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Classification of the Tor-algebras of codimension four Gorenstein local rings. (English) Zbl 0565.13004
Let R,m,k be a Gorenstein local ring in which 2 is a unit and assume that k has square roots. Let K be a grade four Gorenstein ideal in R; and Let \(\Lambda_{\bullet}\) be the graded algebra Tor\({}^ R_{\bullet}(R/K,k)\). We prove that \(\Lambda_{\bullet}\) is a Poincaré duality algebra with one of just four possible forms for the multiplication in non-complementary degrees. In subsequent work with C. Jacobsson this result is used to demonstrate rationality of the Poincaré series \(P_{R/K}\). The main technique used is that of tight double linkage, a notion developed by the authors in earlier work, which enables one to understand the minimal free resolution of R/K in great detail.

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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References:
[1] Avramov, L.: Small homomorphisms of local rings. J. Algebra50, 400-453 (1978) · Zbl 0395.13005
[2] Avramov, L., Golod, E.: On the homology of the Koszul complex of a local Gorenstein ring. Mat. Zametki9, 53-58 (1971); Math. Notes9, 30-32 (1971) · Zbl 0213.04904
[3] Buchsbaum, D., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Amer. J. Math.99, 447-485 (1977) · Zbl 0373.13006
[4] Cartan, H., Eilenberg, S.: Homological Algebra. Princeton: Princeton University Press 1956 · Zbl 0075.24305
[5] Gulliksen, T., Neg?rd, O.: Un complexe r?solvant pours certains id?aux d?terminantiels. C.R. Acad. Sc. Paris (A)274, 16-18 (1972)
[6] Herzog, J., Miller, M.: Gorenstein ideals of deviation two. To appear, Commun. in Algebra · Zbl 0576.13007
[7] Hochster, M.: Topics in the homological theory of modules over commutative rings, Regional Conference Series in Mathematics, v. 24, American Mathematical Society, Providence, 1975 · Zbl 0302.13003
[8] Jacobsson, C.: On the positivity of the deviations of a local ring. Preprint 1983
[9] Jacobsson, C., Kustin, A., Miller, M.: The Poincar? series of a codimension four Gorenstein ring is rational. To appear, J. Pure and Applied Algebra · Zbl 0575.13007
[10] Kunz, E.: Almost complete intersections are not Gorenstein rings. J. Algebra28, 111-115 (1974) · Zbl 0275.13025
[11] Kustin, A.: New examples of rigid Gorenstein unique factorization domains. Comm. in Algebra12, 2409-2439 (1984) · Zbl 0567.13005
[12] Kustin, A.: The minimal free resolutions of the Huneke-Ulrich deviation two Gorenstein ideals. To appear, J. Algebra · Zbl 0646.13011
[13] Kustin, A., Miller, M.: A general resolution for grade four Gorenstein ideals. Manus. Math.35, 221-269 (1981) · Zbl 0495.13004
[14] Kustin, A., Miller, M.: Algebra structures on minimal resolutions of Gorenstein rings, in Commutative Algebra: Analytic Methods (ed. R. Draper), Lecture Notes in Pure and Appl. Math.68, New York: Marcel Dekker 1982 · Zbl 0498.13011
[15] Kustin, A., Miller, M.: Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four. Math. Zeit.173, 171-184 (1980) · Zbl 0435.13009
[16] Kustin, A., Miller, M.: Constructing big Gorenstein ideals from small ones. J. Algebra85, 303-322 (1983) · Zbl 0522.13011
[17] Kustin, A., Miller, M.: Deformation and linkage of Gorenstein algebras. Trans. Amer. Math. Soc.284, 501-534 (1984) · Zbl 0545.13010
[18] Kustin, A., Miller, M.: Multiplicative structure on resolutions of algebras defined by Herzog ideals. J. London Math. Soc. (2)28, 247-260 (1983) · Zbl 0512.13009
[19] Kustin, A., Miller, M.: Structure theory for a class of grade four Gorenstein ideals. Trans. Amer. Math. Soc.270, 287-307 (1982) · Zbl 0495.13005
[20] Kustin, A., Miller, M.: Tight double linkage of Gorenstein algebras. To appear, J. Algebra · Zbl 0568.13006
[21] Lang, S.: Algebra: New York: Addison-Wesley 1971 · Zbl 0193.34701
[22] Peskine, C., Szpiro, L.: Liaison des vari?t?s alg?briques I. Invent. Math.26, 271-302 (1974) · Zbl 0298.14022
[23] Wiebe, H.: ?ber homologische Invarianten lokaler Ringe. Math. Ann.179, 257-274 (1969) · Zbl 0169.05701
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