Poincaré series and deformations of Gorenstein local algebras with low socle degree

2010/05/10 by Gianfranco Casnati, Juan Elias, Casnati, Gianfranco +5
Mathematics · #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG

paper · pdf · doi:10.48550/arxiv.1005.1677

arxiv created 2010/05/10 · arxiv updated 2010/05/12

Abstract

Let K be an algebraically closed field of characteristic 0, and let A be an Artinian Gorenstein local commutative and Noetherian K--algebra, with maximal ideal M. In the present paper we prove a structure theorem describing such kind of K--algebras satisfying M⁴=0. We use this result in order to prove that such a K--algebra A has rational Poincaré series and it is always smoothable in any embedding dimension, if dim_K M²/M³ ≤ 4. We also prove that the generic Artinian Gorenstein local K--algebra with socle degree three has rational Poincaré series, in spite of the fact that such algebras are not necessarily smoothable.

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