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

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

2010-05-10

arxiv.org/abs/1005.1677v1

Mathematics

Commutative Algebra

Scientific paper

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 $M^4=0$. We use this result in order to prove that such a $K$--algebra $A$ has rational Poincar\'e series and it is always smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$--algebra with socle degree three has rational Poincar\'e series, in spite of the fact that such algebras are not necessarily smoothable.

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