# zbMATH — the first resource for mathematics

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)
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.