On Carlson's Depth Conjecture

TL;DR

This paper investigates Carlson's Depth Conjecture in group cohomology, proving it holds when the difference between the dimension and depth of H*(G,k) is exactly one, extending previous special cases.

Contribution

The paper proves Carlson's Depth Conjecture for cases where the dimension minus depth equals one, advancing understanding of the conjecture's validity.

Findings

Conjecture holds when dimension minus depth equals one.

Known counterexamples exist for arbitrary finitely generated algebras.

Extends previous cases where the conjecture was proven.

Abstract

J.F. Carlson conjectured in 1995 that if G is a finite group and k is a field whose characteristic p divides the order of G that the depth of H*(G,k) equals the minimum of the dimensions of associated primes of H*(G,k). This is obviously true if H*(G,k) is Cohen-Macaulay and definitely false for arbitrary finitely generated k-algebras. It was shown by Carlson to be true if the dimension of H*(G,k) equals 2 and by D.J. Green to be true if the depth of H*(G,k) is equal to its possible minimum value, namely the maximum of the dimensions of elementary abelian p-groups contained in the center of G. In this paper we show the conjecture is true if dimension(H*(G,k))-depth(H*(G,k))=1. Known examples show the conjecture to be false for arbitrary finitely generated k-algebras satisfying this condition.