Lyubeznik numbers of monomial ideals

Montaner, Josep; Vahidi, Alireza

doi:10.1090/S0002-9947-2013-05862-X

Lyubeznik numbers of monomial ideals
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by Josep Àlvarez Montaner and Alireza Vahidi PDF

Trans. Amer. Math. Soc. 366 (2014), 1829-1855 Request permission

Abstract:

Let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ independent variables, where $k$ is a field. In this work we will study Bass numbers of local cohomology modules $H^r_I(R)$ supported on a squarefree monomial ideal $I\subseteq R$. Among them we are mainly interested in Lyubeznik numbers. We build a dictionary between the modules $H^r_I(R)$ and the minimal free resolution of the Alexander dual ideal $I^{\vee }$ that allows us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of $I^{\vee }$. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support.

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