We identify all semidualizing modules over certain classes of ladder determinantal rings over a field k. Specifically, given a ladder of variables Y , we show that the ring k[Y ]/It(Y ) has only trivial semidualizing modules up to isomorphism in the following cases: (1) Y is a one-sided ladder, and (2) Y is a two-sided ladder with t = 2 and no coincidental inside corners.
Abstract. Let R be a commutative Noetherian ring, I, J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then the sets Ass R F (M/I n M ), Ass R F (I n−1 M/I n M ) and the values depth J F (M/I n M ), depth J F (I n−1 M/I n M ) become independent of n for large n. Next, we consider several examples in which F is a rather familiar functor, but is not coherent or not even finitely generated in general. In these cases, the sets Ass R F (M/I n M ) still become independent of n for large n. We then show one negative result where F is not finitely generated. Finally, we give a positive result where F belongs to a special class of functors which are not finitely generated in general, an example of which is the zeroth local cohomology functor.
In this note we introduce and study basic properties of two types of modules over a commutative noetherian ring R of positive prime characteristic. The first is the category of modules of finite F -type. These objects include reflexive ideals representing torsion elements in the divisor class group of R. The second class is what we call F -abundant modules. These include, for example, the ring R itself and the canonical module when R has positive splitting dimension. We prove various facts about these two categories and how they are related, for example that Hom R (M, N ) is maximal Cohen-Macaulay when M is of finite F -type and N is F -abundant, plus some extra (but necessary) conditions. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these objects, including complete classifications in many cases of interest, such as complete intersections and invariant subrings.Given S ⊆ mod(R), we use add R (S) to denote the additive subcategory of mod(R) generated by S.Definition 1.1. (1) Let M be an R-module such that Supp(M) = Spec R and is locally free in codimension 1. We let M(e) = F e R (M) * * , the reflexive hull of F e R (M), viewed as an R-module by identifying e R with R. We say that M is of finite F -type if {M(e)} e 0 ⊆ add R (X) for some R-module X (see Lemma 4.3). We let F T (R) denote the category of R-modules of finite F -type.
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