Analytic isomorphisms of compressed local algebras

Elias, J.; Rossi, M.

doi:10.1090/S0002-9939-2014-12313-6

Analytic isomorphisms of compressed local algebras
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by J. Elias and M. E. Rossi PDF

Proc. Amer. Math. Soc. 143 (2015), 973-987 Request permission

Abstract:

In this paper we consider Artin compressed local algebras, that is, local algebras with maximal length in the class of those with given embedding dimension and socle type. They have been widely studied by several authors, including Boij, Iarrobino, Fröberg and Laksov. In this class the Gorenstein algebras play an important role. The authors proved that a compressed Gorenstein $K$-algebra of socle degree $3$ is canonically graded, i.e. analytically isomorphic to its associated graded ring. This unexpected result has been extended to compressed level $K$-algebras of socle degree $3$ in a paper by De Stefani. This paper somehow concludes the investigation proving that Artin compressed Gorenstein $K$-algebras of socle degree $s \le 4$ are always canonically graded, but explicit examples prove that the result does not extend to socle degree $5$ or to compressed level $K$-algebras of socle degree $4$ and type $>1.$ As a consequence of this approach we present classes of Artin compressed $K$-algebras which are canonically graded.

References

Mats Boij, Betti numbers of compressed level algebras, J. Pure Appl. Algebra 134 (1999), no. 2, 111–131. MR 1663785, DOI 10.1016/S0022-4049(97)90163-8
Gianfranco Casnati, Juan Elias, Roberto Notari, and Maria Evelina Rossi, Poincaré series and deformations of Gorenstein local algebras, Comm. Algebra 41 (2013), no. 3, 1049–1059. MR 3037178, DOI 10.1080/00927872.2011.636643
Young Hyun Cho and Anthony Iarrobino, Hilbert functions and level algebras, J. Algebra 241 (2001), no. 2, 745–758. MR 1843323, DOI 10.1006/jabr.2001.8787
A. De Stefani, Short Artinian level local k-algebras, Comm. Algebra 42 (2014), no. 2, 729–754.
J. Elias and A. Iarrobino, The Hilbert function of a Cohen-Macaulay local algebra: extremal Gorenstein algebras, J. Algebra 110 (1987), no. 2, 344–356. MR 910388, DOI 10.1016/0021-8693(87)90050-0
J. Elias and M. E. Rossi, Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4589–4604. MR 2922602, DOI 10.1090/S0002-9947-2012-05430-4
J. Elias and G. Valla, Isomorphism classes of certain Artinian Gorenstein Algebras, Algebra and Representation Theory 14 (2011), 429–448.
Jacques Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), no. 4, 399–416 (French, with English summary). MR 518046
R. Fröberg and D. Laksov, Compressed algebras, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 121–151. MR 775880, DOI 10.1007/BFb0099360
Anthony Iarrobino, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), no. 1, 337–378. MR 748843, DOI 10.1090/S0002-9947-1984-0748843-4
Anthony A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), no. 514, viii+115. MR 1184062, DOI 10.1090/memo/0514
F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612