Topological Hochschild homology and cohomology ofA_∞ring spectra

Abstract

987Topological Hochschild homology and cohomology of A 1 ring spectra VIGLEIK ANGELTVEIT Let A be an A 1 ring spectrum. We use the description from our preprint [1] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the A 1 structure into THH.A/, and allows us to study how THH.A/ varies over the moduli space of A 1 structures on A.As an example, we study how topological Hochschild cohomology of Morava Ktheory varies over the moduli space of A 1 structures and show that in the generic case, when a certain matrix describing the noncommutativity of the multiplication is invertible, topological Hochschild cohomology of 2-periodic Morava K -theory is the corresponding Morava E -theory. If the A 1 structure is "more commutative", topological Hochschild cohomology of Morava K -theory is some extension of Morava E -theory.

The Picard group of topological modular forms via descent theory

“…The maximal ideals in T M F (2) 0 are m = (p, f (s)), where p is an odd prime and f (s) a monic polynomial irreducible modulo p (and not congruent mod p to s, s − 1). For each of these ideals, we have an associative ring spectrum (the "residue field") with homotopy groups T M F (2) * /m by [Ang08]; denote it temporarily by T M F (2)/m. After extending scalars so that f splits, we get that T M F (2)/m is a product of (extensions of) mod p Morava K-theory spectra at height one or two, one for each zero of f .…”

Lifting homotopy T-algebra maps to strict maps

“…Indeed the latter work was a significant source of inspiration for this project. Angeltveit [8] has also constructed an obstruction theory, which appears to be part of a spectral sequence, for computing maps of A ∞ ring spectra.…”

The n -order of algebraic triangulated categories

“…The hypothesis in Proposition 1.5 on the commutative symmetric ring spectrum R can be paraphrased by saying that the R-module spectrum R/n (or rather, some Rmodule spectrum of this homotopy type) can be given the structure of an R-algebra. A theorem of Angeltveit [1,Cor. 3.2] gives a sufficient condition for this in terms of the graded ring π * R of homotopy groups of R: if π * R is n-torsion-free and concentrated in even dimensions, then R/n admits an A ∞ -structure compatible with the R-module structure; equivalently, there is an R-algebra spectrum whose underlying R-module has the homotopy type of R/n.…”

“…Since (0, 1 X ) is a cycle, so is p, which is thus represented by a closed morphismp : Cf −→ X [1]. By definition, a triangle in H(B) is distinguished if it is isomorphic to the image of a triangle of the form…”

“…So on the level of tensor triangulated categories, there does not seem to be any qualitative difference between the Moore spectrum S/p as an object of the triangulated category of finite spectra and Z/p as an object of the bounded derived category of finitely generated abelian groups, as long as p ≥ 5. However, mod-p Moore spectra do not have models as strict ring spectra (or A ∞ -ring spectra); this seems to be folklore, and a proof can be found in [1,Ex. 3.3].…”