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Grothendieck-Neeman duality and the Wirthmüller isomorphism

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Beren Sanders

What
  • Oberseminar
When Jun 10, 2016
from 12:15 pm to 01:15 pm
Where Bonn, Raum 1.008, Mathematik-Zentrum, Endenicher Allee 60
Contact Name sachinidis
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(Joint work with Paul Balmer and Ivo Dell'Ambrogio.) In this talk, I will discuss an intimate relationship between Grothendieck duality in algebraic geometry and the Wirthmüller isomorphism in equivariant stable homotopy theory. To this end, we will make a general study of the existence and properties of adjoints of an arbitrary coproduct-preserving tensor-triangulated functor between rigidly-compactly generated tensor triangulated categories. It turns out that the more adjoints exist, the more strongly related they must be to each other, and the result is a surprising trichotomy: There exist either exactly three adjoints, exactly five or infinitely many. Moreover, this analysis will provide us with purely formal, canonical constructions of Wirthmüller isomorphisms (when they exist) and demonstrate that Grothendieck duality is in fact a necessary condition for the existence of such an isomorphism. If time permits, I will mention some more recent developments which show that the Adams isomorphism can also be constructed in this way as a (suitably generalized) Wirthmüller isomorphism.