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

Lyle, Justin; Montaño, Jonathan

doi:10.1090/tran/8189

Extremal growth of Betti numbers and trivial vanishing of (co)homology
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by Justin Lyle and Jonathan Montaño PDF

Trans. Amer. Math. Soc. 373 (2020), 7937-7958 Request permission

Abstract:

A Cohen–Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname {Tor}_i^R(M,N)=0$ for all large $i$ implies that $M$ 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 that $M$ 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 results 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.

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