Let $(A, \m, K) $ be an Artinian Gorenstein local ring with $K$ an algebraically closed field of characteristic $0. $ In the present paper we prove a structure theorem describing the Artinian Gorenstein local $K$-algebras satisfying $\m^4=0$. We use this result in order to prove that such a $K$-algebra has rational Poincar{\'e} series and it is smoothable in any embedding dimension, provided $\dim_K \m^2/\m^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$-algebra with $\m^4=0$ has rational Poincar{\'e} series.

Poincarè series and deformations of Gorenstein local algebras with low socle degree / Casnati, Gianfranco; Elias, J.; Notari, Roberto; Rossi, M. E.. - In: COMMUNICATIONS IN ALGEBRA. - ISSN 0092-7872. - STAMPA. - 41:3(2013), pp. 1049-1059. [10.1080/00927872.2011.636643]

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

CASNATI, GIANFRANCO;NOTARI, ROBERTO;
2013

Abstract

Let $(A, \m, K) $ be an Artinian Gorenstein local ring with $K$ an algebraically closed field of characteristic $0. $ In the present paper we prove a structure theorem describing the Artinian Gorenstein local $K$-algebras satisfying $\m^4=0$. We use this result in order to prove that such a $K$-algebra has rational Poincar{\'e} series and it is smoothable in any embedding dimension, provided $\dim_K \m^2/\m^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$-algebra with $\m^4=0$ has rational Poincar{\'e} series.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2379814
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