  The article primarily surveys work that followed from the formulas discovered
by Avramov and Iyengar in 2008, which permit one to compute certain Hochschild
homology and cohomology modules as expressions involving dualizing complexes.
  One aspect of the formulas is that (so far) they are only known for maps of
finite Tor-dimension---we specialize even further, for this survey we give the
formulas only for flat maps. Recall that, for general maps of schemes $f:X\to
Y$, the duality functor $f^!$ has traditionally been defined and studied only
on the bounded-below derived category. Alonso, Jeremias and Lipman observed
that, as long as we restrict to morphisms $f$ of finite Tor-dimension, there is
an extension of $f^!$ to the unbounded derived category. But this extension was
so poorly understood that the paper [19], revisiting the formulas of Avramov
and Iyengar, was unable to prove the obvious extensions of the formulas to the
unbounded setting.
  While most of the paper is a survey, sections 4 and 5 are new. They show how
to use the results of [23] to remove the unnatural boundedness hypotheses from
the formulas in [19]. This is the application that originally motivated [23].
