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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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