@ARTICLE{Strickland_gross-hopkinsduality, author = {N. P. Strickland}, title = {Gross-Hopkins duality}, journal = {Topology}, year = {}, pages = {1021--1033} }

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Abstract

In [8] Hopkins and Gross state a theorem revealing a profound relationship between two different kinds of duality in stable homotopy theory. A proof of a related but weaker result is given in [3], and we understand that Sadofsky is preparing a proof that works in general. Here we present a proof that seems rather different and complementary to Sadofsky’s. We thank I-Chiau Huang for help with Proposition 18, and John Greenlees for helpful discussions. We first indicate the context of the Hopkins-Gross theorem. Cohomological duality theorems have been studied in a number of contexts; they typically say that H k (X ∗ ) = H d−k (X) ∨ for some class of objects X with some notion of duality X ↔ X ∗ and some type of cohomology groups H k (X) with some notion of duality A ↔ A ∨ and some integer d. For example, if M is a compact smooth oriented manifold of dimension d we have a Poincaré duality isomorphism H k (M; Q) = Hom(H d−k (M; Q), Q) (so here we just have M ∗ = M). For another example, let S be a smooth complex projective variety of dimension d, and let Ω d be the sheaf of top-dimensional differential forms. Then for any coherent sheaf F on S we have a Serre duality isomorphism H k (S; Hom(F, Ω d)) = Hom(H d−k (S; F), C). This can be seen as a special case of the Grothendieck duality theorem for a proper morphism [7], which is formulated in terms of functors between derived categories. There is a well-known analogy