Let $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\mathfrak{p})$ the residue field of the local ring $R_\mathfrak{p}$. We prove that if an R-module M satisfies $\operatorname{Ext}_R^{n}(k(\mathfrak{p}),M)=0$ for some $n\geqslant\dim R$, then $\operatorname{Ext}_R^i(k(\mathfrak{p}),M)=0$ holds for all $i \geqslant n$. This improves a result of Christensen, Iyengar and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

Let 𝔭 be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k (𝔭), M) = 0 for some n ⩾ R𝔭, where k(𝔭) is the residue field at 𝔭, then ${\rm Tor}_i^R $(k (𝔭), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k (𝔭), M) are proved, and applications to the theory of homological dimensions are explored.

Let $R$ be a commutative ring. Let $M$ respectively $A$ denote a Noetherian respectively Artinian $R$-module, and $\mathfrak{a}$ a finitely generated ideal of $R$. The main result of this note is that the sequence of sets $(\mathrm{Att}_R\mathrm{Tor}_1^R((R/\mathfrak{a}^{n}),A))_{n\in\mathbb{N}}$ is ultimately constant. As a consequence, whenever $R$ is Noetherian, we show that $\mathrm{Ass}_R\mathrm{Ext}_R^1((R/\mathfrak{a}^{n}),M)$ is ultimately constant for large $n$, which is an affirmative answer to the question that was posed by Melkersson and Schenzel in the case $i=1$.

AMS 2000 Mathematics subject classification: Primary 13E05; 13E10