Rings with a local cohomology theorem and applications to cohomology rings of groups

Abstract

Cohomology rings of various classes of groups have curious duality properties expressed in terms of their local cohomology (Benson and Carlson, Trans. Amer. Math. Soc. 342 (1994) 447–488; Bull. London Math. Soc. 26 (1994) 438–448; Greenlees, J. Pure Appl. Algebra 98 (1995) 151–162; Benson and Greenlees, J. Pure Appl. Algebra 122 (1997) 41–53, J. Algebra 192 (1997) 678–700; Symonds, in preparation). We formulate a purely algebraic form of this duality, and investigate its consequences. It is obvious that a Cohen–Macaulay ring of this sort is automatically Gorenstein, and that its Hilbert series therefore satisfies a functional equation, and our main result is a generalization of this to rings with depth one less than their dimension: this proves a conjecture of Benson and Greenlees (1997).