Workshop

Hesselholt–Madsen proved the Lichtenbaum–Quillen conjecture for p-adic local fields K (with p > 2) by proving the stronger statement that V (1)∗TR(OK) is bounded. Bounded TR is now best interpreted as boundedness in the Antieau–
Nikolaus t-structure. The speaker will introduce this t-structure, and note that THH(Fp) is bounded. The main aim of the talk should be to characterize cyclotomic boundedness in more concrete terms, as the combination of the Segal conjecture and canonical vanishing. Bounded cyclotomic rings admit a B ̈okstedt class, and the speaker will discuss its basic properties.

Suggested references: Sections 2.2, 2.3 and 2.4 of [AN21]

This talk will explain the theorem, by Ausoni–Rognes, that V (2)∗TR(l) is bounded for primes p > 3. (The speaker may assume that p > 5, so that V (2) exists as a homotopy commutative and associative ring, or alternatively may make use of the motivic spectral sequence.) Here, l = BP⟨1⟩ is the Adams summand of p-local complex K-theory. This was
the first example of a higher height Lichtenbaum–Quillen theorem, and the speaker will also explain how to deduce chromatic redshift. 

The computations going into the Ausoni–Rognes result will be explained explicitly enough that they may be adapted to compute V (2)∗ TC(lhpkZ) on Thursday (speakers may want to coordinate these two talks).

Suggested references: [AR02], [HRW22] and [BHLS23, §6 & 7]

This talk will explain how many of the results comparing K-theory and TC can be extended beyond the setting of connective rings, and in particular to −1-connective rings, because the universal localizing invariant of −1-connective rings are built out of those of connective rings.In the case of rings that are fixed points of Z-actions on connective rings,
this talk will explain how this can be accomplished using the work of Land–Tamme on the K-theory of pullbacks.

Suggested references: [LT19], [Lev22]

This talk will study the cochains on the circle as an E∞-ring in cyclotomic spectra. In particular, its study is largely controlled by studying the free loop space of the p-adic circle BZp. As a consequence, the coassembly map for the TC of the fixed points by a trivial Z-action is usually not an isomorphism.

Suggested references: [BHLS23, §3]

This talk will disprove the telescope conjecture at height 2 and primes p > 5, by a direct computational approach. Specifically, the speaker will prove that, for k sufficiently large and Z acting by the Adams operation Ψp+1, V (2)∗ TC(l
hpkZ) has non-finite homotopy groups.

Suggested references: Oberwolfach report 34/2023, [BHLS23, §7], final video/notes from Mark Behrens’ graduate course

The key input to seeing that the T(n)-local TC doesn’t commute with taking -fixed points in the case of interest is asymptotic constancy. Asymptotic constancy allows one to reduce checking this to the case of a trivial action. This talk and the next will be dedicated to showing asymptotic constancy for the THH of the ring spectra of interest. The key properties
of the ring spectra that are used to show this are that the Z-actions onthem are locally unipotent, they satisfy the height n Lichtnebaum–Quillen property, and are almost compact.

Suggested references: [BHLS23, §4.1,4.2,A.3]

This is a continuation of the previous talk. Namely, it will be explained how to bootstrap asymptotic constancy at the level of spectra to the level of cyclotomic spectra, and how this can be used to deduce constancy at the level of T(n+1)-homology.

References [BHLS23, §4.3,4.4]