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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].

##### MSC:
 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.)