On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence
| Autor: | Gianfranco Casnati, Roberto Notari, Joachim Jelisiejew |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Hilbert series and Hilbert polynomial Mathematics::Commutative Algebra Series (mathematics) 13H10 General Mathematics Polynomial ring rational Poincar\'e series 13D40 symbols.namesake Artinian Gorenstein local algebra Poincaré series symbols Artinian Gorenstein local algebra rational Poincaré series Maximal ideal rational Poincaré series Ideal (ring theory) Artinian Gorenstein local algebra rational Poincar\'e series Algebra over a field Mathematics |
| Zdroj: | Rocky Mountain J. Math. 46, no. 2 (2016), 413-433 |
| Popis: | Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM $, \[P_A(z) := \sum _{p=0}^{\infty } (\tor _p^A(k,k))z^p \] its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim _k({\fM ^2/\fM ^3}) \leq 4 $ and $ \dim _k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\in [2,c]$ and equal to $1$ for $n =c+1$. The results are obtained due to a decomposition of the apolar ideal $\Ann (F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables. |
| Databáze: | OpenAIRE |
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