The equivariant K-theory of toric varieties
aa r X i v : . [ m a t h . K T ] S e p THE EQUIVARIANT K -THEORY OF TORICVARIETIES SUANNE AU, MU-WAN HUANG, AND MARK E. WALKER
Abstract.
This paper contains two results concerning the equi-variant K -theory of toric varieties. The first is a formula for theequivariant K -groups of an arbitrary affine toric variety, general-izing the known formula for smooth ones. In fact, this result isestablished in a more general context, involving the K -theory ofgraded projective modules. The second result is a new proof ofa theorem due to Vezzosi and Vistoli concerning the equivariant K -theory of smooth (not necessarily affine) toric varieties. Contents
1. Introduction 12. The K -theory of graded projective modules 23. The Equivariant K -theory of Affine Toric Varieties 74. The Vezzosi-Vistoli Theorem 8References 111. Introduction
Let k be a field, suppose U σ is the affine toric k -variety associated toa strongly convex rational polyhedral cone σ in Euclidean n -space, andlet T be the n -dimensional torus that acts on U σ . If U σ is smooth, thenthere is an equivariant isomorphism U σ ∼ = T σ × A r , where r = dim( σ )and T σ is the unique orbit of minimal dimension (namely, dimension n − r ). Using basic properties of equivariant K -theory of smooth varieties(see, for example, [6]), ones obtains natural isomorphisms(1) K Tq ( U σ ) ∼ = K Tq ( T σ ) ∼ = K q ( k ) ⊗ Z Z [ M σ ]where M σ ∼ = Z n − r is the group of characters of T σ . Date : November 20, 2018.2000
Mathematics Subject Classification.
Key words and phrases. equivariant K-theory, toric varieties.Walker’s research was partially supported by NSF grant DMS-0601666.
This paper consists of two main results related to the isomorphism(1). The first, Theorem 4, shows that this isomorphism holds for allaffine toric varieties, not just smooth ones. In fact, this theorem estab-lishes the more general isomorphism(2) K Tq ( U σ × k Spec R ) ∼ = K q ( R ) ⊗ Z Z [ M σ ] , where R is any k -algebra and the action of T on Spec R is trivial.Theorem 4 is actually a consequence of our Theorem 1, concerning the K -theory of graded projective modules.The second main result of this paper is a new proof of a theorem dueto Vezzosi and Vistoli [11, Theorem 6.2] that calculates the equivariant K -theory of an arbitrary smooth toric variety. See our Theorem 6 forthe precise statement. The proof due to Vezzosi and Vistoli uses amore general result, one that applies to arbitrary actions by diagonal-izable groups schemes. However, in the important special case of toricvarieties, we recover their result using only Equation (1), the theoryof sheaf cohomology for fans, and Thomason’s foundational work onequivariant K -theory [9].2. The K -theory of graded projective modules The first main goal of this paper is to establish the isomorphism(2). The action of T on U σ is given by a grading (by the group ofcharacters of T ) of the associated ring of regular functions for U σ , andan equivariant bundle on U σ is given by a graded projective moduleover this ring. Thus, our first theorem is really about the K -theoryof graded projective modules. In this section, we state and prove ageneral theorem of this form.Let R be any commutative ring, M an abelian group (written addi-tively), and A ⊂ M a sub-monoid. We form the associated monoid-ring R [ A ]. As a matter of notation, an element a ∈ A is written as χ a in R [ A ] so that χ a χ b = χ a + b for a, b ∈ A . The commutative ring R [ A ] is an M -graded R -algebra, with elements of R declared to be of degree zeroand for any a ∈ A , deg( χ a ) := a ∈ A ⊂ M . Let P ( R ) denote the cate-gory of finitely generated projective R -modules and let P M ( R [ A ]) de-note the category consisting of finitely generated M -graded projective R [ A ]-modules and with morphisms given by M -graded R [ A ]-modulehomomorphisms. Let K M ∗ ( R [ A ]) denote the K -theory of the exact cat-egory P M ( R [ A ]).Recall that if G is an M -graded R [ A ]-module and m ∈ M , then G [ m ] denotes the same module but with the grading shifted so that G [ m ] w = G w − m for all w ∈ M . In particular, R [ A ][ m ] is graded-free ofrank one generated by an element of degree m . HE EQUIVARIANT K -THEORY OF TORIC VARIETIES 3 Write U ( A ) for the subgroup of units (i.e., elements with additiveinverses) in the monoid A . We fix, once and for all, a set S ( A ) ⊂ M of coset representatives for the subgroup U ( A ) of M . Theorem 1.
For a commutative ring R , an abelian group M , and asub-monoid A of M , we have an isomorphism K q ( R ) ⊗ Z Z [ M/U ( A )] ∼ = K Mq ( R [ A ]) , for all q .Under the identification of K q ( R ) ⊗ Z Z [ M/U ( A )] with L S ( A ) K q ( R ) ,this isomorphism is induced by the collection of exact functors sending ( P, s ) , with P ∈ P ( R ) and s ∈ S ( A ) , to P ⊗ R R [ A ][ s ] . The proof of the Theorem requires the following two lemmas. Through-out the rest of this section, let U = U ( A ) and S = S ( A ). Lemma 2.
The exact functor ψ : M S P ( R ) → P M ( R [ U ]) determined by ( P s ) s ∈ S M s ∈ S P s ⊗ R R [ U ][ s ] is an equivalence of categories.Proof. For
P, P ′ ∈ P ( R ) and s, s ′ ∈ S , we have an isomorphism (3)Hom MR [ U ] ( P ⊗ R R [ U ][ s ] , P ′ ⊗ R R [ U ][ s ′ ]) ∼ = ( Hom R ( P, P ′ ) if s = s ′ and0 otherwise, determined by sending a graded homomorphism from P ⊗ R R [ U ][ s ] to P ′ ⊗ R R [ U ][ s ′ ] to the induced map on the degree s pieces. It followsthat ψ is fully faithful.Given F ∈ P M ( R [ U ]), the M -grading on F gives a decomposition F = L m F m . If m, m ′ ∈ M belong to different cosets of U , then ( R [ U ] · F m ) ∩ F m ′ = 0. Thus we have an internal direct sum decomposition F = M s ∈ S Q s as M -graded R [ U ]-modules, where Q s = L m ∈ s + U F m . Since F isfinitely generated, Q s = 0 for all but a finite number of s . For each s ∈ S , we have F s ∼ = Q s ⊗ R [ U ] R (where R [ U ] → R is the augmentationmap), and hence F s is a finitely generated and projective R -module. If m , m belong to the same coset of U in M , then χ m − m : F m ∼ = −→ F m is an isomorphism of R -modules. Using this, we see that the map F s ⊗ R R [ U ][ s ] → Q s SUANNE AU, MU-WAN HUANG, AND MARK E. WALKER determined by p ⊗ χ u χ u · p is a graded isomorphism of R [ U ]-modules.It follows that F is isomorphic to ψ (( F s ) s ∈ S ), and hence ψ is an equiv-alence. (cid:3) If C, C ′ are M -graded rings, φ : C → C ′ an M -graded ring homo-morphism and F is an M -graded C -module, then the module obtainedfrom F via extension of scalars along φ , namely C ′ ⊗ C F , acquiresthe structure of an M -graded C ′ -module having the property that if c ′ ∈ C ′ m and f ∈ F m then c ′ ⊗ f ∈ ( C ′ ⊗ C F ) m + m (see [7, § C [ m ] by extension of scalarsalong φ is C ′ [ m ]. Lemma 3.
The exact functor P M ( R [ U ]) → P M ( R [ A ]) defined by extension of scalars induces a bijection on isomorphismclasses of objects. In particular, objects of P M ( R [ A ]) are projectivein the category of all M -graded R [ A ] -modules.Proof. For a projective R -module P and an arbitrary M -graded R [ A ]-module G , we have(4) Hom P M ( R [ A ]) ( P ⊗ R R [ A ][ m ] , G ) ∼ = Hom R ( P, G m ) . Since G G m is an exact functor, P ⊗ R R [ A ] is a projective object inthe category of all M -graded R [ A ]-modules. In particular, the secondassertion of the Lemma follows from the first one, using Lemma 2.The M -graded R -algebra map R [ U ] → R [ A ] is split by the M -graded R -algebra map R [ A ] → R [ U ] defined by χ a ( χ a if a ∈ U and0 if a / ∈ U .Since the composition R [ U ] ֒ → R [ A ] ։ R [ U ] is the identity, the functor P M ( R [ U ]) → P M ( R [ A ]) is split injective on isomorphism classes ofobjects.The proof of the surjectivity on isomorphism classes will use thegraded version of Nakayama’s Lemma. Let I ⊂ R [ A ] denote the kernelof the split surjection R [ A ] ։ R [ U ] — it is generated as an R -moduleby { χ a | a / ∈ U } . Clearly I is M -graded and, moreover, every maximal M -graded ideal of R [ A ] contains I . Indeed, if m is a maximal M -graded ideal, then R [ A ] / m is a M -graded ring such that every non-zero homogeneous element is a unit (and whose inverse is, necessarily,homogeneous). For a / ∈ U , if χ a = 0 in R [ A ] / m , then we would have χ a · rχ b = 1 for some r ∈ R and b ∈ A . But then a + b = 0, contrary HE EQUIVARIANT K -THEORY OF TORIC VARIETIES 5 to a / ∈ U . Thus χ a ∈ m for all a / ∈ U . Since I is contained in everymaximal M -graded ideal, the graded version of Nakayama’s Lemma(see, for example, [8, Theorem 3.6] for a proof) gives us: If G is afinitely generated M -graded R [ A ]-module such that IG = G , then G = 0.Given E ∈ P M ( R [ A ]), let F = E ⊗ R [ A ] R [ U ] ∈ P M ( R [ U ]) (withthe map R [ A ] → R [ U ] being the above split surjection) and let ˜ F = F ⊗ R [ U ] R [ A ]. We prove E ∼ = ˜ F in P M ( R [ A ]). As noted above, (4)and Lemma 2 show that ˜ F is a projective object in the category of all M -graded R [ A ]-modules. Thus the canonical map ˜ F ։ F lifts alongthe surjection E ։ F to give a morphism θ : ˜ F → E in P M ( R [ A ]).The map θ induces an isomorphism upon modding out by I and hence,by Nakayama’s Lemma, coker( θ ) = 0. Since E is projective as anungraded R -module, the exact sequence0 → ker( θ ) → ˜ F → E → − ⊗ R [ A ] R [ U ], and hence, usingNakayama’s Lemma again, ker( θ ) = 0. (cid:3) Proof of Theorem 1.
By Lemma 2, we have K Mq ( R [ U ]) ∼ = M S K q ( R ) ∼ = K q ( R ) ⊗ Z Z [ M/U ] . In order to prove the theorem, it therefore suffices to prove the exactfunctor(5) P M ( R [ U ]) → P M ( R [ A ]) , induced by extension of scalars, induces a homotopy equivalence on K -theory spaces.For any finite subset F ⊂ S , let P MF ( R [ A ]) denote the full subcate-gory of those objects in P M ( R [ A ]) isomorphic to one of the form l M i =1 P i ⊗ R R [ A ][ s i ]such that s i ∈ F for i = 1 , . . . , l . Define P MF ( R [ U ]) similarly. Notethat P MF ( R [ U ]) and P MF ( R [ A ]) are closed under direct sum and henceare exact subcategories. Since P M ( R [ A ]) = lim −→ F ⊂ S P MF ( R [ A ]) where F ranges over all finite subsets of S and since K -theory commutes withfiltered colimits, it suffices to prove P MF ( R [ U ]) → P MF ( R [ A ]) SUANNE AU, MU-WAN HUANG, AND MARK E. WALKER induces an equivalence on K -theory for all finite F ⊂ S . We proceed byinduction on F . If F = 1, then by (3) and Lemma 3, P MF ( R [ U ]) →P MF ( R [ A ]) is an equivalence of categories.Define a partial order ≤ on S by declaring s ≤ s ′ if and only if s ′ − s ∈ A . Then for projective R -modules P, P ′ and elements s, s ′ ∈ S ,we have (6) Hom P M ( R [ A ]) ( P ⊗ R R [ A ][ s ] , P ′ ⊗ R R [ A ][ s ′ ]) ∼ = ( Hom R ( P, P ′ ) if s ≤ s ′ and0 otherwise. Now assume
F > s ∈ F be a maximal element. Define F ′ = F \ { s } . We have a commutative diagram of exact functors P MF ′ ( R [ U ]) ⊕ P M { m } ( R [ U ]) (cid:15) (cid:15) / / P MF ′ ( R [ A ]) ⊕ P M { m } ( R [ A ]) (cid:15) (cid:15) P MF ( R [ U ]) / / P MF ( R [ A ])in which the vertical maps are given by direct sum and the horizontalmaps are extensions of scalars. The left-hand vertical map and the tophorizontal map induce equivalences on K -theory using Lemma 2 andinduction, respectively. It therefore suffices to prove that the right-hand vertical map induces an equivalence on K -theory. This followsfrom Waldhausen’s generalization of the Quillen Additivity Theorem,as we now explain.Let E denote the exact category consisting of short exact sequencesof objects of P MF ( R [ A ]) of the form(7) 0 → B → P → C → B ∈ P M { m } ( R [ A ]) and C ∈ P MF ′ ( R [ A ]). By Lemma 3, for any suchshort exact sequence, we have that P is isomorphic to B ⊕ C . Moreover,by (6) there are no non-trivial maps from B to C , and hence this exactsequence is isomorphic to0 → B „ « −→ B ⊕ C (0 , −→ C → . Thus E is equivalent to the full subcategory consisting of such “trivial”exact sequences. A morphism from one such exact sequence to anotheris completely determined by the map on middle objects. That is, thefunctor E → P MF ( R [ A ]) sending the exact sequence (7) to P is anequivalence of categories. On the other hand, Waldhausen’s AdditivityTheorem [13] shows that the functor E → P M { m } ( R [ A ]) ⊕ P MF ′ ( R [ A ])sending (7) to ( B, C ) induces an equivalence on K -theory. (cid:3) HE EQUIVARIANT K -THEORY OF TORIC VARIETIES 7 The Equivariant K -theory of Affine Toric Varieties In this section we provide an interpretation of Theorem 1 for toricvarieties.We adopt the notational conventions for toric varieties found in Ful-ton’s book [4]. An affine toric variety is defined from a strongly convexrational polyhedral cone σ in N ⊗ Z R where N ∼ = Z n is an n dimensionallattice. Let M := Hom Z ( N, Z ) ∼ = Z n be the dual lattice and define thedual cone of σ by σ ∨ := { u ∈ M ⊗ Z R | u ( v ) ≥ v ∈ σ } . We have that σ ∨ ∩ M is a finitely generated abelian monoid, by Gordan’sLemma, and hence, for any commutative ring R , the correspondingmonoid ring R [ σ ∨ ∩ M ] is a finitely generated R -algebra. We let U σ, Z = Spec Z [ σ ∨ ∩ M ] , the affine toric scheme over Z associated to σ .Note that for any commutative ring R , we have U σ,R := U σ, Z × Spec R = Spec R [ σ ∨ ∩ M ] . In particular, for a field k , the affine k -variety U σ,k = Spec k [ σ ∨ ∩ M ]is the classical toric k -variety associated to σ .For any commutative ring R , the R -algebra R [ σ ∨ ∩ M ] is an M -graded R -algebra, and this grading amounts to an action of the n -dimensionaltorus scheme T := Spec Z [ M ] on U σ,R . Viewing U σ,R as U σ, Z × Spec R ,the action of T is given by the usual action on U σ, Z and the trivialaction Spec R . An equivariant vector bundle over U σ,R is identified asa projective module over R [ σ ∨ ∩ M ] that is M -graded. We thereforeobtain K M ∗ ( R [ σ ∨ ∩ M ]) ∼ = K T ∗ ( U σ,R ) . Finally, observe that U ( σ ∨ ∩ M ) = σ ⊥ ∩ M , and we define M σ := M/ ( σ ⊥ ∩ M ). The following is thus an immediate consequence of The-orem 1. Theorem 4.
For any commutative ring R and strongly convex rationalcone σ , there is a natural isomorphism K Tq ( U σ,R ) ∼ = K q ( R ) ⊗ Z Z [ M σ ] . In particular, we see that Equation (1) holds for any affine toricvariety, not only the smooth ones. Observe that M σ , as just defined,coincides with the group of characters on the minimal orbit of U σ . SUANNE AU, MU-WAN HUANG, AND MARK E. WALKER
Remark 5.
The isomorphism of Theorem 1 is natural in R in theobvious sense and is natural in A in the following sense: If A ⊂ A ′ ⊂ M is an inclusion of submonoids of M , then K q ( R ) ⊗ Z Z [ M/U ( A )] ∼ = / / (cid:15) (cid:15) K Mq ( R [ A ]) (cid:15) (cid:15) K q ( R ) ⊗ Z Z [ M/U ( A ′ )]) ∼ = / / K Mq ( R [ A ′ ])commutes, where the left-hand map is the canonical quotient map andthe right-hand map is induced by extension of scalars.Consequently, the isomorphism of Theorem 4 is natural in R andwith respect to the inclusion of a face τ into σ . In the latter case, themap K Tq ( U σ,R ) → K Tq ( U τ,R )is induced by pullback along the equivariant open immersion U τ,R ⊂ U σ,R and the map K q ( R ) ⊗ Z Z [ M σ ] → K q ( R ) ⊗ Z Z [ M τ ]is the map induced by the canonical surjection M σ ։ M τ .4. The Vezzosi-Vistoli Theorem
In this section, we use (1) from the introduction, the theory ofsheaves on fans and the foundational results of Thomason [9] concern-ing equivariant K -theory to recover a result due to Vezzosi and Vistoli[11, 12]: For a field k and a smooth toric k -variety X = X (∆) definedby a fan ∆, the sequence0 −→ K Tq ( X ) −→ M σ ∈ Max (∆) K Tq ( U σ ) ∂ −→ M δ,τ ∈ Max (∆) ,δ<τ K Tq ( U δ ∩ τ )is exact. Here, M ax (∆) is the set of maximal cones in ∆ and we choose,arbitrarily, a total ordering for this set. The map ∂ is given as follows.For f = ( f σ ) σ ∈ Max (∆) in L σ ∈ Max (∆) K Tq ( U σ ), the ( δ < τ )-component ofits image is f τ | U δ ∩ τ − f δ | U δ ∩ τ ∈ K Tq ( U δ ∩ τ ).In fact, we prove that the sequence(8)0 → K Tq ( X ) → M σ K Tq ( U σ ) → M δ<τ K Tq ( U δ ∩ τ ) → M δ<τ<ǫ K Tq ( U δ ∩ τ ∩ ǫ ) → · · · is exact, where L σ K Tq ( U σ ) → L δ<τ K Tq ( U δ ∩ τ ) → · · · is the ˇCechcomplex of the presheaf K Tq for the equivariant open cover V = { U σ | σ is a maximal cone in ∆ } . Using Equation (1) (or our Theorem 4), HE EQUIVARIANT K -THEORY OF TORIC VARIETIES 9 the exactness of this sequence is equivalent to the existence of an exactsequence of the form(9)0 → K Tq ( X ) → M σ K q ( k ) ⊗ Z Z [ M σ ] → M δ<τ K q ( k ) ⊗ Z Z [ M δ ∩ τ ] → · · · . We define a topology on the finite set of cones comprising the fan∆ by declaring the open subsets to be the subfans of ∆; see [2] or [3].In other words, we view ∆ as a poset via face containment, ≺ , and wegive ∆ the “poset topology”, in which an open subset Λ is a subsetsatisfying the condition what whenever x ≺ y and y ∈ Λ, we have x ∈ Λ. For a cone σ ∈ ∆, let h σ i denote the fan consisting of σ andall its faces (i.e., the smallest open subset of ∆ containing σ ). Observethat for a sheaf F on ∆, we have F ( h σ i ) = F σ , the stalk of F at thepoint σ .For this topology, sheaves are uniquely determined by their stalksand the maps between their stalks arising from comparable elements ofthe poset (see [1, § F on the space ∆, the associatedfunctor on the poset ∆ sends σ ∈ ∆ to F σ = F ( h σ i ) and sends a faceinclusion τ ≺ σ to the map induced by h τ i ⊂ h σ i . Given a contravariantfunctor F on the poset ∆, the value of associated sheaf F on an opensubset Λ of ∆ is given by F (Λ) = lim ←− σ ∈ Λ F ( σ ) . Theorem 6.
Assume X = X (∆) is a smooth toric variety defined overan arbitrary field k . Then the presheaf Λ K Tq ( X (Λ)) defined on ∆ is a flasque sheaf. Moreover, there is an isomorphism K Tq ( X ) ∼ = K q ( k ) ⊗ K T ( X ) . and the sequences (8) and (9) are exact.Proof. Let A q be the sheaf on ∆ associated to the functor sending acone σ to K q ( k ) ⊗ Z [ M σ ] and a face inclusion τ ≺ σ to the map inducedby the canonical quotient M σ ։ M τ .The sheaf A is flasque by [1]. Since A is a flasque sheaf of torsionfree abelian groups, the presheaf K q ( k ) ⊗ Z A is actually a sheaf. In-deed, for any open subset U and open covering U = ∪ i V i of it, the mapfrom A ( U ) to the associated ˇCech complex is a quasi-isomorphism by [5, III.4.3], and since A is torsion free, this map remains a quasi-isomorphism upon tensoring by any abelian group. It now followsfrom the correspondence between functors and sheaves that A q ∼ = K q ( k ) ⊗ A . In particular, A q is also flasque.For a subfan Λ of ∆, let V be the Zariski open covering { U σ | σ is a maximal cone in Λ } of X (Λ) and let U be the open covering {h σ i | σ ∈ M ax (∆) } of Λ. By Equation (1) (or Theorem 4), the ˇCechcohomology complex of the presheaf K Tq ( − ) on X (Λ) for the open cov-ering V coincides with the ˇCech cohomology complex of the sheaf A q for the open covering U . Since the higher ˇCech cohomology of flasquesheaves vanishes [5, III.4.3], we have(10) ˇ H p (cid:0) V , K Tq (cid:1) = ˇ H p ( U , A q ) = 0 , for all p > K T coincides with equivariant G -theory (defined from equivariant coherent sheaves) and that the lat-ter satisfies the usual localization property relating X , an equivariantclosed subscheme, and its open complement. From this one deducesthat if X (Λ) = U ∪ V is covering by equivariant open subschemes, then K T ( X (Λ)) / / (cid:15) (cid:15) K T ( U ) (cid:15) (cid:15) K T ( V ) / / K T ( U ∩ V )is a homotopy cartesian square. Arguing just as in [10, § H p (cid:0) V , K Tq (cid:1) = ⇒ K Tq − p ( X (Λ)) . Using (10), this spectral sequence collapses to give(11) ˇ H (cid:0) V , K Tq (cid:1) ∼ = K Tq ( X (Λ)) , for all q .Combining (11) and (10) gives that the complexes0 → K Tq ( X (Λ)) → M σ K Tq ( U σ ) → M δ<τ K Tq ( U δ ∩ τ ) → · · · and0 → A q (Λ) → M σ K q ( k ) ⊗ Z Z [ M σ ] → M δ<τ K q ( k ) ⊗ Z Z [ M δ ∩ τ ] → · · · are exact and isomorphic to each other. In particular, Λ
7→ K Tq ( X (Λ))is isomorphic to the flasque sheaf A q .The remaining assertions of the Theorem follow immediately. (cid:3) HE EQUIVARIANT K -THEORY OF TORIC VARIETIES 11 Acknowledgements:
We are very grateful to Dan Grayson for sug-gesting the use of the Additivity Theorem in the proof of Theorem 1.This allowed us to simplify our original proof and to make the resultmore general.We are also indebted to the anonymous referee for many thoughtfulsuggestions which greatly improved the exposition of this paper.
References [1] Silvano Baggio. Equivariant K-theory of smooth toric varieties.
Tohoku Math.J. (2) , 59(2):203–231, 2007.[2] Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, and Ludger Kaup.Equivariant intersection cohomology of toric varieties. In
Algebraic geometry:Hirzebruch 70 (Warsaw, 1998) , volume 241 of
Contemp. Math. , pages 45–68.Amer. Math. Soc., Providence, RI, 1999.[3] Paul Bressler and Valery A. Lunts. Intersection cohomology on nonrationalpolytopes.
Compositio Math. , 135(3):245–278, 2003.[4] William Fulton.
Introduction to toric varieties , volume 131 of
Annals of Mathe-matics Studies . Princeton University Press, Princeton, NJ, 1993. , The WilliamH. Roever Lectures in Geometry.[5] Robin Hartshorne.
Algebraic geometry . Springer-Verlag, New York, 1977.Graduate Texts in Mathematics, No. 52.[6] Alexander S. Merkurjev. Equivariant K -theory. In Handbook of K -theory. Vol.1, 2 , pages 925–954. Springer, Berlin, 2005.[7] Constantin N˘ast˘asescu and Freddy Van Oystaeyen. Methods of graded rings ,volume 1836 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2004.[8] Markus Perling. Graded rings and equivariant sheaves on toric varieties.
Math.Nachr. , 263/264:181–197, 2004.[9] R. W. Thomason. Algebraic K -theory of group scheme actions. In Algebraictopology and algebraic K -theory (Princeton, N.J., 1983) , volume 113 of Ann.of Math. Stud. , pages 539–563. Princeton Univ. Press, Princeton, NJ, 1987.[10] R. W. Thomason and Thomas Trobaugh. Higher algebraic K -theory of schemesand of derived categories. In The Grothendieck Festschrift, Vol. III , volume 88of
Progr. Math. , pages 247–435. Birkh¨auser Boston, Boston, MA, 1990.[11] Gabriele Vezzosi and Angelo Vistoli. Higher algebraic K -theory for actions ofdiagonalizable groups. Invent. Math. , 153(1):1–44, 2003.[12] Gabriele Vezzosi and Angelo Vistoli. Erratum: “Higher algebraic K -theoryfor actions of diagonalizable groups” [Invent. Math. (2003), no. 1, 1–44; Invent. Math. , 161(1):219–224, 2005.[13] Friedhelm Waldhausen. Algebraic K -theory of spaces. In Algebraic and geo-metric topology (New Brunswick, N.J., 1983) , volume 1126 of
Lecture Notesin Math. , pages 318–419. Springer, Berlin, 1985.
Dept. of Mathematics, University of Nebraska - Lincoln, Lincoln,NE 68588, USA
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