Consider the induced morphism ofstalks ˆ ν z : O X,x → O
Spec F [ T ] ,z = F [ T ]. Since v ( f ) = T − i for i >
0, the element f ∈ O X,η is not contained in O X,x , which shows that f is not a global section. Thisshows that Γ X = { , } , as desired. (cid:3) Vector bundles.
Let X and F be monoid schemes. A fibre bundle on X with fibre F , or simply an F -bundle on X , is a morphism π : E → X ofmonoid schemes such that there is an open covering { U i } of X and isomorphisms ϕ i : E × X U i → F × U i , called trivializations, such that each diagram E × X U i F × U i U iϕ i ∼ π i pr Ui commutes, where π i = E × X U i → U i is the restriction of π to U i . Sometimes wesuppress the morphism π from the notation and say that E is an F -bundle on X .An F -bundle E on X is trivializable if there existis a trivialization E ≃ F × X . A morphism of F -bundles E and E ′ on X is a commutative diagram E E ′ X π π ′ of morphisms of monoid schemes. Remark 3.7. If X = Spec A is affine, then it has a unique closed point and thusevery covering { U i } is trivial in the sense that U i = X for some i . Therefore every F -bundle on X is trivializable.As a consequence, given an F -bundle π : E → X on an arbitrary monoid scheme X , we can find trivializations ϕ i : E × X U i → F × U i for every chosen affine opencovering { U i } of X . In particular, this holds for the minimal affine open covering { Spec O X,x | x ∈ X closed } if X has enough closed points.Let n ∈ Z ≥ . A vector bundle on X of rank n is an A n F -bundle. We denote thecategory of finite rank vector bundles on X , together with all bundle morphisms,by Vect ( X ). Remark 3.8.
In contrast to vector bundles on schemes over a field, we need notrequire any additional datum to describe vector bundles, since the ‘vector spacestructure’ of A n F is intrinsically given, and coordinate changes of an A n F -bundle arenecessarily ‘ F -linear’. This follows from the fact that every A -linear automorphismof A [ T , . . . , T n ] is graded; cf. the proof of Proposition 3.9.As in algebraic geometry over a field, vector bundles correspond to finite locallyfree sheaves. We briefly review the definitions.Let X be a monoid scheme. An O X -module is a sheaf F of pointed sets on X together with a morphism O X × F → F of sheaves such that F ( U ) is an O X ( U )-module and the restriction maps F ( U ) → F ( V ) are pointed O X ( U )-module ho-momorphisms for all open subsets V ⊂ U ⊂ X . A morphism of O X -modules is amorphism ϕ : F → F ′ of sheaves such that F ( U ) → F ′ ( U ) is a pointed O X ( U )-module homomorphism for every open U ⊂ X . This defines the category O X - Mod of O X -modules on X .An O X -module F is said to be finite locally free if every point x ∈ X has an openneighbourhood x ∈ U ⊂ X such that F | U is a free O X | U -module of finite rank. Wedenote by LF ( X ) the full subcategory of O X - Mod on finite locally free sheaves.The relation between finite locally free sheaves and vector bundles uses the sym-metric algebra. Let A be a pointed monoid and M an A -module. The symmetricalgebra of M is the Z ≥ -graded pointed monoidSym( M ) = M i ∈ Z ≥ Sym i ( M )whereSym i ( M ) = M ⊗ i / h a ⊗ · · · ⊗ a i = a σ (1) ⊗ · · · ⊗ a σ ( i ) | a , . . . , a i ∈ M, σ ∈ Σ i i for i > ( M ) = A . The multiplication of Sym( M ) is given by con-catenation of tensors and the inclusion A = Sym ( M ) ֒ → Sym( M ) is a monoidmorphism. ( X / F ) AND GW ( X / F ) 21 Proposition 3.9.
The sheafification of the functor
Spec ◦ Sym ◦ Γ( X, − ) defines anequivalence of categories vect : LF ( X ) → Vect ( X ) .Proof. This is proven similarly to the corresponding fact for schemes over a field.We briefly sketch the key arguments, but forgo to verify all details. First of all,note that vect( F ) is indeed a vector bundle sinceSym( A ⊕ n ) ≃ A [ T , . . . , T n ] = F [ T , . . . , T n ] ⊗ F A and therefore vect( U ) = Spec Sym (cid:0) Γ( U, F ) (cid:1) ≃ A n F × Spec Γ U for every affine open U of X .A quasi-inverse lf : Vect ( X ) → LF ( X ) of vect can be defined as follows. Let π : E → X be a vector bundle of rank n , U = Spec A an affine open of X and V = E × X U . A trivialization ϕ U : V ∼ −→ U × F A n F defines an isomorphism f U : Γ V ≃ A ⊗ F F [ T , . . . , T n ] ≃ A [ T , . . . , T n ] . Since the A -linear automorphisms of A [ T , . . . , T n ] correspond to the images of T , . . . , T n , which must be of the form f U ( T i ) = a i T σ ( i ) for some permutation σ ∈ S n and some a , . . . , a n ∈ A × , the A -invariant subsetsΓ V ,i = f − U (cid:0) { aT i | a ∈ A } (cid:1) and Γ V = n [ i =1 Γ V ,i of Γ V do not depend on the choice of trivialization ϕ U up to a permutation of indices.This yields a canonical representation of Γ V as a symmetric algebra Sym(Γ V ).Define lf( E )( U ) to be the set of A -linear maps s : Γ V → A such that s (Γ V ,i ) = { } for all but one i ∈ { , . . . , n } , which is a free A -module. The sheafificationof the assignment U lf( E )( U ) defines a functor lf : Vect ( X ) → LF ( X ) that isquasi-inverse to vect. (cid:3) In light of Proposition 3.9, we allow ourselves to consider vector bundles assheaves. Note that under the correspondence
Vect ( X ) → LF ( X ), line bundles(vector bundles of rank one) correspond to invertible sheaves, that is, locally freesheaves of rank one.Define Pic X to be the set of isomorphism classes of invertible sheaves on X together with the group operation induced by ⊗ . By abuse of language, we callelements of Pic X line bundles and sometimes identify an isomorphism class witha chosen representative. The neutral element of Pic X is the class of O X and theinverse of a line bundle L is the dual line bundle L ∨ = H om O X ( L , O X ) where H om O X ( L , O X ) is the sheafification of the functor U Hom Γ U (cid:0) L ( U ) , O X ( U ) (cid:1) .3.3. Locally projective sheaves.
Let U = Spec A be an affine monoid scheme.An A -module M defines an O U -module f M with f M ( U h ) = M ⊗ A A [ h − ]. For anarbitrary monoid scheme X , we say that an O X -module F is finite locally projectiveif there exists an open covering { U i } of X such M i = F ( U i ) is a finitely generatedprojective O X ( U i )-module and such that F | U i ≃ f M i as sheaves on U i . We denotethe category of finite locally projective sheaves by LP ( X ). Finite locally free sheavesare locally projective sheaves. The converse implication holds for the following classof monoid schemes. Lemma 3.10.
Let X be a pc monoid scheme. Then every finite locally projectivesheaf is finite locally free. Proof.
Let F be a finite locally projective sheaf on X and { U i } an affine opencovering such that F ( U i ) is a finitely generated projective O X ( U i )-module. Then U i = Spec O X,x i is pc where x i is the unique closed point of U i . If e = e = 1 · e ina pc pointed monoid, then either e = 0 or e = 1. Thus, by Lemma 2.1, F ( U i ) is afree O X ( U i )-module of finite rank, which shows that F is finite locally free. (cid:3) A morphism of O X -modules f : E → F is called normal if the pointed O X,x -module homomorphism f x : E x → F x is normal for each x ∈ X . The category O X - Mod has a proto-exact structure in which conflations are kernel-cokernel pairsof normal morphisms [2, Lemma 5.6]; see also [19, Proposition 3.13]. Normal mor-phisms define a proto-exact subcategory O X - Mod n of O X - Mod .The category LP ( X ) is an extension closed subcategory of O X - Mod and so in-herits a proto-exact structure. The full subcategory LP n ( X ) of O X - Mod n on finitelocally projective sheaves also has aninduced proto-exact structure.We write ⊕ and ⊗ for the direct sum and tensor product on O X - Mod , respec-tively. Both ⊕ and ⊗ induce bifunctors on LP ( X ), making O X - Mod and LP ( X ) intosymmetric bimonoidal categories, as well as the respective subcategories O X - Mod n and LP n ( X ). Lemma 3.11 ([2, Theorem 5.12]) . Let X be a monoid scheme. Then the proto-exact category LP ( X ) is uniquely split and combinatorial. Let hhO X ii ⊂ LP n ( X ) be the full subcategory of objects which are isomorphic to O ⊕ nX for some n ∈ Z ≥ . It is a proto-exact symmetric bimonoidal subcategory. Wecan form the proto-exact symmetric bimonoidal category hhO X ii [Pic( X )] := M M∈ Pic( X ) hhO X ii . See Appendix A.1 for the definition of the right hand side.The following result plays an important role in the remainder of the paper.
Proposition 3.12.
Let X be an integral monoid scheme. Then the functor F := M M∈ Pic( X ) ( − ) M ⊗ M : hhO X ii [Pic( X )] → LP ( X ) is symmetric bimonoidal, proto-exact, essentially surjective and bijective on infla-tions and deflations. In particular, every finite locally projective sheaf on X decom-poses uniquely into a direct sum of line bundles.Proof. That F is symmetric bimonoidal is a direct calculation. It is clear that F isproto-exact. By Proposition 3.5, the scheme X is irreducible and pc. Proposition3.10 therefore implies that finite locally projective sheaves on X are finite locallyfree. Since X is irreducible, it has a unique generic point. The proof of [2, Theorem5.14] then applies to show that any finite locally projective sheaf on X is isomorphicto a direct sum of line bundles, from which it follows that F is essentially surjective.To see that F is bijective on inflations and deflations, consider the mapHom (cid:16) n M k =1 E M ik , m M l =1 F M jl (cid:17) −→ Hom (cid:16) n M k =1 E M ik ⊗ M i k , m M l =1 F M jl ⊗ M j l (cid:17) ( X / F ) AND GW ( X / F ) 23 whose domain and codomain is a set of morphisms in hhO X ii [Pic( X )] and LP ( X ),respectively. The left hand side is Y k,l,i k = j l Hom hhO X ii ( E M ik , F M jl ) . Since LP ( X ) is split (Lemma 3.11), the subset of inflations or deflations in thecodomain isHom inf / def LP ( X ) ( n M k =1 E M ik ⊗ M i k , m M l =1 F L jl ⊗ M j l ) ≃ Y k,li k = j l Hom inf / def LP ( X ) ( E M ik ⊗ M i k , F M jl ⊗ M j l ) . It therefore suffices to consider the case n = m = 1 and i = j . In this case,Hom hhO X ii [Pic( X )] ( E M , F M ) = Hom LP ( X ) ( E M , F M ), which we claim can be identifiedwith Hom LP ( X ) ( E M ⊗ M , F M ⊗ M ). To see this, consider the quasi-inverse auto-equivalences − ⊗ M : O X - Mod ⇆ O X - Mod : − ⊗ M ∨ . Let f : E → F be a normal morphism in O X - Mod . The stalk morphism ( f ⊗ M ) x is naturally identified with f x ⊗ id M x : E x ⊗ O X,x M x → F x ⊗ O X,x M x , which is normal. It follows from this, and the fact that the quasi-inverse of − ⊗ M is of the same form, that − ⊗ M restricts to an autoequivalence of LP ( X ). Thisproves the claim and finishes the proof of the proposition. (cid:3) Example.
The assumption that X is irreducible in Proposition 3.12 cannot bedropped. Indeed, let X = Proj (cid:0) F [ T , T , T ] / h T T T = 0 i (cid:1) , which is the unionof the three coordinate lines in P F . The three canonical opens of P F provide acovering X = U ∪ U ∪ U , where U i ≃ Spec (cid:0) F [ T j , T k ] / h T j T k = 0 i (cid:1) consists ofthe three points h T j i , h T k i and h T j , T k i if { i, j, k } = { , , } and where U i ∩ U j ≃ Spec (cid:0) F [ T ± k ] (cid:1) consists of a single point h T k i . The situation is illustrated in Figure1. h T i h T ih T ih T , T ih T , T i h T , T i U U U Figure 1.
The canonical covering of Proj (cid:0) F [ T , T , T ] / h T T T = 0 i (cid:1) Consider the locally projective sheaf F on X with F ( U i ) ≃ (cid:0) Γ( O X , U i )) ⊕ , i =0 , ,
2, and whose transition functions ϕ i,j : F | U i ( U i ∩ U j ) → F | U j ( U i ∩ U j ) are given by the permutation matrices ϕ , = ϕ , = (cid:18) (cid:19) and ϕ , = (cid:18) (cid:19) . Then F is not isomorphic to a direct sum of line bundles. ⊳ Next, we describe the proto-exact summands of hhO X ii [Pic( X )]. Proposition 3.13.
Let X be an integral monoid scheme. Then Γ X is cancellativeand the global sections functor Γ( X, − ) = Hom LP n ( X ) ( O X , − ) : hhO X ii → Γ X - Mod induces an exact equivalence hhO X ii ∼ −→ Γ X - proj n .Proof. The functor Γ( X, − ) is ⊕ -monoidal and hence exact, since the domain andcodomain categories are uniquely split. The essential image of Γ( X, − ) consists of allfinitely generated free Γ X -modules. Since X is integral, the pointed monoid Γ X iscancellative, being a submonoid of the stalk O X,η of the generic point η of X (Lemma3.4). Hence, projective Γ X -modules are free. That Γ( X, − ) : hhO X ii → Γ X - proj n isfully faithful follows from the observation that both Hom Γ X - mod n (Γ X ⊕ n , Γ X ⊕ m ) andHom hhO X ii ( O ⊕ nX , O ⊕ mX ) can be identified with the set of Γ X -valued m × n -matriceswith at most one non-zero entry in each row and column. (cid:3) K -theory of monoid schemes. In this section we describe the algebraic K -theory space of integral monoid schemes.Let X be a monoid scheme. Denote by K ( X ) = K ( LP ( X )) the algebraic K -theoryspace of LP ( X ) and set K i ( X ) = π i K ( X ). Theorem 3.14.
Let X be an integral monoid scheme. Then there is a homotopyequivalence K ( X ) ≃ Y ′M∈ Pic( X ) Z × B (Γ X × ≀ Σ ∞ ) + . Proof.
By Lemma 1.1, the functor F from Proposition 3.12 defines a homotopyequivalence K ( hhO X ii [Pic( X )]) ∼ −→ K ( X ). Using Propositions A.1 and A.2 and thefact that K -theory commutes with filtered colimits and finite direct sums of cate-gories (see [29, § K ( X ) ≃ lim −→ S ∈ P < ∞ (Pic( X )) K (cid:0) M s ∈ S hhO X ii (cid:1) ≃ lim −→ S ∈ P < ∞ (Pic( X )) Y s ∈ S K ( hhO X ii ) ≃ Y ′M∈ Pic( X ) K ( hhO X ii ) . Using Proposition 3.13, we have K ( hhO X ii ) ≃ K (Γ X - proj n ). The desired homotopyequivalence now follows from Theorem 2.5. (cid:3) Corollary 3.15.
Let X be a proper integral torsion free monoid scheme. Thenthere is a homotopy equivalence K ( X ) ≃ Y ′M∈ Pic( X ) Z × ( B Σ ∞ ) + . Proof.
This follows from Propositions 3.5 and 3.6 and Theorem 3.14. (cid:3) ( X / F ) AND GW ( X / F ) 25 Next, we deduce some results at the level of K -theory groups. In particular, werecover [2, Theorem 5.14]. Corollary 3.16. Le X be an integral monoid scheme with Γ X × finite. Then thereis an isomorphism of graded rings K • ( X ) ≃ π s • ( B (Γ X ) × + )[Pic( X )] , where the right hand side is the group algebra of Pic( X ) with coefficients in thegraded ring π s • ( B (Γ X × ) + ) .Proof. At the level of abelian groups, the statement follows from Theorem 3.14by taking homotopy groups. For the ring structure, we use that the functor F ofProposition 3.12 is symmetric bimonoidal. Denote by hhO X ii M the direct summandof hhO X ii concentrated in degree M ∈
Pic( X ). Then ⊗ restricts to biexact functors ⊗ : hhO X ii M × hhO X ii M → hhO X ii M M , M i ∈ Pic( X ) . In this way we obtain a commutative diagram of pairings of K -theory spectra K ( hhO X ii M ) × K ( hhO X ii M ) K ( hhO X ii M M ) K ( hhO X ii [Pic( X )]) × K ( hhO X ii [Pic( X )]) K ( hhO X ii [Pic( X )]) . Compare with [47, § IV.6.6]. The remaining statements follow. (cid:3)
Example.
Fix n ≥ P n F be the n -dimensional projective space over F . Itis a monoid scheme whose base change P n F × Spec( F ) Spec( Z ) is the n -dimensionalprojective space P n Z over the integers. There is an isomorphism of groups Pic( P n F ) ≃ Z which sends O P n F (1) to 1 ∈ Z ; see [44, Theorem 2.6] or [2, § K ( P n F ) ≃ Z [ Z ]. In this way, we recover [2, Corollary5.15]. In particular, K ( P n F ) is independent of n , in stark contrast to the case ofprojective space over a field.Note, however, that, as explained in [2, Theorem 5.17], the relations for the K -theory of projective space over a field can be recovered by a comparison morphismbetween the K -theory and the G -theory of P n F . Since G -theory does not extendto the Grothendieck–Witt theory, we forgo pursuing this viewpoint in the presentpaper. ⊳ Remark 3.17.
For an arbitrary monoid scheme X , we do not expect K ( X ), asdefined above using the proto-exact category of vector bundles, to be the bestdefinition of the algebraic K -theory space of X . This is for reasons similar to thecase of schemes over a field, where algebraic K -theory defined using vector bundleshas poor cohomological properties without imposing mild assumptions on X . See,for example, [42, Corollary 3.9]. Since we do not study cohomological properties of K ( X ) in this paper, we ignore this issue.3.5. Projective bundles.
Let X be a monoid scheme. A projective bundle on X is a P n F -bundle π : E → X for some n ≥ X . Let E = lf( E ) be a locally free sheaf of rank r on X . The sheafification of the functorProj ◦ Sym ◦ Γ( − , E ) defines a P r − F -bundle π : P E → X ; see [7, §
7] for details on the
Proj construction. Its restriction to an affine open U of X is isomorphic to P r − U such that π U commutes with the structure map P r − U → U . Remark 3.18.
We note without proof that every projective bundle is of the form P E for some finite locally free sheaf E on X . Two finite locally free sheaves E and E ′ define isomorphic projective bundles π : P E → X and π ′ : P E ′ → X if and onlyif there is a locally free sheaf L of rank one such that E ′ ≃ E ⊗ L . Lemma 3.19.
Let A be a pointed monoid and n ≥ . Then Γ P nA = A .Proof. Consider the canonical affine open U i = Spec A [ T j T i | j = 0 , . . . , n ] of P nA =Proj A [ T , . . . , T n ] and U η = T ni =0 U i = Spec A [ T j T i | i, j = 0 , . . . , n ]. Since P nA iscovered by the U i and the restriction maps Γ U i → Γ U η , i = 0 , . . . , n , are injective,we have Γ P nA = T ni =0 Γ U i as an intersection inside Γ U η .In particular, a global section f ∈ Γ P nA is contained in Γ U η and is therefore ofthe form a Q ni =0 T e i i for some a ∈ A and ( e , . . . , e n ) ∈ Z n +1 with P ni =0 e i = 0. Since f ∈ Γ U i , we have e i ≤ i = 0 , . . . , n . Thus e = · · · = e n = 0, which showsthat f ∈ A as claimed. (cid:3) Proposition 3.20.
Let X be an integral monoid scheme and π : P E → X a projec-tive bundle. Then the map π ∗ : Γ X → Γ P E is an isomorphism of pointed monoids.Proof. Since X is integral, it has a unique generic point η , all restriction maps areinjective and Γ X is the intersection T Γ U inside O X,η , where U varies over all affineopen subschemes of X .For every affine open U ⊂ X , the bundle P E × X U ≃ P nU trivializes. Since η iscontained in every open subset of X , we conclude that P E is irreducible with uniquegeneric point ˆ η mapping to η . Moreover, since every affine open subscheme U of X is integral, P E is covered by the integral subschemes P E × X U ≃ P nU , which showsthat P E is integral. We conclude, using Lemma 3.4, that Γ P E embeds into O P E , ˆ η and equals the intersection T U Γ U where U varies over an affine open covering of X .To show that π ∗ is injective, suppose that π ∗ ( s ) = π ∗ ( t ) for s, t ∈ Γ X . Let V ≃ Spec A be an affine open of X and U V,i ≃ Spec A [ T j T i | j = 0 , . . . , n ] the canonicalopen of P E × X V ≃ P nV . Since ( π | V ) ∗ : A → A [ T j T i | j = 0 , . . . , n ] is injective and π ∗ ( s ) | U V,i = π ∗ ( t ) | U V,i , we conclude that s | V = t | V . Covering X with affine opens V yields the equality s = t .We turn to surjectivity of π ∗ . Let s ∈ Γ P E . By Lemma 3.19, the restriction s | U of s to U = P × X V ≃ P nA comes from a global section t V ∈ Γ V for every affineopen V of X . Since ( π | V ) ∗ : Γ V → Γ U is injective, t V is unique, and thereforethe collection { t V } , where V varies through all affine opens of X , glues to a uniqueglobal section t of P E with π ∗ ( t ) = s . This shows that π ∗ is surjective and completesthe proof. (cid:3) Lemma 3.21.
Let X be a monoid scheme and π : P E → X a projective bundle.Then there is a canonical section σ : X → P E such that σ ( x ) is the generic pointof the fibre π − ( x ) for every x ∈ X .Proof. Choose an affine open covering { U i = Spec A i } of X and define V i = U i × X P E . Since P E trivializes over affine opens, V i is isomorphic to P nA i = Proj A i [ T , . . . , T n ], ( X / F ) AND GW ( X / F ) 27 with n the fibre dimension of P E . For each i , the graded A i -linear pointed monoidmorphism s i : A i [ T , . . . , T n ] −→ A i [ b T ] T j b T defines a morphism σ i = s ∗ i : U i = P U i = Proj A i [ b T ] −→ Proj A i [ T , . . . , T n ] ≃ V i . Since the s i are invariant under A i -linear automorphisms of A i [ T , . . . , T n ], theydo not depend on the choice of identification V i ≃ P nA i , and therefore coincide onthe intersections of the U i and glue to a canonical morphism σ : X → P E . Since π ( U i ) ◦ s i is the identity on A i [ b T ] for every i , the composition π ◦ σ is the identity on X , which shows that σ is a section to π . The restriction of σ to σ x : { x } → π − ( x )corresponds to the graded k ( x )-monoid morphism s x : k ( x )[ T , . . . , T n ] −→ k ( x )[ b T ] T j b T where k ( x ) = O X,x / m x is the “residue field” at x . Since s − x ( h i ) = h i , we concludethat σ x ( x ) is the generic point in π − ( x ). This completes the proof. (cid:3) Lemma 3.22.
Let A be a pointed monoid and n ≥ . Then Pic( P nA ) = {O P nA ( m ) | m ∈ Z } .Proof. As explained in the example at the end of Section 3.4, this result is knownfor A = F . The inclusion i : F → A induces a morphism π : P nA → P n F and agroup homomorphism π ∗ : Pic( P n F ) → Pic( P nA ).Choose any pointed monoid morphism p : A → F , such as sending all units to 1and all other elements to 0. Then i ◦ p = id F and p induces a section σ : P n F → P nA of π and a retract σ ∗ : Pic( P nA ) → Pic( P n F ) of π ∗ . We conclude that π ∗ is injective.We turn to the surjectivity of π ∗ . Let U i = Spec F [ T j /T i ] j =0 ,...,n be the canonicalopen subsets of P n F = Proj F [ T , . . . , T n ] and U η = U ∩ . . . ∩ U n = Spec F [ T j /T i ] i,j =0 ...,n . Let V i = U i × P n F P nA and V η = U η × P n F P nA . Then { V i } is an affine open covering of P n and every line bundle L on P nA trivializes over this covering. This means thatwe get, for every i = 0 , . . . , n , a commutative diagram L ( V i ) Γ V i L ( V η ) Γ V ηϕ i ∼ res Vi,Vη ι i ∼ ϕ i,η of Γ V i -linear maps whose right vertical arrow is the canonical inclusion that comesfrom the inclusion Γ U i → Γ U η . Since the Γ V i -linear map ϕ i,η ◦ res V i ,V η is determinedby the image of 1, which is of the form a Q ni =0 T e i i for some a ∈ A and ( e , . . . , e n ) ∈ Z n +1 with P ni =0 e i = 0, and a is invertible (since ι i = ϕ i,η ◦ res V i ,V η ◦ ϕ − i is alocalization of pointed monoids), we can assume that ϕ i,η ◦ res V i ,V η (1) = n Y i =0 T e i i , after replacing ϕ i by a − ϕ − . This shows that we can choose the trivalizations ϕ i so that they restrict to bijections Γ U i → F [ T j /T i ] j =0 ,...,n . This yields a line bundle L ′ on P n F with π ∗ ( L ′ ) = L . Therefore, π ∗ is surjective. (cid:3) The following result is a monoid-theoretic analogue of a well-known result forschemes over algebraically closed fields.
Theorem 3.23.
Let X be an irreducible monoid scheme and π : P E → X a pro-jective bundle. Then the map ϕ : Pic( X ) × Z −→ Pic( P E )( L , m ) π ∗ L ⊗ O P E ( m ) is an isomorphism of abelian groups.Proof. Let η be the generic point of X and U η = Spec O X,η , which comes with acanonical morphism ι η : U η → X . Define V η = U η × X P E , which comes with thecartesian diagram V η P E U η X ιπ η πι η σ where σ : X → P E is the canonical section of π from Lemma 3.21. This yields acommutative diagram of group homomorphismsPic V η Pic P E Pic U η Pic X ι ∗ π ∗ η π ∗ ι ∗ η σ ∗ , where Pic U η is trivial since U η is affine. Thus ι ∗ ◦ π ∗ = π ∗ η ◦ ι ∗ η = 0. Since σ ∗ ◦ π ∗ isthe identity on Pic X , we conclude that π ∗ is injective.By Lemma 3.22, Pic V η = {O V η ( n ) | n ∈ Z } . Thus, the assignment O V η ( n ) P E ( n ) defines a group homomorphism r : Pic V η → Pic P E , which is a section of ι ∗ . This shows that ι ∗ is a surjection.Consider a line bundle L ∈
Pic P E in the kernel of ι ∗ , so that ι ∗ ( L ) ≃ O V η .Choose an affine open covering { U i } of X , so that U i = Spec A i for A i = Γ U i , anddefine V i = U i × X P E . This defines an open covering { V i } of P E . Since the U i areaffine, P E | U i is trivializable, so that P E | U i ≃ P nA i where n is the fibre dimensionof π . By Lemma 3.22, we conclude that L| V i ≃ O V i ( m ) for some m ∈ Z . Since O V η ( m ) ≃ L| V η ≃ O V η , we have m = 0. We therefore obtain a commutativediagram L ( V i ) O P E ( V i ) π ∗ ( O X )( V i ) π ∗ (cid:0) σ ∗ ( L ) (cid:1) ( V i ) L ( V η ) O P E ( V η ) π ∗ ( O X )( V η ) π ∗ (cid:0) σ ∗ ( L ) (cid:1) ( V η ) ∼ res Vi,Vη ∼ res Vi,Vη ∼ res Vi,Vη res
Vi,Vη ∼ ∼ ∼ for every i . Since the V i cover P E , we conclude that L = π ∗ (cid:0) σ ∗ ( L ) (cid:1) is in the imageof π ∗ . ( X / F ) AND GW ( X / F ) 29 Altogether, this shows that there is a canonically split short exact sequence0 Pic X Pic P E Pic V η , π ∗ ι ∗ σ ∗ r which induces the isomorphismPic X × Z ∼ −→ Pic X × Pic V η ( π ∗ ,r ) −→ Pic P E ( L , m ) (cid:0) L , O V η ( m ) (cid:1) P E ( m )of the claim of the theorem. (cid:3) A projective bundle formula.
We combine our earlier results to prove aprojective bundle formula for the K -theory space of a monoid scheme. This givesa monoid-theoretic analogue of Quillen’s projective bundle formula over fields [29,Theorem 2.1 of Section 8]. Theorem 3.24.
Let X be an integral monoid scheme and π : P E → X a projectivebundle. Then there is a homotopy equivalence K ( P E ) ∼ −→ Y ′ n ∈ Z K ( X ) . Proof.
Combining Propositions 3.12 and 3.20 and Theorem 3.23, we obtain a dia-gram of functors LP n ( P E ) ← hhO P E ii [Pic( P E )] ≃ hhO X ii [Pic( X ) × Z ] ≃ ( hhO X ii [Pic( X )])[ Z ] → LP ( X )[ Z ] . By Lemma 1.1, the first and last functors induce homotopy equivalences of K -theoryspaces. We therefore have homotopy equivalences K ( P E ) ≃ K ( LP ( X )[ Z ]) ≃ Y ′ n ∈ Z K ( X ) . (cid:3) Corollary 3.25.
Let X be an integral monoid scheme. Then there is an isomor-phism of graded rings K • ( P E ) ≃ K • ( X ) ⊗ Z Z [ Z ] . Grothendieck–Witt theory of monoid schemes
Duality for locally free sheaves.
Let X be a monoid scheme. There is abifunctor H om O X ( − , − ) : O X - Mod op ×O X - Mod → O X - Mod defined so that, for E , F ∈ O X - Mod and an open set U ⊂ X , we have H om O X ( E , F )( U ) = Hom O X | U ( E | U , F | U ) . Let L be a line bundle on X . As in the local case (Section 2.1), and unlike the caseof schemes over a field, the functor H om O X ( − , L ) does not define a duality functoron LF n ( X ). This can be remedied as follows. Let X be a pc monoid scheme. Given E ∈ O X - Mod , define a sheaf of pointed sets P L ( E ) on X by P L ( E )( U ) = Hom n O X | U ( E | U , L | U ) , the right hand side being the set of normal O X | U -module homomorphisms E | U →L | U . Because X is pc, P L ( E )( U ) has a natural O X ( U )-module structure, as followsfrom a local calculation using Lemma 2.2. Proposition 4.1.
Let L be a line bundle on a reversible pc monoid scheme X . Let Θ L : id LF n ( X ) ⇒ P L ◦ ( P L ) op be the natural isomorphism with components Θ LE ( s )( f ) = f ( s ) , s ∈ E ( U ) , f ∈ P L ( E )( U ) where U ⊂ X is an open subset. Then ( LF n ( X ) , P L , Θ L ) is a uniquely split com-binatorial proto-exact category with duality. Moreover, the Reduction Assumption(see Section 1.3) holds.Proof. That P L sends locally free sheaves to locally free sheaves follows from thefact that X is reversible and a local calculation using Lemma 2.2. Let f : E → F bea morphism in LF n ( X ). For each x ∈ X , the stalk morphism P L ( f ) x : P L ( F ) x → P L ( E ) x can be identified with( − ) ◦ f x : Hom O X,x - Mod n ( F x , L x ) → Hom O X,x - Mod n ( E x , L x ) . Since the composition of normal morphisms is normal, P L ( f ) is well-defined. That P L is compatible with ⊕ follows from Lemma 2.2. That Θ L is a natural isomorphismfollows from a local calculation using Proposition 2.6.It remains to verify the Reduction Assumption. Whether or not the induced map ψ N//U : N//U → P ( N//U ) is an isomorphism can be checked locally, in which caseit reduces to Proposition 2.6. (cid:3)
Remark 4.2.
Proposition 4.1 admits the following generalization, which can beseen as the natural commutative globalization of the setting of Proposition 2.6. Let σ : X → X be an involution and ǫ ∈ Γ( X, O X ) × . Assume that σ ∗ L ≃ L and ǫσ ( ǫ ) = 1. Let P L ,σ : LF n ( X ) op → LF n ( X ) be the functor E 7→ H om n O X ( σ ∗ E , L )and define Θ L ,σ,ǫ by Θ L ,σ,ǫ E ( s )( f ) = ǫσ ( f ( s )). Then the analogue of Proposition 4.1holds for ( LF n ( X ) , P L ,σ , Θ L ,σ,ǫ ). The results which follow hold also at this level ofgenerality, with essentially the same proofs. We note only that the involution of thepointed monoid Γ X is determined through the isomorphism Γ X ≃ End O X - Mod ( M )via the formula f ψ M ◦ P L ,σ ( f ) ◦ ψ − M . However, for ease of exposition, we restrictto the case σ = id X and ǫ = 1.When L = O X we omit it from the notation so that, for example, P O X = P . Lemma 4.3.
Let X be a reversible pc monoid scheme. If line bundles L , L ′ areequal in Pic( X ) / Pic( X ) , then there is an equivalence of proto-exact categories withduality ( LF n ( X ) , P L , Θ L ) ≃ ( LF n ( X ) , P L ′ , Θ L ′ ) . Proof.
Under the assumption of the lemma, there exists a line bundle ˜
L ∈ LF n ( X )and an isomorphism L ⊗ ˜ L ∼ −→ ˜ L ∨ ⊗ L ′ . (2)Let T be the exact autoequivalence − ⊗ ˜ L : LF n ( X ) → LF n ( X ); see the proof ofProposition 3.12. Then ( T, η ) is an equivalence of categories with duality, where η : T ◦ P L ⇒ P L ′ ◦ T op is the natural isomorphism with components η E : P ( E ) ⊗ L ⊗ ˜ L → P ( E ) ⊗ ˜ L ∨ ⊗ L ′ , E ∈ LF n ( X )defined using the chosen isomorphism (2). (cid:3) ( X / F ) AND GW ( X / F ) 31 Lemma 4.4.
Let X be a reversible pc monoid scheme and E ∈ LF n ( X ) a linebundle. Then there is a canonical isomorphism H om O X ( E , O X ) ≃ P ( E ) .Proof. This follows from the local observation that, for a pc pointed monoid A , any A -module homomorphism A → A is normal. (cid:3) The functor P is ⊗ -monoidal, while P L is not in general. Instead, P L is P -monoidal, that is, there are coherent isomorphisms P L ( E ⊗ F ) ≃ P ( E ) ⊗ P L ( F ) , E , F ∈ LF n ( X )which are natural in E and F . The functor P L induces an involution P L : Pic( X ) → Pic( X ) . (3)By Lemma 4.4, this map agrees with that induced by H om O X ( − , L ). We emphasizethat, since P L is only P -monoidal, the map (3) is not a group homomorphism unless L ∈
Pic( X ) is trivial. Denote by Pic( X ) P L the Z / X ). Thecomplement Pic( X ) ∗ = Pic( X ) \ Pic( X ) P L has a free Z / hhO X ii ⊂ LF n ( X ) is P -stable and so inherits from LF n ( X ) a proto-exact duality,again denoted by ( P, Θ). Define a proto-exact duality ( P L , Θ L ) on hhO X ii [Pic( X )]as follows. The functor P L is defined on basic objects by P L ( E M ) = P ( E ) P L ( M ) , E ∈ hhO X ii , M ∈
Pic( X )and extended to hhO X ii [Pic( X )] by additivity. The natural isomorphism Θ L hascomponents Θ LE M = Θ LE .Note that, by Remark 3.2 and Proposition 3.5, a monoid scheme X is integral ifand only if it is irreducible, pc and reversible. Lemma 4.5.
Let X be an integral monoid scheme. The functor F of Proposition3.12 lifts to an exact form functor ( F, µ ) : ( hhO X ii [Pic( X )] , P L , Θ L ) → ( LF n ( X ) , P L , Θ L ) . Proof.
Define the natural isomorphism µ : F ◦ P L ⇒ P L ◦ F op so that its components µ E M : P ( E ) ⊗ P L ( M ) → P ( E ⊗ M ) ⊗ L , E ∈ LF n ( X )are determined by the monoidal data of P . It is straightforward to verify that µ iscompatible with Θ L and Θ L . We omit the details. (cid:3) Define a proto-exact category hhO X ii [Pic( X ) P L ] = M M∈ Pic( X ) P L hhO X ii . (4)Since Pic( X ) P L ⊂ Pic( X ) is P L -stable, hhO X ii [Pic( X ) P L ] is a P L -stable proto-exact subcategory of hhO X ii [Pic( X )]. The induced duality on hhO X ii [Pic( X ) P L ] isequivalent to that induced by ( P, Θ) on each summand of hhO X ii . In other words,equation (4) defines hhO X ii [Pic( X ) P L ] as a proto-exact category with duality.Define a second proto-exact category by hhO X ii [Pic( X ) ∗ /P L ] = M M∈ Pic( X ) ∗ /P L hhO X ii . The choice of a set-theoretic section of the quotient Pic( X ) ∗ → Pic( X ) ∗ /P L embeds hhO X ii [Pic( X ) ∗ /P L ] as a proto-exact subcategory of LF n ( X ) which, however, is not P L -stable. The following result is immediate. Proposition 4.6.
Let X be an integral monoid scheme. There is an equivalence ofproto-exact categories with duality hhO X ii [Pic( X )] ≃ hhO X ii [Pic( X ) P L ] ⊕ H (cid:0) hhO X ii [Pic( X ) ∗ /P L ] (cid:1) . Grothendieck–Witt theory of monoid schemes.
Let L be a line bundleon a reversible pc monoid scheme X . Let GW ( X ; L ) = GW ( LF n ( X ) , P L , Θ L ), de-fined by either via the hermitian Q -construction or group completion. Set GW i ( X ; L ) = π i GW ( X ; L ). Proposition 4.7.
Let X be an integral monoid scheme. Then the homotopy typeof GW ( X ; L ) depends on L only through its class in Pic( X ) / Pic( X ) .Proof. This follows from Proposition 1.6 and Lemma 4.3. (cid:3)
We have the following analogue of Theorem 3.14 for Grothendieck–Witt theory.
Theorem 4.8.
Let L be a line bundle on an integral monoid scheme X . Then thereis a natural homotopy equivalence GW ( X ; L ) ≃ Y ′M∈ Pic( X ) P L GW ( hhO X ii ) × Y ′M∈ Pic( X ) ∗ /P L K ( hhO X ii ) . Proof.
By Lemma 1.7, Proposition 4.6 and the fact that GW commutes with directsums of categories, there is a natural homotopy equivalence GW ( X ; L ) ≃ GW ( hhO X ii [Pic( X ) P L ]) × GW (cid:0) H ( hhO X ii [Pic( X ) ∗ /P L ]) (cid:1) . The second factor is GW (cid:0) H ( hhO X ii [Pic( X ) ∗ /P L ]) (cid:1) ≃ K ( hhO X ii [Pic( X ) ∗ /P L ]) ≃ Y ′M∈ Pic( X ) ∗ /P L K ( hhO X ii ) , where the first homotopy equivalence follows from Proposition 1.5 and the secondfrom the proof of Theorem 3.14.Turning to the factor GW ( hhO X ii [Pic( X ) P L ]), note that since equation (4) re-spects the duality structures, there is a homotopy equivalence GW ( hhO X ii [Pic( X ) P L ]) ≃ Y ′M∈ Pic( X ) P L GW ( hhO X ii ) . This completes the proof. (cid:3)
Using Proposition 3.13, we can combine Theorems 2.5 and 2.11 with Theorem4.8 to obtain an explicit description of GW ( X ; L ). Corollary 4.9.
Let X be a proper integral monoid scheme. Then there is a homo-topy equivalence GW ⊕ ( X ; L ) ≃ Y ′M∈ Pic( X ) P L Z × B (Σ ∞ × ( Z / ≀ Σ ∞ )) + × Y ′M∈ Pic( X ) ∗ /P L (cid:0) Z × B Σ + ∞ (cid:1) . Proof.
The only additional piece of information needed is Proposition 3.6. (cid:3) ( X / F ) AND GW ( X / F ) 33 Without the properness assumption, there is an analogue of Corollary 4.9 iswritten in terms of Pic sym (Γ X ) and the isometry groups I ( ξ ). Since we will not usethis, we omit its formulation.Specializing to direct sum (Grothendieck–)Witt groups, we obtain the followingresults. Theorem 4.10.
Let L be a line bundle on an integral monoid scheme X .(i) There is an isomorphism of abelian groups GW ⊕ ( X ; L ) ≃ Pic sym (Γ X )[Pic( X ) P L ] × Z [Pic( X ) /P L ] . (ii) There is an isomorphism of abelian groups W ⊕ ( X ; L ) ≃ Pic sym (Γ X )[Pic( X ) P L ] . Proof.
This follows from Theorem 4.8, after using Theorems 2.5 and 2.11. We omitthe details. (cid:3)
Example.
Let A F = Spec( F [ t ]). Since F [ t ] has no non-trivial idempotents andonly the trivial automorphism, the functor − ⊗ F F [ t ] : Vect F → LF n ( A F ) in-duces an equivalence on maximal groupoids. Moreover, this equivalence respectsdualities. It follows that there are homotopy equivalences K ( A F ) ≃ K ( Vect F ) and GW ( A F ) ≃ GW ( Vect F ). ⊳ Example.
Let X = P n F . Fix d ∈ Z and set L = O P n F ( d ). The involution P L ofPic( X ) ≃ Z is k
7→ − k + d . In particular, Pic( X ) P L is non-empty if and only if d iseven, in which case Pic( X ) P L = { d } . Let [ d ] = 0 if d is even and [ d ] = 1 otherwise.With this notation, we obtain from Theorem 4.10 an isomorphism Z [ d ] × Z [ Z ≥ d ] ≃ GW ⊕ ( P n F ; d ) , bl d + X i ≥ d a i l i b [ O P n F ( d )] + X i ≥ d a i [ H ( O P n F ( i ))] . Note that O P n F ( d ) admits a unique symmetric form, which is omitted from thenotation. In particular, GW ⊕ ( P n F ; d ) is independent of n and, as guaranteed byLemma 4.3, depends on d only through its parity. Moreover, we have W ⊕ ( P n F ; d ) ≃ Z [ d ] with generator O P n F ( d ). ⊳ Example.
Using Theorem 4.8 and the isomorphism Pic( P n F ) ≃ Z , we find GW Q ( P n F ) ≃ GW Q ( Vect F ) × Y ′ k ∈ Z > K ( Vect F ) ≃ G n ∈ Z ≥ B Σ n × Z × B ( Z / ≀ Σ ∞ ) + × Y ′ k ∈ Z > Z × B (Σ ∞ ) + . Again, the result is independent of n . ⊳ A projective bundle formula.
We begin with a lemma.
Lemma 4.11.
Let π : P E → X be a projective bundle on an integral monoidscheme. Then the isomorphism ϕ : Pic( X ) × Z → Pic( P E ) , ( M , m ) π ∗ ( M ) ⊗ O P E ( m ) from Theorem 3.23 is Z / -equivariant, where Z / acts on Pic( X ) × Z by P L andnegation on the first and second factors, respectively, and on Pic( P E ) by P π ∗ L . Proof.
This is a direct calculation. (cid:3)
Theorem 4.12.
Let π : P E → X be a projective bundle on an integral monoidscheme and L a line bundle on X . Then there is a homotopy equivalence GW ( P E ; π ∗ L ) ≃ GW ( X ; L ) × Y ′ ( M ,i ) ∈ (Pic( X ) × Z ∗ ) / h ( P L , − i K ( hhO X ii ) . Proof.
By Lemma 4.11, the map ϕ induces a bijectionPic( P E ) P π ∗L → (Pic( X ) × Z ) ( P L , − = Pic( X ) P L × { } (5)and a Z / P E ) ∗ ≃ Pic( X ) P L × Z ∗ ⊔ Pic( X ) ∗ × Z = Pic( X ) ∗ × { } ⊔ Pic( X ) × Z ∗ . Together with Theorem 4.8, these bijections yield homotopy equivalences GW ( P E ; π ∗ L ) = Y ′M∈ Pic( X ) P L GW ( hhO P E ii ) × Y ′ ( M ,i ) ∈ (Pic( X ) × Z ∗ ) / h ( P L , − i K ( hhO P E ii ) × Y ′M∈ Pic( X ) ∗ /P L K ( hhO P E ii ) ≃ GW ( X ; L ) × Y ′ ( M ,i ) ∈ (Pic( X ) × Z ∗ ) / h ( P L , − i K ( hhO P E ii ) , as claimed. (cid:3) Passing to homotopy groups, we obtain F -linear analogues of previously knownresults over fields in which 2 is invertible [45], [35], [31]. Corollary 4.13.
Let π : P E → X be a projective bundle on an integral monoidscheme and L a line bundle on X .(i) The map ϕ : GW ( X ; L ) × ( K ( X ) ⊗ Z Z [ Z ∗ ]) ( π ∗ P L , − → GW ( P E ; π ∗ L ) defined by ϕ ( M ) = π ∗ M and ϕ ( W , m ) = H L ( π ∗ W ⊗ O P E O P E ( m )) is an isomorphismof abelian groups.(ii) There is an isomorphism of abelian groups W ( X ; L ) ≃ W ( P E ; π ∗ L ) . Proof.
The first statement follows from Theorem 4.12 by taking connected compo-nents. Alternatively, we could use Theorems 3.16 and 4.10 and Lemma 4.11.Turning to the second statement, Theorem 4.10 gives W ( P E ; π ∗ L ) ≃ W (Γ( P E , O P E ) - proj n )[Pic( P E ) P π ∗L ] . Using Theorem 3.23 and the bijection (5), we conclude. (cid:3)
Example.
We have GW (Spec( F )) × K (Spec( F )) ⊗ Z Z [ Z ∗ ] Z / ≃ Z × Z ⊗ Z Z [ Z ∗ ] Z / ≃ Z × Z [ Z > ]which is isomorphic to GW ( P n F ) ≃ Z × Z [ Z ≥ ], as required by Corollary 4.13. ⊳ ( X / F ) AND GW ( X / F ) 35 Appendix
A.A.1.
Direct sums of categories.
Let {C i } i ∈ I be a family of categories indexedby a set I . Definition.
The direct sum category L i ∈ I C i has objects which are finite lists V i ∈C i , . . . , V i n ∈ C i n labelled by distinct i , . . . , i n ∈ I . Write L nj =1 V i j for such anobject. Morphisms are given by Hom L i ∈ I C i ( n M k =1 V i k , m M l =1 W j l ) = Y k,li k = j l Hom C ik ( V i k , W j l ) . Many properties and structures of the individual categories C i extend in a point-wise fashion to L i ∈ I C i . For example, if all C i are proto-exact (with exact directsum), then so too is L i ∈ I C i .One can realize L i ∈ I C i as a filtered colimit of finite direct sums. Let P < ∞ ( I ) bethe partially ordered set of finite subsets of I , ordered by inclusion. Consider thefunctor P < ∞ ( I ) → Cat which assigns to a finite subset S ⊂ I the category L s ∈ S C s and to an inclusion S ֒ → T the obvious functor L s ∈ S C s ֒ → ⊕ t ∈ T C t . Proposition A.1.
There is an equivalence of categories lim −→ S ∈ P < ∞ ( I ) M s ∈ S C s ≃ M i ∈ I C i . If C = C i is a constant family of categories with a symmetric bimonoidal structure( ⊕ , ⊗ ) and ( I, · ) is an abelian group, then we denote the direct sum by C [ I ] := M i ∈ I C . Define a symmetric bimonoidal structure on C [ I ] by extending ⊕ componentwiseand defining ⊗ using the convolution product M a V a ⊗ M b W b := M c ( M a,bab = c V a ⊗ W b ) . A.2.
Restricted products.
Let { ( Y i , ∗ i ) } i ∈ I be family of pointed topologicalspaces indexed by a set I . Definition.
The restricted product of { ( Y i , ∗ i ) } i ∈ I is Y ′ i ∈ I Y i = { ( y i ) | y i = ∗ i for only finitely many i ∈ I } ⊆ Y i ∈ I Y i equipped with the subspace topology. The restricted product can be realized as a filtered colimit of finite products asfollows.
Proposition A.2.
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