Local cohomology and Matlis duality.
Let be an ideal of a commutative Noetherian ring . It is shown that the -modules are -cofinite for all finitely generated -modules and all if and only if the -modules and are -cofinite for all finitely generated -modules , and all integers .
Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that kills the general local cohomology module for every integer j less than a fixed integer n, where , then there exists an integer k such that for every j < n.
Let be a commutative Noetherian local ring, be an ideal of and a finitely generated -module such that and , where is the cohomological dimension of with respect to and is the -grade of . Let be the Matlis dual functor, where is the injective hull of the residue field . We show that there exists the following long exact sequence where is a non-negative integer, is a regular sequence in on and, for an -module , is the th local cohomology module of with respect...
Let be a complete local ring, an ideal of and and two Matlis reflexive -modules with . We prove that if is a finitely generated -module, then is Matlis reflexive for all and in the following cases: (a) ; (b) ; where is the cohomological dimension of in ; (c) . In these cases we also prove that the Bass numbers of are finite.
We study matrix factorizations of a potential W which is a section of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.
Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition , defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class consisting of all modules satisfying . If and are ideals of R, we get a necessary and sufficient condition for to satisfy and simultaneously. We also find some sufficient...