Talk:Algebraic K-theory

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Mathematics rating:	B Class	Mid Priority	Field: Algebra
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. 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)

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)

[edit] 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.