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Isomorphisms between left and right adjoints

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H. Fausk, P. Hu, and J.P. May

There are many contexts in algebraic geometry, algebraic topology, and
homological algebra where one encounters a functor that has both a left
and right adjoint, with the right adjoint being isomorphic to a shift of
the left adjoint specified by an appropriate `dualizing object'.
Typically the left adjoint is well understood while the right adjoint is
more mysterious, and the result identifies the right adjoint in familiar
terms. We give a categorical discussion of such results. One essential
point is to differentiate between the classical framework that arises in
algebraic geometry and a deceptively similar, but genuinely different,
framework that arises in algebraic topology. Another is to make clear
which parts of the proofs of such results are formal. The analysis
significantly simplifies the proofs of particular cases, as we illustrate
in a sequel discussing applications to equivariant stable homotopy theory.

Keywords:
Grothendieck duality, Verdier duality, Wirthmuller
isomorphism, symmetric monoidal category, triangulated category

2000 MSC:
Primary 14A10, 18D99, 18F99; Secondary 55U99

*Theory and Applications of Categories*
, Vol. 11, 2003,
No. 4, pp 107-131.

http://www.tac.mta.ca/tac/volumes/11/4/11-04.dvi

http://www.tac.mta.ca/tac/volumes/11/4/11-04.ps

http://www.tac.mta.ca/tac/volumes/11/4/11-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/4/11-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/4/11-04.ps

Revised 2003-05-21. Original version at

http://www.tac.mta.ca/tac/volumes/11/4/11-04a.dvi

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