On Carlson's depth conjecture

[摘要] Let G be a finite group and k a finite field whose characteristic divides the order of G. Recall that an associated prime of H* (G, k) is a prime ideal of H* (G, k) which is the annihilator of some non-zero element of H* (G, k), and that depth H* (G, k) refers to the maximal length of a regular sequence in H* (G, k). J.F. Carlson conjectured that depth H* (G, k) = min{dim p} where p ranges over the associated primes of H* (G, k). We show that for any finitely generated connected k-algebra A (A(n) = 0 for n < 0, A(0) = k) and for any finitely generated A-module M, the statement depth M = min{dim p vertical bar p an associated prime of M} is equivalent to several other statements concerning the module M, in particular that there exists a non-zero A-homomorphism M-V -> H-m(d) where M-V is the k-dual of M and d is the depth of M. Then a spectral sequence due to Greenlees shows this last condition is true when the edge homomorphism in this spectral sequence is nontrivial. This is trivially true if the spectral sequence collapses, which happens when dim H* (G, k) = depth H* (G, k) + 1. (C) 2019 Elsevier B.V. All rights reserved.

[发布日期] 2019-12-01

[关键词] Group cohomology;Depth;Associated primes;Local cohomology