Ausoni–Bökstedt duality for topological Hochschild homology

Abstract

Abstract. 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 Bökstedt's calculation of T HH(k), this gives a non-calculational proof of dualities observed by Bökstedt [9]

Homotopy Invariant Commutative Algebra over Fields

“…Indeed, we can exploit the fact that the hypothesis on ν in Lemma 19.3 only depends on R as a module over S to show that ν is an equivalence when R is suitably approximated as an inverse limit. This is a central ingredient in proving Gorenstein duality for many topological Hochschild homology spectra [54].…”

“…(Topological modular forms) Precisely similar statements hold for the ring spectrum tmf of topological modular forms at various primes. This uses results of Hopkins-Mahowald as proved by Matthew [82] (see [54] for a slightly expanded discussion).…”

“…Sources. Much of this account comes from joint work [33,32,21,22,55,54] and other conversations with D.J.Benson, W.G.Dwyer, K.Hess, S.B. Iyengar and S. Shamir, and I would like to thank them for their collaboration and inspiration.…”

“…• Hochschild homology. In[54] several classes of examples are identified where the Hochschild homology with field coefficients inherits Gorenstein properties. The context is that we are given maps S −→ R −→ k of commutative rings, so that we can define the Hochschild homologyHH • (R|S; k) = R ⊗ R⊗ S R k,and it is a ring spectrum with a map to k. The results then say that (under substantial additional hypotheses) if R is Gorenstein of shift a and HH • (k|S; k) is Gorenstein of shift b then HH • (R|S; k) is Gorenstein of shift b − a.…”

The local cohomology spectral sequence for topological modular forms

“…There is a growing list [Gre93], [Gre95], [BG97], [BG97b], [BG03], [DGI06], [BG08], [BG10], [Gre16], [GM17], [GS18] of examples known to enjoy Gorenstein duality. Many of them are of equivariant origin, or have R = C * (X) for a manifold X, or arise from Serre duality in derived algebraic geometry.…”