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 finite projective 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 finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic 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 $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke.

中文翻译：

---

Betti 数的极值增长和（共）同源性的微不足道的消失

如果所有大 $i$ 的 $\operatorname{Tor}_i^R(M,N)=0$ 意味着 $M$ 或 $N$ 具有有限的投影维数，则 Cohen-Macaulay 局部环 $R$ 满足平凡消失。如果 $R$ 满足平凡消失，那么我们也有 $\operatorname{Ext}^i_R(M,N)=0$ 对于所有大 $i$ 意味着 $M$ 具有有限的射影维数或 $N$ 具有有限的射影维数. 在本文中，我们根据所涉及模块的 Betti 和 Bass 数的渐近增长，为平凡消失的失败建立了障碍。这些，连同 Gasharov 和 Peeva 的结果，为 $R$ 提供了满足平凡消失的充分条件；当 $R$ 是广义黄金时，我们提供了尖锐的条件。我们的方法使我们能够在几个新案例中解决 Auslander-Reiten 猜想。在论文的最后一部分，我们提供基于 Ext 的连续消失的 Gorenstein 属性的标准。由于 Ulrich、Hanes-Huneke 和 Jorgensen-Leuschke，后者的结果改进了类似的陈述。