aa r X i v : . [ m a t h . AG ] M a r FORMAL GROUP RINGS OF TORIC VARIETIES
WANSHUN WONG
Abstract.
In this paper we use formal group rings to construct an alge-braic model of the T -equivariant oriented cohomology of smooth toric vari-eties. Then we compare our algebraic model with known results of equivariantcohomology of toric varieties to justify our construction. Finally we constructthe algebraic counterpart of the pull-back and push-forward homomorphismsof blow-ups. Introduction
Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel[14], where examples include the Chow group of algebraic cycles modulo rationalequivalence, algebraic K -theory, connective K -theory, elliptic cohomology, and auniversal such theory called algebraic cobordism. It is known that to any h one canassociate a one-dimensional commutative formal group law F over the coefficientring R = h (pt), given by c ( L ⊗ L ) = F ( c ( L ) , c ( L ))for any line bundles L , L on a smooth variety X , where c is the first Chernclass.Let T be a split algebraic torus which acts on a smooth variety X . An objectof interest is the T -equivariant cohomology ring h T ( X ) of X , and we would like tobuild an algebraic model for it. More precisely, instead of using geometric methodsto compute h T ( X ), we want to use algebraic methods to construct another ring A (our algebraic model), such that A should be easy to compute and it gives infor-mation about h T ( X ). Of course the best case is that A and h T ( X ) are isomorphic.Let T ∗ be the character lattice of our torus T . A new combinatorial object,called a formal group ring and denoted by R J T ∗ K F , can then be defined by us-ing R, F and T ∗ . It serves as an algebraic model of the completed T -equivariantcohomology ring h T (pt) ∧ of a point, or of the cohomology ring of the classifyingspace h ( BT ) of T . Various computations can then be performed on formal grouprings to provide algebraic models of the (usual) cohomology ring h ( G/B ) and the T -equivariant cohomology ring h T ( G/B ) of a homogeneous space
G/B , where G isa split semisimple linear algebraic group with a maximal split torus T , and B is aBorel subgroup containing T . We refer to [6], [7] and [8] for details.The main goal of this paper is to apply the techniques of formal group ringsto a smooth toric variety X , so that we obtain algebraic models of the usual co-homology and the T -equivariant cohomology of X . Since any toric variety can be Mathematics Subject Classification.
Key words and phrases.
Formal Group Law, Toric Variety, Oriented Cohomology.The author is supported by the NSERC Discovery grant 385795-2010 and NSERC DAS grant396100-2010 of Kirill Zaynullin. constructed by gluing affine toric varieties together, our idea is to find a model ofthe T -equivariant cohomology for each affine toric variety and then “glue” themtogether. This paper is organized as follows. First we establish notation and recallbasic facts on toric varieties and formal group rings in Section 2. Then in Section3 we prove our main result of gluing formal group rings together, and the outputof gluing will be our models of the usual cohomology and the T -equivariant co-homology of X . Next we compare our models with known results of equivariantcohomology in Section 4. Finally in Section 5 we define the algebraic counterpartof the pull-back and push-forward homomorphisms of blow-ups.2. Notation and Preliminaries
Toric Varieties.
Our references for the theory of toric varieties are [5] and[10], and also [9] for the definition of a toric variety over an arbitrary base.Let T be a split torus over our base field k . The character and cocharacterlattices of T are denoted by T ∗ and T ∗ respectively, and there is a perfect pairing h , i : T ∗ × T ∗ → Z . A toric variety X is a normal variety on which the split torus T acts faithfully with an open dense orbit. Recall that X is determined by its fan Σin the lattice T ∗ . In this paper we will always assume X is smooth unless otherwisestated, so that every cone σ in Σ is generated by a subset of a basis of T ∗ . LetΣ max be the set of maximal cones of Σ. The set of all rays (i.e. one-dimensionalcones) in Σ is denoted by Σ(1), and similarly the set of all rays in σ is denoted by σ (1) for every cone σ ∈ Σ.For every cone σ ∈ Σ, let U σ be the associated open affine subscheme of X ,and O σ be the T -orbit corresponding to σ under the Orbit-Cone Correspondence.The stabilizer of any geometric point of O σ is a subtorus T σ ⊆ T , so that O σ ∼ = T /T σ . The character and cocharacter lattices of T σ are given by T ∗ σ = T ∗ /σ ⊥ and T σ ∗ = h σ i = σ + ( − σ ) ⊆ N respectively, and dim T σ = dim σ . Finally, for every ray ρ ∈ Σ(1), the unique generator of the monoid ρ is denoted by v ρ ∈ T ∗ .2.2. Formal Group Rings.
Our main reference for formal group rings is [6].Let R be a commutative ring, and F be a formal group law over R which wealways assume to be one-dimensional and commutative, i.e. F ( x, y ) ∈ R J x, y K is aformal power series such that F ( x,
0) = 0 , F ( x, y ) = F ( y, x ) , and F ( x, F ( y, z )) = F ( F ( x, y ) , z ) , see [14, p.4]. For any nonnegative integer n we use the notation x + F y = F ( x, y ) , n · F x = x + F · · · + F x | {z } n copies , and ( − n ) · F x = − F ( n · F x )where − F x denotes the formal inverse of x , i.e. x + F ( − F x ) = 0.Let M be an abelian group, and let R [ x M ] denote the polynomial ring over R with variables indexed by M . Let ǫ : R [ x M ] → R be the augmentation map whichsends x λ to 0 for every λ ∈ M . We denote R J x M K the ker( ǫ )-adic completion ofthe polynomial ring R [ x M ]. Let J F ⊆ R J x M K be the closure of the ideal generatedby x and x λ + µ − ( x λ + F x µ ) over all λ, µ ∈ M . The formal group ring (also calledformal group algebra) is then defined to be the quotient R J M K F = R J x M K /J F . ORMAL GROUP RINGS OF TORIC VARIETIES 3
By abuse of notation the class of x λ in R J M K F is also denoted by x λ . By definition R J M K F is a complete Hausdorff R -algebra with respect to the ker( ǫ ′ )-adic topology,where ǫ ′ : R J M K F → R is the induced augmentation map.An important subring of R J M K F is the image of R [ x M ] under the composition R [ x M ] → R J x M K → R J M K F , denoted by R [ M ] F . Then R J M K F is the completionof R [ M ] F at the ideal ker( ǫ ′ ) ∩ R [ M ] F . Example 2.1. (see [6, Example 2.19]) The additive formal group law over R isgiven by F ( x, y ) = x + y . In this case we have R -algebra isomorphisms R [ M ] F ∼ = Sym R ( M ) and R J M K F ∼ = Sym R ( M ) ∧ , where Sym R ( M ) is the ring of symmetric powers of M over R , and the completionis at the kernel of the augmentation map x λ
0. The isomorphisms are given by x λ λ ∈ Sym R ( M ). Example 2.2. (see [6, Example 2.20]) The multiplicative periodic formal grouplaw over R is given by F ( x, y ) = x + y − βxy , where β is a unit in R . Let R [ M ] be the(usual) group ring R [ M ] = { P r i e λ i | r i ∈ R, λ i ∈ M } written in the exponentialnotation, and tr : R [ M ] → R be the trace map which sends e λ to 0 for every λ ∈ M .Then there are the following R -algebra isomorphisms R [ M ] F ∼ = R [ M ] and R J M K F ∼ = R [ M ] ∧ where the completion is at ker(tr), and the isomorphisms are given by sending x λ to β − (1 − e λ ).Finally we remark that given φ : M → M ′ a homomorphism of abelian groups,it induces ring homomorphisms R J M K F → R J M ′ K F and R [ M ] F → R [ M ′ ] F bysending x λ to x φ ( λ ) .3. Formal Group Rings of Toric Varieties
Let X be the smooth toric variety of the fan Σ ⊆ T ∗ . For every cone σ ∈ Σ, since σ is smooth we have U σ = A dim σ × T /T σ . Therefore if F is the formal group lawassociated to some oriented cohomology theory h over the coefficient ring R = h (pt),by homotopy invariance the formal group ring R J T ∗ σ K F can be viewed as an algebraicsubstitute of the completed equivariant cohomology ring h T ( U σ ) ∧ , see [6, Remark2.22]. It is known that a topology can be given to the fan Σ by defining the opensets to be subfans of Σ (see for example [11, Section 7.2]). Our goal is to “glue” R J T ∗ τ K F over all maximal cones τ ∈ Σ max together as a sheaf on Σ, and the ring ofglobal sections of this sheaf will then be an algebraic model of h T ( X ) ∧ .For any toric variety X , there is a natural isomorphism(1) CDiv T ( X ) ∼ = ker M τ ∈ Σ max T ∗ τ −→ M τ = τ ′ T ∗ τ ∩ τ ′ where CDiv T ( X ) is the group of T -invariant Cartier divisors, and the map on theright hand side is given by the difference of the two natural projections T ∗ τ , T ∗ τ ′ → T ∗ τ ∩ τ ′ on each summand. The idea is that for every T -invariant Cartier divisor, itsrestriction to U τ is equal to the divisor of a character, and the obvious compatabilitycondition holds (see [5, Chapter 4.2]). WANSHUN WONG
For every ray ρ ∈ Σ(1), the closure of the corresponding orbit O ρ is a T -invariantprime divisor on X , and we will denote it by D ρ . The group of T -invariant Weildivisors Div T ( X ) is a lattice generated by D ρ , i.e.Div T ( X ) = M ρ ∈ Σ(1) Z · D ρ . Since we assume X is smooth, all Weil divisors are Cartier. HenceCDiv T ( X ) = Div T ( X ) = M ρ ∈ Σ(1) Z · D ρ . For every character α ∈ T ∗ , it determines a T -invariant principal Cartier divisordiv( α ) = P ρ ∈ Σ(1) h α, v ρ i D ρ . This defines a group homomorphism T ∗ → CDiv T ( X ).Passing to the formal group rings, we have R J T ∗ K F −→ R J CDiv T ( X ) K F x α x P h α,v ρ i D ρ = P [ h α, v ρ i ] F x D ρ .Therefore R J CDiv T ( X ) K F is a R J T ∗ K F -algebra. Clearly, R J T ∗ σ K F is also a R J T ∗ K F -algebra under the natural map for every cone σ ∈ Σ.Consider the group homomorphism CDiv T ( X ) → T ∗ σ defined by P ρ ∈ Σ(1) n ρ D ρ ✤ / / P ρ ∈ σ (1) n ρ α σ,ρ where { α σ,ρ | ρ ∈ σ (1) } is the basis of T ∗ σ dual to { v ρ | ρ ∈ σ (1) } . We remark thatthe above homomorphism is different from the one in literature by a minus sign.Again passing to formal group rings we obtain a map R J CDiv T ( X ) K F → R J T ∗ σ K F . Lemma 3.1.
The map R J CDiv T ( X ) K F → R J T ∗ σ K F constructed above is a R J T ∗ K F -algebra homomorphism.Proof. By functoriality of formal group rings it suffices to show thatCDiv T ( X ) / / T ∗ σ T ∗ e e ❏❏❏❏❏❏❏❏❏❏ > > ⑤⑤⑤⑤⑤⑤⑤ is commutative. For every α ∈ T ∗ , we have P ρ ∈ Σ(1) h α, v ρ i D ρ ✤ / / P ρ ∈ σ (1) h α, v ρ i α σ,ρ α ✍ g g ◆◆◆◆◆◆◆◆◆◆◆ ✵ ♣♣♣♣♣♣♣♣♣♣♣ which is clearly commutative. (cid:3) Now we are ready to do the “gluing”. Motivated by the isomorphism (1), weconsider the following sequence(2) R J CDiv T ( X ) K F ψ / / Y τ ∈ Σ max R J T ∗ τ K F π / / Y τ = τ ′ R J T ∗ τ ∩ τ ′ K F where ψ is the product of the maps in Lemma 3.1, π is given by the product of thedifferences of pr τ,τ ∩ τ ′ : R J T ∗ τ K F → R J T ∗ τ ∩ τ ′ K F , pr τ ′ ,τ ∩ τ ′ : R J T ∗ τ ′ K F → R J T ∗ τ ∩ τ ′ K F . ORMAL GROUP RINGS OF TORIC VARIETIES 5 (Here it involves a choice between pr τ,τ ∩ τ ′ − pr τ ′ ,τ ∩ τ ′ and pr τ ′ ,τ ∩ τ ′ − pr τ,τ ∩ τ ′ . How-ever we are only interested in the kernel so it does not matter.) Note that π is onlya module homomorphism but not an algebra homomorphism. Proposition 3.2.
The sequence (2) is an exact sequence of R J T ∗ K F -modules.Proof. i) First, it follow from the functoriality of formal group rings that im( ψ ) ⊆ ker( π ).ii) To show ker( π ) ⊆ im( ψ ), let τ , . . . , τ d be the maximal cones of Σ, and let( f i ) be any element in ker( π ) ⊆ Q i R J T ∗ τ i K F . Then we define f ij to be the image of f i in R J T ∗ τ i ∩ τ j K F (which is the same as the image of f j ), f ijk to be the image of f i in R J T ∗ τ i ∩ τ j ∩ τ k K F , and so on.By [6, Corollary 2.13], for every cone σ we identify R J T ∗ σ K F with the ring of powerseries R J x α σ,ρ K with variables x α σ,ρ , ρ ∈ σ (1), where we recall that { α σ,ρ | ρ ∈ σ (1) } is the basis of T ∗ σ dual to { v ρ | ρ ∈ σ (1) } . Since X is smooth, under this identifica-tion for every face µ of σ , the natural maps R J T ∗ σ K F → R J T ∗ µ K F coincides with thecanonical projection of the rings of power series R J x α σ,ρ K → R J x α µ,ρ K , x α σ,ρ x α µ,ρ if ρ ∈ µ (1), x α σ,ρ ρ / ∈ µ (1). Similarly we identify R J CDiv T ( X ) K F with R J x D ρ K with variables x D ρ , ρ ∈ Σ(1). Then ψ coincides with product of the pro-jections R J x D ρ K → R J x α τi,ρ K .For every i , let g i be the unique preimage of f i under the projection R J x D ρ K → R J x α τi,ρ K , such that g i does not involve any x D ρ for ρ / ∈ τ i (1). Informally speaking, g i is obtained from f i by replacing all x α τi,ρ with corresponding x D ρ . We define g ij , . . . , g ··· d in the same way.Finally we define g = X i g i − X i Remark 3.4. I Σ is power series version of the standard Stanley-Reisner ideal.The underlying geometric meaning follows from the Orbit-Cone Correspondence:It is known that ρ is a face of σ if and only if O σ ⊆ D ρ . Therefore given S = { ρ , . . . , ρ t } ⊆ Σ(1), we have S * σ (1) for any cone σ if and only if D ρ ∩· · ·∩ D ρ t = ∅ in X . Corollary 3.5. ker( π ) = im( ψ ) = R J x D ρ K /I Σ . By “gluing” we have constructed an algebraic model for the completed equi-variant cohomology ring h T ( X ) ∧ . Recall that by an algebraic model we mean aring that can be computated by purely algebraic methods and is closely related to h T ( X ) ∧ . Theorem 3.6. R J CDiv T ( X ) K F /I Σ is our algebraic model for the completed equi-variant cohomology ring h T ( X ) ∧ .Proof. By Proposition 3.2 and Corollary 3.5 we have the following exact sequenceof R J T ∗ K F -modules0 / / R J CDiv T ( X ) K F /I Σ / / Y τ ∈ Σ max R J T ∗ τ K F / / Y τ = τ ′ R J T ∗ τ ∩ τ ′ K F . Notice that this is precisely the exact sequence of the sheaf axiom, where the sub-fans induced by τ , τ varies over Σ max , form an open covering of Σ. Hence the R J T ∗ K F -algebra R J CDiv T ( X ) K F /I Σ is the ring of global sections and is our alge-braic model of h T ( X ) ∧ . We remark that similar exact sequences for equivariantsingular cohomology and equivariant K -theory can be found in [5, Chapter 12] and[1] respectively. (cid:3) Remark 3.7. Let k be a field of characteristic 0 and h = Ω ∗ be the algebraiccobordism. The formal group law associated to Ω ∗ is the univeral formal grouplaw, and the coefficient ring is the Lazard ring L . As we see in Example 4.6, ouralgebraic model L J CDiv T ( X ) K F /I Σ is isomorphic to Ω ∗ T ( X ). Since Ω ∗ is the uni-versal oriented cohomology theory, it follows from the functoriality of formal grouprings that we will have isomorphisms between R J CDiv T ( X ) K F /I Σ and h T ( X ) ∧ forall other oriented cohomology theories as well.One of the main advantages of our construction is that it still works when k is ofcharacteristic p > 0. We are still able to construct an algebraic model for h T ( X ) ∧ ,while on the other hand there is no universal theory for us to specialize from.Next we would like to study the usual cohomology of X . Recall that R is a R J T ∗ K F -algebra via the augmentation map. Then we have an isomorphism( R J CDiv T ( X ) K F /I Σ ) ⊗ R J T ∗ K F R ∼ = R J CDiv T ( X ) K F /J Σ where J Σ is the ideal generated I Σ and P ρ ∈ Σ(1) [ h α, v ρ i ] F x D ρ over all α ∈ T ∗ . Thisconstruction corresponds to the idea that the usual cohomology ring is a quotient ofthe equivariant cohomology ring, where the corresponding results for Chow group,algebraic K -theory and algebraic cobordism are proved in [4, Corollary 2.3], [16,Proposition 28] and [13, Theorem 8.1] respectively.It is known that there is the following exact sequence T ∗ / / CDiv T ( X ) / / Pic( X ) / / ORMAL GROUP RINGS OF TORIC VARIETIES 7 where the first homomorphism is defined before Lemma 3.1, and the second homo-morphism sends a T -invariant Cartier divisor to its class in the Picard group. Lemma 3.8. R J CDiv T ( X ) K F ⊗ R J T ∗ K F R is isomorphic to R J Pic( X ) K F .Proof. First, if we identify the lattices T ∗ and CDiv T ( X ) with Z m and Z m ′ re-spectively, the homomorphism T ∗ → CDiv T ( X ) is given by a m ′ × m matrix withcoefficients in Z . Since every matrix with coefficients in Z has a Smith normal form,it means that we can choose a new Z -basis { u , . . . , u m } of T ∗ and a new Z -basis { u ′ , . . . , u ′ m ′ } of CDiv T ( X ) such that the homomorphism is given by u i ( a i u ′ i if 1 ≤ i ≤ s s + 1 ≤ i ≤ m for some integer s , and a | · · · | a s are positive integers (notice that the value 1 isallowed). Then Pic( X ) is isomorphic to Z /a Z ⊕ · · · ⊕ Z /a s Z ⊕ Z m ′ − s .By using Theorem 2.11, Corollary 2.13 and Example 2.15 of [6], R J T ∗ K F ∼ = R J x , . . . , x m K R J CDiv T ( X ) K F ∼ = R J x ′ , . . . , x ′ m ′ K R J Pic( X ) K F ∼ = R J x ′ , . . . , x ′ m ′ K / h a i · F x ′ i | i = 1 , . . . s i . The formal group ring homomorphism induced by T ∗ → CDiv T ( X ) is given by x i ( a i · F x ′ i if 1 ≤ i ≤ s s + 1 ≤ i ≤ m, and our lemma follows immediately. (cid:3) Corollary 3.9. The following R -algebras are isomorphic:1. ( R J CDiv T ( X ) K F /I Σ ) ⊗ R J T ∗ K F R .2. R J CDiv T ( X ) K F /J Σ .3. R J Pic( X ) K F /I Σ , where I Σ is the image of I Σ under the surjective homomorphism R J CDiv T ( X ) K F → R J Pic( X ) K F .The R -algebras above are our algebraic model of h ( X ) ∧ . Remark 3.10. If F is a polynomial formal group law, the subring R [ T ∗ σ ] F isan algebraic substitute of the equivariant cohomology ring h T ( U σ ) by homotopyinvariance and the fact that X is smooth. Then we want to “glue” R [ T ∗ τ ] F overall maximal cones τ ∈ Σ max together. By the definition of R [ T ∗ σ ] F and Remark3.6, the R [ T ∗ ] F -algebra obtained from “gluing” is R [CDiv T ( X )] F /I Σ , which willbe our algebraic model of h T ( X ). As a result, ( R [CDiv T ( X )] F /I Σ ) ⊗ R [ T ∗ ] F R ∼ = R [CDiv T ( X )] F /J Σ is an algebraic model of h ( X ). Example 3.11. (see [13, Example 8.3]) As a first example we let F to be anyformal group law, and we consider X = P n , where Σ ⊆ T ∗ ∼ = Z n is the complete fanconsisting of the n + 1 rays ρ , . . . , ρ n +1 generated by v = e , . . . , v n = e n , v n +1 = − e − · · · − e n . Here { e , . . . , e n } is the standard basis of Z n . Then it is easy to seethat R J CDiv T ( X ) K F /I Σ ∼ = R J x , . . . , x n +1 K / h x · · · x n +1 i . Although the right hand side is independent of the formal group law F , the iso-morphism depends on F , see [6, Remark 2.14]. WANSHUN WONG Let { α , . . . , α n } be the basis of T ∗ dual to { e , . . . , e n } . The R J T ∗ K F -algebrastructure of R J x , . . . , x n +1 K / h x · · · x n +1 i is given by x α i n +1 X j =1 [ h α i , v j i ] F x j = x i − F x n +1 = x i − x n +1 + x i x n +1 f ( x i , x n +1 )= x i − u i x n +1 where f is some power series determined by the formal group law F , and u i =1 − x i f ( x i , x n +1 ) is a unit in R J x , . . . , x n +1 K / h x · · · x n +1 i . Therefore R J CDiv T ( X ) K F /J Σ ∼ = R J x , . . . , x n +1 K h x · · · x n +1 , x − u x n +1 , . . . , x n − u n x n +1 i∼ = R J x n +1 K / h x n +1 n +1 i∼ = R [ x n +1 ] / h x n +1 n +1 i . Comparison Results In the present section we compute the formal group rings of a smooth toricvariety X for different formal group laws. Then we compare them with knownresults of equivariant cohomology of smooth toric varieties to justify the validity ofour models, and we also obtain new results. Example 4.1. When F is the additive formal group law F ( x, y ) = x + y over R ,we recall that R [ M ] F is isomorphic to the ring of symmetric powers Sym R ( M ) over R . The corresponding oriented cohomology theory is the Chow ring of algebraiccycles modulo rational equivalence, with coefficient ring CH ∗ (pt) = Z .Take R = Z . For every maximal cone τ , Z [ T ∗ τ ] F is isomorphic to Sym Z ( T ∗ τ ),which can be viewed as the ring of integral polynomial functions on τ . Then theabove “gluing” process means that the following Sym Z ( T ∗ )-algebras are isomorphic:(a) Z [CDiv T ( X )] F /I Σ .(b) Z [ x D ρ ] / h Q ρ ∈ S x D ρ i , where x D ρ are indeterminates over ρ ∈ Σ(1), and the idealis generated over all subsets S ⊆ Σ(1) such that S * σ (1) for any cone σ .(c) the algebra of integral piecewise polynomial functions on Σ.The isomorphism between (b) and (c) is given by mapping x D ρ to the uniquepiecewise polynomial function ϕ ρ satisfying(i) ϕ ρ is homogeneous of degree 1,(ii) ϕ ρ ( v ρ ) = 1, ϕ ρ ( v ρ ′ ) = 0 for all ρ ′ ∈ Σ(1) , ρ ′ = ρ .This coincides with the description of the equivariant Chow ring CH ∗ T ( X ) by [4]and [17]. Remark 4.2. It is known that for any smooth variety X , the natural homomor-phism Pic( X ) → CH n − ( X ) is an isomorphism, where CH n − ( X ) is the (usual)Chow group of ( n − X is a smoothtoric variety, this isomorphism can be recovered as follows:First note that since X is smooth, we have CH ( X ) = CH n − ( X ). When F isthe additive formal group law, we can modify the proof of Lemma 3.8 to show that R [CDiv T ( X )] F ⊗ R [ T ∗ ] F R ∼ = R [Pic( X )] F . Therefore Z [Pic( X )] F /I Σ coincides with CH ∗ T ( X ) ⊗ Sym Z ( T ∗ ) Z = CH ∗ ( X ), the usual Chow ring of X . Then our result follows ORMAL GROUP RINGS OF TORIC VARIETIES 9 by comparing the degree 1 elements of the two graded rings, where R [Pic( X )] F /I Σ is given the natural grading as a quotient of a polynomial ring. Example 4.3. When F is the multiplicative periodic formal group law F ( x, y ) = x + y − βxy over R , where β ∈ R × , we have seen that R [ M ] F is isomorphicto the group ring R [ M ] = { P r i e λ i | r i ∈ R, λ i ∈ M } . The corresponding ori-ented cohomology theory is the K -theory that assigns every smooth variety Y to K ( Y )[ β, β − ], where K ( Y ) denotes the Grothendieck group of vector bundles on Y . The coefficient ring is K (pt)[ β, β − ] = Z [ β, β − ].Take R = Z and β = 1. For every maximal cone τ , Z [ T ∗ τ ] F can be viewed as thering of integral exponential functions on τ . Therefore the following Z [ T ∗ ]-algebrasare isomorphic:(a) Z [CDiv T ( X )] F /I Σ .(b) Z [ e ± D ρ ] / h Q ρ ∈ S (1 − e D ρ ) i , where the ideal is generated over all subsets S ⊆ Σ(1)such that S * σ (1) for any cone σ .(c) the algebra of integral piecewise exponential functions on Σ.The isomorphism between (b) and (c) is given by mapping 1 − e D ρ to the piecewisefunction 1 − e ϕ ρ , where the notation means that on each cone σ ∈ Σ,(1 − e ϕ ρ ) σ = 1 − e ( ϕ ρ ) σ , and ϕ ρ is the piecewise polynomial function defined in the previous example. Thisdescription agrees with that of the Grothendieck group of equivariant vector bundles K T ( X ) by [2] and [18, Theorem 6.4]. Example 4.4. Let F be the multiplicative formal group law over R , given by F ( x, y ) = x + y − vxy , where v is not required to be a unit. If v = β ∈ R × ,then clearly we obtain the multiplicative periodic formal group law of the previousexample. If v / ∈ R × , then the multiplicative formal group law is non-periodic. Inparticular, if v = 0 we get the additive formal group law.The oriented cohomology theory corresponding to F is the connective K -theory.It is the universal oriented cohomology theory for Chow ring and K -theory, byspecializing at v = 0 and v = β ∈ R × respectively. The coefficient ring for theconnective K -theory is Z [ v ].The following construction is motivated by the result in [12]. Consider the groupring R [ M ] = { P r i e λ i | r i ∈ R, λ i ∈ M } , and let tr : R [ M ] → R be the trace map,the R -linear map defined by mapping any e λ to 1. The ideal I = ker(tr) is generatedby 1 − e λ over λ ∈ M . Then we consider the Rees ring of R [ M ] with respect to IR = Rees( R [ M ] , I ) = ∞ X n = −∞ I n t − n = R [ M ][ t, I t − ] ⊆ R [ M ][ t, t − ]where t is an indeterminate, and I n = R [ M ] if n ≤ 0. We have the R -algebraisomorphisms R [ M ] F ∼ = R / ( t − v ) R and R J M K F ∼ = ( R / ( t − v ) R ) ∧ induced by x λ (1 − e λ ) t − , and e λ − vx λ , (1 − e λ ) t − x λ . Here the barmeans the image of an element in the quotient ring R / ( t − v ) R , and ( R / ( t − v ) R ) ∧ is the completion of R / ( t − v ) R at the ideal generated by (1 − e λ ) t − . Specializing at v = 0 and v = β ∈ R × , we have R /t R ∼ = gr I R [ M ] R / ( t − β ) R ∼ = R [ M ]where gr I R [ M ] is the associated graded ring of R [ M ] with respect to I . Noticethat gr I R [ M ] is also isomorphic to Sym R ( M ) via 1 − e λ λ . Therefore we recoverthe previous two examples.As a simple, concrete example for the case v / ∈ R × and v = 0, consider R = Z , v = 2, and M = Z . By direct computation we see that Z [ Z ] F ∼ = Rees( Z [ Z ] , I ) / ( t − 2) Rees( Z [ Z ] , I ) ∼ = Z [ x, x ′ ] / h x + x ′ − xx ′ i where the second isomorphism is induced by (1 − e ) t − x, (1 − e − ) t − x ′ .Back to our study of toric varieties. Our “gluing” process above shows that Z [ v ][CDiv T ( X )] F /I Σ is isomorphic to a ring of tuples of elements in the quotientof Rees rings, where the compatability condition for the tuples of elements hold.This provides a conjecture for the equivariant connective K -theory ring of X . Example 4.5. Let char( R ) = 2, and let F be the Lorentz formal group law over R , given by F ( x, y ) = x + y u xy for some u ∈ R . If u = 0, then we just recover the additive formal group law.If u = 0, then F is an elliptic formal group law, and the corresponding orientedcohomology theory is an elliptic cohomology with coefficient ring Z [ u ]. We remarkthat F also appears in the theory of special relativity as the formula of relativisticaddition of parallel velocities, where u is taken to be c , the reciprocal of the speedof light.Even though F is not a polynomial formal group law, we can still study R [ M ] F .Once again we consider the group ring R [ M ]. Denote by S the multiplicative subsetof R [ M ] generated by e λ + e − λ over all λ ∈ M , and we let H to be the R -subalgebraof S − R [ M ] generated by 1 and the “hyperbolic tangents” e λ − e − λ e λ + e − λ over all λ ∈ M .Finally we let I be the ideal of H generated by all e λ − e − λ e λ + e − λ , and similar to theprevious example we consider the Rees ring of H with respect to IR = Rees( H , I ) = ∞ X n = −∞ I n t − n = H [ t, t − I ] ⊆ H [ t, t − ] . We have the R -algebra isomorphisms R [ M ] F ∼ = R / ( t − u ) R and R J M K F ∼ = ( R / ( t − u ) R ) ∧ induced by x λ e λ − e − λ e λ + e − λ t − and e λ − e − λ e λ + e − λ ux λ , e λ − e − λ e λ + e − λ t − x λ , ORMAL GROUP RINGS OF TORIC VARIETIES 11 where R / ( t − u ) R ) ∧ is the completion at the ideal generated by e λ − e − λ e λ + e − λ t − . Spe-cializing at u = 0 and u = β ∈ R × , we get R /t R ∼ = gr I HR / ( t − β ) R ∼ = H . Notice that the associated graded ring gr I H is naturally isomorphic to the ring ofsymmetric powers Sym R ( M ) by construction, hence we recover the example for theadditive formal group law again.Just like the previous example, we see that for a smooth toric variety X andthe Lorentz formal group law F , Z [ u ][CDiv T ( X )] F /I Σ is isomorphic to a ring oftuples of elements in the quotient of Rees rings, where the compatability conditionfor the tuples of elements hold. Example 4.6. When F is the universal formal group law, the corresponding ori-ented cohomology theory is the algebraic cobordism (defined over a base field ofcharacteristic 0). The coefficient ring is the Lazard ring L .Similar to the first two examples, our “gluing” process shows that the following L J T ∗ K F -algebras are isomorphic:(a) L J CDiv T ( X ) K F /I Σ .(b) L J x D ρ K / h Q ρ ∈ S x D ρ i , where x D ρ are indeterminates over ρ ∈ Σ(1), and the idealis generated over all subsets S ⊆ Σ(1) such that S * σ (1) for any cone σ .(c) the algebra of piecewise power series on Σ with coefficients in L .Our result agrees with the description of the equivariant cobordism ring Ω ∗ T ( X ) in[11] and [13].To conclude this section we have a concrete example demonstrating how theabove comparison results apply. Example 4.7. Consider the del Pezzo surface of degree 6 dP , obtained by blowing-up the three T -fixed points p , p , p of P . We begin by recalling some classicalresults of dP (for example, see [15]). The fan Σ of dP , Σ ⊆ T ∗ ∼ = Z , consists ofsix 2-dimensional maximal cones and all their faces, and the rays are generated by(0 , , (1 , , (1 , , (0 , − , ( − , − , ( − , L E L E L E The T -invariant divisors corresponding to the six rays are precisely the six excep-tional curves E , E , E , L , L , L on dP : E i is the exceptional curve inducedby blowing-up p i , and L i is the strict transform of the unique line in P passingthrough p j and p k , i, j, k all distinct. { E , E , E , L , L , L } is a basis of the lattice CDiv T ( dP ), and T ∗ injects intoCDiv T ( dP ) by x E + L − L − E y L + E − E − L where { x, y } is the basis of T ∗ dual to the standard basis { (1 , , (0 , } of T ∗ ∼ = Z .It follows that Pic( dP ) is a rank 4 lattice with basis { ℓ, E , E , E } , where ℓ = L + E + E = L + E + E = L + E + E . The intersection pairing h , i : Pic( dP ) × Pic( dP ) → Z is determined by h ℓ, ℓ i = 1 , h ℓ, E i i = 0 , h E i , E j i = − δ ij where δ ij is the usual Kronecker delta function.Now we go back to our study of equivariant cohomology on dP . For all theexpressions below, we always assume the subindices i, j ∈ { , , } , i = j . (i) When F is the additive formal group law over Z , the following Z [ x, y ]-algebrasare isomorphic:(a) CH ∗ T ( dP ).(b) Z [CDiv T ( dP )] F /I Σ .(c) Z [ L , L , L , E , E , E ] / h L i L j , E i E j , L i E i i .(d) the algebra of integral piecewise polynomial functions on Σ.As a corollary, the following rings are isomorphic:(a) CH ∗ ( dP ).(b) ( Z [CDiv T ( dP )] F /I Σ ) ⊗ Z [ x,y ] Z .(c) Z [Pic( dP )] F /I Σ .(d) Z [ ℓ, E , E , E ] / h ℓ + E i , E i E j , ℓE i i .Notice that the relations ℓ + E i = E i E j = ℓE i = 0 in the last ring agree with thevalues of the intersection pairing on Pic( dP ). (ii) When F is the multiplicative periodic formal group law over Z with β = 1, thefollowing Z [ e ± x , e ± y ]-algebras are isomorphic:(a) K T ( dP ).(b) Z [CDiv T ( dP )] F /I Σ .(c) Z [ e ± L , e ± L , e ± L , e ± E , e ± E , e ± E ] h (1 − e L i )(1 − e L j ) , (1 − e E i )(1 − e E j ) , (1 − e L i )(1 − e E i ) i .(d) the algebra of integral piecewise exponential functions on Σ.Similar to part (i), we see that the following rings are isomorphic:(a) K ( dP ).(b) ( Z [CDiv T ( dP )] F /I Σ ) ⊗ Z [ e ± x ,e ± y ] Z .(c) Z [ e ± ℓ , e ± E , e ± E , e ± E ] h (1 − e ℓ ) + (1 − e − E i ) , (1 − e E i )(1 − e E j ) , (1 − e ℓ )(1 − e E i ) i . (iii) In general, for arbitrary formal group law F over a ring R , we have(3) R J CDiv T ( dP ) K F /I Σ ∼ = R J x L , x L , x L , x E , x E , x E K h x L i x L j , x E i x E j , x L i x E i i . ORMAL GROUP RINGS OF TORIC VARIETIES 13 There are a couple of useful arithmetic identities in this algebra. Recall that anyformal group law F can be expressed as F ( x, y ) = x + y − xy · g ( x, y )for some power series g ( x, y ). Then for example we see that x E i + E j = x E i + F x E j = x E i + x E j − x E i x E j g ( x E i , x E j ) = x E i + x E j , and similarly x E i − E j = x E i − F x E j = x E i + F χ ( x E j )= x E i + χ ( x E j ) − x E i χ ( x E j ) g ( x E i , χ ( x E j ))= x E i + x − E j , where χ ( z ) is the unique power series such that z + F χ ( z ) = 0. It follows that wehave ( x E i + E j + E k ) n = ( x E i + x E j + x E k ) n = x nE i + x nE j + x nE k for any positive integer n . Clearly we have the corresponding identities for the x L i ’sas well.Tensoring the isomorphism (3) with R over R J T ∗ K F , we obtain(4) R J Pic( dP ) K F /I Σ ∼ = R J x ℓ , x E , x E , x E K / h x ℓ + χ ( x E i ) , x E i x E j , x ℓ x E i i . By direct computation we see that x ℓ = x E = x E = x E = 0in this ring, which is an expected result as dP is a surface. This allows us tosimplify the expression in (4). Let a i,j ∈ R be the coefficients of the formal grouplaw F F ( x, y ) = X i,j a i,j x i y j . Then by [14, Equation (2.3)] the power series χ ( z ) is of the form χ ( z ) = − z + a , z − ( a , ) z + terms of degree ≥ . Hence we have R J Pic( dP ) K F /I Σ ∼ = R J x ℓ , x E , x E , x E K / h x ℓ + x E i , x E i x E j , x ℓ x E i i . We remark that the isomorphism depends on the formal group law F , even thoughthe right hand side is independent of F .5. Pull-back and push-forward formula of blow-up Let X be the smooth toric variety of the fan Σ, σ ∈ Σ \ Σ(1) be a cone, and X ′ be the blow-up of X along the orbit closure O σ . Then X ′ is smooth toric varietywhose fan Σ ′ is equal to the star subdivision of Σ relative to σ ,Σ ′ = { θ ∈ Σ | σ * θ } ∪ [ σ ⊆ θ Σ ∗ ( θ )where we let v σ = P ρ ∈ σ (1) v ρ , andΣ ∗ ( θ ) = { cone( S ) | S ⊆ { v σ } ∪ θ (1) , σ (1) * S } . The fan Σ ′ is a refinement of Σ, and the induced toric morphism π : X ′ → X is projective. Therefore for every equivariant cohomology theory h T , π inducesthe pull-back homomorphism π ∗ : h T ( X ) → h T ( X ′ ) and also the push-forward homomorphism π ∗ : h T ( X ′ ) → h T ( X ). Notice that π ∗ is a h T (pt)-algebra homo-morphism, and π ∗ is a h T ( X )-module homomorphism, where h T ( X ) acts on h T ( X ′ )via π ∗ . Our goal in this section is to define two homomorphisms of formal grouprings that will serve as algebraic substitutes of the pull-back and the push-forwardhomomorphisms. We remark that formulas for the pull-back and the push-forwardof equivariant Chow rings are proved in [3, Theorem 2.3].First, let E be the T -invariant prime divisor on X ′ corresponding to the ray e ρ generated by v σ . Then we haveCDiv T ( X ′ ) ∼ = CDiv T ( X ) ⊕ Z · E, where the T -invariant prime divisor D ρ, Σ ′ on X ′ corresponding to ρ ∈ Σ ′ is thestrict transform of the divisor D ρ, Σ on X corresponding to ρ ∈ Σ. We denote both D ρ, Σ ′ and D ρ, Σ by D ρ if the ambient fan is clear from the context.Next, we want to define the pull-back homomorphism for formal group rings.Informally speaking from the point of view of piecewise functions on fans, π ∗ : h T ( X ) → h T ( X ′ ) is given by treating piecewise functions on Σ as piecewise func-tions on the refinement Σ ′ . Translating this to the language of formal group rings,we define π ∗ : R J CDiv T ( X ) K F /I Σ −→ R J CDiv T ( X ′ ) K F /I Σ ′ x D ρ ( x D ρ if ρ / ∈ σ (1) x D ρ + E = x D ρ + F x E if ρ ∈ σ (1) . The underlying geometric meaning can again be explained by the Orbit-Cone Cor-respondence: ρ ∈ σ (1) if and only if O σ ⊆ D ρ .For the push-forward homomorphism, we impose the condition that F is the as-sociated formal group law of a birationally invariant theory h , i.e the push-forwardof the fundamental class satisfies f ∗ (1 Y ) = 1 X for any birational projective mor-phism f : Y → X between smooth irreducible varieties. Examples of birationallyinvariant theories include Chow ring over an arbitrary field, K -theory over a fieldof characteristic 0. It is known that the connective K -theory over a field of char-acteristic 0 is univeral among all birationally invariant theories, see [14, Theorem4.3.9] and [6, Example 8.10]. Therefore from now on we assume F is of the form F ( x, y ) = x + y − vxy for some v ∈ R .Now we begin the construction of the push-forward π ∗ : R J CDiv T ( X ′ ) K F /I Σ ′ → R J CDiv T ( X ) K F /I Σ , which is a homomorphism of R J CDiv T ( X ) K F /I Σ -modules. Asthe blow-up morphism π is birational and projective, by our assumption on F wedefine π ∗ (1) = 1. Since D ρ, Σ ′ is the strict transform of D ρ, Σ , we define π ∗ ( x D ρ, Σ ′ ) = x D ρ, Σ .From here we can deduce that π ∗ ( − F x E ) = 0, and the proof is as follows: Let ρ ∈ σ (1) be a ray in σ ⊆ Σ, and consider x D ρ, Σ ′ + F x E = π ∗ ( x D ρ, Σ ). By theprojection formula,(5) π ∗ ( x D ρ, Σ ′ + F x E ) = π ∗ ( π ∗ ( x D ρ, Σ )) = π ∗ (1) x D ρ, Σ = x D ρ, Σ . On the other hand, π ∗ (cid:16) ( x D ρ, Σ ′ + F x E ) − F x E (cid:17) = π ∗ ( x D ρ, Σ ′ ) = x D ρ, Σ ORMAL GROUP RINGS OF TORIC VARIETIES 15 as well. By subtracting the two equations, we get π ∗ (cid:16) ( − F x E ) − v ( x D ρ, Σ ′ + F x E )( − F x E ) (cid:17) = π ∗ (cid:16)(cid:0) − v ( x D ρ, Σ ′ + F x E ) (cid:1) ( − F x E ) (cid:17) = π ∗ (cid:16) π ∗ (cid:0) − vx D ρ, Σ (cid:1) ( − F x E ) (cid:17) = (1 − vx D ρ, Σ ) π ∗ ( − F x E )= 0 . As 1 − vx D ρ, Σ is a unit in R J CDiv T ( X ) K F /I Σ , we have π ∗ ( − F x E ) = 0. Notice thatin general π ∗ ( x E ) = 0.If dim σ = 2, then by the property of a R J CDiv T ( X ) K F /I Σ -module homomor-phism we can already determine the image of every element in R J CDiv T ( X ′ ) K F /I Σ ′ .Let σ (1) = { ρ , ρ } ⊆ Σ(1). After the star subdivision { ρ , ρ } * θ (1) for any cone θ ∈ Σ ′ , hence x D ρ , Σ ′ x D ρ , Σ ′ = 0 in R J CDiv T ( X ′ ) K F /I Σ ′ . Then for example, π ∗ ( x D ρ , Σ ) x D ρ , Σ ′ = ( x D ρ , Σ ′ + x E − vx D ρ , Σ ′ x E ) x D ρ , Σ ′ (6) = x E x D ρ , Σ ′ π ∗ ( π ∗ ( x D ρ , Σ ) x D ρ , Σ ′ ) = x D ρ , Σ x D ρ , Σ . It follows from equations (5) and (6) that π ∗ ( x D ρ , Σ ′ + F x E ) = x D ρ , Σ + π ∗ ( x E ) − vx D ρ , Σ x D ρ , Σ = x D ρ , Σ , therefore π ∗ ( x E ) = vx D ρ , Σ x D ρ , Σ . Theorem 5.1. Let dim σ = 2 and σ (1) = { ρ , ρ } ⊆ Σ(1) . We use x i to denoteboth x D ρi, Σ ′ and x D ρi, Σ , i = 1 , . Then the push-forward homomorphism definedabove satisfies π ∗ : R J CDiv T ( X ′ ) K F /I Σ ′ −→ R J CDiv T ( X ) K F /I Σ x E vx x x nE v n X i =1 x n +1 − i x i − n − X i =1 x n − i x i x sa x tE x a x tb ( x a − F x b ) s − ,where n ≥ , s ≥ , t ≥ , and { a, b } = { , } .Proof. First, by induction on t we see that π ∗ ( x a x tE ) = π ∗ (( x b + x E − vx b x E ) x a x t − E )(7) = π ∗ ( π ∗ ( x b ) x a x t − E )= x a x tb , where the first equality follows from the fact that x a x b = 0 in R J CDiv T ( X ′ ) K F /I Σ ′ .To compute the image of x nE , we use induction on n , and compare the image of π ∗ ( x a ) x n − E computed by two different methods: π ∗ ( π ∗ ( x a ) x n − E ) = x a π ∗ ( x n − E )and π ∗ ( π ∗ ( x a ) x n − E ) = π ∗ (( x a + x E − vx a x E ) x n − E ) = x a x n − b + π ∗ ( x nE ) − vx a x nb . Next we want to compute the image of x a . Let χ ( z ) be the unique power seriessuch that z + F χ ( z ) = 0. Then π ∗ ( x a ) = π ∗ (( π ∗ ( x a ) − F x E ) x a )= π ∗ (( π ∗ ( x a ) + χ ( x E ) − vπ ∗ ( x a ) χ ( x E )) x a )= x a + χ ( x b ) x a − vx a χ ( x b ) x a = x a ( x a − F x b ) , where the third equality follows from equation (7). By using the same trick for π ∗ ( x a x tE ), we see that π ∗ ( x a x tE ) = x a x tb ( x a − F x b )for every t ≥ 0. This allows us to compute π ∗ ( x a ), and then π ∗ ( x a x tE ), by the sameargument as above. The general case now follows from induction. (cid:3) Remark 5.2. In our case where the formal group law F is of the form F ( x, y ) = x + y − vxy , we have an explicit description of the power series χ ( z ): χ ( z ) = − z ∞ X i =0 ( vz ) i = − ∞ X i =0 v i z i +1 . Hence − F x E = − P ∞ i =0 v i x i +1 E . Then by using the formula in Proposition 5.1, werecover the result π ∗ ( − F x E ) = 0 when dim σ = 2.If dim σ = 3, let σ (1) = { ρ , ρ , ρ } ⊆ Σ(1) and use x i to denote both x D ρi, Σ ′ and x D ρi, Σ , i = 1 , , 3. We further define π ∗ ( x i x j ) = x i x j for every i, j ∈ { , , } such that i = j . Then the image for the rest of the elementsin R J CDiv T ( X ′ ) K F /I Σ ′ can be computed by the same technique as above. 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Department of Mathematics and Statistics, University of Ottawa, Ontario, Canada E-mail address ::