Related papers: Syzygy modules for quasi $k$-Gorenstein rings

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…

We study Cohen-Macaulay non-Gorenstein local rings $(R,\mathfrak{m},k)$ admitting certain totally reflexive modules. More precisely, we give a description of the Poincar\'{e} series of $k$ by using the Poincar\'{e} series of a non-zero…

Let R be a commutative noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M eventually stabilize to the depth of R. In this paper, we…

For a given class of modules $\mathcal{A}$, we denote by $\widetilde{\mathcal{A}}$ the class of exact complexes $X$ having all cycles in $\mathcal{A}$, and by $dw(\mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in…

Finitely generated reflexive modules over commutative Noetherian rings form a key component of Auslander and Bridger's stable module theory and are likewise essential in the study of Cohen--Macaulay representations. Recently, H. Dao…

Let $k$ be a field, and let $\Lambda$ be a finite dimensional $k$-algebra. We prove that if $\Lambda$ is a self-injective algebra, then every finitely generated $\Lambda$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a…

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the…

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the…

Given a finitely generated module $M$ over a Noetherian local ring $R$, we give a characterization for the first syzygy of the associated graded module $G_{\mathfrak{m}}(M)$ to be equigenerated. As an application of this, we identify a…

Let $\Lambda$ be an artin algebra and let $\mathcal{P}^{<\infty}_\Lambda$ the category of finitely generated right $\Lambda$-modules of finite projective dimension. We show that $\mathcal{P}^{<\infty}_\Lambda$ is contravariantly finite in…

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous…

A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C) \simeq A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call…

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the…

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective…

Let $K$ be a field of characteristic two, and let $\lambda$ be a two-part partition of some natural number $r$. Denote the permutation module corresponding to the (maximal) Young subgroup $\Sigma_\lambda$ in $\Sigma_r$ by $M^\lambda$. We…

Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost…

Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical $J$ vanishes. We determine the irreducible components of the module variety $\text{Mod}_{\bf…

Let $\Lambda$ be the path algebra of a finite quiver $Q$ over a finite-dimensional algebra $A$. Then $\Lambda$-modules are identified with representations of $Q$ over $A$. This yields the notion of monic representations of $Q$ over $A$. If…

Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of a…

In this paper we give versions of Hilbert's syzygy theorem for finitely generated modules over polynomial rings over direct product of principal ideal rings.