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Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nn-graded 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 Zn-graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon-Reiner and Terai are generalized to all Nn-graded modules: the projective dimension of M equals the supportregularity of its Alexander dual, and M is Cohen-Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the Zn-graded 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

We introduce a new variant of tight closure associated to any fixed ideal a, which we call a-tight closure, and study various properties thereof. In our theory, the annihilator ideal τ(a) of all a-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal τ(a) and the multiplier ideal associated to a (or, the adjoint of a in Lipman’s sense) in normal Q-Gorenstein rings reduced from characteristic zero to characteristic p ≫ 0. Also, in fixed prime characteristic, we establish some properties of τ(a) similar to those of multiplier ideals (e.g., a Briançon-Skoda type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal τ(a) and the F-rationality of Rees algebras.

Abstract. In this paper we show how, given a complex of graded modules and knowing some partial Castelnuovo-Mumford regularities for all the modules in the complex and for all the positive homologies, it is possible to get a bound on the regularity of the zero homology. We use this to prove that if dim TorR 1 (M, N) ≤ 1 then reg(M ⊗N) ≤ reg(M)+reg(N), generalizing results of Chandler, Conca and Herzog, and Sidman. Finally we give a description of the regularity of a module in terms of the postulation numbers of filter regular hyperplane restrictions. 1. introduction Let R = K[X1,..., Xn] be a polynomial ring over a field K, M a finitely generated graded R-module and let I ⊂ R be an ideal. Recently some work has been done to study when the Castelnuovo-Mumford regularity of Ir can be bounded by r times the regularity of I, and more generally when the regularity of IM can be bounded by the sum of the regularity of I and M. This is not always the case; see the papers of Sturmfels [St], and Conca, Herzog [CH] for counterexamples. On the

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric

UW dedicates this paper to the memory of his father, Hansjoachim Walther. ABSTRACT. We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems HA(β) arising from a d × n integer matrix A and a parameter β ∈ Cd. To do so we introduce an Euler–Koszul functor for hypergeometric families over Cd, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter β ∈ Cd is rank-jumping for HA(β) if and only if β lies in the Zariski closure of the set of Zd-graded degrees α where the local cohomology L i<d Hi m(C[NA])α of the semigroup ring C[NA] supported at its maximal graded ideal m is nonzero. Consequently, HA(β) has no rank-jumps over Cd if and only if C[NA] is Cohen–Macaulay of dimension d. CONTENTS

Abstract. We show that the method of moving quadrics for implicitizing surfaces in P 3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit

this paper we reexamine the Gale transform in the light of modern

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann’s regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes. 1.

Abstract. Each partition λ = (λ1, λ2,..., λn) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulæ for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory. 1.