Workshop on topological Hochschild homology
LAGA, Wednesday 18th and Thursday 19th of January 2017
This small workshop is organized around the visit of Lars Hesselholt and Thomas
Nikolaus. It is financed by the ANR-project Chromatic
homotopy and K-Theory
and the LAGA
Introductory talks on THH are scheduled on Wednesday afternoon, before more
specialized lectures by Thomas Nikolaus on Thursday morning and Lars Hesselholt on Thursday
LAGA - Institut Galilée
Université Paris 13
99 avenue Jean Baptiste Clément
The talks will be in the Institut Galilée. Here is a description on
to come to the LAGA
No registration necessary. Please contact Christian Ausoni
if you have any
questions or want to join in for lunch on Thursday.
Wednesday 18th of January 2017 (Amphi Copernic, Institut Galilée)
Bruno Stonek (Paris 13),
Introduction to (topological) Hochschild Homology.
We define Hochschild homology and topological Hochschild homology as a certain
derived tensor product, highlighting how it is the same construction but carried
either in a category of chain complexes or in a category of spectra. We give a
relationship between them. Then, we give an alternative definition via the
cyclic bar construction. We sketch a computation of THH(KU), where KU is the
periodic complex topological K-theory spectrum.
Christian Ausoni (Paris 13),
A sample of older and newer computations of THH.
Topological Hochschild has been computed for many spectra using a large range of
techniques. In this lecture, I will sketch several examples of computations,
including the Eilenberg-Mac Lane specrum of a finite field, and the second
Johnson-Wilson spectrum (joint work with Birgit Richter).
Eva Höning (Paris 13),
The topological Hochschild homology of algebraic K-theory of finite fields
Angeltveit and Rognes calculated the mod p homology of THH(K(F_q)) via
the Bökstedt spectral sequence, when q is a topological generator of the
p-adic units. We mention when their methods apply to further cases and
when other methods are necessary. I explain how a spectral sequence of
M. Brun for the calculation of THH of rings generalizes to S-algebras.
We show how this spectral sequence can be used for an alternative
V(1)_* THH(ku_p), where ku_p is p-completed connected complex K-theory.
Finally, we apply the spectral sequence for K(F_q) and explain first
Geoffroy Horel (Paris 13),
Introduction to factorization homology.
I will explain the construction of factorization homology. It is
a pairing between d-dimensional manifolds and algebras over the little
d-disks operad. When d=1, we obtain a model for THH that makes the circle
action more transparent than the standard model. I will try to explain the
proof of this fact (with more or less details depending on the time I have).
Thursday 19 January 2017 (Amphi Euler, Institut Galilée)
- 9h15-10h15 and 10:30-11:45
Thomas Nikolaus (MPIM Bonn),
On topological cyclic homology .
In the first part of the talk I will give an overview of the result of
joint work with P. Scholze. We establish a new model for the cyclotomic
structure on topological Hochschild homology and in particular a resulting
simplification of the formula for topological cyclic homology in relatively
basic terms (the Tate spectrum will come up, so I will also explain that).
Finally we indicate basic applications and results.
In the second half we explain how to prove these results, in particular the
computational key lemma (which we call the Tate orbit lemma). Another key to
understand the cyclotomic structure will be a generalization of the Segal
conjecture which is in the finite type case due to Rognes-Nielsen and whose
generalization to the bounded below case we will explain. If time permits we
will also indicate how to construct the cyclotomic trace and a refinement
through "K-theory of endomorphisms".
- 14h00-15h00 and 15h30-16h30 (Colloquium du LAGA)
Lars Hesselholt (Nagoya/Copenhagen),
Topological Hochschild homology and the Hasse-Weil zeta function .
In the nineties, Deninger gave a detailed description of a
conjectural cohomological interpretation of the (completed) Hasse-Weil zeta
function of a regular scheme proper over the ring of rational integers. He
envisioned the cohomology theory to take values in countably infinite
dimensional complex vector spaces and the zeta function to emerge as the
regularized determinant of the infinitesimal generator of a Frobenius flow.
In this talk, I will explain that for a scheme smooth and proper over a
finite field, the desired cohomology theory naturally appears from the Tate
cohomology of the action by the circle group on the topological Hochschild
homology of the scheme in question.
- 16:30 Thé du LAGA.