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22
The Alexander duality functors and local duality with monomial support
 J. Algebra
"... Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as co ..."
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Cited by 32 (13 self)
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Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual, and contains all of the maps in the minimal free resolution of M over every Zngraded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcmlattices. Using injective resolutions, theorems of EagonReiner and Terai are generalized to all Nngraded modules: the projective dimension of M equals the supportregularity of its Alexander dual, and M is CohenMacaulay if and only if its Alexander dual has a supportlinear free resolution. Alexander duality is applied in the context of the Zngraded local cohomology functors Hi I (−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I = m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai’s formula for the Hilbert series of H i I (S) is equivalent to Hochster’s for Hn−i m (S/I). 1
Multigraded CastelnuovoMumford Regularity
 J. REINE ANGEW. MATH
, 2003
"... We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric ..."
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Cited by 26 (6 self)
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We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric
Bass numbers of semigroupgraded local cohomology
 Pacific J. Math. 209
, 2003
"... Abstract. Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H i I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semi ..."
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Cited by 9 (4 self)
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Abstract. Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H i I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q sat is simplicial, and a finitely generated module M over k[Q] that is graded by Q gp, the Bass numbers of H i I(M) are finite for any Qgraded ideal I of k[Q]. Conversely, if Q sat is not simplicial, we find a graded ideal I and graded k[Q]module M such that the local cohomology module H i I(M) has infinitedimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup. 2000 Mathematics Subject Classification: 13 (primary); 05, 14 (secondary)
Homogeneous coordinates and quotient presentations for toric varieties
 Math. Nachr
"... Abstract. Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas ..."
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Cited by 6 (3 self)
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Abstract. Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas QCartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.
Algorithms for graded injective resolutions and local cohomology over semigroup rings
 J. Symbolic Computation
, 2004
"... Let Q be an affine semigroup generating Z d, and fix a finitely generated Z dgraded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z dgraded injective resolution of M up to any desired cohomological degree. As an application, we derive an algor ..."
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Cited by 6 (3 self)
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Let Q be an affine semigroup generating Z d, and fix a finitely generated Z dgraded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z dgraded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules H i I (M) supported on any monomial (that is, Zdgraded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them.
Graded Greenlees–May duality and the Čech hull
 LECT. NOTES IN PURE AND
, 2001
"... The duality theorem of Greenlees and May relating local cohomology with support on an ideal I and the left derived functors of Iadic completion [GM92] holds for rather general ideals in commutative rings. Here, simple formulas are provided for both local cohomology and derived functors of Z ngra ..."
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Cited by 5 (5 self)
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The duality theorem of Greenlees and May relating local cohomology with support on an ideal I and the left derived functors of Iadic completion [GM92] holds for rather general ideals in commutative rings. Here, simple formulas are provided for both local cohomology and derived functors of Z ngraded completion, when I is a monomial ideal in the Z ngraded polynomial ring k[x1,..., xn]. GreenleesMay duality for this case is a consequence. A key construction is the combinatorially defined Čech hull operation on Zngraded modules [Mil98, Mil00, Yan00]. A simple selfcontained proof of GM duality in the derived category is presented for arbitrarily graded noetherian rings, using methods motivated by the Čech hull.
Asymptotic cohomological functions of toric divisors
, 2005
"... We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the selfintersection ..."
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Cited by 5 (3 self)
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We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the selfintersection number of a TCartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing. Suppose D is an ample divisor on an ndimensional algebraic variety. The sheaf cohomology of O(D) does not necessarily reflect the positivity of D; O(D) may have few global sections and its higher cohomology groups may not vanish. However, for m ≫ 0, O(mD) is globally generated and all of its higher cohomology groups vanish. Moreover, the rate of growth of the space of global sections of O(mD) as m increases carries information on the positivity of D. Indeed, if we write h 0 (mD) for the dimension of H 0 (X, O(mD)), then by asymptotic RiemannRoch [La1, Example 1.2.19], (D n) = lim
Exact matrix formula for the unmixed resultant in three variables
 Journal of Pure and Applied Algebra
"... We give the first exact determinantal formula for the resultant of an unmixed sparse system of four Laurent polynomials in three variables with arbitrary support. This follows earlier work by the author on exact formulas for bivariate systems and also uses the exterior algebra techniques of Eisenbud ..."
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Cited by 5 (0 self)
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We give the first exact determinantal formula for the resultant of an unmixed sparse system of four Laurent polynomials in three variables with arbitrary support. This follows earlier work by the author on exact formulas for bivariate systems and also uses the exterior algebra techniques of Eisenbud and Schreyer. Along the way we will prove an interesting new vanishing theorem for the sheaf cohomology of divisors on toric varieties. This will also allow us to describe some supports in four or more variables for which determinantal formulas for the resultant exist.