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The Adams isomorphism as a generalized Wirthmueller isomorphism Sanders, Beren


In a recent paper, joint with Paul Balmer and Ivo Dell'Ambrogio, we made a general study of the existence and properties of adjoints to an arbitrary coproduct-preserving tensor-triangulated functor between rigidly-compactly generated tensor triangulated categories. One of the highlights of this work was the recognition that such a functor has a left adjoint if and only if it satisfies Grothendieck-Neeman duality, in which case there is a Wirthmueller isomorphism between its left and right adjoint (twisted by the relative dualizing object). In particular, this work provided a purely formal canonical construction of the classical Wirthmueller isomorphism in equivariant stable homotopy theory. In this talk, I will review aspects of the above story before explaining how the Adams isomorphism can also be obtained purely formally by an extension of the theory. The main punch-line is that every such functor -- even one which does not have a left adjoint -- gives rise to a "Wirthmueller type" isomorphism (properly understood). This construction generalizes the Wirthmueller isomorphism of the earlier paper, and includes the Adams isomorphism as a special case.

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