Growth of Betti numbers of modules over local rings of small embedding codimension or small linkage number (Q1336813)

Let \(R\) denote a commutative Noetherian local ring and \(M\) a finitely generated \(R\)-module. Suppose the radius of convergence \(\rho\) of the Poincaré series \(\sum b_n X^n\) of \(M\) is \(<1\). Assume that \(R\) is one of the following types of rings: (a) \(R\) is one link from a complete intersection. (b) \(R\) is two links from a complete intersection and Gorenstein. (c) \(\text{edim} R - \text{depth} R \leq 3\). (d) \(\text{edim} R - \text{depth} R = 4\) and \(R\) is Gorenstein. (e) \(\text{edim} R - \text{depth} R = 4\) and \(R\) is a Cohen-Macaulay almost complete intersection which contains \({1 \over 2}\). Using results by \textit{L. L. Avramov}, \textit{C. Jacobsson}, \textit{A. R. Kustin}, \textit{M. Miller} and \textit{S. M. Palmer}, the author shows via direct computations (sometimes with the help of a computer): \[ \lim_{n \to \infty} \rho^n b_n > 0. \] As a consequence of this he gets: The sequence \((b_n)\) of Betti numbers of \(M\) is eventually nondecreasing and its growth is strongly exponential.