Let (R,m) be a commutative Noetherian local ring. Suppose that M and N are finitely generated modules over R such that M has finite projective dimension and such that TorRi (M,N) = 0 for all i > 0. The main result of this note gives a condition on M which is necessary and sufficient for the tensor product of M and N to be a Cohen–Macaulay module over R, provided N is itself a Cohen–Macaulay mod...

We consider a class of hypergraphs called hypercycles and we show that a hypercycle $C_n^{d,alpha}$ is shellable or sequentially the Cohen--Macaulay if and only if $nin{3,5}$. Also, we characterize Cohen--Macaulay hypercycles. These results are hypergraph versions of results proved for cycles in graphs.

A few years ago, Huneke and Leuschke proved a theorem which solved a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is consider...

In the late 1960s Auslander and Bridger [2] introduced a notion of approximation which they used to prove that every module whose n syzygy is n-torsionfree can be described as the quotient of an n-spherical module by a submodule of projective dimension less than n. About two decades later Auslander and Buchweitz [3] introduced the notion of Cohen-Macaulay approximation which they used to show t...

we consider a class of hypergraphs called hypercycles and we show that a hypercycle $c_n^{d,alpha}$ is shellable or sequentially the cohen--macaulay if and only if $nin{3,5}$. also, we characterize cohen--macaulay hypercycles. these results are hypergraph versions of results proved for cycles in graphs.