Let kk be an algebraically closed field and A be a finitely generated, centrally finite,
nonnegatively graded (not necessarily commutative) k k-algebra. In this note we construct a …

Let K be a field and sets of indetorminates, and f∈ k [[x]] g∈ k [[Y]] two non-zero formal
power series with f∈(x) g∈(Y). Set and suppose that δg is a (y)-primary ideal (eg, if char K …

Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of
maps from modules to local cohomology modules and the work by the third author with …

We give a necessary condition of degeneration via matrix representations, and consider
degenerations of indecomposable Cohen–Macaulay modules over hypersurface …

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0,
and let M be an A-module. We consider the following question: Under what conditions is it …

I include a complete proof of this result in this survey article. My emphasis lies on the
conscruction of maximal Cohen-Macaulay modules via matrix factorizations (section 2), from …

For any $ d\ge 4$, by deformation theory of schemes, we present examples of (complete or
excellent) $ d $-dimensional mixed characteristic normal local domains admitting no small …

For a skew version of a graded (A∞) hypersurface singularity A, we study the stable
category of graded maximal Cohen-Macaulay modules over A. As a consequence, we see …

We show that the number of parameters for CM-modules of prescribed rank is semi-
continuous in families of CM rings of Krull dimension 1. This transfers a result of Knoerrer …

The canonical modules of Cohen-Macaulay rings give a useful technical tool in commutative
algebra, algebraic geometry, and singularity theory via various duality theorems. But not …