Ulrich ideals and almost Gorenstein rings (Q2802103)

Let \(R\) be a Cohen-Macaulay local ring of Krull dimension \(d\) and \(I\) be an Ulrich ideal of \(R\). In other words, the associated graded ring of I is a Cohen-Macaulay ring with \(a\)-invariant at most \(1 - d\) and \(I/I^2\) is a free \(R/I\)-module. The first condition is equivalent to the statement that \(R\) has maximal embedding dimension in the sense of [\textit{J. D. Sally}, J. Algebra 56, 168--183 (1979; Zbl 0401.13016)]. The main theorem in this paper is a calculation of the structure of the cohomology modules \(\text{Ext}_R^i(R/I,R)\). One of the consequences of this theorem is a characterization of Gorenstein rings in terms of Ulrich ideals. A further consequence is the proof that in a one-dimensional local ring which is almost Gorenstein, but not Gorenstein, the only possible Ulrich ideal is the maximal ideal. (The authors call the ring \(R\) \textit{almost Gorenstein} if there is an exact sequence of \(R\)-modules, NEWLINE\[NEWLINE0\to R\to \omega_R \to C\to 0,NEWLINE\]NEWLINEwhere \(\omega_R\) is the canonical module of \(R\) and \(C\) is an Ulrich \(R\)-module.) It is also studied when all Ulrich ideals \(I\) in a given ring \(R\) have the same minimal number of generators.