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 $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\text{f}{{\text{d}}_{R}}X,\,\text{Gf}{{\text{d}}_{R}}X$ and ${{\text{G}}_{\text{C}}}-\text{f}{{\text{d}}_{R}}\,X\,\text{by}\,\text{T}\left( X \right)$. Let $M$ be an $R$-module such that $\text{H}_{\mathfrak{a}}^{i}\left( M \right)\,=\,0$ for all $i\,\ne \,n$. It is proved that if $T\left( M \right)\,<\,\infty$, then $\text{T}\left( \text{H}_{\mathfrak{a}}^{n}\left( M \right) \right)\,\le \,\text{T}\left( M \right)\,+\,n$, and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen–Macaulay modules, dualizing modules, and Gorenstein rings.

Let $\left( R,\,m \right)$ be a non-zero commutative Noetherian local ring (with identity) and let $M$ be a non-zero finitely generated $R$-module. In this paper for any $\mathfrak{p}\,\in \,\text{Spec}\left( R \right)$ we show that

1

$$\text{injdi}{{\text{m}}_{{{R}_{\mathfrak{p}}}}}\,H_{\mathfrak{p}{{R}_{\mathfrak{p}}}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$

1

$$\text{f}{{\text{d}}_{{{R}_{\mathfrak{p}}}}}H_{\mathfrak{p}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$

are bounded from above by $\text{injdi}{{\text{m}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ and $\text{f}{{\text{d}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ respectively, for all integers $i\,\ge \,\dim\left( {R}/{\mathfrak{p}}\; \right)$.

In this paper we extend the notion of FP-injective modules to that of complexes and characterize such complexes. We show that some characterizations similar to those for injective complexes exist for FP-injective complexes. We also introduce and study the notion of an FP-injective dimension associated to every complex of left R-modules over an arbitrary ring. We show that there is a close connection between the FP-injective dimension of complexes and flat dimension.