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Local Cohomology Modules Supported on Monomial Ideals

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Monomial Ideals, Computations and Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2083))

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

Josep Àlvarez Montaner was partially supported by SGR2009-1284 and MTM2010-20279- C02-01.

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Notes

  1. 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. 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. 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. 4.

    After completion we can always assume that \(R = k[[x_{1},\ldots,x_{n}]]\) is the formal power series ring.

  5. 5.

    One has to interpret the non-zero entries in the matrix as inclusions of the corresponding components of the Čech complex.

  6. 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. 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. 8.

    In positive characteristic we apply the functor \({}^{{\ast}}\mathrm{Hom}_{R}(\cdot,E_{\mathbf{1}})\).

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Acknowledgements

Many thanks go to Anna M. Bigatti, Philippe Gimenez, and Eduardo Sáenz-de-Cabezón for the invitation to participate in the MONICA conference and the great enviroment they created there. I am also indebted with Oscar Fernández Ramos who agreed to develop the Macaulay 2 routines that not only allowed us to perform many computations but also enlightened part of my research.

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  1. Departamento de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, Barcelona, 08028, Spain

    Josep Àlvarez Montaner

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  1. Universitá degli Studi di Genova Dipartimento di Matematica, Genova, Italy

    Anna M. Bigatti

  2. Geometría y Topología, University of Valladolid, Dpto. de Álgebra, Análisis Matemático, Valladolid, Spain

    Philippe Gimenez

  3. University of La Rioja Matemáticas y Computación, Logroño, Spain

    Eduardo Sáenz-de-Cabezón

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Montaner, J.À. (2013). Local Cohomology Modules Supported on Monomial Ideals. In: Bigatti, A., Gimenez, P., Sáenz-de-Cabezón, E. (eds) Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, vol 2083. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38742-5_5

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