Betti numbers of multigraded modules (Q805679)

Horrocks' question concerns the comparison of the i-th Betti number of a finite length module over a d-dimensional regular local ring R with the binomial coefficient \(\left( \begin{matrix} d\\ i\end{matrix} \right)\). When k is a field and \(R=k[x_ 1,...,x_ d]\), \textit{Santoni} extended the work of Evans and Griffith to show that \(\beta^ R_ i(M)\geq \left( \begin{matrix} d\\ i\end{matrix} \right)\) for every finite length module M graded by monomials. Here the author uses a slightly different method to prove the result for any finite length multigraded module M. He shows that \(\beta^ R_ i(M)\geq \left( \begin{matrix} d\\ i\end{matrix} \right)\) and that for (i\(\neq 0,d)\) equality holds only if M is isomorphic to R modulo an R- sequence. When M is not isomorphic to R modulo an R-sequence, then \(\beta^ R_ i(M)\geq \left( \begin{matrix} d\\ i\end{matrix} \right)+\left( \begin{matrix} d-1\\ i-1\end{matrix} \right)\) or \(\beta^ R_ i(M)\geq \left( \begin{matrix} d\\ i\end{matrix} \right)+\left( \begin{matrix} d-1\\ i\end{matrix} \right).\) If M is an arbitrary multigraded module and s is the length of a maximal R-sequence in the annihilator of M, then similar inequalities hold for the Betti numbers of M with d replaced by d-s.