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2013 | OriginalPaper | Chapter

Local Cohomology Modules Supported on Monomial Ideals

Author : Josep Àlvarez Montaner

Published in: Monomial Ideals, Computations and Applications

Publisher: Springer Berlin Heidelberg

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Abstract

Local cohomology was introduced by A. Grothendieck in the early 1960s and quickly became an indispensable tool in Commutative Algebra. Despite the effort of many authors in the study of these modules, their structure is still quite unknown. C

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Footnotes
1
The presentation given in [135] slightly differs from the one given in [7] at the E 1-page but they coincide at the E 2-page.
 
2
One may also consider filtrations associated to other weight vectors \((u,v) \in {\mathbb{Z}}^{2n}\) with u + v ≥ 0, but then the corresponding associated graded ring \(\mathit{gr}_{(u,v)}D_{R}\) is not necessarily a polynomial ring.
 
3
Given the relation \(x_{i}\partial _{i} + 1 = 0\) one may interpret i as the fraction \(\frac{1} {x_{i}}\) in the localization.
 
4
After completion we can always assume that \(R = k[[x_{1},\ldots,x_{n}]]\) is the formal power series ring.
 
5
One has to interpret the non-zero entries in the matrix as inclusions of the corresponding components of the Čech complex.
 
6
We consider this point of view since the same results are true if we consider the defining ideal of any arrangement of linear subspaces.
 
7
In the language of [157] we would say that the n-hypercube has the same information as the frame of the r-linear strand.
 
8
In positive characteristic we apply the functor \({}^{{\ast}}\mathrm{Hom}_{R}(\cdot,E_{\mathbf{1}})\).
 
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Metadata
Title
Local Cohomology Modules Supported on Monomial Ideals
Author
Josep Àlvarez Montaner
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-642-38742-5_5

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