### Isomorphisms between left and right adjoints, by 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.

H. Fausk <fausk@math.northwestern.edu>

P. Hu <poh@math.uchicago.edu>

J.P. May <may@math.uchicago.edu>