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On Gorenstein ideals of codimension four. (English) Zbl 0665.13011
R will be a local Gorenstein ring, and I is an ideal of height g and finite projective dimension. We denote by \(W=Ext^ g(R/I,R)\) the canonical module of \(S=R/I\). - We shall say that I (or S) is strongly unobstructed if \(I\otimes W\) is Cohen-Macaulay. We denote by \(H_ i(I)\) the homology modules of a Koszul complex built on a set of generators of the ideal. I is said to be strongly Cohen-Macaulay if all the \(H_ i(I)\) are Cohen-Macaulay. Finally, the deviation of the ideal I is the deficit n-g, where n is the minimum number of generators of I and g is its height. The notation \(S_ 2(M)\) will stand for the symmetric square of the R-module M.
(1.1) Theorem. Let R be a Gorenstein local ring, and let I be a Gorenstein ideal of codimension four. If \(H_ 1(I)\) is Cohen-Macaulay then I is strongly unobstructed.
(1.2) Theorem. Let R be a Gorenstein local ring in which 2 is a unit. Let I be a Gorenstein ideal of codimension four and deviation two. If I is a generic complete intersection, then I is a hypersurface section of a Gorenstein ideal of height three. That is, \(I=(J,f)\), where J is the ideal generated by the \(4\times 4\) Pfaffians of an alternating \(5\times 5\) matrix and f is a regular element on R/J.
(1.3) Theorem. Let R be a Gorenstein local ring, and let I be a Cohen- Macaulay ideal of height three. Then \(H_ 1(I)\) is Cohen-Macaulay if and only if \(S_ 2(W)\) is Cohen-Macaulay.

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
14M07 Low codimension problems in algebraic geometry
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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