Depth of symmetric algebras of certain ideals (Q2706567)

Let \(R\) be a Cohen-Macaulay local ring with infinite residue field and let \(I\) be an ideal with \(\text{height} I \geq 2\). The purpose of the paper under review is to compute the depth of symmetric algebras \(S(I)\) and \(S(I/I^2)\) in terms of the depth of \(R/I_1(\phi)\) with certain assumptions, where \(\phi\) is a minimal presentation matrix of \(I\). As a corollary, conditions for \(S(I)\) or \(S(I/I^2)\) to be Cohen-Macaulay are given and it is found that \(S(I)\) and \(S(I/I^2)\) can never be simultaneously Cohen-Macaulay with the mentioned assumptions. The author constructs some regular local rings and generically complete intersection perfect ideals such that \(S(I/I^2)\) is Cohen-Macaulay and \(S(I)\) is not Cohen-Macaulay, which answers a question of \textit{J. Herzog, A. Simis} and \textit{W. V. Vasconcelos} [in: ``Commutative algebra'', Proc. Conf., Trento 1981, Lect. Notes Pure Appl. Math. 84, 79-169 (1983; Zbl 0499.13002)].