Cohomology groups of toric varieties

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1 Cohomology groups of toric varieties Masanori Ishida Mathematical Institute, Tohoku University 1 Fans and Complexes Although we treat real fans later, we begin with fans consisting of rational cones which define toric varieties. Let r be a nonnegative integer, N Z r, M = Hom Z (N, Z), N R = N Z R and M R = M Z R. Let, : M R N R R be the natural perfect pairing. We assume that cones in N R are strongly convex rational polyhedral cones. We consider only finite fans here. Definition 1.1 A subset Φ of a fan Σ is said to be (1) star closed if σ Φ, τ Σ and σ τ imply τ Φ, (2) star open if τ Φ, σ Σ and σ τ imply σ Φ and (3) locally star closed if σ, ρ Φ, τ Σ and σ τ ρ imply τ Φ. A subset Φ Φ of a lcally star closed subset Φ Σ of a fan is defined to be star closed, star open and locally star closed similarly. In any of these cases, Φ is a locally closed subset of Σ. For a cone σ in N R, we define σ = {x M R ; x, a = 0, a σ}, M[σ] = M σ and N[σ] = N/(N (σ + ( σ))). Then N[σ] and M[σ] are mutually dual free Z-module of rank codim σ = r dim σ. We set Z(σ) = r dim σ M[σ], which is a free Z-module of rank one. Here we understand Z(σ) = Z if r dim σ = 0. For a nonnegative integer p, we define Σ(p) = {σ Σ ; dim σ = p} and Φ(p) = Φ Σ(p) if Φ Σ is a locally star closed subset. For a locally star closed subset Φ and elements σ, ρ Σ, we define (1) (2) (3) Φ(σ ) = {τ Φ ; σ τ}, Φ( ρ) = {τ Φ ; τ ρ}, Φ(σ ρ) = {τ Φ ; σ τ ρ}, respectively. When σ Σ(r p), τ Σ(r p + 1) and σ τ, the isomorphism q σ/τ : Z(σ) Z(τ) is defined as follows. Since M[τ] M[σ] and M[σ]/M[τ] Z by definition, we can take an element n 1 N which is zero on M[τ] and maps M[σ] τ onto Z 0 = {c Z ; c 0}. 1

2 Here τ is the dual cone of τ defined by {x M R ; x, a 0, a τ}. Then, for m 1 M[σ] and m 2,..., m p M[τ], we define q σ/τ (m 1 m 2 m p ) = m 1, n 1 m 2 m p. This definition does not depend on the choice of n 1. Lemma 1.2 ([I1]) If σ Σ(r p) is a face of ρ Σ(r p + 2), there exist exactly two cone τ Σ(r p + 1) with σ τ ρ. If we denote them τ 1, τ 2, the the equality q τ1 /ρq σ/τ1 + q τ2 /ρq σ/τ2 = 0 holds. Let A be an abelian category. For simplicity, we sometimes assume that objects of A are modules or sheaves of modules. Each fan Σ is regarded as a category where the morphisms are only inclusions i σ/τ : σ τ of cones. For a covarinat functor G : Σ A and a locally closed subset Φ Σ, the complex C (Φ, G) in A is defined as follows. For each integer p, we set C p (Φ, G) = G(σ) Z(σ). σ Φ(r+p) Note that C i (Φ, G) is the direct sum for σ Φ which corresponds to the orbit O(σ) of dimension i in the toric variety Z(Σ) associated to the fan Σ. If p < r or 0 < p, then Φ(r + p) = and hence C p (Φ, G) = {0}. The component of the coboundary map d p : C p (Φ, G) C p+1 (Φ, G) for G(σ) Z(σ) and G(τ) Z(τ) is G(i σ/τ ) q σ/τ if σ τ and the zero map otherwise. By Lemma 1.2, we have d p+1 d p = 0 for every p. For a contravariant functor G : Σ A, the complex C (Φ, G ) is defined similarly. For each σ Σ, we set Z(σ) = Hom Z (Z(σ), Z). If σ Σ(r p 1) and τ Σ(r p) satisfy σ τ, then the isomorphism q τ/σ : Z(τ) Z(σ) is defined by q τ/σ = ( 1)r p 1 (q σ/τ ). For each integer p, we set C p (Φ, G ) = The component of the coboundary map τ Φ(r p) G (τ) Z(σ). d p : C p (Φ, G ) C p+1 (Φ, G ) for G (τ) Z(τ) and G (σ) Z(σ) is defined to be G (i σ/τ ) qτ/σ if σ τ and the zero map otherwise. In case of contravarinat functor, C p (Φ, G ) = {0} for p < 0 and r < p. We denote simply Z the covariant functor G such that G(σ) = Z for every σ Σ and G(i σ/τ) is the identity map of Z for every pair of cones σ, τ Σ with σ τ. Then C (Φ, Z) is a finite complex of free Z-modules of finite rank. 2

3 For ρ Σ, we set Σ(ρ ) = {σ Σ ; ρ σ}. We denote by σ[ρ] the image of σ Σ(ρ ) in N[ρ] R = N R /(ρ + ( ρ)). Then we get a bijection from Σ(ρ ) to Σ[ρ] = {σ[ρ] ; σ Σ(ρ )} which preserve the orders. In particular, if Φ is a locally star closed subset of Σ(ρ ), then we get a locally star closed subset Φ[ρ] = {σ[ρ] ; σ Φ} of Σ[ρ]. Since M[σ[ρ]] = M[σ] M[ρ] if ρ σ, we have Z(σ[ρ]) = Z(σ). Since the degrees of the complex is defined by the codimensions of the cones, the complex C (Φ, G) is canonically isomorphic to C (Φ[ρ], G[ρ]) for any covariant or contravariant functor G from F (π) to an abelian category. From here, we set V = N R and V = M R, and we consider cones in V without the condition of rationality. Namely, we assume cone in V to be polyhedral and strongly convex but not necessarily rational. A set Σ of cones in V is said to be a real fan if (1) Σ is not empty, (2) σ Σ and ρ σ imply ρ Σ and (3) σ, τ Σ implies that σ τ is a common face of σ and τ. A subset Φ of a real fan Σ is said to be star closed, star open or locally star closed similarly as in the case of a fan with the rationality condition. For a cone σ in V, we set V [σ] = σ + ( σ) and V [σ] = σ. If d = dim σ, then V [σ] and V [σ] are mutually dual real vector spaces of dimension r d. Since det V [σ] = r d V [σ] is a real vector of dimension one, det V [σ] \ {0} has exactly two connected components. Each connected component is called an orientation of V [σ]. For an orientation e σ, we define Z(σ) = Ze σ. By denoting the other orientation by e σ, we get the definition of Z(σ) which does not depend on the choice of the orientation. When σ τ and dim τ = dim σ+1, we take x (τ σ )\τ and y det V [τ] \{0}. Let e σ be the orientation of V [σ] which contains x y and e τ be that of V [τ] which contains y. Then the isomorphism of the modules q σ/τ : Z(σ) Z(τ) is defined by q σ/τ (ae σ ) = ae τ for a Z. By this definition, the complex C (Φ, G) is defined for a locally star closed subset Φ and a covariant or contravariant functor G from Φ to an abelian category similarly as in the case of fans with the rationality condition. If Σ is rational, the complex C (Φ, G) defined here is equal to the previous one by identifying both Z(σ) for every σ Φ. Definition 1.3 A locally closed subset Φ Σ is said to be homologically trivial if the cohomology group H i (C (Φ, Z)) is zero for every i Z. Since C (Φ, Z) is a finite complex of free modules, C (Φ, Z) is a split exact sequence if Φ is homologically trivial. In particular, the all cohomologies of the complex C (Φ, Z) Z A is zero for any abelian group A. If a locally star closed subset Φ is contained in Σ(ρ ), then Φ is homologically trivial if and only if so is Φ[ρ] Σ[ρ] since C (Φ, G) is isomorphic to C (Φ[ρ], G[ρ]). Let π be a cone of V and F (π) be the real fan consisting of all faces of π, which we call affine fan or affine real fan. For a cone π and x V, we define F (π) x = {σ F (π) ; x, u 0, u σ}, which is a star open subset of F (π). We set F (π) x (1) = F (π) x F (π)(1). If x, u 0 is 3

4 satisfied for all points of one-dimensional faces of σ, then it holds for all points of σ. Hence F (π) x consists of the cones of F (π) such that all one-diomensional face of it is contained in F (π) x (1). Hence F (π) x is determined by the set F (π) x (1). Theorem 1.4 F (π) x is homologically trivial if x int π. Since F (π) = F (π) x for x π and π int π = if dim π > 0, we get the following corollary as a special case. Corollary 1.5 If dim π > 0, then F (π) homologically trivial. Lemma 1.6 For a cone π and x V, we define Φ(π, x) = {σ F (π) ; x int σ }. Then Φ(π, x) is locally star closed in F (π), and Φ(π, x) is equal to {0} if x π and is homologically trivial otherwise. The following lemma is a generalization of Lemma 1.6, and is important in the proof of Theorem 1.8. Lemma 1.7 For a cone π, a face ρ F (π) and x V, we define Φ(π, ρ, x) = {σ F (π)(ρ ) ; x rel. int σ ρ }. Then Φ(π, ρ, x) is locally star closed in F (π), and is homologically trivial if x π. Let N be a free Z-module of rank r which contains N as a direct summand, and let U = N R. Then V is a real subspace of U. Since we do not use the lattice, it is equivalent to consider a real vector space U and a subspace V. Since V U, there exists a natural surjection φ : U V from the dual vector space U of U to the dual V of V. Let α be an r -dimensional cone in U. We assume that V intersects the interior of α, and we set π = α N R. The map λ : F (α) F (π) is defined by λ(β) = β π for each face β of α. Since γ β implies λ(γ) λ(β), λ is a covariant functor. Let G be a covariant functor from F (π) to an abelian category A. The pull-back λ G is a covariant functor from F (α) to A. The following theorem is fairly hard to prove but it can be done by induction on r r. Namely, we can show that, if r r = 1, there exists a complex W and quasiisomorphisms C (F (α), λ G) W and C (F (π), G) W. Theorem 1.8 The complex C (F (α), λ G) is quasi-isomorphic to C (F (π), G). 2 Yanagawa s theory and Fujino s vanishing theorem There is a relation between Yanagawa s result on the local cohomologies of squarefree modules and Fujino s cohomology vanishing theorems of differential modules on projective toric varieties and toric polyhedra. 4

5 Let S M be a finitely generated additive subsemigroup with 0 S. We assume S + ( S ) = M. For a field k of any characteristic, the semigroup ring S = k[s ] is defined. We denote the k-basis of S by {e(m) ; m S }. This is a k-subalgebra of the group ring k[m] with the basis {e(m) ; m M}. For a subset A M, we denote by A k the vector space with the basis {e(m) ; m A}. We investigate this ring combinatorially by using the associated cone. Let C(S ) M R be the closed convex cone generated by S. Then C(S ) is a rational polyhedral cone of dimension r. We denote by π the dual cone of C(S ) in the dual space N R of M R. The normalization of S is k[m C(S )]. In particular, S is normal if and only if S = M C(S ) (= M π ). The definition of squarefree modules on a normal S is given in the next section. If C(S ) is strongly convex, i.e., if C(S ) ( C(S )) = {0}, then the vector subspace m = S \ {0} k is the M-homogeneous maximal ideal of S. If E is an M-graded S module, then each local cohomology group H i m(e) is an M-graded S-module. Theorem 2.1 (Yanagawa[Y]) Assume that S = C(S ) M and C(S ) is strongly convex. Let E be a finitely generated M-graded S-module. If E is squarefree, then each local cohomology group H i m(e) is the Matlis dual of a squarefree module. In particular H i m(e)(m) = 0 if m S. On the other hand, Fujino proved the following theorem by his method of multiplication maps which is analogous to that of Frobenius morphisms. Theorem 2.2 (Fujino[F1]) Let X be a projective toric variety, L an ample line bundle, B a reduced torus invariant Weil divisor and i a nonnegative integer. Then for all j > 0. H j (X, Ω i X(log B) L) = 0 This theorem is generalized for the modules of differentials on a projective toric polyhedron [F2]. Here toric polyhedron is a torus action invariant subvariety of a toric variety defined by a squarefree ideal. Yanagawa used the description of the local cohomology groups by Burns and Herzog [BH] for the proof of his theorem. There is a similar description of the cohomology groups of coherent sheaves on a projective toric variety (cf. [I3]). Then we can understand the relation between these two theorems. 3 Squarefree modules Let π be a strongly convex rational polyhedral cone of N R. We denote by S π the normal semigroup ring k[m π ]. Let E be an M-graded S π -module. For each m in M, we denote by E(m) the homogeneous component of degree m. If m is in M and m is in M π, the multiplication of e(m) defines a k-linear map µ E (m, m ) : E(m) E(m + m ). 5

6 Definition 3.1 An M-graded S π -module E is said to be squarefree if the following conditions are satisfied. (1) E(m) = 0 if m M π. (2) µ E (m, m ) is an isomorphism if m M π and π m = π (m + m ). Note that for an element m in M π, σ = π m is a face of π and m is in the relative interior of the face π σ of π. If m 1, m 2 are in the relative interior of a face of π, then m 1 + m 2 is also in the relative interior. Hence if E is squarefree, then both E(m 1 ) and E(m 2 ) are isomorphic to E(m 1 + m 2 ). This implies that there exists a k-vector space E(σ) for each σ F (π) such that E(m) is identified with E(σ)e(m) for all m in M rel. int(π σ ). If σ and τ are in F (π) and σ τ, then for m, m with m M rel. int(π τ ) M π σ and m M rel. int(π σ ), we have m + m M rel. int(π σ ). Hence the multiplication of e(m ) induces a k-linear map f E (σ/τ) : E(τ) E(σ), which does not depend on the choice of m, m. Namely, we get a contravariant functor f E from F (π) to k-vector spaces defined by f E (σ) = E(σ). Conversely, if a contravariant functor f from F (π) to k-vector spaces is given, then we define a squarefree M-graded S π -module E f by E f = f(π m )e(m). m M π The multiplication map e(m ) : E f (m) E f (m + m ) for the above m, m is defined by f(σ/τ) : f(τ) f(σ). The following proposition is proved easily (cf. [Y]). Proposition 3.2 Let E be a squarefree M-graded S π -module. (1) E is finitely generated if and only if the dimension of E(σ) is finite for every σ in F (π). (2) E is a free S π -module if and only if f E (σ/π) is an isomorphism for every σ in F (π). In this case, E is isomorphic to S π k E(π). (3) E is a quotient of a squarefree free S π -module if and only if f E (σ/π) is surjective for every σ in F (π). In this case, E is a quotient of S π k E(π). (4) E is an S π -submodule of a squarefree free S π -module if and only if f E (0/σ) is injective for every σ in F (π). In this case, E is an S π -submodule of S π k E(0). Example 3.3 We denote by k F (π) or simply k the constant functor defined by k(σ) = k. Then the squarefree module E k is equal to S π. For σ F (π), the M-homogeneous prime ideal P (σ) S π is defined by P (σ) = M (π \ σ ) k. For a star closed subset Φ of F (π), the ideal I(Φ) = σ Φ P (σ) is a squarefree module which corresponds to the functor { 0 if σ Φ G(σ) = k if σ F (π) \ Φ. Example 3.4 Let V be a k-vector space of finite dimension. Suppose that a subspace A(γ) V is given for every γ F (π)(1). We define the functor A by A(σ) = A(γ) γ F (σ)(1) 6

7 for every σ F (π). For σ, τ F (π) with σ τ, the morphism A(σ/τ) is defined to be the inclusion map A(τ) A(σ). Then the squarefree module E A is a reflective submodule of S π k V. When char k = 0, the modules of differentials on an affine toric variety defined in Danilov s paper [D] belong to Example 3.4. Let V = M k = M Z k and Ω(σ) = M[σ] k for every σ in F (π) where M[σ] = M σ. Then E Ω is equal to Danilov s sheaf Ω 1 X(π) of 1-forms on the affine toric variety X(π) = Spec S π. More generally, for any p with 0 p r, the contravariant functor Ω p defined by Ω p (σ) = p M[σ] k defines a squarefree module E Ω p which is equal to Danilov s sheaf Ω p X(π) of p-forms on X(π). Let ρ be a face of π. If m is an element of M with π m = ρ, then the localization S π [e(m) 1 ] is equal to S ρ = k[m ρ ]. Proposition 3.5 Let E be a squarefree M-graded S π -module and ρ a face of π. Then E ρ = E Sπ S ρ is a squarefree S ρ -modules. The corresponding contravariant functor f Eρ from F (ρ) to k-vector spaces is equal to the restriction f E F (ρ). Let Σ be a fan of N R and Z = Z(Σ) the associated toric variety. For a T N -equivariant O Z -module E and for an element σ in Σ, the O Z (X(σ))-module E(X(σ)) has M-grading corresponding to the T N -action. We call E a squarefree sheaf if E(X(σ)) is a squarefree S σ -module for every σ in Σ. A quasicoherent squarefree O Z -module E corresponds to a contravariant functor from Σ to k-vector spaces. We denote the functor by f E. The local cohomology groups of a squarefree S π -module E is described as follows. For the contravariant functor E defined by E (σ) = E Sπ S σ = E σ, we get a complex C (F (π), E ). The result by Bruns and Herzog on local cohomologies [BH, Theorem 6.2.5] implies H i m(e) = H i (C (F (π), E )) for every i Z. Since this complex is M-graded, the m-component of H i m(e) is equal to the i-th cohomology of the m-component of C (F (π), E ) for every m M. Since the m-component E σ (m) is equal to E(σ m ) if m σ and 0 otherwise, we know the component C (F (π), E )(m) is equal to the complex C (F (π), A(E, m)) of k-vector spaces, where A(E, m) is the contravariant functor defined by A(E, m)(σ) = { 0 if m σ f E (σ m ) if m σ. If σ τ and m τ, then σ m τ m and the morphism A(E, m)(σ/τ) is defined to be f E (σ m /τ m ). The result of Yanagawa [Y, Theorem 3.10] implies that H i (C (F (π), A(E, m))) = 0 if m π. This vanishing of cohomologies can also be proved by using Lemma 1.7. Note that Φ(π, ρ, m) in Lemma 1.7 is the set of σ F (π) with m σ and ρ = σ m. The homological trivialities of these sets imply the exactness of the complex C (F (π), A(E, m)) when m π. Now we consider the cohomology groups of a projective toric variety by its homogeneous coordinate ring. 7

8 Let P M R be an integral convex polytope of dimension r. The set of cones (P ) = {(P x) ; x P } is a projective fan of N R, and the associated projective toric variety Z(P ) has the tautological line bundle O Z(P ) (1) such that H 0 (Z(P ), O Z(P ) (1)) = M P k. Set M = M Z and let Ñ be its dual Z-module. Denote by C(P ) the closed convex cone generated by P {1} M R = M Z R and ω the dual cone in ÑR = Ñ Z R. Then Z(P ) = Proj S ω for S ω = k[ M ω ], where the degree of the monomial e((m, d)) is defined to be d for every (m, d) in M = M Z. There exists a natural bijection between (P ) and F (ω) \ {ω}. Namely, if σ = (P x), the corresponding face σ of ω is defined by ω (x, 1). The projection ÑR = N R R N R induces bijection σ σ for every σ. For a finitely generated graded S ω -module E, we denote by E the coherent O Z(P ) -module E. For the tautological ample line bundle O Z(P ) (1), we denote E(d) = E O Z(P ) (1) d. For a contravariant functor f from (P ) to k-vector spaces, the functor f from F (ω) is defined by f( σ) = f(σ) for every σ in (P ) while f(ω) is defined to be the projective limit of {f(σ) ; σ (P )}. Proposition 3.6 Let E be a finitely generated squarefree O Z(P ) -module, and E the squarefree M-graded S ω -module associated to the contravariant functor f E. Then E is isomorphic to the associated sheaf E. For every σ in (P ), there exists an exact sequence which induces the exact squence 0 M[σ] M[ σ] Z 0, 0 p M[σ] p M[ σ] p 1 M[σ] 0 (1) for every integer 0 p r. Let F p P be the functor from (P ) to k-vector spaces defined by F p P (σ) = Ω p ( σ) for every σ in (P ). If char k = 0, the sequence 0 Ω p (σ) F p P ( σ) Ω p 1 (σ) 0 induced by (1) is exact. Hence we get an exact squence 0 Ω p Z(P ) E F p P Ω p 1 Z(P ) 0 of squarefree modules on Z(P ) for every integer 0 p r. Well-known exact sequence 0 Ω 1 P n O P n( 1) n+1 O P n 0 8

9 on a projective space is a special case. Here O P n( 1) n+1 can be understand as a squarefree module n i=0 O P ndx i on P n with respect to the homogeneous coordinates [x 0, x 1,, x n ]. For a finitely generated graded S ω -module E, the local cohomology group H i m(e) is equal to that of the complex C (F (ω), E ) by [BH], where E is the contravariant functor defined by E (σ) = E Sω S σ. Here we need not assume that E is M-graded. We set F (ω) = F (ω) \ {ω}. For the cohomology groups of the coherent sheaf E(d), we get the following theorem. Theorem 3.7 For any 0 i r, we have H i (Z(P ), E(d)) = H i+1 (C (F (ω), E )). d Z Since the cohomology groups of C (F (ω), E ) and C (F (ω), E ) are equal in degree greater than one, we get the following corollary. Corollary 3.8 H i m(e) is a graded module. For any 2 i r + 1 and d Z, we have H i m(e) d = H i (C (F (ω), E )) d = H i 1 (Z(P ), E(d)). We also have the following. Corollary 3.9 There exists an exact sequence of k-vector spaces 0 H 0 m(e) d E d H 0 (Z(P ), E(d)) H 1 m(e) d 0. If E is a squarefree S ω -module, then Hm(E) i is an M-graded S ω -module. Its m- component is zero if m is outside M ( ω ) by Yanagawa s theorem. This implies the vanishing of H i (Z(P ), E(d)) for i > 0. This shows the relation between Yanagawa s theory and Fujino s vanishing theorem of differential modules on projective toric varieties. We can discuss the case of toric polyhedra similarly. References [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in advanced mathematics, 39, Cambridge University Press, Cambridge, [D] V.I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33, (1978), [F1] [F2] O. Fujino, Multiplication maps and vanishing theorems for Toric varieties, Mathematische Zeitschrift 257, (2007), O. Fujino, Vanishing theorems for toric polyhedra, Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu B9, (2008),

10 [I1] M.-N. Ishida, Torus embeddings and dualizing complexes, Tohoku Math. J. 32(1980), [I2] [I3] [O] [Y] M.-N. Ishida, The local cohomology groups of an affine semigroup ring, Algebraic geometry and commutative algebra, in honor of Masayoshi Nagata, (1987), (a book written by Ishida in Japanese). T. Oda, Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties, Ergebnisse der Math. (3), 15, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, K. Yanagawa, Sheaves on finite posets and modules over normal semigroup rings, Journal of Pure and Applied Algebra 161 (2001), Mathematical Institute Tohoku University Sendai, JAPAN 10