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Lyubeznik numbers of monomial ideals

Authors: Josep Àlvarez Montaner and Alireza Vahidi
Journal: Trans. Amer. Math. Soc. 366 (2014), 1829-1855
MSC (2010): Primary 13D45, 13N10, 13F55
Published electronically: November 25, 2013
MathSciNet review: 3152714
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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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Additional Information

Josep Àlvarez Montaner
Affiliation: Department Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, Barcelona 08028, Spain

Alireza Vahidi
Affiliation: Department of Mathematics, Payame Noor University, 19395-4697 Tehran, I.R. of Iran

Received by editor(s): August 14, 2011
Received by editor(s) in revised form: February 7, 2012, and April 17, 2012
Published electronically: November 25, 2013
Additional Notes: The first author was partially supported by MTM2010-20279-C02-01 and SGR2009-1284
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.