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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.

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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[1] Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. · Zbl 0204.36402
[2] Lâcezar Avramov and Jürgen Herzog, The Koszul algebra of a codimension 2 embedding, Math. Z. 175 (1980), no. 3, 249 – 260. · Zbl 0461.14014
[3] David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), no. 3, 447 – 485. · Zbl 0373.13006
[4] R. O. Buchweitz and B. Ulrich, Homological properties which are invariant under linkage, preprint, 1983.
[5] Jürgen Herzog, Deformationen von Cohen-Macaulay Algebren, J. Reine Angew. Math. 318 (1980), 83 – 105 (German). · Zbl 0425.13005
[6] J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), no. 13-14, 1627 – 1646. · Zbl 0543.13008
[7] Jürgen Herzog and Ernst Kunz , Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, Vol. 238, Springer-Verlag, Berlin-New York, 1971. Seminar über die lokale Kohomologietheorie von Grothendieck, Universität Regensburg, Wintersemester 1970/1971. · Zbl 0231.13009
[8] Jürgen Herzog and Matthew Miller, Gorenstein ideals of deviation two, Comm. Algebra 13 (1985), no. 9, 1977 – 1990. · Zbl 0576.13007
[9] J. Herzog, W. V. Vasconcelos, and R. Villarreal, Ideals with sliding depth, Nagoya Math. J. 99 (1985), 159 – 172. · Zbl 0561.13014
[10] J. Herzog, A. Simis, and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661 – 683. · Zbl 0541.13005
[11] Craig Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043 – 1062. · Zbl 0505.13003
[12] -, Determinantal ideals of linear type, preprint, 1985.
[13] Craig Huneke and Matthew Miller, A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions, Canad. J. Math. 37 (1985), no. 6, 1149 – 1162. · Zbl 0579.13012
[14] Andrew R. Kustin and Matthew Miller, Deformation and linkage of Gorenstein algebras, Trans. Amer. Math. Soc. 284 (1984), no. 2, 501 – 534. · Zbl 0545.13010
[15] Karsten Lebelt, Zur homologischen Dimension äusserer Potenzen von Moduln, Arch. Math. (Basel) 26 (1975), no. 6, 595 – 601. · Zbl 0335.13007
[16] Karsten Lebelt, Freie Auflösungen äusserer Potenzen, Manuscripta Math. 21 (1977), no. 4, 341 – 355 (German, with English summary). · Zbl 0365.13004
[17] A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), no. 2, 203 – 224. · Zbl 0467.13009
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