Tate cohomology was originally defined over finite groups. More recently, Avramov and Martsinkovsky showed how to extend the definition so that it now works well over Gorenstein rings. This paper improves the theory further by giving a new definition that works over more general rings, specifically, those with a dualizing complex. The new definition of Tate cohomology retains the desirable prop...

Remark 1.2. As a note on terminology, according to Milne, an affine variety over a field k is a variety isomorphic to the spectrum of a finitely generated k-algebra R such that R ⊗k k that has no nonzero nilpotents. A variety is a separated scheme admitting a covering by finitely many affine schemes [Mil, p. 5]. This is slightly different from Hartshorne’s definition; he requires the base field...

In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of “rational Tate cla...