On Carlson’s Depth Conjecture in Group Cohomology

Depth, detection and associated primes in the cohomology of finite groups

For a fixed prime p, let H∗(G) denote the mod p cohomology ring of a finite group G, that is, ExtFpG(Fp,Fp) interpreted algebraically and H ∗(K(G, 1); Fp) interpreted topologically. The cohomology ring is a graded commutative Noetherian ring, or equivalently, a finitely generated graded commutative Fp-algebra. Although the ring is not strictly commutative, the usual concepts from commutative ri...

On the Castelnuovo-mumford Regularity of the Cohomology Ring of a Group

Dave Benson, in [2], conjectured that for any finite group G and any prime p the Castelnuovo-Mumford regularity of the cohomology ring, H(G,Fp), is zero. He showed that reg(H(G,Fp)) ≥ 0 and succeeded in proving equality when the difference between the dimension and the depth is at most two. The purpose of this paper is to prove Benson’s Regularity Conjecture as a corollary of the following result.

ANNIHILATOR OF LOCAL COHOMOLOGY MODULES UNDER THE RING EXTENSION R⊂R[X]

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...

On Reducing sequences and an Application to Local Cohomology Modules

Isoperimetric Inequalities and the Friedlander–milnor Conjecture

We prove that Friedlander’s generalized isomorphism conjecture on the cohomology of algebraic groups, and hence Milnor’s conjecture on the cohomology of the complex algebraic Lie group G(C) made discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G(F ), where F is the algebraic closure of a finite field.