# 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).

# Serre-Swan Theorem and some K groups

This is an overview on Serre-Swan theorem and some ideas on the construction of K-groups for a Banach category. Serre-Swan theorem establishes equivalences between the categories of topological vector bundles over a compact Hausdorff space $X$, the category of finitely generated projective $C(X)$-modules and the categories of algebraic vector bundles of finite rank over
the affine scheme $\mathrm{Spec}C(X)$. This theorem connects different objects of interest in K-theory.
It also introduces some ideas on the construction of K-groups for a Banach category and
in particular for compact topological spaces and Banach algebras.

# 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):

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.)

# [Short Notes] Non-compactness of the closed unit ball in an infinite-dimensional Banach space

This is about an exercise in [Bass]:

Exercise 19.5. Prove that if $H$ is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in $H$ is not compact.

Proof. Choose an orthonormal basis $\{x_i\}$, then $||x_i-x_j||^2=||x_i||^2+||x_j||^2=2$. This means the sequence is not Cauchy hence has no convergent subsequence.

For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.

[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.

# Lusin’s Theorem and Continuous Extension

Here we give proofs for two versions of Lusin’s Theorem, one from Exercise 44, Ch2 in Folland’s Real Analysis and the other from the textbook used for my mathematical analysis course in Beijing.  The latter version is a stronger result which in addition discusses the condition for a real-valued function defined on a subset of $\mathbb{R}^n$ to be extended to the whole of $\mathbb{R}^n$. A more general result in topology is the Tietze Extension Theorem.

See the full post here: Lusin’s Theorem and Continuous Extension

Here we let $\mu$ denote the Lebesgue measure on $\mathbb{R}$.

Lusin’s Theorem (Version 1)[Exercise 2.44, Folland]. Suppose $E\subset \mathbb{R}^n$ is Lebesgue  measurable, $f: E\to \mathbb{R}$ is Lebesgue measurable and $\epsilon> 0$, there is a compact set $F\subset E$ such that $\mu(F^c)<\epsilon$ and $f|_F$ is continuous.

Lusin’s Theorem(Version 2)[Huan]. Suppose $E\subset \mathbb{R}^n$ is Lebesgue measurable and $f: E\to \bar{\mathbb{R}}$ is a Lebesgue measurable extended real valued function with $\mu(|f|=+ \infty)=0$, then  $\forall \epsilon >0$, $\exists g\in C(E)$ such that $\mu(f\neq g)<\epsilon$, where $C(E)$ denotes the space of continuous function on $E$

Continuous Extension Theorem[Huan]. Suppose $E\subset \mathbb{R}^n$, then $f$ can be extended to a continuous function on $\mathbb{R}^n$ if and only if $f$ can be extended to a continuous function on the closure $\bar{E}$ of $E$.

Tietze Extension Theorem. Let $X$ be normal and $F \subset X$ be closed and let $f: F \to R$ be continuous. Then there is a map $g: X \to R$ such that
$g(x) = f(x)$ for all $x\in F$. (Note that in topology, by a map we mean a continuous function. )

# Krull’s Principal Ideal Theorem in Dimension Theory and Regularity

This post is about some applications of Krull’s Principal Ideal Theorem and regular local rings in dimension theory and regularity of schemes [Part IV, Vakil], with the aim of connecting the 2018-2019 Warwick course MA4H8 Ring Theory with algebraic geometry. The lecture notes/algebraic references are here:  2018-2019 Ring Theory.  Note that the algebraic results included here follow the notes. Alternatively, one can also find them in [Vakil] either as exercises or proved results for which I have included the references.

Besides including results in both their geometric and algebraic statements, I have given proofs to a selection of exercises in Part IV, [Vakil] to illustrate more applications and other connections to the contents in the Ring Theory course. The indexes for exercises follow those in [Vakil].

See here for the full post: Application of Krull’s Principal Ideal Theorem

Please also let me know if you find any errors or have suggestions on any of my posts.