# Brauer Lifting in Algebraic K-Theory of Finite Fields

#### Abstract:

Quillen’s calculation of algebraic $K$-theory of finite fields incorporates a wide range of techniques from group cohomology to representation theory. This exposition focuses on the proof of Green’s theoerm: For any finite group $G$ and any representation of $G$ over $\mathbb{F}_q$ ($q$ is a power of some prime $p$), one can construct a character called the Brauer character which is a virtual complex character. We will also introduce its application in algebraic K-theory. The Brauer character of the n-dimensional standard representation of $GL_n(\mathbb{F}_q)$ is a virtual complex character, thus it induces a map $GL_n(\mathbb{F}_q) \to GL(\mathbb{C})$, hence a map $BGL\mathbb{F}_q^+\to BGL(\mathbb{C}) \cong BU$. Taking the n-th homotopy group gives a map $\theta: K_n(\mathbb{F}_q)\to \pi_nBU$. This map is a key construction that allows one to identify $K_n(\mathbb{F}_q)$ with the homotopy groups of a better understood space.

This writing is part of the course project for MATH 506 Group Representation Theory:

# Counterexamples about Infinitely Generated Modules over Commutative Rings

This post is about examples of some nice things that could fail for modules that are not finitely generated (or not finitely presented when the ring is not Noetherian).

# Counterexamples that Tensor Product of a Non-finitely Generated R-module M with the Completion of R is not the Completion of M

Let $A$ be a commutative noetherian ring, $m$ an ideal of $A$ and $M$ a finitely generated $A$-module, then we have an isomorphism $M\otimes_A \hat{A} \to\hat{M}$ where the completion is with respect to the $m$-adic topology [III. 3.4 Theorem 3, Commutative Algebra, Bourbaki].

Note that when $A$ is not noetherian, $M$ being finitely presented would suffice.

Here are two counterexamples showing that the map above need not be injective or surjective when $M$ is not finitely generated: https://math.stackexchange.com/q/1143557

# Depth and Dimension of the Fibre of a Local Ring Homomorphism

An important information of a homomorphism of local rings $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ is its fibre $S/\mathfrak{n}S$. For example it relates the dimensions and depth of $R$ and $S$. We introduce two main theorems that relate the depth and dimensions and some corollaries. Finally we apply them to show that the polynomial ring or formal power series ring over a Cohen-Macaulay ring is Cohen-Macaulay. See also Theorem A.11 in [Bruns, Herzog] and Theorem 23.2 in [Matsumura].

# Checking the ring k[x,y,u,v]/(xy-uv) is a normal domain

Let $A= k[x,y,u,v]/(xy=uv)$, is this ring normal (integral closed in its field of fraction)?

Edit (see comments below): Because $p$ is of height 1, then there exists one among $x,y,u,v,$ say $x$ such that $f(x)\notin p$ for any $f(x)\in k[x]$. Otherwise, if there are polynomials in one variable for each of the variables $x,y,u,v,$ contained in $p$, then $p$ is of height $>1$. Thus $f(x)$ becomes invertible in $A_p$ for any $f(x)\in k[x]$.

# Exploring the Grothendieck ring K0 of endomorphisms

Abstract: The aim of this note is to compute the Grothendieck group $K_0$ of the category of endomorphisms. This computation mostly plays with linear algebra. The main result is that in $K_0$, every endomorphism $f:P\to P$ is uniquely characterized by $P$ and its characteristic polynomial $\lambda_t(f)$. This computation was due to [1]. We will explain how to think about this computation, the reason for certain constructions and the “diagonalization” in this computation.

Edit history: fixed some mislabeling of diagrams (Mar 21 2022)

[1] Almkvist, G. (1974). The Grothendieck ring of the category of endomorphisms. Journal of Algebra28(3), 375-388.

# On the Lichtenbaum-Quillen Conjectures (Updated 10/30/2021)

### 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, updated on 10/30/2021. I thank Prof Grayson for comments and pointing out some typos.

Updates (10/30/2021): Fixing a few typos, adding reference to the claim about the fixed point spectrum $K_*F= (K_*E)^G$ for a Galois extension $E\to F$.

### 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

# 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.