Local Cohomology at Monomial Ideals

Details Local Cohomology at Monomial Ideals Local Cohomology at Monomial Ideals

2000-01-26

arxiv.org/abs/math/0001153v1

J. Symbolic Comput. 29 (2000), no. 4-5, 709-720.

Mathematics

Commutative Algebra

13 pages, 2 figures, to appear in Journal of Symbolic Computation

Scientific paper

For a reduced monomial ideal B in R=k[X_1,...,X_n], we write H^i_B(R) as the union of {Ext^i(R/B^[d],R)}_d, where {B^[d]}_d are the "Frobenius powers of B". We describe H^i_B(R)_p, for every p in Z^n, in the spirit of the Stanley-Reisner theory. As a first application we give an isomorphism Tor_i(B', k)_p\iso Ext^{|p|-i}(R/B,R)_{-p} for all p in {0,1}^n, where B' is the Alexander dual ideal of B. We deduce a canonical filtration of Ext^i(R/B,R) with succesive quotients of the form R/(X_{j_1},...,X_{j_i}) suitably shifted, the multiplicities being computed from the Betti numbers of B'. As a final application, we give a topological description for the associated primes of Ext^i(R/B,R).

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