Related papers: Modules that detect finite homological dimensions

We study Gorenstein dimension and grade of a module $M$ over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded…

We give characterizations of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over the group algebra for large families of infinite groups and show that every weak Gorenstein projective, weak Gorenstein flat and weak…

The depth of tensor product of modules over a Gorenstein local ring is studied. For finitely generated modules M and N over a Gorenstein local ring R, under some assumptions on the vanishing of finite number of Tate and relative homology…

Distinctive characteristics of Iwanaga--Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even…

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology…

For any ring $R$ and any positive integer $n$, we prove that a left $R$-module is a Gorenstein $n$-syzygy if and only if it is an $n$-syzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of…

In this paper we prove the following generalization of a result of Hartshorne: Let $(S,\n)$ be a regular local ring of dimension $4$. Assume that $x,y,u,v$ is a regular system of parameters for $S$ and $a:=xu+yv$. Then for each finitely…

Let $R$ be a commutative noetherian ring, and let $C$ be a semidualizing $R$-module. In this paper, we study levels of bounded complexes of finitely generated $R$-modules with respect to the full subcategory $\mathsf{G}_{C}(R)$ consisting…

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl…

An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show…

Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n\geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and…

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a…

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and…

Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if…

Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules…

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring admitting a canonical module, and assume that $R$ is maximal Cohen--Macaulay over the base ring. We provide a characterization of when $R$ is left weakly Gorenstein. We further…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the…

Over Cohen--Macaulay rings admitting a pointwise dualizing module, we show that the class of modules of restricted projective dimension bounded by any integer is finitely deconstructible and that the class of modules of restricted flat…

It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here…

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a…