Let , is this ring normal (integral closed in its field of fraction)?

# Category: Mathematics

# Exploring the Grothendieck ring K0 of endomorphisms

**Abstract:** The aim of this note is to compute the Grothendieck group of the category of endomorphisms. This computation mostly plays with linear algebra. The main result is that in , every endomorphism is uniquely characterized by and its characteristic polynomial . 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.

[1] Almkvist, G. (1974). The Grothendieck ring of the category of endomorphisms. *Journal of Algebra*, *28*(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 for a Galois extension .

### 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 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 , the category of finitely generated projective -modules and the categories of algebraic vector bundles of finite rank over

the affine scheme . 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.

# A simple visual editor for creating commutative diagrams

A simple visual editor for creating commutative diagrams.

Announcing quiver: a new commutative diagram editor for the web: https://varkor.github.io/blog/2020/11/25/announcing-quiver.html

# 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.)

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

# [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 is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in is not compact.

Proof. Choose an orthonormal basis , then . 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.

# [Soft Question] Category theory and set theory: just a different language, or different foundation of mathematics?

A question about the philosophy of mathematics for casual reading:

Category theory and set theory: just a different language, or different foundation of mathematics?