Binomial behavior of Betti numbers for modules of finite length (Q1107585)

Let (R,m,k) be a regular local ring of dimension n, M an R-module of finite length, then dim k(Tor\({}\) \(R_ i(k,M))\equiv \beta_ i(M)\) are called the Betti numbers of M, where \(Tor\) \(R_ i(k,M)\) is the i-th syzygy module in a minimal free resolution of M. In the first section of this paper the author establishes fairly general lower bounds on the \(\beta_ i(M)\) if R is isomorphic to the power series ring \(k[[ x_ 1,...,x_ n]]\). In the second section the following result is obtained: If M is an R-module of monomial type and of finite length, then \(\beta_ j(M)\geq \left( \begin{matrix} \dim (R)\\ j\end{matrix} \right)\) for each j.