Complete intersections and Gorenstein rings. (English) Zbl 0860.13012

Coates, John (ed.) et al., Elliptic curves, modular forms, and Fermat’s last theorem. Proceedings of the conference on elliptic curves and modular forms held at the Chinese University of Hong Kong, December 18-21, 1993. Cambridge, MA: International Press. Ser. Number Theory. 1, 99-109 (1995).
Let \({\mathfrak O}\) be a complete discrete valuation ring, \(R\) a complete noetherian local \({\mathfrak O}\)-algebra, \(B\) a finite flat local \({\mathfrak O}\)-algebra, and \(\phi: R\to B\), \(\pi:B\to{\mathfrak O}\) surjective \({\mathfrak O}\)-algebra homomorphisms. The authors main result is the following theorem: If the length of the \({\mathfrak O}\)-module \((\text{ker } \pi\phi)/(\text{ker } \pi\phi)^2\) is finite and less than or equal to the length of \({\mathfrak O}/\pi(\text{Ann}_B \text{ ker } \pi)\), then \(B\) is a complete intersection, \(\pi(\text{Ann}_B \text{ ker } \pi)\neq 0\), and \(\pi\) is an isomorphism. (Actually the converse is also true.) As the author points out, the theorem is a generalization of a result due to Wiles in the case in which \(B\) is a Gorenstein ring. The reviewer’s impression is that the topic treated in the paper is closely related to an investigation by G. Scheja and U. Storch [J. Reine Angew. Math. 278/279, 174-190 (1975; Zbl 0316.13003) and Manuscr. Math. 19, 75-104 (1976; Zbl 0329.13011)]. At least the following proposition at the end of the paper seems to be contained in the articles just quoted (see section 5 in the first article and section 6 in the second): Let \({\mathfrak O}\) and \(B\) be as above, having the same residue class field, and denote by \(\mu\) the multiplication \(B^e= B\otimes_{\mathfrak O}B\to B\). Suppose that the \(B\)-module \(\Omega_{B/{\mathfrak O}}\) of Kähler differentials has finite length. Then \(B\) is a complete intersection if and only if \(\mu(\text{Ann}_{B^e} (\text{Ker }\mu))\) is a principal ideal and equal to the Kähler different \(F_B(\Omega_{B/{\mathfrak O}})\).
For the entire collection see [Zbl 0824.00025].


13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C40 Linkage, complete intersections and determinantal ideals