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 ...

We study Gorenstein dimension and grade of a module M over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dimM ≤ G-dimgrM and an equality gradeM = grade grM , whe...

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...

The main aim of this paper is to find a large class of rings for which there are indecomposable maximal Cohen-Macaulay modules of arbitrary high multiplicity (or rank in the case of domains).

Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1, . . . , xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R/ in(I), where in(I) denotes the initial ideal of I with respect to <. In this note we study the question when the local cohomology modules of R/I and R/ in(I) hav...