We show that an excellent local domain of characteristic p has a separable big Cohen–Macaulay algebra. In the course of our work we prove that an element which is in the Frobenius closure of an ideal can be forced into the expansion of the ideal to a module-finite separable extension ring.

The so-called ’change-of-ring’ results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.

A new \compressed straight" basis for the polynomial ring Z[ti;j ] is constructed. This basis is then employed to study the lattice of representations generated by the representations associated with row-convex shapes under the operations of intersection and linear span. Applications to ascertaining the Cohen-Macaulay property for rings associated to elements of this lattice are also given.

The so-called ’change-of-ring’ results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with trace closure. We use this fact to compute ideals various rings respect closures coming from their indecomposable maximal Cohen–Macaulay modules. also give an easier way hypersurface ring in three variables particular type matrix factorization.

The nilpotent endomorphisms over a finite free module over a domain with principal ideal are characterized. One may apply these results to the study of the maximal Cohen-Macaulay modules over the ring R := A[[x]]/(x), n ≥ 2, where A is a DVR. Subject Classification: 15A21, 13C14.

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

Let R be a local ring and C semidualizing module of R. We investigate the behavior certain classes generalized Cohen-Macaulay R-modules under Foxby equivalence between Auslander Bass with respect to C. In particular, we show that are invariant this if M is finitely generated R-module in class such C\otimes_RM surjective Buchsbaum, then also Buchsbaum.

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y : If they intersect properly in an arithmetically Cohen-Macaulay subsc...

A certain inequality is shown to hold for the values of the Möbius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen– Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjec...