Betti numbers of modules of exponent two over regular local rings (Q1364309)

Let \((R, {\mathfrak m},k)\) be a regular local ring of dimension \(n\) and \(M\) an \(R\)-module of finite length. Horrocks posed two questions: (1) Is the \(i\)-th Betti number of \(M\) at least \(n\choose i\)? (2) Is the rank of the \(i\)-th syzygy of \(M\) at least \({n-1 \choose i-1}\)? Of course the affirmative answer of the second question gives the affirmative answer of the first one and we have the affirmative answer of these questions for modules annihilated by \({\mathfrak m}\). The present paper gives the affirmative answer of these questions for modules annihilated by \({\mathfrak m}^2\). This paper contains a further result. Let \(r,s\) and \(i\) be positive integers. Then there is an integer \(a\geq{n-1 \choose i-1}\), depending only on \(n,i,r\) and \(s\), such that the rank of the \(i\)-th syzygy of \(M\geq a\) for any \(R\)-module \(M\) with exact sequence \(0\to k^s \to M \to k^r \to 0\) and the equality holds for ``almost all'' \(M\).