Hopf algebra structure on topological Hochschild homology

Abstract

The topological Hochschild homology THH (R) of a commutative S -algebra (E ∞ ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Bökstedt spectral sequence converging to the mod p homology of THH (R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S -algebras ku, ko, tmf , ju and j , and to calculate the homotopy groups of THH (ku) and THH (ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K -theory of S -algebras, by means of the cyclotomic trace map to topological cyclic homology.

Differentials in the homological homotopy fixed point spectral sequence

“…The homology of E(ξ 3 ) ⊗ P (σξ 3 ) is F 2 . So E 4 * * = P (y) ⊗ P (ξ See [AnR,6.2(b)]. This gives the E 2 -term of the homological spectral sequence, and as before its homology with respect to the σ -operator is E 4 * * = P (y) ⊗ P (ξ By Theorem 5.1(a), all of these algebra generators are in fact infinite cycles, so the homological spectral sequence collapses, as claimed.…”

Relative Loday constructions and applications to higher THH-calculations

“…The examples ko, ku, ℓ and tmf . Angeltveit and Rognes calculate in [AR,5.13,6.2] H * (THH(ko); F 2 ), H * (THH(tmf ); F 2 ), H * (THH(ku); F 2 ) and for any odd prime p they determine…”

“…The explicit description of the generators in [AR,Theorem 6.2] for p = 2 and µ 2 in Bökstedt's calculation corresponds to σξ 1 . Therefore the right-hand Tor is isomorphic to…”

“…In the case of the Adams summand, ℓ, we work at odd primes and here in [AR,proof of 5.12] the relation…”

“…However, if we use the bar spectral sequence we get as the E 2 -term [AR,6.1]). Hence the spectral sequence collapses because…”