Talk:Algebraic K-theory

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Crediting the initial theory to Serre rather than Grothendieck is strange and just not true. That Bass defined K1 should be here. Suslin's proof of the Karoubi conjecture showing topological K is a special case of algebraic K should also be here, and the Q construction should be mentioned. Milnor's definition wasn't only for fields.--John Z 16:47, 17 Jun 2005 (UTC)

While I don't know of an explicit counter-example, nor do I have a reference to hand, I'm fairly sure the following statement is incorrect: "When A is a Dedekind domain (e.g. the ring of algebraic integers in an algebraic number field), SK1(A) is zero."

It IS true for the ring of integers in a number field, but not for a general Dedekind domain. In fact, I'm sure I remember reading somewhere that there were even PID's with non-zero SK1, although again I don't know an example. (I believe this is in Rosenberg's Algebraic K-Theory and Its Applications, but like I said i don't have it to hand) 81.76.125.163 22:04, 13 April 2006 (UTC)

Wrong historical comment removed.

i have just added a sentence about the FACT that grothendieck started the whole theory. Topological K-theory was invented by atiyah and hirzebruch by replacing "algebraic vector bundles by ""topological bundles" in the definition of K_0 by grothendieck. The conjecture of serre on projective modules on polynomial rings, had initially nothing to do with K-theory.

Description of Q construction is wrong?

The section on the Q construction doesn't seem to be correct (unless I've made a mistake of course). For A a ring and P_A the category of finitely-generated projectives, the category QP_A as described seems to have a null object, the zero module. This would make BQP_A contractible.

Motivation

I wish there was some information on the properties of K-groups and their applications. Why are they so important? At the moment the talk about higher groups is completely incomprehensible. Sure, now I know a grotesque definition, but what's in it for me? --Anton (talk) 22:22, 5 September 2011 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Algebraic K-theory/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

explain why the higher K-groups are the "right" extension of the lower ones. properties of K-groups would be good (like homotopy invariance etc.) Jakob.scholbach 00:54, 13 May 2007 (UTC)

Last edited at 00:54, 13 May 2007 (UTC). Substituted at 01:45, 5 May 2016 (UTC)

Drafting of "Basic theorems of algebraic K-theory"

Would someone independently knowledgeable please comment at Wikipedia:Miscellany for deletion/Draft:Basic theorems of algebraic K-theory. The question, I think, is whether "Basic theorems of algebraic K-theory" is an appropriate new topic to draft, or whether it should be included in this article first. --SmokeyJoe (talk) 00:48, 26 July 2016 (UTC)

K_1 elementary matrices mistake

In the section K_1 the statement E(A) = [GL(A), GL(A)] might be considered wrong, since the article Elementary_matrix also row-multiplication is allowed (multiplying a row with scalar), while for "our" elementary matrices it is not allowed (cf. http://www.maths.ed.ac.uk/~aar/papers/jhcw2.pdf p. 5 (2.4.) ) as far as I understand it. Since I am quite new to this area, can someone confirm this and mark that difference in the article(s).ChristianTS (talk) 18:51, 9 January 2017 (UTC)

The terminology is inconsistent between K-theory and linear algebra. I've tried to make the article clearer. Ozob (talk) 03:50, 10 January 2017 (UTC)

Computations

This page should discuss computations in algebraic K-theory. The following two papers provide excellent tools for this

Negative Algebraic K-theory

Since negative algebraic K-theory is readily computable for a variety over a field of characteristic 0, this should be included on this page. Check out theorem 0.2 of http://annals.math.princeton.edu/wp-content/uploads/annals-v167-n2-p04.pdf for this.

Cones of Smooth Varieties

Check out the theorem on page 1 of https://arxiv.org/pdf/0905.4642.pdf

Yes, this article can and should definitely be expanded. Please go ahead and add whatever you see fit! If you need help with the Wikipedia editing purpose, feel free to contact me on my talk page. Jakob.scholbach (talk) 09:25, 10 September 2017 (UTC)

Computations of Etale K-theory

This page should have a section dedicated to defining etale K-theory and giving some computations. The section of the Quillen-Lichtenbaum conjecture in https://faculty.math.illinois.edu/K-theory/0403/2pw.pdf can be used to compute most of the higher algebraic K-theory with $\mathbb {Z} /\ell$-coefficients for a smooth complete intersection. This is just higher topological K-theory which can be easily computed using the Atiyah-Hirzebruch spectral sequence (and the euler characteristic for smooth complete intersections). The smooth complete intersection page should explain doing these cohomology computations though...