  We consider the Gorenstein condition for topological Hochschild homology, and
show that it holds remarkably often. More precisely, if R is a commutative ring
spectrum and and R----->k is a ring map to a field of characteristic p then,
provided k is small as an R-module, THH(R;k) is Gorenstein in the sense of
Dwyer-Greenlees-Iyengar. In particular, this holds if R is a (conventional)
regular local ring with residue field k of characteristic p.
  Using only Bokstedt's calculation of THH(k), this gives a non-calculational
proof of dualities observed in calculations by Bokstedt, McClure-Staffeldt,
Ausoni-Rognes, Ausoni, Lindenstrauss-Madsen, Angeltweit-Rognes and others.
  A lemma of Dundas shows that THH(R;k) is remarkably computable.
