Bach on the Classical Guitar

Bach didn’t compose for the guitar. The pieces in this collection are some arrangements which are now standard repertoires for the classical guitar.

Ana Vidovic plays Prelude BWV 998 by J. S. Bach on a classical guitar – guitare classique

Johann Sebastian Bach — Prelude in E major BWV 1006a — Mateusz Kowalski:

https://www.youtube.com/watch?v=badl5MWzSs4

Chaconne in d minor by J.S.Bach (Arr. John Feeley): https://www.youtube.com/watch?v=JNEnzNHTkd8

Marcelo Kayath – Gigue & Double – Suite BWV 997 A Minor – Bach: https://www.youtube.com/watch?v=sHi3e8zCOHE

Johannes Monno spielt Johann Sebastian Bach: Fuge BWV 997: https://www.youtube.com/watch?v=Hpjw7SAuH28

Błażej Sudnikowicz plays Bach Partita No. 1 BWV 825: https://www.youtube.com/watch?v=XMUg2bkHPTI&t=140s

J.S.Bach, Prelude BWV 997 played by Thu Le, classical guitar: https://www.youtube.com/watch?v=5JfxpiunYoM

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).

1. Nakayama’s Lemma (Krull’s Intersection Theorem)

2. \mathrm{Supp}M =V(\mathrm{ann M}).

3. \mathrm{Hom}_A(M,N)\otimes B= \mathrm{Hom}_B(M\otimes_A B, N\otimes_A B) for B a flat A-module

4. M\otimes _R \hat{R} =\hat{M}

5. Finitely Generated Nil Ideal is Nilpotent

6. Finitely presented flat module is projective

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.

Our freedom does not lie outside us- Carl Jung

Our freedom does not lie outside us, but within us. One can be bound outside, and yet one will still feel free since one has burst inner bonds. One can certainly gain outer freedom through powerful actions, but one creates inner freedom only through the symbol.

The symbol is the word that goes out of the mouth, that one does not simply speak, but that rises out of the depths of the self as a word of power and great need and places itself unexpectedly on the tongue. It is an astonishing and perhaps seemingly irrational word, but one recognizes it as a symbol since it is alien to the conscious mind. If one accepts the symbol, it is as if a door opens leading into a new room whose existence one previously did not know. But if one does not accept the symbol, it is as if one carelessly went past this door; and since this was the only door leading to the inner chambers, one must pass outside into the streets again, exposed to everything external. But the soul suffers great need, since outer freedom is of no use to it. Salvation is a long road that leads through many gates. These gates are symbols. Each new gate is at first invisible; indeed, it seems at first that / it must be created, for it exists only if one has dug up the spring’s root, the symbol.

P.311, The red book

[Jung] Jung, C. G., Shamdasani, S. E., Kyburz, M. T., & Peck, J. T. (2009). The red book: Liber novus. WW Norton & Co.

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.