We prove a Cohen-Macaulay version of result by Avramov-Golod and Frankild-J{\o}rgensen about Gorenstein rings, showing that if noetherian ring $A$ is Cohen-Macaulay, $a_1,\dots,a_n$ any sequence elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ DG-ring. further generalize this result, it also holds for commutative DG-rings. In process proving this, we develop new technique to study ...

Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open pr...

let $mathcal {a}$ be an abelian category with enough projective objects and $mathcal {x}$ be a full subcategory of $mathcal {a}$. we define gorenstein projective objects with respect to $mathcal {x}$ and $mathcal{y}_{mathcal{x}}$, respectively, where $mathcal{y}_{mathcal{x}}$=${ yin ch(mathcal {a})| y$ is acyclic and $z_{n}yinmathcal{x}}$. we point out that under certain hypotheses, these two g...

Chow rings of matroids were instrumental in the resolution Heron–Rota–Welsh Conjecture by Adiprasito, Huh, and Katz Top-Heavy Braden, Matherne, Proudfoot, Wang. The ring a matroid is commutative, graded, Artinian, Gorenstein algebra with linear quadratic relations defined matroid. Dotsenko conjectured that any Koszul. purpose this paper to prove Dotsenko’s conjecture. We also show augmented As ...

Let R be a polynomial ring in r variables over an infinite field K, and denote by D a corresponding dual ring, upon which R acts as differential operators. We study type two graded level Artinian algebras A = R/I , having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to ...

Let (R,m, k) be a d-dimensional Noetherian reduced local ring of prime characteristic p such that R1/p e are finite over R for all e ∈ N (i.e. R is F -finite). Consider the sequence { ae qα(R)+d }e=0, in which α(R) = logp[k : kp], q = pe, and ae is the maximal rank of free R-modules appearing as direct summands of R-module R1/q . Denote by s−(R) and s+(R) the liminf and limsup respectively of t...

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants of m copies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds...

We investigate the behavior of the test ideal of an excellent reduced ring of prime characteristic under base change. It is shown that if h : A ?! D is a smooth homomorphism, then A D = D , assuming that all residue elds of A at maximal ideals are perfect and that formation of the test ideal commutes with localization. It is also shown that if h : (A; m) ?! D is a nite at homomorphism of Gorens...