zbMATH — the first resource for mathematics

On the Gorensteinness of graded rings associated to ideals of analytic deviation one. (English) Zbl 0816.13004
Heinzer, William J. (ed.) et al., Commutative algebra: syzygies, multiplicities, and birational algebra: AMS-IMS-SIAM summer research conference on commutative algebra, held July 4-10, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 159, 51-72 (1994).
Let \(I\) be an ideal in a Cohen-Macaulay local ring \(A\) with infinite residue class field. This paper studies conditions for the Rees algebra \(R(I)\) and the associated graded ring \(G(I)\) to be either Cohen-Macaulay or Gorenstein. For example, suppose in addition that \(I\) is generically a complete intersection with analytic deviation one, \(A/I\) is Cohen- Macaulay, and \(J\) is a minimal reduction of \(I\). The authors then prove that \(G(I)\) is Gorenstein if and only if \(A\) is Gorenstein and \(r_ J(I) = 0\), and that if \(r_ J(I) \leq 2\), then \(G(I)\) is Cohen-Macaulay if and only if \[ \text{depth}(A/I^ 2) \geq \dim (A/I) - 1. \]
For the entire collection see [Zbl 0790.00007].

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C14 Cohen-Macaulay modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)