Bass series of local ring homomorphisms of finite flat dimension

Abstract:

Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism $\phi :R \to S$ which makes $S$ into an $R$-module of finite flat dimension. In particular, it is shown that an inequality $\mu _R^{i + {\text {depth}}\;R} \leq \mu _s^{i + {\text {depth}}\;S}$ holds for all $i \in \mathbb {Z}$. This is a consequence of an equality involving the Bass series $I_R^M(t) = \sum \nolimits _{i \in \mathbb {Z}} {\mu _R^i(M){t^i}}$ of a complex $M$ of $R$-modules which has bounded above and finite type homology and the Bass series of the complex of $S$-modules $M{\underline {\underline \otimes } _R}S$, where $\underline {\underline {\otimes }}$ denotes the derived tensor product. It is proved that there is an equality of formal Laurent series $I_s^{M{{\underline {\underline \otimes } }_R}S}(t) = I_R^M(t){I_{F(\phi )}}(t)$, where $F(\phi )$ is the fiber of $\phi$ considered as a homomorphism of commutative differential graded rings. Coefficientwise inequalities are deduced for $I_S^{M{{\underline {\underline \otimes } }_R}S}(t)$, and Golod homomorphisms are characterized by one of them becoming an equality.