On the Lichtenbaum-Quillen Conjectures

Abstract

Starting with some motivations and brief expositions on algebraic K theory, I’ll introduce some early important computations of algebraic K-theory, including computations of K theory of finite fields and of rings of integers for which I will briefly outline the proofs. Then we’ll move on to K-theory with finite coefficients of separably closed fields. With the motivation of recovering some information of K-theory of an arbitrary field from its separable closure, we introduce a few versions of the Lichtenbaum-Quillen conjectures as descent spectral sequences of \’etale Cohomology groups. If time permits, I’ll mention relation to motivic Cohomology that a key tool is some “motivic-to-K-theory” spectral sequence.

These are notes based on my talk on Oct 01, 2021 in UIUC Graduate Homotopy Seminar. The main references are [1] and [2]. The outline of the proof of Quillen’s K-theory of finite fields has been moved to Appendix A.

Here are the notes:

References

[1] Mitchell, Stephen A. “On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint.” Algebraic topology and its applications. Springer, New York, NY, 1994. 163-240.

[2] Weibel, Charles A. The K-book: An introduction to algebraic K-theory. Vol. 145. Providence, RI: American Mathematical Society, 2013.

A Summary of Quillen’s K-theory of Finite Fields

Abstract

This is an outline of Quillen’s proof for the calculation of K-theory of finite fields, originally done by Quillen in [1], see also [2] for a slightly different presentations with more background materials included.

Here are the full post:

References

[1] Quillen, Daniel. “On the cohomology and K-theory of the general linear groups over a finite field.” Annals of Mathematics 96.3 (1972): 552-586.

[2] Mitchell, S. Notes on K theory of nite elds. Available online:
https://sites.math.northwestern.edu/ jnkf/Mitchell-niteeldsKtheory.pdf

K theory of finite fields (mod l homology)

Recently I have been reading about K theory of finite fields. Here I write some summaries about this calculation and calculate the homology ring of F\psi ^q which is one step in the computation of finite fields, see section 4, {Mitchell}. This post may be continued.

This post has been continued on A Summary of Quillen’s K-theory of Finite Fields (Oct 16, 2021).

Calculate (co)limits as (co)equalisers (two examples)

There is a general formulation for constructing limits as equalisers: see Theorem 1 in Section V.2, Maclane. For the dual version, see Theorem A.2.1 in Appendix A written by me.

The constructions look like these (see the links above for details):

limitscolimits

But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.

The connected components [Background]

For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)

Two examples of (co)limits as (co)equalisers

Pdf here: Two examples of (co)limits as (co)equalisers

exampleexample1example2example3example4example5

Notes and Remarks on Unstable Motivic Homotopy Theory

I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.

Affine representability results: 

For remarks on section 2.3 and Lemma 2.3.2 of [2] (also quoted as Proposition 5.5 and Proposition 5.6 in [1]), we take the approach of stack and descent.  By relating hyperdescent condition for simplicial presheaves with descent for stacks [3], we show that how this implies classifying space of a stack satisfies hyperdescent.

After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.

Exercises on the construction of motivic homotopy theory: 

I also have some informal notes about the exercises in [1]. The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.

 

 

[1] Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry95, 305-370.

[2] Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology22(2), 1181-1225.

[3] Jardine, J. F. (2015). Local homotopy theory. Springer.

Descent, Affine Representability and Classifying Space in A1-Homotopy Theory

This is my master thesis in Warwick, supervised by Marco Schlichting.

Abstract
We relate hyperdescent condition for simplicial presheaves, with hyperdescent in $$\infty$$-topos and descent for stacks. One consequence is that the classifying space of a stack satises hyperdescent which simplies existing proofs in some cases. We show an algebraic topological approach to some properties of singular constructions for sites with interval. Using results of affine representability and A1-algebraic topology over a field, we show that the A1-homotopy theory is not a model topos. As an extension of this result, we prove that an A1-local monoid is strongly A1-invariant if and only if its 0-th Nisnevich homotopy sheaf is strongly A1-invariant. This can be used for calculation of A1-loop space and we apply it to BBGm. Finally we present some calculations of A1-homotopy sheaves of BGLn and BSLn and in particular
the first a few A1-homotopy sheaves of BGL2.