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...

Let Fq be a finite field of characteristic p, K a field containing it, and R = K[X1, . . . , Xn] a polynomial ring in n variables. The general linear group GLn(Fq) has natural action on R by degree preserving ring automorphisms. L. E. Dickson showed that the subring of elements which are fixed by this group action is a polynomial ring, [Di], though for an arbitrary subgroupG of GLn(Fq), the str...

In this paper, we consider a finite, torsion-free module E over Gorenstein local ring. We provide sufficient conditions for to be of linear type and the Rees algebra R ( ) Cohen-Macaulay. Our results are obtained by constructing generic Bourbaki ideal I exploiting properties residual intersections .

We extend the “linearly exponential” bound for the Castelnuovo-Mumford regularity of a graded ideal in a polynomial ring K[x1, . . . , xr] over a field (established by Galligo and Giusti in characteristic 0 and recently, by Caviglia-Sbarra for abitrary K) to graded submodules of a graded module over a homogeneous Cohen-Macaulay ring R = ⊕n≥0Rn with artinian local base ring R0. As an application...

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $\omega$. Attached to each such ring R, when $\omega$ is an ideal, there are integers--the type the reduction number $\omega$--that provide valuable metrics express deviation R from being Gorenstein ring. In arXiv:1701.05592 and arXiv:1711.09480 we enlarged this list with degree bi-canonical degree. work extend wh...

Let S be an unramified regular local ring having mixed characteristic p > 0 and R the integral closure of S in a pth root extension of its quotient field. We show that R admits a finite, birational module M such that depth(M) = dim(R). In other words, R admits a maximal Cohen-Macaulay module.

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.

In this paper, we explore the almost Cohen-Macaulayness of associated graded ring stretched ${\mathfrak m}$ -primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local $(A,{\mathfrak m})$ . particular, structure satisfying equality e1(I) = e0(I) − ℓA(A/I) + 4, where and denote multiplicity coefficient, respectively.

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.