Asymptotic behavior of dimensions of syzygies (Q2839295)

In this interesting and well-written paper the authors study the asymptotic behavior of supports of syzygies under the next (general) assumptions. \(R\) is a commutative Noetherian (not necessarily unmixed or equidimensional) local ring, \(M\) is a finitely generated \(R\)-module with \textit{eventually non-decreasing Betti numbers}.NEWLINENEWLINELet \(\Omega_i(M)\) be the \(i\)'th \textit{syzygy module} of \(M\). The main result of this paper states that, for \(n \gg 0\), the following hold: Min\((\Omega_n(M))\subseteq \)Min\((R)\), Supp\((\Omega_n(M))\subseteq \)Supp\((\Omega_{n+2i}(M))\) for all \(i\geq 0\); and if Supp\((\Omega_n(M))\neq\)Spec\((R)\), then \(\beta_{n+2i}(M)=\beta_{n+2i+1}(M)\) for all \(i\geq 0\).NEWLINENEWLINEThe authors also determine how quickly Supp\((\Omega_{2i}(M))\) stabilizes.NEWLINENEWLINEIt should be mentioned that according to a conjecture of \textit{L. Avramov} [Homotopie algébrique et algébre locale, Journ. Luminy/France 1982, Astérisque 113--114, 15--43 (1984; Zbl 0552.13003)], the Betti numbers of any finitely generated module over an arbitrary noetherian local ring are eventually non-decreasing.