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Local Cohomology at Monomial Ideals

✍ Scribed by Mircea Mustaţa

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
266 KB
Volume
29
Category
Article
ISSN
0747-7171

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✦ Synopsis

We prove that if B ⊂ R = k[X 1 , . . . , Xn] is a reduced monomial ideal, then d] , R), where B [d] is the dth Frobenius power of B. We give two descriptions for H i B (R) in each multidegree, as simplicial cohomology groups of certain simplicial complexes. As a first consequence, we derive a relation between Ext R (R/B, R) and Tor R (B ∨ , k), where B ∨ is the Alexander dual of B. As a further application, we give a filtration of Ext i R (R/B, R) such that the quotients are suitable shifts of modules of the form R/(X i 1 , . . . , X ir ). We conclude by giving a topological description of the associated primes of Ext i R (R/B, R). In particular, we characterize the minimal associated primes of Ext i R (R/B, R) using only the Betti numbers of B ∨ .

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