Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bass series of local ring homomorphisms of finite flat dimension

Authors: Luchezar L. Avramov, Hans-Bjørn Foxby and Jack Lescot
Journal: Trans. Amer. Math. Soc. 335 (1993), 497-523
MSC: Primary 13D03; Secondary 13C11, 13D25, 18G15, 55T20
MathSciNet review: 1068924
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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.

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Article copyright: © Copyright 1993 American Mathematical Society