zbMATH — the first resource for mathematics

On graded rings associated to analytic deviation one ideals. (English) Zbl 0803.13002
The purpose of the paper under review is to prove an “analytic deviation one” version of a result of S. Goto and Y. Shimoda [cf. Commutative algebra: Analytical methods, Conf. Fairfax 1979, Lect. Notes Pure Appl. Math. 68, 201-231 (1982; Zbl 0482.13011); theorem 3.1]. More precisely, given a local Cohen-Macaulay ring \(R\) and a generically complete intersection ideal \(I \subseteq R\) of analytic deviation one, the main theorem characterizes the Cohen-Macaulayness of the Rees algebra \(R[It]\) by the Cohen-Macaulayness of the graded ring \(G(I)\) and the conditions \(\text{ht}(I)>0\), \(r_ J(I) \leq \text{ht}(I)\), where \(J\) is a minimal reduction of \(I\).
The heart of the proof is a formula for the a-invariant of \(G(I)\) in terms of \(r_ J(I)\) and \(\text{ht}(I)\) which generalizes a result of M. Herrmann, J. Ribbe and S. Zarzuela [cf. Trans. Am. Math. Soc. 342, No. 2, 631-643 (1994); proposition 2.5]. An application to the relation type of an ideal, i.e. to the largest degree of an equation in a minimal system defining the Rees algebra, is presented in the last section.
Meanwhile, the main result has been generalized to ideals of analytic deviation two by S. Goto and Y. Nakamura [cf. “Cohen- Macaulay Rees algebras of ideals having analytic deviation two” (preprint)], and to ideals having higher analytic deviation by Z. Tang [cf. Commun. Algebra 22, No. 12, 4855-4898 (1994; see the following review)].

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13A02 Graded rings
Full Text: DOI