Ausoni–Bökstedt duality for topological Hochschild homology

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 R −→ k is a map to a field of characteristic p then, provided k is small as an R-module, T HH(R; k) is Gorenstein in the sense of [11]. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p. Using only B¨okstedt’s calculation of T HH(k), this gives a non-calculational proof of dualities observed by B¨okstedt [9] and Ausoni [3], Lindenstrauss-Madsen [17], AngeltweitRognes [4] and others.