Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\mathfrak{a},\,\mathfrak{b},\,\mathfrak{a}\,\cap \,\mathfrak{b}$ and $\mathfrak{a}\,+\,\mathfrak{b}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen–Macaulay if there exists an ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer such that $0\,\le \,r\,<\,{{\dim}_{R}}(M)$, any maximal element q of the non-empty set of ideals {$\mathfrak{a}\,:\,\text{H}_{\mathfrak{a}}^{i}(M)$ is not artinian for some $i$, $i\,\ge \,r$} is a prime ideal, and all Bass numbers of $\text{H}_{\mathfrak{q}}^{i}(M)$ are finite for all $i\,\ge \,r$.

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings.

2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

The local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes. A concept of finite Gorenstein dimension, which substantially weakens the notion of finite flat dimension, is introduced for homomorphisms. It is shown to impose structural constraints, due to a remarkable equivalence of subcategories of the derived category of all modules.

An essential part of this study is the development of relative notions of dualizing complexes and Bass numbers. It is proved that the Bass numbers of local homomorphisms are rigid, extending a known result for local rings. Quasi-Gorenstein homomorphisms are introduced as local homomorphisms that base-change a dualizing complex for the source ring into one for the target. They are shown to have the stability properties of the Gorenstein homomorphism that they generalize.

1991 Mathematics Subject Classification: primary 13H10, 13D23, 14E40; secondary 13C15.