Talk:K-theory

message: I find it hard to believe that a page on K theory doesn't mention Bott periodicity (except as a link), or mentions the Index Theorem only in passing.  The page seems to consist far too much of vague assertions without substantiation (so the K group is never really defined, we are told about Quillen's construction but only in passing, and with no detail).    —Preceding unsigned comment added by Jdstmporter (talk • contribs) 22:34, 14 November 2009 (UTC)
thread_title: Strange omissions

message: Hi,  thanks for this informative entry.  I am looking for a reference on K-theory for sheaves.  (the references given are all about C^*-algebras and topological K-theory, in my impression).  I think that it would enrich this entry if such a reference was given.
10:20, 28 July 2008 (UTC)
thread_title:  Reference 

message: It is written:
Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that projective modules over the ring of polynomials over a field are free modules;
This seems like the wrong definition of Serre's Conjecture. The page for the conjecture says:
every finitely generated projective module over a polynomial ring is free.
For the first version, a ring of polynomial over a field is a PID, and it is known that every projective module over a PID is a free. For this, there is a characterization of projective modules as direct summand of a free module, and over a PID every submodule of a free module is free. (Ref: Jacobson, Basic Algebra II, or Anderson & Fuller: Rings & Categories of Modules). 
Since this is somewhat easy to show, I doubt this was the statement of Serre's conjecture. I have changed it to the definition on the page of Serre's conjecture to reflect this. —Preceding unsigned comment added by 24.200.85.102 (talk) 22:05, 30 April 2011 (UTC)
thread_title:  Is the definition of Serre's Conjecture wrong? 

message: I noticed that there's a page on Algebraic K theory which seemingly covers the same subject. If the two subjects are different perhaps it would be useful to explain in what sense.
132.70.66.9 (talk) 09:28, 23 August 2016 (UTC)
thread_title: Possibly repeated entry

message: * Add examples from https://web.archive.org/web/20181221072346/http://mcfaddin.github.io/documents/Curves_Talk.pdf and correct computation with the curve
Park's Book:
* Define K^n for topological K-theory
* Add computations in topological K-theory of \mathbb{R}^n,S^n,S^1\times S^1
* Add Mayer-Vietoris w/ computations for orientable genus-g surfaces
* Discuss Bott-Periodicity
* Discuss Hopf Invariant
Algebraic Geometry:
* Add explicit computations of K-theory for dedekind domain
* Given examples of intersections in K_0(\mathbb{P}^n)
* Define chern character/characteristic classes for algebraic geometry
* Give explicit computations of characteristic classes and give derived pushforward
* Give examples for GRR (Look in 3264 and all that...)  — Preceding unsigned comment added by 71.212.234.13 (talk) 04:00, 20 June 2017 (UTC)
thread_title:  To do 

message: There are several articles on this topic: K-theory, Algebraic K-theory, Topological K-theory, K-theory of a category, etc. I think the introduction to this article (the most general one) should clarify the relations between them, and explain the way mathematicians use these words. Here're some suggestions (maybe I'll get to them some day). I'm too lazy to support my judgments with precise references, but they are mostly observations of the linguistic nature, not something controversial.
* The words 'K-theory' may mean both an object K(X) (for whatever thing X is -- a topological space, a scheme, a Waldhausen category) and the branch of mathematics which studies (constructs, compares, etc) such objects. This double meaning is natural for mathematicians, but in an encyclopedia we should probably remark on this ambiguity. The algebraic K-theory term also has double meaning like that. Though topologists may just say cohomology theory and wash their hands.
* The very fact that 'xyz theory' may refer to a specific object instead of a "type of thinking or a result of such thinking" (definition of a theory) is traditional in mathematics (perfect obstruction theory, extraordinary cohomology theory), but not a standard English usage. It may lead to a confusion. Personal anecdote: I remember being confused by the fact that obstruction theory in algebraic topology is a general theory (a collection of propositions which deal with the question when can maps be continuously extended), while obstruction theory in algebraic geometry is an object (a pair of vector space plus some functorial maps). The article on cohomology mentions something to this end.
* Note: there is an article mathematical theory, which doesn't indicate this usage, and anyway a hidden redefinition of an English word wouldn't be a good idea. For more technical pieces like K-theory of a category we can surely ignore this nuance, but for broad articles like cohomology or K-theory something should be said.
::Honestly I don't think any of the preceding three points is really a problem. The readership of this article will consist to a large majority, if not completely, of professional mathematicians. A linguistic digression into the naming of "K-theory" should, if any where, take place at an article about "theory". Jakob.scholbach (talk) 09:37, 8 December 2017 (UTC)
replies: text: Yes, the typical motivation is the exact sequence K_0(Z) \to K_0(X) \to K_0(U) \to 0, where U \subset X is an open subscheme and Z is its complement and Z and X are regular. The left map is not usually injective, which motivates introducing K_1. Jakob.scholbach (talk) 09:37, 8 December 2017 (UTC)
replies: text: I disagree that introducing higher K-theory is what motivates splitting algebraic and topological K-theory. The formalism is in both cases exactly the same. What is different is the fact that in topological K-theory one often only has two interesting K-groups, K_0 and K_1, which makes the story completely different from algebraic K-theory. Another key difference is that in the topological context short exact sequences of vector bundles usually split, unlike for algebraic K-theory over a scheme. Jakob.scholbach (talk) 09:37, 8 December 2017 (UTC)
thread_title:  Suggestions concerning various K-theory articles (mostly for the introductions)