Bass series of local ring homomorphisms of finite flat dimension

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Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism φ: R → S which makes S into an R-module of finite flat dimension. In particular, it is shown that an inequality μ R i+depth R ≤ μ s i+depth S holds for all i ∈ Z.This is a consequence of an equality involving the Bass series I R M (t) = Σ i ∈ z μ 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⊗ R S, where ⊗ denotes the derived tensor product. It is prove that there is an equa lity of formal Laurent series I s M⊗RS (t) = I R M (t)I F(φ) (t), where F(φ) is the fiber of φ considered as a homomorphism of commutative differential graded rings