Let R be a ring and M a right R-module. M is called n-FP-projective if Ext M N = 0 for any right R-module N of FP-injective dimension ≤n, where n is a nonnegative integer or n = . R M is defined as sup n M is n-FP-projective and R M = −1 if Ext M N = 0 for some FP-injective right R-module N. The right -dimension r -dim R of R is defined to be the least nonnegative integer n such that R M ≥ n im...

We study the ∆-filtered modules for the Auslander algebra of k[T ]/Tn ⋊C2 where C2 is the cyclic group of order two. The motivation for this is the bijection between parabolic orbits in the nilradical of a parabolic subgroup of SLn and certain ∆-filtered modules for the Auslander algebra of k[T ]/Tn as found by Hille and Röhrle and Brüstle et al., cf. [HR99] [BHRR99]. Under this bijection, the ...

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of th...

For a simplicial complex on {1, 2, . . . , n} we define enriched homology and cohomology modules. They are graded modules over k[x1, . . . , xn] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein∗ complexes , and also orientable homology manifolds in terms of the enriched modules. We ...

This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way their Cohen-Macaulayness and minimal Betti numbers are preserved. is achieved by showing finitely generated modules over power series rings manner preserves diagrams of initial exponents numbers. These results also applied to obtain an approximation result for flat homomorphisms from algebras.

Let R be a Cohen-Macaulay ring and M a maximal CohenMacaulay R-module. Inspired by recent striking work by Iyama, BurbanIyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension versio...

We extend the results of Adcock, Carlsson, and Carlsson ([ACC13]) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson, Singh, and Zomorodian in [CSZ10]. The use of topology to study point cloud data has been well established ([Car09], [Car14]). Given a finite metric space (e.g., a finite set in R n), one first constructs a filter...

we investigate the relative cohomology and relative homology theories of $f$-gorenstein modules, consider the relations between classical and $f$-gorenstein (co)homology theories.