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A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if$\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finiteprojective dimension. If $R$ satisfies trivial vanishing then we also have that$\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finiteprojective dimension or $N$ has finite injective dimension. In this paper, weestablish obstructions for the failure of trivial vanishing in terms of theasymptotic growth of the Betti and Bass numbers of the modules involved. These,together with a result of Gasharov and Peeva, provide sufficient conditions for$R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ isgeneralized Golod. Our methods allow us to settle the Auslander-Reitenconjecture in several new cases. In the last part of the paper, we providecriteria for the Gorenstein property based on consecutive vanishing of Ext. Thelatter results improve similar statements due to Ulrich, Hanes-Huneke, andJorgensen-Leuschke.

Extremal growth of Betti numbers and trivial vanishing of (co)homology