aa r X i v : . [ m a t h . AG ] O c t HIGHER K-THEORY OF TORIC STACKS
ROY JOSHUA AND AMALENDU KRISHNA
Abstract.
In this paper, we develop several techniques for computing thehigher G-theory and K-theory of quotient stacks. Our main results for com-puting these groups are in terms of spectral sequences. We show that thesespectral sequences degenerate in the case of many toric stacks, thereby provid-ing an efficient computation of their higher K-theory. We apply our main resultsto give explicit description for the higher K-theory for many smooth toric stacks.As another application, we describe the higher K-theory of toric stack bundlesover smooth base schemes. Introduction
Toric varieties form a good testing ground for verifying many conjectures in alge-braic geometry. This becomes particularly apparent when one wants to understandcohomology theories for algebraic varieties. Computations of cohomology rings ofsmooth toric varieties such as the Grothendieck ring of vector bundles, the Chowring and the singular cohomology ring have been well-understood for many years.These computations facilitate predictions on the structure of various cohomologyrings of a general algebraic variety.Just like toric varieties, one would like to have a class of algebraic stacks on whichvarious cohomological problems about stacks can be tested. The class of toricstacks , first introduced and studied in [4] by Borisov, Chen and Smith, in terms ofcombinatorial data called stacky fans , is precisely such a class of algebraic stacks.These stacks are expected to be the toy models for understanding cohomologytheories of algebraic stacks, a problem which is still very complicated in general.Such a point of view probably accounts for the recent flurry of activity in thisarea with several groups considering various forms of toric stacks. (See for example,[12], [31], [16] in addition to [4].) In [12], Fantechi, Mann and Nironi study thestructure of toric Deligne-Mumford stacks in detail. Recently, in [16], Geraschenkoand Satriano consider in detail toric stacks which may not be Deligne-Mumford.A class of toric stacks of this kind and their cohomology were earlier considered byLafforgue [31] in the study of geometric Langlands correspondence. All examplesof toric stacks studied before, including those in [4], [12] and [31] are shown to bespecial cases of the stacks introduced in [16]. These stacks appear naturally whilesolving certain moduli problems and computation of their cohomological invariantsallows us to understand these invariants for many moduli spaces.In [4], Borisov, Chen and Smith computed the rational Chow ring and orbifoldChow ring of toric Deligne-Mumford stacks. The integral version of this resultfor certain type of toric Deligne-Mumford stacks is due to Jiang and Tseng [25],and Iwanari [22]. Jiang and Tseng also extend some of these results to certaintoric stack bundles in [24] and [26]. Borisov and Horja [5] computed the integralGrothendieck ring K ( X ) of a toric Deligne-Mumford stack X . See also [40].On the other hand, almost nothing has been worked out till now regardingthe higher K-groups of toric stacks, even when they are Deligne-Mumford stacks, Mathematics Subject Classification.
Key words and phrases. toric, fan, stack, K-theory.The first author was supported by a grant from the National Science Foundation. The secondauthor was supported by the Swarnajayanti fellowship, Govt. of India, 2011. though the higher K-groups of toric varieties had been well understood ( cf. [46])and the higher Chow groups of toric varieties have been computed recently in [30].Furthermore, we still do not know how to compute even the Grothendieck K-theoryring of a general smooth toric stack.One goal of this paper is to develop general techniques for computing the (inte-gral) higher K-theory of smooth toric stacks. In fact, our results apply to a muchbigger class of stacks than just toric stacks. In particular, these results can beused to describe the higher equivariant K-theory of many spherical varieties. Ourgeneral results are in terms of spectral sequences which we show degenerate invarious cases of interest. This allows us to give explicit description of the higherK-theory of toric stacks.As a consequence of this degeneration of spectral sequences, we show how onecan recover (the integral versions of) and generalize all the previously known com-putations of the Grothendieck group of toric Deligne-Mumford stacks. As furtherapplications of the main results, we completely describe the (integral) higher K-theory of weighted projective spaces. As another application, we give a completedescription of the higher K-theory of toric stack bundles over a smooth base scheme.1.1.
Overview of the main results.
The following is an overview of our mainresults. We shall fix a base field k throughout this text. A scheme in this paperwill mean a separated and reduced scheme of finite type over k . A linear algebraicgroup G over k will mean a smooth and affine group scheme over k . By a closedsubgroup H of an algebraic group G , we shall mean a morphism H → G ofalgebraic groups over k which is a closed immersion of k -schemes. In particular,a closed subgroup of a linear algebraic group will be of the same type and hencesmooth. An algebraic group G will be called diagonalizable if it is a product of asplit torus over k and a finite abelian group of order prime to the characteristicof k . In particular, we shall be dealing with only those tori which are split over k . Unless mentioned otherwise, all products of schemes will be taken over k . A G -scheme will mean a scheme with an action of the algebraic group G .For a G -scheme X , let G G ( X ) (resp. K G ( X )) denote the spectrum of the K-theory of G -equivariant coherent sheaves (resp. vector bundles) on X . Let R ( G )denote the representation ring of G . This is canonically identified with K G ( k ). If X denotes an algebraic stack, we let K ( X ) ( G ( X )) denote the Quillen K-theory (G-theory) of the exact category of vector bundles (coherent sheaves, respectively) onthe stack X . For a quotient stack X = [ X/G ], the spectrum K ( X ) (resp. G ( X )) iscanonically weakly equivalent to the equivariant K-theory K G ( X ) (resp. G-theory G G ( X )) of X . See § stacky toric stack X is of the form [ X/G ] where X isa toric variety with dense torus T and G is a diagonalizable group with a givenmorphism φ : G → T .Our first result is the construction of a spectral sequence which allows one tocompute the higher K-theory of the stack [ X/G ] from the K-theory of the stack[
X/T ], whenever a torus T acts on a scheme X and φ : G → T is a morphism ofdiagonalizable groups. This is related to the spectral sequence of Merkurjev ([34,Theorem 5.3]), whose E -terms are expressed in terms of G ∗ ([ X/T ]) and whichconverges to G ∗ ( X ) (See also [32] for related constructions).We also prove the degeneration of our spectral sequences in many cases, whichprovides an efficient tool for computing the higher K-theory of many quotientstacks, including toric stacks. Theorem 1.1.
Let T be a split torus acting on a scheme X and let φ : G → T bea morphism of diagonalizable groups so that G acts on X via φ . Then, there is aspectral sequence: (1.1) E s,t = Tor R ( T ) s ( R ( G ) , G t ([ X/T ])) ⇒ G s + t ([ X/G ]) . IGHER K-THEORY OF TORIC STACKS 3
Moreover, the edge map G ([ X/T ]) ⊗ R ( T ) R ( G ) → G ([ X/G ]) is an isomorphism.The spectral sequence (1.1) degenerates at the E -terms if X is a smooth toricvariety with dense torus T such that K ([ X/T ]) is a projective R ( T ) -module andwe obtain the ring isomorphism: (1.2) K ∗ ([ X/T ]) ⊗ R ( T ) R ( G ) ∼ = −→ K ∗ ([ X/G ]) . In particular, this isomorphism holds when X is a smooth and projective toricvariety. If X = [ X/G ] is a generically stacky toric stack associated to the data X =( X, G φ −→ T ), then the above results apply to the G-theory and K-theory of X .We shall apply Theorem 1.1 in Subsection 4.1 to give an explicit presentationof the Grothendieck K-theory ring of a smooth toric stack. If we specialize tothe case of smooth toric Deligne-Mumford stacks, this recovers the main result ofBorisov–Horja [5].Another useful application of Theorem 1.1 is that it tells us how we can readoff the T ′ -equivariant G-groups of a T -scheme X in terms of its T -equivariant G-groups, whenever T ′ is a closed subgroup of T . The special case of the isomorphism G ([ X/T ]) ⊗ R ( T ) R ( G ) ∼ = −→ G ([ X/G ]) when G is the trivial group and X is a smoothtoric variety, recovers the main result of [35].We should also observe that Theorem 1.1 applies to a bigger class of schemes thanjust toric varieties. In particular, one can use them to compute the equivariant K -theory of many spherical varieties. Another special case of the isomorphism G ([ X/T ]) ⊗ R ( T ) R ( G ) ∼ = −→ G ([ X/G ]) when G is the trivial group and X is a sphericalvariety, recovers the main result of [42]. Theorem 1.2.
Let T be a split torus acting on a smooth and projective scheme X which is T -equivariantly linear ( cf. Definition 3.1). Let φ : G → T be a morphismof diagonalizable groups so that G acts on X via φ . Then the map (1.3) ρ : K ([ X/G ]) ⊗ Z K ∗ ( k ) ∼ = K ([ X/G ]) ⊗ R ( G ) K G ∗ ( k ) → K ∗ ([ X/G ]) is a ring isomorphism. It turns out that all smooth and projective spherical varieties ( cf. § X is a smooth projective toric varietyand G = T , then the above theorem recovers a result of Vezzosi–Vistoli ([46,Theorem 6.9]).As an illustration of how the spectral sequence (1.1) degenerates in the cases notcovered by Theorems 1.1 and 1.2, we prove the following result which describesthe higher K-theory of toric stack bundles. Theorem 1.3.
Let B be a smooth scheme over a perfect field k and let [ X/G ] be a toric stack where X is smooth and projective. Let R G ( K ∗ ( B ) , ∆) denotethe Stanley-Reisner algebra ( cf. Definition 7.1) over K ∗ ( B ) associated to a closedsubgroup G of T . Let π : X → B be a toric stack bundle with fiber [ X/G ] . Thenthere is a ring isomorphism (1.4) Φ G : R G ( K ∗ ( B ) , ∆) ∼ = −→ K ∗ ( X ) . ROY JOSHUA AND AMALENDU KRISHNA
When G is the trivial group, the Grothendieck group K ( X ), was computed in[38, Theorem 1.2(iii)]. When [ X/G ] is a Deligne-Mumford stack, a computation of K ( X ) appears in [26].The focus of this paper was to describe the higher K-theory of toric stacks.Similar description of the motivic cohomology (higher Chow groups) of such stackswill appear in [28].Here is an outline of the paper . The second section is a review of toric stacksand their K-theory. In §
3, we define the notion of equivariantly linear schemesand study their G-theory. We prove Theorem 1.1 in §
4, which is the most generalresult of this paper. We conclude this section with a detailed description of theGrothendieck K-theory ring of general smooth toric stacks.In §
5, we prove a derived K¨unneth formula and prove Theorem 1.2 as a conse-quence. We conclude this section by working out the higher K-theory of (stacky)weighted projective spaces. We study the K-theory of toric stack bundles oversmooth base schemes in the last two sections and conclude by providing a com-plete determination of these.2.
A review of toric stacks and their K-theory
In this section, we review the concept of toric stacks from [16] and set up thenotations for the G-theory and K-theory of such stacks. This is done in some detailfor the convenience of the reader.In what follows, we shall fix a base field k and all schemes and algebraic groupswill be defined over k . Let V k denote the category of k -schemes and let V Sk denotethe full subcategory of smooth k -schemes. If G is an algebraic group over k , weshall denote the category of G -schemes with G -equivariant maps by V G . The fullsubcategory of smooth G -schemes will be denoted by V SG .2.1. Toric stacks.Definition 2.1.
Let T be a torus and let X be a toric variety with dense torus T .According to [16], a toric stack X is an Artin stack of the form [ X/G ] where G isa subgroup of T .A generically stacky toric stack is an Artin stack of the form [ X/G ] where G isa diagonalizable group with a morphism φ : G → T . In this case, the stack X hasan open substack of the form [ T /G ] which acts on it. The action of T = [ T /G ] on X is induced from the torus action on X . The stack T is often called the stacky dense torus of X . A generically stacky toric stack [ X/G ] as above will often bedescribed by the data X = ( X, G φ −→ T ). Examples 2.2.
Generically stacky toric stacks arise naturally while one studiestoric stacks. This is because a toric variety X with dense torus T has many T -invariant subvarieties which are toric varieties and whose dense tori are quotientsof T . If Z ( X is such a subvariety, and G is a diagonalizable subgroup of thetorus T , then [ Z/G ] is not a toric stack but only a generically stacky toric stack.A (generically stacky) toric stack X is called a toric Deligne-Mumford stack if itis a Deligne-Mumford stack after forgetting the toric structure. It is called smoothif X is a smooth scheme. As pointed out in the introduction, Deligne-Mumfordtoric stacks were introduced for the first time in [4] using the notion of stacky fans.A geometric description of the stacks considered in [4] was given in [12] wheremany nice properties of such stacks were proven. It turns out that all these stacksare special cases of the ones defined above.One extreme case of a toric stack is when G is the trivial group, in which case X is just a toric variety. The other extreme case is when G is all of T : clearly suchtoric stacks are Artin. Toric stacks of this form were considered before by Lafforgue[31]. In general, a toric stack occupies a place between these two extreme cases. If IGHER K-THEORY OF TORIC STACKS 5 X is a toric Deligne-Mumford stack, then the stacky torus T is of the form T ′ × B µ ,where T ′ is a torus and B µ is the classifying stack of a finite abelian group µ .In general, every generically stacky toric stack can be written in the form X × B G ,where X is a toric stack and B G is the classifying stack of a diagonalizable group G . This decomposition often reduces the study of the cohomology theories ofgenerically stacky toric stacks to the study the cohomology theories of toric stacksand the classifying stacks of diagonalizable groups.The coarse moduli space π : X → X of a Deligne-Mumford toric stack is asimplicial toric variety whose dense torus is the moduli space of T . Conversely,every simplicial toric variety is the coarse moduli space of a canonically definedtoric Deligne-Mumford stack ( cf. [12, § Toric stacks via stacky fans.
In [16], Geraschenko and Satriano showedthat all (generically stacky) toric stacks are obtained from stacky fans in muchthe same way toric varieties are obtained from fans. They describe in detail thedictionary between toric stacks and stacky fans.Associated to the toric variety X is a fan Σ on the lattice of 1-parameter sub-groups of T , L = Hom gp ( G m , T ) (see [14, § § T → T /G corresponds to the homomorphism of lattices of 1-parameter subgroups, β : L → N = Hom gp ( G m , T /G ). The dual homomorphism, β ∗ : hom( N, Z ) → hom( L, Z ), is the induced homomorphism of characters. Since T → T /G is surjec-tive, β ∗ is injective, and the image of β has finite index. Therefore, one may definea stacky fan as a pair (Σ , β ), where Σ is a fan on a lattice L , and β : L → N is ahomomorphism to a lattice N such that β ( L ) has finite index in N . Conversely,any stacky fan (Σ , β ) gives rise to a toric stack as follows.Let X Σ be the toric variety associated to Σ. The dual of β , β ∗ : N ∨ → L ∨ ,induces a homomorphism of tori T β : T L → T N , naturally identifying β with theinduced map on lattices of 1-parameter subgroups. Since β ( L ) is of finite index in N , β ∗ is injective, so T β is surjective. Let G β = ker( T β ). Note that T L is the torusof X Σ and G β ⊆ T L is a subgroup. If (Σ , β ) is a stacky fan, the associated toricstack X Σ ,β is defined to be [ X Σ /G β ], with the torus T N = T L /G β .A generically stacky fan is a pair (Σ , β ), where Σ is a fan on a lattice L , and β : L → N is a homomorphism to a finitely generated abelian group. If (Σ , β ) is agenerically stacky fan, the associated generically stacky toric stack X Σ ,β is definedto be [ X Σ /G β ], where the action of G β on X Σ is induced by the homomorphism G β → D ( L ∗ ) = T L .One can give a more explicit description of X Σ ,β considered above which willshow that it is a generically stacky toric stack. Let (Σ , β : L → N ) be a genericallystacky fan and let C ( β ) denote the complex L β −→ N . Let Z s Q −→ Z r → N → N , and let B : L → Z r be a lift of β (which exists). One definesthe fan Σ ′ on L ⊕ Z s as follows. Let τ be the cone generated by e , . . . , e s ∈ Z s .For each σ ∈ Σ, let σ ′ be the cone spanned by σ and τ in L ⊕ Z s . Let Σ ′ bethe fan generated by all the σ ′ . Corresponding to the cone τ , we have the closedsubvariety Y ⊆ X Σ ′ , which is isomorphic to X Σ since Σ is the star (sometimescalled the link ) of τ [7, Proposition 3.2.7]. One defines β ′ = B ⊕ Q : L ⊕ Z s / / Z r ( l, a ) ✤ / / B ( l ) + Q ( a ) . Then (Σ ′ , β ′ ) is a generically stacky fan and we see that X Σ ,β ∼ = [ Y /G β ′ ]. Note that C ( β ′ ) is quasi-isomorphic to C ( β ), so G β ′ ∼ = G β . ROY JOSHUA AND AMALENDU KRISHNA
Toric stacks and generically stacky toric stacks arise naturally, especially in thesolution of certain moduli problems. Any toric variety naturally gives rise to atoric stack. In fact, it is shown in [17, Theorem 6.1] that if k is algebraically closedfield of characteristic zero, then every Artin stack with a dense open torus substackis a toric stack under certain fairly general conditions. We refer the readers to [16]where many examples of toric and generically stacky toric stacks are discussed. In the rest of this paper, a toric stack will always mean any generically stackytoric stack. A toric stack as in Definition 2.1 will be called a reduced toric stackor a toric orbifold .2.3.
K-theory of quotient stacks.
Let G be a linear algebraic group acting ona scheme X . The spectrum of the K-theory of G -equivariant coherent sheaves(resp. vector bundles) on X is denoted by G G ( X ) (resp. K G ( X )). We will let K G denote K G ( Spec k ). The direct sum of the homotopy groups of these spectraare denoted by G G ∗ ( X ) and K G ∗ ( X ). The latter is a graded ring. The natural map K G ( X ) → G G ( X ) is a weak equivalence if X is smooth. For a quotient stack X of the form [ X/G ], one writes K G ( X ) and K ( X ) interchangeably. The ring K G ( k )will be denoted by R ( G ). This is same as the representation ring of G .The functor X G G ( X ) on V G is covariant for proper maps and contravariantfor flat maps. It also satisfies the localization sequence and the projection formula.It satisfies the homotopy invariance property in the sense that if f : V → X isa G -equivariant vector bundle, then the map f ∗ : G G ( X ) → G G ( V ) is a weakequivalence. The functor X K G ( X ) on V G is a contravariant functor withvalues in commutative graded rings. For any G -equivariant morphism f : X → Y , G G ( X ) is a module spectrum over the ring spectrum K G ( Y ). In particular, G G ∗ ( X )is an R ( G )-module. We refer to [43, §
1] to verify the above properties.3.
Equivariant G-theory of linear schemes
We will prove Theorem 1.1 as a consequence of a more general result (The-orem 4.1) on the equivariant G-theory of schemes with a group action. In thissection, we study the equivariant G-theory of a certain class of schemes which wecall equivariantly linear. Such schemes in the non-equivariant set-up were earlierconsidered by Jannsen [23] and Totaro [45]. The G-theory of such schemes in thenon-equivariant set-up was studied in [27]. We end this section with a proof ofTheorem 4.1 for equivariantly linear schemes.
Definition 3.1.
Let G be a linear algebraic group over k and let X ∈ V G .(1) We will say X is G -equivariantly 0-linear if it is either empty or isomorphicto Spec (Sym( V ∗ )) where V is a finite-dimensional rational representationof G .(2) For a positive integer n , we will say that X is G -equivariantly n -linearif there exists a family of objects { U, Y, Z } in V G such that Z ⊆ Y is a G -invariant closed immersion with U its complement, Z and one of theschemes U or Y are G -equivariantly ( n − X is the othermember of the family { U, Y, Z } .(3) We will say that X is G -equivariantly linear (or simply, G -linear) if it is G -equivariantly n -linear for some n ≥ G → G ′ is a morphism ofalgebraic groups then every G ′ -equivariantly linear scheme is also G -equivariantlylinear. Definition 3.2.
Let G be a linear algebraic group over k . A scheme X ∈ V G iscalled G -equivariantly cellular (or, G -cellular) if there is a filtration ∅ = X n +1 ( X n ( · · · ( X ( X = X IGHER K-THEORY OF TORIC STACKS 7 by G -invariant closed subschemes such that each X i \ X i +1 is isomorphic to arational representation V i of G . These representations of G are called the (affine) G - cells of X .It is obvious that a G -equivariantly cellular scheme is cellular in the usual sense( cf. [13, Example 1.9.1]).Before we collect examples of equivariantly linear schemes, we state the followingtwo elementary results which will be used throughout this paper. Lemma 3.3.
Let G be a diagonalizable group over k and let H ⊆ G be a closedsubgroup. Then H is also defined over k and is diagonalizable. If T is a split torusover k , then all subtori and quotients of T are defined over k and are split over k .Proof. The first statement follows from [3, Proposition 8.2]. If T is a split torusover k , then any of its subgroups is defined over k and is split by the first assertion.In particular, all quotients of T are defined over k . Furthermore, all such quotientsare split over k by [3, Corollary 8.2]. (cid:3) Lemma 3.4.
Let T be a split torus acting on a scheme X with finitely many orbits.Then: (1) Any T -orbit in X of minimal dimension is closed. (2) Any T -orbit in X of maximal dimension is open.Proof. The first assertion is well known and can be found in [3, Proposition 1.8].We prove the second assertion.Let f : S → X be the inertia group scheme over X for the T -action and let γ : X → S denote the unit section. Then for a point x ∈ X , the fiber S x of themap f is the stabilizer subgroup of x and the dimension of S x is its dimension atthe point γ ( x ). For any s ≥
0, let X ≤ s denote the set of points x ∈ X such thatdim( S x ) ≤ s . It follows from Chevalley’s theorem ( cf. [18, § X ≤ s is open in X (see also [46, § U ⊆ X denote a T -orbit of maximal dimension (say, d ) and let x ∈ U .Suppose s = dim( S x ) ≥
0. Notice that all points in a T -orbit have the samestabilizer subgroup because T is abelian. We claim that there is no point on X whose stabilizer subgroup has dimension less than s . If there is such a point y ∈ X , then T y is a T -orbit of X of dimension bigger than the dimension of U ,contradicting our choice of U . This proves the claim.It follows from this claim that X ≤ s is a T -invariant open subscheme of X whichis a disjoint union of its T -orbits such that the stabilizer subgroups of all pointsof X ≤ s have dimension s . In particular, all T -orbits in X ≤ s have dimension d . Weconclude from the first assertion of the lemma that all T -orbits (of closed points)in X ≤ s (including U ) are closed in X ≤ s . Since there only finitely many orbits in X , the same is true for X ≤ s . We conclude that X ≤ s is a finite disjoint union ofits closed orbits. Hence these orbits must also be open in X ≤ s . In particular, U isopen in X ≤ s and hence in X . (cid:3) Remark 3.5.
The reader can verify that Lemma 3.4 is true for the action of anydiagonalizable group. But we do not need this general case.The following result yields many examples of equivariantly linear schemes.
Proposition 3.6.
Let T be a split torus over k and let T ′ be a quotient of T . Let T act on T ′ via the quotient map. Then the following hold. (1) T ′ is T -linear. (2) A toric variety with dense torus T is T -linear. (3) A T -cellular scheme is T -linear. (4) If k is algebraically closed, then every T -scheme with finitely many T -orbitsis T -linear. ROY JOSHUA AND AMALENDU KRISHNA
Proof.
We first prove (1). It follows from Lemma 3.3 that T ′ is a split torus. Hence,it is enough to show using the remark following the definition of T -linear schemesthat a split torus T is T -linear under the multiplication action.We can write T = ( G m ) n and consider A n as the toric variety with the densetorus T via the coordinate-wise multiplication so that the complement of T is theunion of the coordinate hyperplanes. Since A n is T -linear, it suffices to show thatthe union of the coordinate hyperplanes is T -linear.We shall prove by induction on the rank of T that any union of the coordinatehyperplanes in A n is T -linear. If n = 1, then this is obvious. So let us assumethat n > Y be a union of some coordinate hyperplanes in A n . Afterpermuting the coordinates, we can write Y as Y n { , ··· ,m } = H ∪ · · · ∪ H m where H i = { ( x , · · · , x n ) ∈ A n | x i = 0 } . If m = 1, then Y n { } is T -equivariantly 0-linear.So we assume by an induction on m that Y n { , ··· ,m } is T -linear.Set U = Y n { , ··· ,m } \ Y n { , ··· ,m } . Then U is the complement of a union of hyperplanes W n − { , ··· ,m } in H ∼ = A n − . Notice that T acts on H through the product T of itslast ( n −
1) factors. By induction on n , we conclude that W n − { , ··· ,m } is T -linear.Since H is clearly T -linear, we conclude that U is T -linear and hence T -linear.Thus we have concluded that both Y n { , ··· ,m } and U are T -linear. It follows fromthis that Y n { , ··· ,m } is T -linear too.The assertion (2) easily follows from (1) and an induction on the number of T -orbits in a toric variety. The assertion (3) is immediate from the definitions, usingan induction on the length of the filtration of a T -cellular scheme. To prove (4), let X be a T -scheme with only finitely many T -orbits. It follows from Lemma 3.4 that X has an open T -orbit U . Since k is algebraically closed, such an open T -orbitmust be isomorphic to a quotient of T . In particular, it is T -linear by the firstassertion. An induction of the number of T -orbit implies that X \ U is T -linear.We conclude that X is also T -linear. (cid:3) Spherical varieties.
Recall that if G is a connected reductive group over k ,then a normal variety X ∈ V G is called spherical if a Borel subgroup of G has adense open orbit in X . The spherical varieties constitute a large class of varietieswith group actions, including toric varieties, flag varieties and all symmetric vari-eties. It is known that a spherical variety X has only finitely many fixed pointsfor the T -action where T is a maximal torus of G contained in B .It follows from a theorem of Bialynicki-Birula [2] (generalized to the case of non-algebraically closed fields by Hesselink [20]) that if T is a split torus over k and if X is a smooth projective variety with a T -action such that the fixed point locus X T is isolated, then X is T -equivariantly cellular. We conclude that a smooth andprojective spherical variety is T -cellular and hence T -linear. We do not know if allspherical varieties are T -linear.3.1. Equivariant G-theory of equivariantly linear schemes.
Recall that ifa linear algebraic group G acts on a scheme X , then the G-theory and K-theoryof the quotient stack [ X/G ] are same as the equivariant G-theory and K-theory of X for the action G . We shall use this identification throughout this text withoutfurther mention.The following result from [43, § Theorem 3.7.
Let G be a linear algebraic group over k and let H ⊆ G be a closedsubgroup of G . Then for any X ∈ V H , the map of spectra G ([( X H × G ) /G ]) → G ([ X/H ]) is a weak-equivalence. In particular, the map of spectra G ([( X × G/H ) /G ]) → G ([ X/H ]) IGHER K-THEORY OF TORIC STACKS 9 is a weak-equivalence if X ∈ V G . These are weak equivalences of ring spectra if X is smooth. Recall that for a stack X , the K-theory spectrum K ( X ) is a ring spectrum and G ( X ) is a module spectrum over K ( X ). In the following results, we make essentialuse of the derived smash products of module spectra over ring spectra. This isthe derived functor of the smash product of spectra in their homotopy category.We refer to [39] (see also [11] and [27, § R is a ring spectrum and M, N are module spectra over R , the derivedsmash product of M and N over R will be denoted by M L ∧ R N . We shall now provethe following special case of Theorem 4.1. The proof follows a trick used in [27,Theorem 4.1] in a different context. Proposition 3.8.
Let T be a split torus and let X ∈ V T be T -linear. Let φ : G → T be a morphism of diagonalizable groups such that G acts on X via φ . Then thenatural map of spectra (3.1) K ([Spec ( k ) /G ]) L ∧ K ([Spec ( k ) /T ]) G ([ X/T ]) → G ([ X/G ]) is a weak-equivalence.Proof. We assume that X is T -equivariantly n -linear for some n ≥
0. We shallprove our result by an ascending induction on n . If n = 0, then X ∼ = A n andhence by the homotopy invariance, we can assume that X = Spec ( k ), and theresult is immediate in this case. We now assume that n >
0. By the definition of T -linearity, there are two cases to consider:(1) There exists a T -invariant closed subscheme Y of X with complement U such that Y and U are T -equivariantly ( n − T -scheme Z which contains X as a T -invariant open sub-scheme such that Z and Y = Z \ X are T -equivariantly ( n − : K G L ∧ K T G ([ Y /T ]) / / (cid:15) (cid:15) K G L ∧ K T G ([ X/T ]) / / (cid:15) (cid:15) K G L ∧ K T G ([ U/T ]) (cid:15) (cid:15) G ([ Y /G ]) / / G ([ X/G ]) / / G ([ U/G ]) . The left and the right vertical maps are weak equivalences by the induction. Weconclude that the middle vertical map is a weak equivalence too.In the second case, we obtain as before, a commutative diagram of fiber sequencesin the homotopy category of spectra: K G L ∧ K T G ([ Y /T ]) / / (cid:15) (cid:15) K G L ∧ K T G ([ Z/T ]) / / (cid:15) (cid:15) K G L ∧ K T G ([ X/T ]) (cid:15) (cid:15) G ([ Y /G ]) / / G ([ Z/G ]) / / G ([ X/G ]) . The first two vertical maps are weak equivalences by induction and hence the lastvertical map must also be a weak equivalence. This completes the proof of theproposition. (cid:3) For spectra, this is same as a cofiber sequence
We end this section with the following (rather technical) result which will beused in the proof of Theorem 4.1. Taking V to be Spec ( k ), this becomes a specialcase of what is considered in the last Proposition. Lemma 3.9.
Let T be a split torus over k and let T ′ be a quotient of T . Let T acton T ′ via the quotient map and let it act trivially on an affine scheme V . Considerthe scheme X = V × T ′ where T acts diagonally. Let φ : G → T be a morphismof diagonalizable groups such that G acts on any T -scheme via φ . Then the mapof spectra (3.2) K ([Spec ( k ) /G ]) L ∧ K ([Spec ( k ) /T ]) G ([ X/T ]) → G ([ X/G ]) is a weak-equivalence.Proof. Let H denote the image of G in T ′ under the composite map G φ −→ T ։ T ′ and let H ′ = T ′ /H . Notice that T ′ is a split torus by Lemma 3.3. Since T (andhence G ) acts trivially on the scheme V , it follows that T and G act on X via theirquotients T ′ and H , respectively. Since X is affine and all the underlying groupsare diagonalizable, it follows from [43, Lemma 5.6] that the maps of spectra(3.3) G ([ X/T ′ ]) L ∧ K T ′ K T → G ([ X/T ]); G ([ X/H ]) L ∧ K H K G → G ([ X/G ])are weak equivalences. Using the first weak equivalence, we obtain(3.4) G ([ X/T ]) L ∧ K T K G ∼ = (cid:18) G ([ X/T ′ ]) L ∧ K T ′ K T (cid:19) L ∧ K T K G ∼ = G ([ X/T ′ ]) L ∧ K T ′ K G ∼ = G ([ X/T ′ ]) L ∧ K T ′ (cid:18) K H L ∧ K H K G (cid:19) ∼ = (cid:18) G ([ X/T ′ ]) L ∧ K T ′ K H (cid:19) L ∧ K H K G . On the other hand, we have(3.5) G ([ X/T ′ ]) L ∧ K T ′ K H ∼ = G ([ X/T ′ ]) L ∧ K T ′ G ([ H ′ /T ′ ]) ∼ = G ([( X × H ′ ) /T ′ ]) ∼ = G ([ X/H ]) , where the isomorphisms ∼ = and ∼ = follow from Theorem 3.7. The isomorphism ∼ = follows from Propositions 3.6 and 5.1. Combining (3.3), (3.4) and (3.5), we getthe weak equivalences G ([ X/T ]) L ∧ K T K G ∼ = G ([ X/H ]) L ∧ K H K G ∼ = G ([ X/G ])and this proves the lemma. (cid:3)
IGHER K-THEORY OF TORIC STACKS 11 G-theory of general toric stacks
This section is devoted to the determination of the G-theory of a general (gener-ically stacky) toric stack. We prove our main results in a much more general set-upwhere the underlying scheme with a T -action need not be a toric variety.Our first result is a spectral sequence that computes the G -equivariant G-theoryof a T -scheme X in terms of its T -equivariant G-theory and the representation ringof G whenever there is a morphism of diagonalizable groups φ : G → T . Whenthe underlying scheme is assumed to be smooth, these conclusions may be statedin terms of K-theory instead of G-theory.This result specializes to the case of all (generically stacky) toric stacks when X is assumed to be a toric variety. We conclude this section with an explicitpresentation of the Grothendieck K-theory ring of a smooth toric stack which maynot necessarily be Deligne-Mumford.We now prove the following main result of this section and derive its conse-quences. Theorem 4.1.
Let T be a split torus acting on a scheme X and let φ : G → T be a morphism of diagonalizable groups such that G acts on X via φ . Then thenatural map of spectra (4.1) K ([Spec ( k ) /G ]) L ∧ K ([Spec ( k ) /T ]) G ([ X/T ]) → G ([ X/G ]) is a weak-equivalence. In particular, one obtains a spectral sequence: (4.2) E s,t = Tor K T ∗ ( k ) s,t ( K G ∗ ( k ) , G ∗ ([ X/T ])) ⇒ G s + t ([ X/G ]) . Proof.
We shall prove the theorem by the noetherian induction on T -schemes. Thestatement of the theorem is obvious if X is the empty scheme so that both sidesof (4.1) are contractible. Suppose X is any T -scheme such that (4.1) holds when X is replaced by all its proper T -invariant closed subschemes. We show that (4.1)holds for X . This will prove the theorem.By Thomason’s generic slice theorem [44, Proposition 4.10], there exists a T -invariant dense open subset U ⊆ X which is affine. Moreover, T acts on U via itsquotient T ′ which in turn acts freely on U with affine geometric quotient U/T suchthat there is a T -equivariant isomorphism U ∼ = ( U/T ) × T ′ . Here, T acts triviallyon U/T , via the quotient map on T ′ and diagonally on U . The weak equivalenceof (4.1) holds for U by Lemma 3.9.We now set Y = X \ U . Then Y is a proper T -invariant closed subscheme of X .The localization sequence induces the commutative diagram of the fiber sequencesin the homotopy category of spectra: K G L ∧ K T G ([ Y /T ]) / / (cid:15) (cid:15) K G L ∧ K T G ([ X/T ]) / / (cid:15) (cid:15) K G L ∧ K T G ([ U/T ]) (cid:15) (cid:15) G ([ Y /G ]) / / G ([ X/G ]) / / G ([ U/G ]) . We have shown above that the right vertical map is a weak equivalence. The leftvertical map is a weak equivalence by the noetherian induction. We conclude thatthe middle vertical map is a weak equivalence too.The existence of the spectral sequence now follows along standard lines (see forexample, [11, Theorem IV.4.1]). (cid:3)
Proof of Theorem 1.1:
To obtain the spectral sequence (1.1), it is enough toidentify this spectral sequence with the one in (4.2).
To see this, we recall from [43, Lemma 5.6] that the maps R ( T ) ⊗ Z K ∗ ( k ) → K ∗ ([Spec ( k ) /T ]) and R ( G ) ⊗ Z K ∗ ( k ) → K ∗ ([Spec ( k ) /G ]) are ring isomorphisms.Since R ( T ) and R ( G ) are flat Z -modules, these isomorphisms can be written as(4.3) R ( T ) L ⊗ Z K ∗ ( k ) ∼ = −→ K ∗ ([Spec ( k ) /T ]) and R ( G ) L ⊗ Z K ∗ ( k ) ∼ = −→ K ∗ ([Spec ( k ) /G ]) , where L ⊗ denotes the derived tensor product.Let M • ∼ −→ R ( G ) be a flat resolution of R ( G ) as an R ( T )-module. Since R ( T )is a flat Z -module, we see that M • ∼ −→ R ( G ) is a flat resolution of R ( G ) also as a Z -module. In particular, we obtain(4.4) K G ∗ ( k ) L ⊗ K T ∗ ( k ) G ∗ ([ X/T ]) ∼ = (cid:18) R ( G ) L ⊗ Z K ∗ ( k ) (cid:19) L ⊗ R ( T ) ⊗ Z K ∗ ( k ) G ∗ ([ X/T ]) ∼ = (cid:18) M • L ⊗ Z K ∗ ( k ) (cid:19) L ⊗ R ( T ) ⊗ Z K ∗ ( k ) G ∗ ([ X/T ]) ∼ = (cid:18) M • ⊗ Z K ∗ ( k ) (cid:19) L ⊗ R ( T ) ⊗ Z K ∗ ( k ) G ∗ ([ X/T ]) ∼ = (cid:18) M • ⊗ Z K ∗ ( k ) (cid:19) ⊗ R ( T ) ⊗ Z K ∗ ( k ) G ∗ ([ X/T ]) ∼ = M • ⊗ R ( T ) (cid:18) R ( T ) ⊗ Z K ∗ ( k ) (cid:19) ⊗ R ( T ) ⊗ Z K ∗ ( k ) G ∗ ([ X/T ]) ∼ = M • ⊗ R ( T ) G ∗ ([ X/T ]) ∼ = M • L ⊗ R ( T ) G ∗ ([ X/T ]) ∼ = R ( G ) L ⊗ R ( T ) G ∗ ([ X/T ]) , where the isomorphism ∼ = follows because M • is a complex of flat Z -modules, ∼ = follows because M • ⊗ Z K ∗ ( k ) is a complex of flat R ( T ) ⊗ Z K ∗ ( k )-modules and theisomorphism ∼ = follows because M • is a complex of flat R ( T )-modules. Takingthe homology groups on the both sides, we obtainTor K T ∗ ( k ) s,t ( K G ∗ ( k ) , G ∗ ([ X/T ])) ∼ = Tor R ( T ) s,t ( R ( G ) , G ∗ ([ X/T ]))which yields the spectral sequence (1.1). The isomorphism of the edge map G ([ X/T ]) ⊗ R ( T ) R ( G ) → G ([ X/G ]) follows immediately from (1.1) and the factthat the equivariant G-theory spectra appearing in Theorem 1.1 are all connected(have no negative homotopy groups).Let us now assume that X is a smooth toric variety with dense torus T suchthat K ([ X/T ]) is a projective R ( T )-module. In this case, we can identify the G-theory and the K-theory. To show the degeneration of the spectral sequence (1.1),it suffices to show that the map(4.5) G ∗ ([ X/T ]) ⊗ R ( T ) R ( G ) → G ∗ ([ X/T ]) L ⊗ R ( T ) R ( G ) IGHER K-THEORY OF TORIC STACKS 13 is an isomorphism. However, we have G ∗ ([ X/T ]) L ⊗ R ( T ) R ( G ) ∼ = (cid:18) K T ∗ ( k ) ⊗ R ( T ) G ([ X/T ]) (cid:19) L ⊗ R ( T ) R ( G ) ∼ = (cid:18) K T ∗ ( k ) L ⊗ R ( T ) G ([ X/T ]) (cid:19) L ⊗ R ( T ) R ( G ) ∼ = (cid:18) K ∗ ( k ) L ⊗ Z R ( T ) (cid:19) L ⊗ R ( T ) (cid:18) G ([ X/T ]) L ⊗ R ( T ) R ( G ) (cid:19) ∼ = (cid:18) K ∗ ( k ) L ⊗ Z R ( T ) (cid:19) L ⊗ R ( T ) (cid:18) G ([ X/T ]) ⊗ R ( T ) R ( G ) (cid:19) ∼ = K ∗ ( k ) L ⊗ Z (cid:18) G ([ X/T ]) ⊗ R ( T ) R ( G ) (cid:19) ∼ = K ∗ ( k ) ⊗ Z (cid:18) G ([ X/T ]) ⊗ R ( T ) R ( G ) (cid:19) ∼ = (cid:18) K ∗ ( k ) ⊗ Z G ([ X/T ]) (cid:19) ⊗ R ( T ) R ( G ) ∼ = G ∗ ([ X/T ]) ⊗ R ( T ) R ( G ) . The isomorphism ∼ = follows from [46, Proposition 6.4] in general and also fromTheorem 1.2 when X is projective. The isomorphism ∼ = follows because G ([ X/T ])is projective R ( T )-module. The isomorphism ∼ = follows from [43, Lemma 5.6]because R ( T ) is flat Z -module. The isomorphism ∼ = follows again from theprojectivity of G ([ X/T ]) as an R ( T )-module. The isomorphisms ∼ = and ∼ = are the associativity of the ordinary and derived tensor products. The isomor-phism ∼ = follows because R ( G ) is a free Z -module and G ([ X/T ]) ⊗ R ( T ) R ( G ) is aprojective R ( G )-module and hence is flat as a Z -module. The isomorphism ∼ = follows again from [46, Proposition 6.4] in general and also from Theorem 1.2when X is projective. This proves (4.5). The projectivity of G ([ X/T ]) as R ( T )-module when X is a smooth and projective toric variety, is shown in [46, Propo-sition 6.9] (see also Lemma 6.1). The proof of Theorem 1.1 is now complete. (cid:3) Remark 4.2.
The spectral sequence (1.1) is basically an Eilenberg-Moore typespectral sequence. A spectral sequence similar to the one in (1.1) had been con-structed by Merkurjev [34] in the special case when G is the trivial group. Theconstruction of that spectral sequence is considerably more involved. This specialcase ( G = { e } ) of the above construction yields a completely different and simplerproof of Merkurjev’s theorem in the setting of schemes with the action of split tori. Remark 4.3.
It was shown by Baggio [1] that there are examples of non-projectivesmooth toric varieties X such that G ([ X/T ]) is a projective R ( T )-module. Thisshows that there are smooth non-projective toric varieties for which the spectralsequence in Theorem 1.1 degenerates. In all these cases, one obtains a completedescription of the K-theory of the toric stack [ X/G ]. We shall see in Section 5.2that there are examples where the spectral sequence of Theorem 1.1 degenerateseven if G ([ X/T ]) is not a projective R ( T )-module.4.1. Grothendieck group of toric stacks.
In [5], Borisov and Horja had com-puted the Grothendieck K -theory ring K ([ X/G ]) when [
X/G ] is a smooth toricDeligne-Mumford stack. Recall from § T ′ × B µ where T ′ is a torus and µ is a finite abeliangroup. The following consequence of Theorem 1.1 generalizes the result of [5] to the case of all smooth toric stacks, not necessarily Deligne-Mumford. Even in thislatter case, we obtain a simpler proof. Theorem 4.4.
Let X = [ X/G ] be a smooth and reduced toric stack associated tothe data X = ( X, G φ −→ T ) . Let ∆ be the fan defining X and let d be the number ofrays in ∆ . Let I G ∆ denote the ideal of the Laurent polynomial algebra Z [ t ± , · · · , t ± d ] generated by the relations: (1) ( t j − · · · ( t j l − , ≤ j p ≤ d such that the rays ρ j , · · · , ρ j l do not spana cone of ∆ . (2) d Q j =1 ( t j ) < − χ,v j > ! − , χ ∈ ( T /G ) ∨ .Then there is a ring isomorphism (4.6) φ : Z [ t ± , · · · , t ± d ] I G ∆ ∼ = −→ K ( X ) . Proof.
It follows from Theorem 1.1 that the map K ([ X/T ]) ⊗ R ( T ) R ( G ) ∼ = −→ K ([ X/G ])is a ring isomorphism. Since G is a diagonalizable subgroup of T ([ X/G ] is re-duced), the ring R ( G ) is a quotient of R ( T ) by the ideal J G ∆ = ( χ − , χ ∈ ( T /G ) ∨ )( cf. Lemma 7.8). This implies that(4.7) K ([ X/G ]) ∼ = K ([ X/T ]) J G ∆ K ([ X/T ]) . If we let ∆(1) = { ρ , · · · , ρ d } , then for each 1 ≤ j ≤ d , there is a unique T -equivariant line bundle L j on X which has a T -equivariant section s j : X → L j and whose zero locus is the orbit closure V j = O ρ j . Then every character χ ∈ T ∨ acts on K ([ X/T ]) by multiplication with the element ( d Q j =1 ([ L j ]) <χ,v j > ) ( cf. [38,Proposition 4.3]). We conclude that there is a ring isomorphism(4.8) K ([ X/T ]) d Q j =1 ([ L ∨ j ]) < − χ,v j > − , χ ∈ ( T /G ) ∨ ! ∼ = −→ K ([ X/G ]) . If I T ∆ denotes the ideal of Z [ t ± , · · · , t ± d ] generated by the relations (1) above,then it follows from [46, Theorem 6.4] that there is a ring isomorphism(4.9) Z [ t ± , · · · , t ± d ] I T ∆ ∼ = −→ K ([ X/T ]) . Setting φ ( t j ) = [ L ∨ j ], we obtain the isomorphism (4.6) by combining (4.8)and (4.9). (cid:3) Remark 4.5.
If [
X/G ] is not a reduced stack and there is an exact sequence0 → H → G → F → F = Im( φ ), then the stack [ X/G ] is isomorphic to [
X/F ] × B H . Inthis case, one obtains an isomorphism K ∗ ([ X/G ]) ∼ = K ∗ ([ X/F ]) ⊗ R ( F ) R ( G ) ( cf. [43, Lemma 5.6]). In particular, if H is a torus, one obtains K ∗ ([ X/G ]) ∼ = K ∗ ([ X/F ]) ⊗ Z R ( H ). Thus, we see that the calculation of the K-theory of a (gener-ically stacky) toric stack can be easily reduced to the case of reduced stacks. IGHER K-THEORY OF TORIC STACKS 15 A K¨unneth formula and its consequences
Our goal in this section is to prove Theorem 1.2 and give applications. Weshall deduce this theorem from a K¨unneth spectral sequence for the equivariantK-theory for the action of diagonalizable groups. A similar spectral sequence fortopological K-theory was constructed long time ago by Hodgkin [21] and Snaith[41]. A spectral sequence of this kind in the non-equivariant setting was constructedby the first author in [27, Theorem 4.1].5.1.
K¨unneth formula.
Suppose that X and X ′ are schemes acted upon by alinear algebraic group G . In this case, the flatness of X and X ′ over k implies thatthe spectra G ([ X/G ]) and G ([ X ′ /G ]) are module spectra over the ring spectrum K ([Spec ( k ) /G ]). This flatness also ensures that the external tensor product ofcoherent O -modules induces a pairing G ([ X/G ]) ∧ G ([ X ′ /G ]) → G ([( X × X ′ ) /G ]),where the action of G on X × X ′ is the diagonal action. This pairing is compatiblewith the structure of the above spectra as module spectra over the ring spectrum K ([Spec ( k ) /G ]) so that one obtains the induced pairing: p ∗ ∧ p ∗ : G ([ X/G ]) L ∧ K ([Spec ( k ) /G ]) G ([ X ′ /G ]) → G ([( X × X ′ ) /G ]) . This is a map of ring spectra if X and X ′ are smooth. Proposition 5.1.
Let T be a split torus and let X, X ′ be in V T such that X is T -linear. Let φ : G → T be a morphism of diagonalizable groups such that G actson X and X ′ via φ . Then the natural map of spectra (5.1) G ([ X/G ]) L ∧ K ([Spec ( k ) /G ]) G ([ X ′ /G ]) → G ([( X × X ′ ) /G ]) is a weak-equivalence.In particular, there exists a first quadrant spectral sequence (5.2) E s,t = Tor K G ∗ ( k ) s,t ( G ∗ ([ X/G ]) , G ∗ ([ X ′ /G ])) ⇒ G s + t ([( X × X ′ ) /G ]) . Proof.
We assume that X is T -equivariantly n -linear for some n ≥
0. This propo-sition is proved by an ascending induction on n , along the same lines as the proofof Proposition 3.8. We sketch the argument.If n = 0, then X ∼ = A n and hence by the homotopy invariance, we can assumethat X = Spec ( k ), and the result is immediate in this case. We now assume that n >
0. By the definition of T -linearity, there are two cases to consider:(1) There exists a T -invariant closed subscheme Y of X with complement U such that Y and U are T -equivariantly ( n − T -scheme Z which contains X as a T -invariant open sub-scheme such that Z and Y = Z \ X are T -equivariantly ( n − G ([ X ′ /G ]) L ∧ K G G ([ Y /G ]) / / (cid:15) (cid:15) G ([ X ′ /G ]) L ∧ K G G ([ X/G ]) / / (cid:15) (cid:15) G ([ X ′ /G ]) L ∧ K G G ([ U/G ]) (cid:15) (cid:15) G ([( Y × X ′ ) /G ]) / / G ([( X × X ′ ) /G ]) / / G ([( U × X ′ ) /G ]) . The left and the right vertical maps are weak equivalences by the induction on n .We conclude that the middle vertical map is a weak equivalence. The second caseis proved in the same way where we now use induction on Y and Z (see the proofof Proposition 3.8). The existence of the spectral sequence now follows along standard lines (see forexample, [11, Theorem IV.4.1]). (cid:3)
Remark 5.2.
As an application of Proposition 5.1, one can obtain another proofof the special case of the spectral sequence (1.1) when G is a closed subgroup of T . This is done by taking G = T , X ′ = T /G in (5.2) and using the Morita weakequivalences G ([ X ′ /T ]) ∼ = G ([Spec ( k ) /G ]) and G ([( X × X ′ ) /T ]) ∼ = G ([ X/G ]).Notice that X ′ = T /G is T -linear by Proposition 3.6. Corollary 5.3 (K¨unneth decomposition) . Let T be a split torus over k and let X be a T -linear scheme. Then the class of the diagonal [∆] ∈ G ([( X × X ) /G ]) admits a strong K¨unneth decomposition, i.e., may be written as n Σ i =1 p ∗ ( α i ) ⊗ p ∗ ( β i ) ,where α i , β i ∈ G ([ X/G ]) .Proof. The spectral sequence of Proposition 5.1 shows in general that(5.3) G ([( X × X ′ ) /G ]) ∼ = G ([ X/G ]) ⊗ R ( G ) G ([ X ′ /G ]) . The K¨unneth decomposition now follows by taking X = X ′ . (cid:3) Proof of Theorem 1.2:
Let X be a smooth and projective T -linear scheme.Since the group G is diagonalizable, we apply [43, Lemma 5.6] to obtain the iso-morphism:(5.4) R ( G ) ⊗ Z K ∗ ( k ) ∼ = −→ K G ∗ ( k )and this provides the first isomorphism of (1.3). Since X is smooth, we can identify G ∗ ([ X/G ]) with K ∗ ([ X/G ]).Let [ x ] ∈ K ∗ ([ X/G ]). Then [ x ] = p ∗ (∆ ◦ p ∗ ([ x ])). Now we use the K¨unnethdecomposition for ∆ obtained in Corollary 5.3 and the projection formula (since X is projective) to identify the last term with n P i =1 α i ◦ p ∗ p ∗ ( β i ◦ [ x ]). The Cartesiansquare X × X p / / p (cid:15) (cid:15) X p ′ (cid:15) (cid:15) X p ′ / / Spec ( k )and the flat base-change for the equivariant G-theory show that p ∗ ( p ∗ ( β i ◦ [ x ]))identifies with p ′ ∗ p ′ ∗ ( β i ◦ [ x ]) so that(5.5) [ x ] = n X i =1 α i ◦ p ′ ∗ ( p ′ ∗ ( β i ◦ [ x ])) . The class p ′ ∗ ( β i ◦ [ x ]) ∈ G ([Spec ( k ) /G ]). It follows that the classes { α i } generate π ∗ ( G [ X/G ]) as a module over K G ∗ ( k ). This shows that the map in question issurjective.Next we prove the injectivity of the map ρ . The key is the following diagram:(5.6) K ∗ ([ X/G ]) µ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ K ([ X/G ]) ⊗ K G ( k ) K G ∗ ( k ) ρ o o α (cid:15) (cid:15) Hom K G ( k ) ( K ([ X/G ]) , K G ∗ ( k )) IGHER K-THEORY OF TORIC STACKS 17 where α ( x ⊗ y ) (resp. µ ( x ), x ∈ K ∗ ([ X/G ])) is defined by α ( x ⊗ y ) = the map x ′ f ∗ ( x ′ ◦ x ) ◦ y (resp., the map x ′ f ∗ ( x ′ ◦ x )). Here, f denotes the projectionmap X → Spec ( k ) and x ′ ◦ x denotes the product in the ring K ∗ ([ X/G ]). Thecommutativity of the above diagram is an immediate consequence of the projectionformula: observe that ρ ( x ⊗ y ) = x ◦ f ∗ ( y ). Therefore, to show that ρ is injective,it suffices to show that the map α is injective. For this, we define a map β to bea splitting for α as follows.If φ ∈ Hom K G ( k ) ( K ([ X/G ]) , K G ∗ ( k )), we let β ( φ ) = n P i =1 α i ⊗ ( φ ( β i )). Observethat β ( α ( x ⊗ y )) = β ( the map x ′ → f ∗ ( x ′ ◦ x ) ◦ y )= ( n P i =1 α i ⊗ f ∗ ( β i · x )) ◦ y. We next observe that f ∗ ( β i · x ) ∈ K G ( k ), so that we may write the last termas ( n P i =1 α i .f ∗ f ∗ ( β i · x )) ◦ y . By (5.5), the last term = x ◦ y . This proves that α isinjective and hence that so is ρ . This completes the proof. (cid:3) The following result generalizes (1.2) to a bigger class of schemes.
Corollary 5.4.
Let T be a split torus over k and let X be a smooth and projective T -linear scheme. Let φ : G → T be a morphism of diagonalizable groups such that G acts on X via φ . Then the map K ∗ ([ X/T ]) ⊗ R ( T ) R ( G ) → K ∗ ([ X/G ]) is an isomorphism. In particular, K ([ X/G ]) is a free R ( G ) -module ( and hence afree Z -module ) if X is T -cellular.Proof. To prove the first part of the corollary, we trace through the sequence ofisomorphisms: K ∗ ([ X/T ]) ⊗ R ( T ) R ( G ) ∼ = (cid:18) K T ∗ ( k ) ⊗ R ( T ) K ([ X/T ]) (cid:19) ⊗ R ( T ) R ( G ) ∼ = (cid:18) K ∗ ( k ) ⊗ Z R ( T ) (cid:19) ⊗ R ( T ) (cid:18) K ([ X/T ]) ⊗ R ( T ) R ( G ) (cid:19) ∼ = † K ∗ ( k ) ⊗ Z K ([ X/G ]) ∼ = K ∗ ([ X/G ]) . The first and the last isomorphisms in this sequence follow from Theorem 1.2and the isomorphism ∼ = † follows from Theorem 1.1. This proves the first part ofthe corollary. If X is T -cellular, the freeness of K ([ X/G ]) as an R ( G )-modulefollows from Lemma 6.1. (cid:3) Remark 5.5.
In the special case when [
X/G ] is a smooth toric Deligne-Mumfordstack (with X projective), the freeness of K ([ X/G ]) as Z -module was earliershown in [19, Theorem 2.2] and independently in [15] using symplectic methods.It is known ( cf. [19, Example 4.1]) that the freeness property may fail if X is notprojective.5.2. K-theory of weighted projective spaces.
In the past, there have beenmany attempts to study the K-theory and Chow rings of weighted projectivespaces. However, there are only a few explicit computations in this regard. Weend this section with an explicit description of the integral higher K-theory ofstacky weighted projective spaces. These are examples of toric stacks, where thespectral sequence (1.1) degenerates even though K ([ X/T ]) is not a projective R ( T )-module. We also describe the rational higher G-theory of weighted projec-tive schemes as another application of Theorem 1.2.5.2.1. Weighted projective spaces.
Let q = { q , · · · , q n } be an ordered set of posi-tive integers and let d = gcd ( q , · · · , q n ). This ordered set of positive integers givesrise to a morphism of tori φ : G m → ( G m ) n +1 given by φ ( λ ) = ( λ q , · · · , λ q n ).The (stacky) weighted projective space P ( q , · · · , q n ) is the stack [( A n +1 k \ { } ) / G m ],where G m acts on A n +1 k by λ · ( a , · · · , a n ) = ( λ q a , · · · , λ q n a n ). Notice that A n +1 \{ } is a toric variety with dense torus T = ( G m ) n +1 acting by the coordinate-wise multiplication. We see that P ( q ) is the toric stack associated to the data(( A n +1 k \ { } ) , G m φ −→ T ). It is known that P ( q ) is a Deligne-Mumford toric stackand is reduced (an orbifold) if and only if d = 1.5.2.2. K -theory of P ( q ) . To describe the higher K-theory of P ( q ), we consider A n +1 as the toric variety with dense torus T = ( G m ) n +1 acting by the coordinate-wisemultiplication. Let V be the ( n + 1)-dimensional representation of T which rep-resents A n +1 as the toric variety. Let ι : Spec ( k ) → A n +1 and j : U → A n +1 bethe T -invariant closed and open inclusions, where we set U = A n +1 \ { } . Observethat V is the T -equivariant normal bundle of Spec ( k ) sitting inside A n +1 as theorigin.We have the localization exact sequence:(5.7) · · · → K i ([Spec ( k ) / G m ]) ι ∗ −→ K i ([ A n +1 / G m ]) j ∗ −→ K i ([ U/ G m ]) → · · · . Our first claim is that this sequence splits into short exact sequences(5.8) 0 → K i ([Spec ( k ) / G m ]) ι ∗ −→ K i ([ A n +1 / G m ]) j ∗ −→ K i ([ U/ G m ]) → i ≥ λ − ( V ) = n P i =0 ( − i [ ∧ i ( V )] isnot a zero-divisor in the ring K ∗ ([Spec ( k ) / G m ]). However, we can write V = n ⊕ i =0 V i ,where G m acts on V i ∼ = k by λ · v = λ q i v . Since each q i is positive, we see thatno irreducible factor of V is trivial. It follows from [46, Lemma 4.2] that λ − ( V )is not a zero-divisor in the ring K ∗ ([Spec ( k ) / G m ]), and hence (5.8) is exact. Wehave thus proven our claim.We can now use (5.8) to compute K ∗ ([ U/ G m ]). We first observe that the map K ∗ ([Spec ( k ) / G m ]) → K ∗ ([ A n +1 / G m ]) induced by the structure map is an isomor-phism by the homotopy invariance. So we can identify the middle term of (5.8)with K i ([Spec ( k ) / G m ]). Furthermore, it follows from the Self-intersection for-mula ([46, Theorem 2.1]) that the map ι ∗ is multiplication by λ − ( V ) under thisidentification.Since V = n ⊕ i =0 V i , we get λ − ( V ) = n Q i =0 λ − ( V i ). Furthermore, since the class of V i in R ( G m ) = Z [ t ± ] is t q i , we see that λ − ( V i ) = 1 − t q i . We conclude that λ − ( V ) = n Q i =0 (1 − t q i ). We have thus proven: Theorem 5.6.
There is a ring isomorphism K ∗ ( k )[ t ± ] n Q i =0 (1 − t q i ) ∼ = −→ K ∗ ( P ( q )) . IGHER K-THEORY OF TORIC STACKS 19
Remark 5.7.
In the above calculations, we can replace G m by the dense torus T to get a similar formula. In this case, the exact sequence (5.8) shows that K ([( A n +1 \ { } ) /T ]) is a quotient of R ( T ) and hence is not a projective R ( T )-module.5.2.3. G -theory of weighted projective scheme. The weighted projective scheme isthe scheme theoretic quotient of A n +1 \ { } by the above action of G m . This isthe coarse moduli scheme of P ( q ). We shall denote this scheme by g P ( q ). It isknown that this is a normal (but singular in general) projective scheme. Therewas no computation available for the higher G-theory or K-theory of this schematicweighted projective space. As an application of Theorem 1.2, we now give a simpledescription of the rational higher G-theory of g P ( q ). We still do not know how tocompute its K-theory.In order to describe the higher G-theory of g P ( q ), we shall use the followingpresentation of this scheme which allows us to use our main results. We assumethat the characteristic of k does not divide any q i .The torus T = G nm acts on P nk as the dense open torus by ( λ , · · · , λ n ) ⋆ [ z , · · · , z n ] = [ z , λ z , · · · , λ n z n ]. Let G = µ q × · · · × µ q n be the product of finitecyclic groups. Then G acts on P nk by ( a , · · · , a n ) • [ z , · · · , z n ] = [ a z , · · · , a n z n ].It is then easy to see that g P ( q ) is isomorphic to the scheme P nk /G .Define φ : G → T by φ ( a , · · · , a n ) = ( a /a , · · · , a n /a ). Then one checks that H := Ker( φ ) = { ( a , · · · , a n ) ∈ G | a = · · · = a n } = { λ ∈ G m | λ q = 1 = · · · = λ q n } = { λ ∈ G m | λ d = 1 }∼ = µ d . Moreover, it is easy to see that( a , · · · , a n ) • [ z , · · · , z n ] = [ a z , · · · , a n z n ]= [ a − ( a z ) , · · · , a − ( a n z n )]= [ z , ( a /a ) z , · · · , ( a n /a ) z n ]= φ ( a , · · · , a n ) ⋆ [ z , · · · , z n ] . In particular, G acts on P nk through φ . We conclude that X = [ P nk /G ] is a smoothtoric Deligne-Mumford stack associated to the data ( P nk , G φ −→ T ) and there is anisomorphism X ∼ = [ P nk /F ] × B µ d , where F = Im( φ ). Theorem 5.8.
There is a ring isomorphism (5.9) K ∗ ( k ) ⊗ Z [ t, t , · · · , t n ](( t − n +1 , t q − , · · · , t q n n − ∼ = −→ K ∗ ( X ) . Proof.
It follows from Corollary 5.4 and Theorem 1.2 that there is a ring isomor-phism K ∗ ( k ) ⊗ Z K ([ P nk /T ]) ⊗ R ( T ) R ( G ) ∼ = −→ K ∗ ( X )) . On the other hand, the projective bundle formula implies that the left side ofthis isomorphism is same as K ∗ ( k ) ⊗ Z R ( T )[ t ](( t − n +1 ) ⊗ R ( T ) R ( G ) which in turn is isomorphicto K ∗ ( k ) ⊗ Z R ( G )[ t ](( t − n +1 ) . The theorem now follows from the isomorphism R ( G ) ∼ = Z [ t , ··· ,t n ]( t q − , ··· ,t qnn − . (cid:3) Corollary 5.9.
There is an isomorphism G ∗ ( k )[ t ](( t − n +1 ) ∼ = −→ G ∗ (cid:16) g P ( q ) (cid:17) with the rational coefficients.Proof. All the groups in this proof will be considered with rational coefficients. Let π : P n +1 k → g P ( q ) be the quotient map. The assignment F 7→ ( π ∗ ( F )) G defines acovariant functor from the category of G -equivariant coherent sheaves on P n +1 k tothe category of ordinary coherent sheaves on g P ( q ). Since the characteristic of k does not divide the order of G , this functor is exact and gives a push-forward map π ∗ : G G ∗ ( P n +1 k ) → G ∗ (cid:16) g P ( q ) (cid:17) .Let CH G ∗ ( P n +1 k ) denote the equivariant higher Chow groups of P n +1 k ([9]). By [9,Theorem 3], there is a push-forward map π ∗ : CH G ∗ ( P n +1 k ) → CH ∗ (cid:16) g P ( q ) (cid:17) whichis an isomorphism. It follows from [29, Theorem 9.8, Lemma 9.1] (see also [10,Theorem 3.1]) that there is a commutative diagram G G ∗ ( P n +1 k ) ⊗ R ( G ) Q τ G / / π ∗ (cid:15) (cid:15) CH G ∗ ( P n +1 k ) π ∗ ∼ = (cid:15) (cid:15) G ∗ (cid:16) g P ( q ) (cid:17) τ / / CH ∗ (cid:16) g P ( q ) (cid:17) , where the horizontal arrows are the Riemann-Roch maps which are isomorphisms([29, Theorem 8.6]). It follows that the left vertical arrow is an isomorphism. Thecorollary now follows by combining this isomorphism with Theorem 5.8. (cid:3) Toric stack bundles and the stacky Leray-Hirsch theorem
Toric bundle schemes and their cohomology were first studied by Sankaran andUma in [38]. They computed the Grothendieck group of a toric bundle over asmooth base scheme. Jiang [24] studied smooth and simplicial Deligne-Mumfordtoric stack bundles over schemes and computed their Chow rings. These bun-dles are relative analogues of toric Deligne-Mumford stacks. A description of theGrothendieck group of toric Deligne-Mumford stack bundles was given by Jiangand Tseng in [26].In this section, we give a general definition of toric stack bundles over a basescheme in such a way that every fiber of this bundle is a (generically stacky) toricstack in the sense of [16]. We prove a stacky version of the Leray-Hirsch theoremfor the algebraic K-theory of stack bundles. This Leray-Hirsch theorem will beused in the next section to describe the higher K-theory of toric stack bundles.6.1.
Toric stack bundles.
Let T be a split torus of rank n and let X be a schemewith a T -action. Let G be a diagonalizable group over k and let φ : G → T be amorphism of algebraic groups over k .Let p : E → B be a principal T -bundle over a scheme B . Let G act on E × X by g ( e, x ) = ( e, gx ) := ( e, φ ( g ) x ) and let T act on E × X via the diagonal action. Itis easy to see that these two actions commute and the projection map E × X → E is equivariant with respect to these actions.The commutativity of the actions ensures that the G -action descends to thequotients E ( X ) := E T × X and E/T = B such that the induced map of quotients p : E ( X ) → B is G -equivariant. Since E has trivial G -action, so does B and wesee that G acts on E ( X ) fiber-wise and the map p canonically factors through IGHER K-THEORY OF TORIC STACKS 21 the stack quotient π : [ E ( X ) /G ] → B . Notice that E is a Zariski locally trivial T -bundle and so are E ( X ) → B and [ E ( X ) /G ] → B . Setting X = [ E ( X ) /G ],we conclude that the map π : X → B is a Zariski locally trivial fibration each ofwhose fiber is the stack [ X/G ]. The morphism π will be called a stack bundle over B .If X is a toric variety with dense torus T , then π : X → B will be called a toricstack bundle over B . In this case, each fiber of π is the toric stack [ X/G ] in thesense of [16]. If [
X/G ] is a Deligne-Mumford stack, this construction recovers thenotion of toric stack bundles used in [24] and [26].6.2.
Leray-Hirsch Theorem for stack bundles.
First we prove the followinglemma.
Lemma 6.1.
Let X be a T -equivariantly cellular scheme with the T -equivariantcellular decomposition (6.1) ∅ = X n +1 ( X n ( · · · ( X ( X = X and let U i = X \ X i for ≤ i ≤ n + 1 . Let G be a diagonalizable group providedwith a morphism of algebraic groups φ : G → T . Then for any ≤ i ≤ n , thesequence (6.2) 0 → G G ∗ ( U i +1 \ U i ) → G G ∗ ( U i +1 ) → G G ∗ ( U i ) → is exact. In particular, G G ( X ) is a free R ( G ) -module of rank equal to the numberof T -invariant affine cells in X with basis given by the closures of the affine cells.Proof. To prove the exactness part of the proposition, we first make the followingclaim. Suppose X is a G -scheme and j : U ֒ → X is a G -invariant open inclusionwith complement Y . Suppose that U is isomorphic to a representation of G . Thenthe localization sequence(6.3) 0 → G G ∗ ( Y ) → G G ∗ ( X ) j ∗ −→ G G ∗ ( U ) → α : X → Spec ( k ) and β : U → Spec ( k ) be the structuremaps (which are G -equivariant) so that β = α ◦ j . The homotopy invariance ofequivariant K-theory shows that β ∗ is an isomorphism. Let γ = α ∗ ◦ ( β ∗ ) − . Thenone checks that γ is a section of j ∗ and hence the localization sequence splits intoshort exact sequences. This proves the claim.We shall prove (6.2) by induction on the number of T -invariant affine cells in X .For i = 0, (6.2) is immediate. So we assume i ≥ (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / G G ∗ ( X i \ X i +1 ) / / G G ∗ ( X \ X i +1 ) / / (cid:15) (cid:15) G G ∗ ( X \ X i ) / / (cid:15) (cid:15) / / G G ∗ ( X i \ X i +1 ) / / G G ∗ ( X \ X i +1 ) / / (cid:15) (cid:15) G G ∗ ( X \ X i ) / / (cid:15) (cid:15) G G ∗ ( X \ X ) (cid:15) (cid:15) G G ∗ ( X \ X ) / / (cid:15) (cid:15)
00 0 . The top row is exact by induction on the number of affine cells since X is T -equivariantly cellular with fewer number of cells. The two columns are exact bythe above claim. It follows that the middle row is exact, which proves (6.2).To prove the last (freeness) assertion, we apply (6.3) to the inclusion X ⊂ X and see that G G ( X ) ∼ = G G ( X ) ⊕ R ( G ). An induction on the number of affine G -cells now finishes the proof. (cid:3) Proposition 6.2.
Let X be a T -equivariantly cellular scheme and let B be anyscheme with trivial T -action. Then the external product map (6.5) G ∗ ( B ) ⊗ Z G G ( X ) → G G ∗ ( B × X ) is an isomorphism. In particular, the natural map K ∗ ( k ) ⊗ Z G G ( X ) → G G ∗ ( X ) isan isomorphism.Proof. Since the map(6.6) G ∗ ( B ) ⊗ Z R ( G ) ∼ = −→ G G ∗ ( B )is an isomorphism ( cf. [43, Lemma 5.6]), the lemma is equivalent to the assertionthat the map(6.7) G G ∗ ( B ) ⊗ R ( G ) G G ( X ) → G G ∗ ( B × X )is an isomorphism.Consider the cellular decomposition of X as in Lemma 6.1. Then each U i = X \ X i is also a T -equivariantly cellular scheme. It suffices to show by inductionon i ≥ X is any of these U i ’s. There is nothing to provefor i = 0 and the case i = 1 follows by the homotopy invariance since U is anaffine space.To prove the general case, we use the short exact sequence(6.8) 0 → G G ( U i +1 \ U i ) → G G ( U i +1 ) → G G ( U i ) → G G ( U i ) was shown to befree over R ( G ) in Lemma 6.1. Tensoring this with G G ∗ ( B ) over R ( G ), we obtain acommutative diagram0 / / G G ∗ ( B ) ⊗ G G ( U i +1 \ U i ) / / (cid:15) (cid:15) G G ∗ ( B ) ⊗ G G ( U i +1 ) / / (cid:15) (cid:15) G G ∗ ( B ) ⊗ G G ( U i ) / / (cid:15) (cid:15) / / G G ∗ ( B × ( U i +1 \ U i )) i ∗ / / G G ∗ ( B × U i +1 ) j ∗ / / G G ∗ ( B × U i ) / / where the top row remains exact since the short exact sequence in (6.8) is split.The bottom row is the localization exact sequence. The left vertical arrow isan isomorphism by the homotopy invariance and the right vertical arrow is anisomorphism by the induction. In particular, j ∗ is surjective in all indices. Weconclude that i ∗ is injective in all indices and the middle vertical arrow is anisomorphism. (cid:3) Theorem 6.3. ( Stacky Leray-Hirsch theorem ) Suppose that k is a perfect field and B is a smooth scheme over k . Let X be a T -equivariantly cellular scheme. Let F i −→ X π −→ B be a Zariski locally trivial stack bundle ( § F is a smooth stack of the form [ X/G ] . Assume that there are elements { e , · · · , e r } in K ( X ) such that { f = i ∗ ( e ) , · · · , f r = i ∗ ( e r ) } is an R ( G ) -basis of K ( X b ) foreach fiber X b = F of the fibration. Then the map (6.9) Φ : K ( F ) ⊗ R ( G ) K G ∗ ( B ) → K ∗ ( X ) IGHER K-THEORY OF TORIC STACKS 23 Φ X ≤ i ≤ r f i ⊗ b i ! = X ≤ i ≤ r π ∗ ( b i ) e i is an isomorphism of R ( G ) -modules. In particular, K ∗ ( X ) is a free K G ∗ ( B ) -moduleand the map π ∗ : K G ∗ ( B ) → K ∗ ( X ) is injective.Proof. Since k is perfect and since the fibration p is Zariski locally trivial, we canfind a filtration(6.10) ∅ = B n +1 ( B n ( · · · ( B ( B = B of B by closed subschemes such that for each 0 ≤ i ≤ n , the scheme B i \ B i +1 is smooth and the given fibration is trivial over it. We set U i = B \ B i and V i = U i \ U i − = B i − \ B i . Observe then that each of U i ’s and V i ’s is smooth.Set X i = π − ( U i ) and W i = π − ( V i ) = V i × F . Let η i : X i ֒ → X and ι i : W i ֒ → X be the inclusion maps. We prove by induction on i that the map K ( F ) ⊗ R ( G ) K G ∗ ( U i ) → K ∗ ( X i ) is an isomorphism, which will prove the theorem.Since U = ∅ and X = U × F , the desired isomorphism for i ≤ U × F ∼ = [( U × X ) /G ]. We now considerthe commutative diagram:(6.11) K G ∗ ( U i ) ⊗ K ( F ) / / (cid:15) (cid:15) K G ∗ ( V i +1 ) ⊗ K ( F ) (cid:15) (cid:15) / / K G ∗ ( U i +1 ) ⊗ K ( F ) (cid:15) (cid:15) / / K G ∗ ( U i ) ⊗ K ( F ) / / (cid:15) (cid:15) K G ∗ ( V i +1 ) ⊗ K ( F ) (cid:15) (cid:15) K ∗ ( X i ) / / K ∗ ( W i +1 ) / / K ∗ ( X i +1 ) / / K ∗ ( X i ) / / K ∗ ( W i +1 ) . The top row in this diagram is obtained by tensoring the K-theory long exactlocalization sequence with K ( F ) over R ( G ), and the bottom row is just the local-ization exact sequence. Since K ( F ) is a free R ( G )-module ( cf. Lemma 6.1), thetop row is also exact.It is easily checked that the second and the third squares commute using thecommutativity property of the push-forward and pull-back maps of K-theory ofcoherent sheaves in a Cartesian diagram of proper and flat maps. We show that theother squares also commute. It is enough to show that the first square commutesas the fourth one is same as the first. Let δ denote the connecting homomorphismin a long exact localization sequence for higher K-theory.If we start with an element b ⊗ i ∗ ( e j ) ∈ K ∗ ( U i ) ⊗ K ( F ) and map this horizontally,we obtain δb ⊗ i ∗ ( e j ) which maps vertically down to π ∗ ( δb ) · ι ∗ i +1 ( e j ). On the otherhand, if we first map vertically, we obtain π ∗ ( b ) · η ∗ i ( e j ) which maps horizontallyto δ ( π ∗ ( b ) · η ∗ i ( e j )).Now, we recall that these elements in the higher K-theory of coherent sheaves arerepresented by the elements in the higher homotopy groups of the various infiniteloop spaces. Moreover, if we have a closed immersion of smooth stacks F ֒ → X with open complement U , then we have a fibration sequence of ring spectra(6.12) K ( F ) → K ( X ) → K ( U ) . The homotopy groups of these ring spectra form graded rings and the connectinghomomorphism in the long exact sequence of the homotopy groups associated tothe above fibration sequence satisfies the Leibniz rule (e.g., see [6, Appendix A]and [36, § δ ( π ∗ ( b ) · η ∗ i ( e j )) is same as δπ ∗ ( b ) · ι ∗ i +1 ( e j ) = π ∗ ( δb ) · ι ∗ i +1 ( e j ) since δ ( η ∗ i ( e j )) = 0. We have shown that theabove diagram commutes. The first and the fourth vertical arrows in (6.11) are isomorphisms by induction.The second and the fifth vertical arrows are isomorphisms by Proposition 6.2.Hence the middle vertical arrow is also an isomorphism by 5-lemma.To show that π ∗ is injective, consider the T -invariant filtration of X as in (6.1)and let j : [ E ( U ) /G ] = X → X be the open inclusion. If we apply (6.9) to themap X → B , we see that the composite map K G ∗ ( B ) → K ∗ ( X ) → K ∗ ( X ) is anisomorphism (since U is a T -invariant cell of X ). We conclude that π ∗ is splitinjective. (cid:3) Higher K-theory of toric stack bundles
In this section, we give explicit descriptions of the higher K-theory of toric stackbundles in terms of the higher K-theory of the base scheme.Let T be a split torus of rank n . Let N = Hom( G m , T ) be the lattice of one-parameter subgroups of T and let M = Hom( T, G m ) = N ∨ be its character group.Let X = X (∆) be a smooth projective toric variety associated to a fan ∆ in N R .Let(7.1) 0 → G → T → T ′ → → T ′∨ → T ∨ → G ∨ → . The Stanley-Reisner algebra associated to a subgroup of T . We fix anordering { σ , · · · , σ m } of ∆ max and let τ i ⊂ σ i be the cone which is the intersectionof σ i with all those σ j such that j ≥ i and which intersect σ i in dimension n − τ ′ i ⊂ σ i be the cone such that τ i ∩ τ ′ i = { } and dim( τ i ) + dim( τ ′ i ) = n for1 ≤ i ≤ m . It is easy to see that τ ′ i is the intersection of σ i with all those σ j such that j ≤ i and which intersect σ i in dimension n −
1. Since X is smooth andprojective, it is well known that we can choose the above ordering of ∆ max suchthat(7.3) τ i ⊂ σ j ⇒ i ≤ j and τ ′ i ⊂ σ j ⇒ j ≤ i. Let ∆ = { ρ , · · · , ρ d } be the set of one-dimensional cones in ∆ and let { v , · · · , v d } be the associated primitive elements of N . We choose { ρ , · · · , ρ n } to be a set ofone dimensional faces of σ m such that { v , · · · , v n } is a basis of N . Let { χ , · · · , χ n } be the dual basis of M . Let { χ ′ , · · · , χ ′ r } be a chosen basis of T ′∨ = M ′ . We willdenote the group operations in all the lattices additively. Definition 7.1.
Let A be a commutative ring with unit and let { r , · · · , r n } be aset of invertible elements in A . Let I T ∆ denote the ideal of the Laurent polynomialalgebra A [ t ± , · · · , t ± d ] generated by the elements(7.4) ( t j − · · · ( t j l − , ≤ j p ≤ d such that ρ j , · · · , ρ j l do not span a cone of ∆. Let J G ∆ denote the ideal of A [ t ± , · · · , t ± d ] generated by the relations(7.5) s i := d Y j =1 ( t j ) < − χ ′ i ,v j > ! − r i , ≤ i ≤ r. We define the A -algebras R T ( A, ∆) and R G ( A, ∆) to be quotients of A [ t ± , · · · , t ± d ]by the ideals I T ∆ and I G ∆ = I T ∆ + J G ∆ , respectively. IGHER K-THEORY OF TORIC STACKS 25
The ring R G ( A, ∆) will be called the Stanley-Reisner algebra over A associated tothe subgroup G . Every character χ ∈ M acts on R T ( A, ∆) via multiplication by theelement t χ = d Q j =1 ( t j ) < − χ,v j > ! and this makes R T ( A, ∆) (and hence R G ( A, ∆))an (cid:18) A ⊗ Z R ( T ) (cid:19) -algebra.7.2. The K-theory of toric stack bundles.
Let T be a split torus over a perfectfield k and let G be a closed subgroup of T (which may not necessarily be atorus). Let X be a smooth projective toric variety with dense torus T and let π : X = [( E ( X ) /G ] → B be a toric stack bundle over a smooth k -scheme B associated to a principal T -bundle p : E → B . We wish to describe the K-theoryof X in terms of the K-theory of B .Any T -equivariant line bundle L → X uniquely defines a G -equivariant linebundle E ( L ) = E T × L on E ( X ), where the G -action on E ( L ) is given exactly ason E ( X ). Every ρ ∈ ∆ defines a unique T -equivariant line bundle L ρ on X witha T -equivariant section s ρ : X → L ρ which is transverse to the zero-section andwhose zero locus is the orbit closure V ρ = O ρ .For any σ ∈ ∆, let u σ denote the fundamental class [ O V σ ] of the T -invariantsubscheme V σ in K T ( X ) and let y σ denote the fundamental class of [ E ( V σ )] in K G ( E ( X )) = K ( X ).Notice that p σ : E ( V σ ) → B is a G -equivariant smooth projective toric sub-bundle of p : E ( X ) → B with fiber V σ . In particular, π σ : [ E ( V σ ) /G ] → B is atoric stack sub-bundle of π : X → B with fiber [ V σ /G ]. We set X σ = [ E ( V σ ) /G ].Suppose that ρ j , · · · , ρ j l do not span a cone in ∆. Then s = ( s j , · · · , s j l ) yieldsa G -equivariant nowhere vanishing section of E ( L ρ j ) ⊕ · · · ⊕ E ( L ρ jl ) and hencethe Whitney sum formula for Chern classes in K-theory implies that(7.6) y ρ j · · · y ρ jl = 0 in K G ( E ( X )) . We now consider the commutative diagram(7.7) X l ι / / π l (cid:15) (cid:15) E ( X ) p (cid:15) (cid:15) E × X p E (cid:15) (cid:15) p X / / p ′ o o X π X (cid:15) (cid:15) Spec( l ) / / B E p o o π E / / Spec( k ) , where Spec( l ) is any point of B . It is clear that all squares are Cartesian and allthe maps in the right square are T -equivariant.We define ( T × G )-actions on any T -invariant subscheme Y ⊆ X and on E by( t, g ) · y = tg · y and ( t, g ) · e = t · e , respectively. An action of ( T × G ) on E × X is defined by ( t, g ) · ( e, x ) = ( t · e, tg · x ). It is clear that these are group actionssuch that the square on the right in (7.7) is ( T × G )-equivariant. This implies thatthe middle square is also ( T × G )-equivariant and the map p is G -equivariant withrespect to the trivial action of G on B . The square on the left is G -equivariant.Let L χ denote the T -equivariant line bundle on Spec( k ) associated to a character χ of T . Let ( T × G ) act on L χ by ( t, g ) · v = χ ( t ) χ ( g ) · v . If χ ∈ M ′ = T ′∨ , then G acts trivially on L χ and hence it acts trivially on π ∗ E ( L χ ). Recall that ( T × G )acts on E via T . Hence π ∗ E ( L χ ) → E is a ( T × G )-equivariant line bundle on which G -acts trivially. Since the T -equivariant line bundles on E are same as ordinary line bundles on B , we find that for every χ ∈ M ′ , there is a unique ordinary linebundle ζ χ on B such that π ∗ E ( L χ ) = p ∗ ( ζ χ ).Since G acts trivially on B , there is a canonical ring homomorphism c B : K ∗ ( B ) → K G ∗ ( B ) such that the composite K ∗ ( B ) c B −→ K G ∗ ( B ) → K ∗ ( B ) is identity.These maps are simply the maps K ∗ ( B ) c B −→ K G ∗ ( B ) = K ∗ ( B ) ⊗ Z R ( G ) → K ∗ ( B ).Since p ∗ X ◦ π ∗ X ( L χ ) = p ∗ E ◦ π ∗ E ( L χ ) and since the ( T × G )-equivariant vector bundleson E × X are same as G -equivariant vector bundles on E ( X ), we conclude thatfor every χ ∈ M ′ , there is a unique ordinary line bundle ζ χ on B such that(7.8) E ( π ∗ X ( L χ )) = p ∗ ( ζ χ ) = p ∗ ( c B ( ζ χ )) . Notice also that on each open subset of B where the bundle p is trivial, therestriction of ζ χ is the trivial line bundle since ζ χ is obtained from the T -linebundle L χ on Spec ( k ).We define a homomorphism of K ∗ ( B )-algebras K ∗ ( B )[ t ± , · · · , t ± d ] → K ∗ ( X ) bythe assignment t i [ E ( L ∨ ρ i ) /G ] for 1 ≤ i ≤ d . If we let r i = ζ χ ′ i for 1 ≤ i ≤ r ( cf. § K ∗ ( B )-algebra homomorphism(7.9) Φ G : R G ( K ∗ ( B ) , ∆) → K ∗ ( X ) . Given a sequence γ = { i , · · · , i d } of integers, set E ( γ ) = E (cid:0) ( L ∨ ρ ) i ⊗ · · · ⊗ ( L ∨ ρ d ) i d (cid:1) .We then see that for a monomial γ ( t ) = t i · · · t i d d , we have(7.10) Φ G ( γ ( t )) = [ E ( γ ) /G ] . The following result describes the higher K-theory of the toric stack bundle π : X → B . Theorem 7.2.
The homomorphism Φ G is an isomorphism. Before we prove this theorem, we consider some special cases which will be usedin the final proof. The following observations will be used throughout the proofs.The first observation is that the cell closures of X are the T -equivariant sub-schemes V τ i . So the classes of O V τi form an R ( G )-basis of K G ( X ) by Lemma 6.1.Since ι ∗ ( y τ i ) = [ O V τi ], we see that Theorem 6.3 applies to the toric stack bundle π : X → B .Second observation is that G is a diagonalizable group which acts triviallyon B . Hence the map K ∗ ( B ) ⊗ Z R ( G ) → K G ∗ ( B ) is a ring isomorphism by [43,Lemma 3.6]. This identification will be used without further mention. Since anycharacter χ ∈ M acts on R T ( K ∗ ( B ) , ∆) and K G ∗ ( E ( X )) via multiplication by t χ and Φ G ( t χ ) respectively ( cf. [38, Proposition 4.3]), we observe that the compositemap R T ( K ∗ ( B ) , ∆) → R G ( K ∗ ( B ) , ∆) → K G ∗ ( E ( X )) is K T ∗ ( B )-linear. Remark 7.3.
We remark that the result of Thomason in [43, Lemma 3.6] is statedfor affine schemes, but his proof works for all schemes. Another way to deduce thegeneral case from the affine case is to get a stratification of B by affine subschemesas in (6.10), use induction on the number of affine strata, the localization sequenceand the fact that R ( G ) is free over Z . Lemma 7.4.
The homomorphism Φ G is an isomorphism when G = T .Proof. In this case, we first notice that the map R T ( Z , ∆) φ −→ K T ( X ) which takes t i to [ L ∨ ρ i ], is an isomorphism of R ( T )-algebras by [46, Theorem 6.4]. On the otherhand, we have the maps(7.11) K ∗ ( B ) ⊗ Z R T ( Z , ∆) ∼ = −→ R T ( K ∗ ( B ) , ∆) Φ T −→ K T ∗ ( E ( X )) , IGHER K-THEORY OF TORIC STACKS 27 where the first map takes α ⊗ t i to α · t i for 1 ≤ i ≤ d . This map is clearly anisomorphism (see (7.4)). It is clear from the definition of Φ T that the compositemap is same as the map Φ in (6.9) (with G = T ). It follows from Theorem 6.3that the composite map in (7.11) is an isomorphism. We conclude that Φ T is anisomorphism. (cid:3) Corollary 7.5.
For any closed subgroup G ⊆ T , the ring R G ( K ∗ ( B ) , ∆) is a free K ∗ ( B ) -module.Proof. We have seen above that the image of a character χ ∈ M in R T ( K ∗ ( B ) , ∆) is t χ . If we let J G denote the ideal (cid:0) χ ′ − ζ χ ′ , · · · , χ ′ r − ζ χ ′ r (cid:1) in K T ∗ ( B ), then it followsfrom (7.5) that J G ∆ = J G R T ( K ∗ ( B ) , ∆) under the map K T ∗ ( B ) → R T ( K ∗ ( B ) , ∆).It follows from Lemma 7.4 and Theorem 6.3 (with G = T ) that R T ( K ∗ ( B ) , ∆) isa free K T ∗ ( B )-module. This implies that R G ( K ∗ ( B ) , ∆) = R T ( K ∗ ( B ) , ∆) /J G ∆ is afree K T ∗ ( B ) /J G -module. Thus, it suffices to show that K T ∗ ( B ) /J G is a free K ∗ ( B )-module. Since K T ∗ ( B ) is isomorphic to a Laurent polynomial ring K ∗ ( B )[ x ± , · · · , x ± n ]and since each character χ ∈ M ′ is a monomial in this ring, the desired freenessfollows from Lemma 7.7. (cid:3) Lemma 7.6.
The homomorphism Φ G is an isomorphism when p : E → B is atrivial principal bundle.Proof. Since p : E → B is a trivial bundle, we have observed before that ζ χ ′ i = 1 foreach 1 ≤ i ≤ r . In particular, the map K T ∗ ( B ) /J G → K G ∗ ( B ) is an isomorphismby Lemma 7.8, where J G is as in Corollary 7.5.It follows from Theorem 6.3 and Lemma 7.4 that Φ T is an isomorphism of free K T ∗ ( B )-modules. This implies that R G ( K ∗ ( B ) , ∆) = R T ( K ∗ ( B ) , ∆) /J G ∆ is a free K T ∗ ( B ) /J G = K G ∗ ( B )-module. It follows from this and Theorem 6.3 that Φ G is abasis preserving homomorphism of free K G ∗ ( B )-modules of same rank. Hence, itmust be an isomorphism. (cid:3) Lemma 7.7.
Let S = A [ x ± , · · · , x ± n ] be a Laurent polynomial ring over a com-mutative ring A with unit. Let { t , · · · , t r } be a set of monomials in S and let { u , · · · , u r } be a set of units in A . Then the ring S ( t − u , ··· ,t r − u r ) is free over A .Proof. This is left as an easy exercise using the fact that S is a free A -module onthe monomials. (cid:3) Lemma 7.8.
Let A be a commutative ring with unit and let → L → M → N → be a short exact sequence of finitely generated abelian groups. Let I L be the idealof the group ring A [ M ] generated by the set { s − | s ∈ S } , where S is a generatingset of L . Then the map of group rings A [ M ] I L → A [ N ] is an isomorphism.Proof. This is an elementary exercise and a proof can be found in [33, Proposi-tion 2]. (cid:3)
Proof of Theorem 7.2:
We shall prove this theorem along the same lines asthe proof of Theorem 6.3. Recall that our base field k is perfect. We consider thestratification of B by smooth locally closed subschemes as in (6.10). We shall followthe notations used in the proof of Theorem 6.3. It suffices to show by inductionon i that the theorem is true when B is replaced by each U i . Since U = ∅ andsince E p −→ B is trivial over U , the desired isomorphism for i ≤ Given a smooth locally closed subscheme j : U ֒ → B , let ζ Ui = j ∗ ( ζ χ ′ i ) ∈ K ∗ ( U )for 1 ≤ i ≤ r and set J GU = (cid:0) χ ′ − ζ U , · · · , χ ′ r − ζ Ur (cid:1) .We have seen in the proof of Lemma 7.4 that for any such inclusion U ⊆ B , R T ( K ∗ ( U ) , ∆) is same as K T ( X ) ⊗ Z K ∗ ( U ). Moreover, the maps(7.12) R G ( K ∗ ( B ) , ∆) ⊗ K ∗ ( B ) K ∗ ( U ) ∼ = R T ( K ∗ ( B ) , ∆) J G R T ( K ∗ ( B ) , ∆) ⊗ K ∗ ( B ) K ∗ ( U ) → R T ( K ∗ ( U ) , ∆) J GU R T ( K ∗ ( U ) , ∆) → R G ( K ∗ ( U ) , ∆)are all isomorphisms.We now consider the diagram:(7.13) R G ( K ∗ ( U i ) , ∆) / / Φ UiG (cid:15) (cid:15) R G ( K ∗ ( V i +1 ) , ∆) / / Φ Vi +1 G (cid:15) (cid:15) R G ( K ∗ ( U i +1 ) , ∆) / / Φ Ui +1 G (cid:15) (cid:15) R G ( K ∗ ( U i ) , ∆) / / Φ UiG (cid:15) (cid:15) R G ( K ∗ ( V i +1 ) , ∆) Φ Vi +1 G (cid:15) (cid:15) K ∗ ( X i ) / / K ∗ ( W i +1 ) / / K ∗ ( X i +1 ) / / K ∗ ( X i ) / / K ∗ ( W i +1 ) . Using (7.12), we see that the top row of (7.13) is obtained by tensoring thelocalization exact sequence · · · → K ∗ ( U i ) → K ∗ ( V i +1 ) → K ∗ ( U i +1 ) → K ∗ ( U i ) → K ∗ ( V i +1 ) → · · · of K ∗ ( B )-modules with R G ( K ∗ ( B ) , ∆). Hence, this row is exact by Corollary 7.5.The bottom row is anyway a localization exact sequence.We now show that the diagram (7.13) commutes. It is clear that the third squarecommutes and the fourth square is same as the first. So we need to check that thefirst two squares commute.Let α : V i +1 ֒ → U i +1 and β : M i +1 ֒ → X i +1 be the closed immersions of smoothschemes and stacks. Following the notations in the proof of Theorem 6.3, we seethat for any u ∈ K ∗ ( U i ) and for any monomial γ ( t ) = t i · · · t i d d ,(7.14) δ ◦ Φ U i G ( u ⊗ γ ( t )) = δ (cid:0) π ∗ U i ( u ) · η ∗ i ([ E ( γ ) /G ]) (cid:1) = δ ( π ∗ U i ( u )) · ι ∗ i +1 ([ E ( γ ) /G ])= π ∗ V i +1 ( δ ( u )) · ι ∗ i +1 ([ E ( γ ) /G ])= Φ V i +1 G ( δ ( u ) ⊗ γ ( t ))= Φ V i +1 G ◦ δ ( u ⊗ γ ( t )) , where E ( γ ) ∈ K ∗ ( X ) is as in (7.10). The second equality follows from the Leibnizrule and the third equality follows from the commutativity of (6.11). This showsthat the first (and the last) square commutes.To show the commutativity of the second square, let v ∈ K ∗ ( V i +1 ). We thenhave(7.15) β ∗ ◦ Φ V i +1 G ( v ⊗ γ ( t )) = β ∗ (cid:16) π ∗ V i +1 ( v ) · ι ∗ i +1 ([ E ( γ ) /G ]) (cid:17) = β ∗ (cid:16) π ∗ V i +1 ( v ) · β ∗ ◦ η ∗ i +1 ([ E ( γ ) /G ]) (cid:17) = β ∗ ( π ∗ V i +1 ( v )) · η ∗ i +1 ([ E ( γ ) /G ])= π ∗ U i +1 ( α ∗ ( v )) · η ∗ i +1 ([ E ( γ ) /G ])= Φ U i +1 G ( α ∗ ( v ) ⊗ γ ( t ))= Φ U i +1 G ◦ α ∗ ( v ⊗ γ ( t )) , IGHER K-THEORY OF TORIC STACKS 29 where third equality follows from the projection formula and the fourth equalityfollows from the commutativity of (6.11). This shows that the second squarecommutes.The first and the fourth vertical arrows are isomorphisms by induction. The sec-ond and the fifth vertical arrows are isomorphisms by Lemma 7.6. Hence the middlevertical arrow is also an isomorphism by 5-lemma. This concludes the proof of The-orem 7.2. (cid:3)
Remark 7.9.
It was assumed in Theorem 7.2 that G is subgroup of T . Since X isjust the toric stack [ E ( X ) /G ] associated to the data ( E ( X ) , G φ −→ T ), the generalcase can always be reduced to the case of Theorem 7.2. We refer to Remark 4.5for how this can be done. Acknowledgments.
Parts of this work were carried out while the first author wasvisiting the Tata Institute of Fundamental Research, while the second author wasvisiting the Mathematics department of Ohio state university, Columbus and alsowhile both the authors were visiting the Mathematics department of the HarishChandra Research Institute, Allahabad. The first author was also supported by anadjunct professorship at the same institute. They would would like to thank thesedepartments for the invitation and financial support during these visits. They alsowould like to thank Hsian-Hua Tseng for helpful comments on an earlier versionof this paper.
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Department of Mathematics, Ohio State University, Columbus, Ohio, 43210,USASchool of Mathematics, Tata Institute of Fundamental Research, Homi BhabhaRoad, Colaba, Mumbai, India
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