Huneke-Ulrich almost complete intersections of Cohen-Macaulay type two
Let \((R, {\mathfrak m}, k)\) be a commutative, local, noetherian ring, \(X_{(2n + 1) \times (2n + 1)}\) be an alternating matrix and \(Y_{1 \times (2n + 1)}\) be a matrix, both with entries from \({\mathfrak m}\), and \(I\) be the ideal generated by the entries of \(YX\). Assume \(n \geq 2\), and \(I\) has grade at least \(2n\). Then \(I\) has grade exactly \(g = 2n\) and \(I\) is minimally generated by \(g + 1\) elements (hence \(I\) is a perfect almost complete intersection). If \(R\) is Gorenstein, then \(I\) has Cohen-Macaulay type two and is in the linkage class of a complete intersection. The author shows that the minimal free resolution of \(R/I\) over \(R\) is a \(DG \Gamma\)-algebra, that is, an associative, differential, graded algebra with divided powers. -- The proof is an explicit construction of a suitable multiplication on a minimal free resolution of \(R/I\). As an application, if \(R\) is regular and \(\text{char} k \geq (n + 2)/2\) or \(\text{char} k = 0\), then the Poincaré series \(P^M_{R/I} (z) = \sum_{i \geq 0} \dim_k \text{Tor}_i^{R/I} (M,k)z^i\) is a rational function for every finitely generated module \(M\). It follows that if \(\text{pd} M\) is infinite, then the Betti numbers of \(M\) eventually increase with strong exponential growth.
scientific article; zbMATH DE number 783869