Related papers: On the vanishing of Ext and Tor

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…

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…

We consider homomorphisms of complete, separated right or two-sided linear topological rings with countable bases of neighborhoods of zero $\mathfrak f\colon\mathfrak R\to\mathfrak S$. Taut maps of right linear topological rings, strongly…

It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and…

We consider vanishing of Ext and Tor, especially over Artinian rings. In particular, we prove the Auslander-Reiten conjecture for all commutative local rings in which the cube of the maximal ideal is zero.

We investigate stable homology of modules over a commutative noetherian ring $R$ with respect to a semidualzing module $C$, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of…

We show the vanishing of higher extension groups and torsion groups between linearisation of additive functors from a semi-additive category satisfying some conditions to a category of vector spaces. In particular, we apply our results to…

Let $\varphi\colon R \rightarrow A$ be a finite ring homomorphism, where $R$ is a two-sided Noetherian ring, and let $M$ be a finitely generated left $A$-module. Under suitable homological conditions on $A$ over $R$, we establish a close…

Via the construction of a functor from $\mathsf{C}_{u}(H)$ to an auxiliary category we associate, with any triplet $(G,F,\rho)$, two natural transformations, $\mathfrak{m}_{\star}$ morphism of…

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting…

We reprove and generalize the result that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a…

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the…

Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander…

In this article we study cohomological properties of the category of polynomial outer functors on free groups, which are the functors from the category of finitely generated free groups to the category of rational vector spaces which send…

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)]…

Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines…

Let $R$ be a commutative noetherian ring and $f: X \to \mathrm{Spec} R$ a proper smooth morphism, of relative dimension $n$. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map $\mathrm{Tr}_f :…

If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective…

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global…

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely…