The Alexander Duality Functors and Local Duality with Monomial Support

Abstract

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated ‫ގ‬ n -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 ‫ޚ‬ n -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 ‫ގ‬ n -graded modules: the projective dimension of M equals the support-regularity 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 n i Ž . ‫ޚ‬ -graded local cohomology functors H y for squarefree monomial ideals I in I the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I s ᒊ is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, i Ž . that Terai's formula for the Hilbert series of H S is equivalent to Hochster's for

Local Cohomology Modules Supported on Monomial Ideals

“…This construction is generalized [69] in order to describe cellular (cocellular) injective complexes.…”

“…In [69], a convenient modification of the notations of the source and the target of such a matrix is introduced in order to work with injective resolutions.…”

“…Roughly speaking, their corresponding monomial matrices have the same entries. In [69], a bridge among both complexes is given by means of the so-calledČech hull functor. In fact, this transition works for all free resolutions of R/I so we can get some different complexes which can replace theČech complex.…”

Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions

“…In general, any injective resolution of a Z d -graded module yields an irreducible resolution of its Q-graded part, although the indecomposable injective summands with zero Q-graded part get erased. In particular, the "cellular injective resolutions" of [Mil00] become what should be called "cellular irreducible resolutions" here.…”

Free resolutions for multigraded modules: a generalization of Taylor’s construction

“…Soon after that [GaPeWe99] discuss the significance of the LCM-lattice in determining the minimal free resolution of an ideal. For multigraded modules, [ChDe01] discuss the second syzygies, while [Ya99] and [Mi00] among others generalize results concerning homological invariants of monomial ideals to multigraded modules.…”