A note on Artinian Gorenstein algebras of codimension three
The famous structure theorem for Gorenstein ideals of codimension 3 of \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Am. J. Math. 99, 447-485 (1977; Zbl 0373.13006)] allows one to derive necessary conditions for the graded Betti numbers of a Gorenstein algebra of codimension 3. Several authors like \textit{J. Herzog, Ngô Viêt Trung} and \textit{G. Valla} [J. Math. Kyoto Univ. 34, No.~1, 47-72 (1994; Zbl 0836.14031)] or \textit{S. J. Diesel} [Pac. J. Math. 172, No.~2, 365-397 (1996; Zbl 0882.13021)] showed, by explicit examples, that these conditions are also sufficient. In this paper, the author proposes a new explicit construction of Artinian Gorenstein algebras of codimension 3, achieving all the possible graded Betti numbers. The idea is, roughly speaking, the following: Let \(X\), \(Y\) be disjoint finite sets of points in \(\mathbb{P}^2\) geometrically linked, i.e., such that \(X\cup Y\) is a complete intersection. Let \(R=k[x,y,z]\) be the homogeneous coordinate ring of \(\mathbb{P}^2\). Then \(R/(I(X)+I(Y))\) is an Artinian Gorenstein algebra of codimension 3 and using the exact sequence: \[ 0\to R/I(X)\cap I(Y)\to R/I(X)\oplus R/I(Y)\to R/I(X)+I(Y)\to 0 \] one can deduce, from free resolutions of \(R/I(X)\) and \(R/I(Y)\), a free resolution of \(R/I(X)+I(Y)\) as a cone complex. As an application, the author shows that, for any Artinian Gorenstein algebra of codimension 3, there is an Artinian Gorenstein algebra of codimension 3 with the weak Stanley property which has the same Hilbert function.
scientific article; zbMATH DE number 1308298