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).
Bibliographical noteFunding Information:
(This work began whilst the second author was visiting She eld, supported by a Visiting Fellowship from the EPSRC. ∗Corresponding author. E-mail addresses: j.greenlees@she eld.ac.uk (J.P.C. Greenlees), firstname.lastname@example.org (G. Lyubeznik).