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Abstract:This paper investigates the vanishing and non-vanishing of Betti and Bass numbers for non-finitely generated modules. We prove that for \(d\)-dimensional Cohen--Macaulay local rings, every non-zero \(\mathfrak{m}\)-torsion module satisfies \(\beta_d(M)\neq 0\), and we establish the Betti number behavior of the injective hull \(E_R(k)\). We study Tor-rigidity for \(H^d_{\mathfrak{m}}(R)\). We also provide partial positive answers to Schoutens' question on whether the vanishing of sufficiently high Betti numbers of a big Cohen--Macaulay algebra forces the Cohen--Macaulay property of \(R\). For the absolute integral closure \(R^+\), we establish both Tor and Ext results. On the Tor side, we prove that \(\operatorname{Tor}_i^R(R^+,k)=0\) for some \(i>0\) implies regularity in a series cases including quotient singularities. On the Ext side, we prove that \(\operatorname{Ext}^i_R(k,R^+)=0\) for some \(i\geq d\) forces regularity for Gorenstein domains of prime characteristic, and we obtain analogous results for graded normal domains of dimension \(2\) and also for quotient and isolated singularities in any dimension.

Submission history

From: Mohsen Asgharzadeh [view email]
[v1] Thu, 18 Jun 2026 15:45:54 UTC (19 KB)