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Analytic isomorphisms of compressed local algebras


Authors: J. Elias and M. E. Rossi
Journal: Proc. Amer. Math. Soc. 143 (2015), 973-987
MSC (2010): Primary 13H10; Secondary 13H15, 14C05
DOI: https://doi.org/10.1090/S0002-9939-2014-12313-6
Published electronically: November 5, 2014
MathSciNet review: 3293715
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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.


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Additional Information

J. Elias
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
Email: elias@ub.edu

M. E. Rossi
Affiliation: Dipartimento di Matematica, Università di genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: rossim@dima.unige.it

DOI: https://doi.org/10.1090/S0002-9939-2014-12313-6
Received by editor(s): August 6, 2012
Received by editor(s) in revised form: July 2, 2013
Published electronically: November 5, 2014
Additional Notes: The first author was partially supported by MTM2013-40775-P
The second author was partially supported by PRIN 2010-11 “Geometria delle varieta’ algebriche”
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society