Non-Noetherian Bass and Betti numbers

Mathematics > Commutative Algebra [Submitted on 18 Jun 2026] Title:Non-Noetherian Bass and Betti numbers View PDF HTML (experimental)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. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.