Talk:K-theory

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class High Priority  Field: Algebra
Note: also topology


In mathematics, the topic of K-theory spans the subjects of algebraic topology, abstract algebra and some areas of application like operator algebras and algebraic geometry. It leads to the construction of families of K-functors, which contain useful but often hard-to-compute information.

All well and good, but what exactly is K-Theory? Any article on a maths topic should begin with a defintion of some kind. I would add one myself, but am not up to the task. Tompw 15:25, 25 July 2005 (UTC)

K-theory is an extraordinary (this adjective was invented for K-theory I believe) cohomology theory. It constructs so-called K-groups from topological spaces (topological K-theory) and from C^* algebras (algebraic K-theory).--MarSch 12:20, 2 September 2005 (UTC)
Topological K-theory is indeed an extraordinary cohomology theory (ie the usual axioms without the dimension axiom). I don't think one can say that algebraic K-theory is an ECT. Charles Matthews 21:10, 2 September 2005 (UTC)
Given that 'extraordinary' was invented for K-theory, this seems something of a tautology. However... I still don't feel that the opening sentence is a definition. It's more of a classification. Tompw 16:17, 16 October 2005 (UTC)
OK, K-theory as of 2005 is a whole big area. It's a wretched name, but we don't have any power over that. From a top-down look, it is something like the homotopy theory of the general linear groups over any old ring. If you look at applications it is not obviously that at all; it is implicated in number theory and algebraic geometry in the biggest way. Not often I say this, but I'm not competent to give an expert discussion of how it fits together. The topological stuff is not so bad. The algebraic K-theory stuff is about trying to get good invariants in module theory (and then finding that even the K-theory of the integers Z looks very deep). Charles Matthews 17:08, 16 October 2005 (UTC)

Using Quillen's definitions, K-theory can be defined for any exact category in a way that captures the algebraic and topological theories (using categories of projective modules or vector bundles). This approach also allows the study of other interesting objects in this context, such as the K-theory of a scheme. I can write an overview of these ideas for the article to give it some sense of 'what K-theory is' if desired. MarcHarper 02:52, 30 September 2006 (UTC)

Having a survey would be very good. In line with the concentric style we favour, it should not 'write over' some gentler explanations at the start. We don't want people to have to cope with the axiomatics of any one perspective immediately. Charles Matthews 09:17, 3 October 2006 (UTC)
Some of the aximomatic work is already laid out in the Algebraic K-theory article, though it only discusses the application to the case of rings. Perhaps it more naturally belongs here? Marc Harper 20:09, 25 October 2006 (UTC)
How about an softer introduction, such along the lines of:
K-theory is an extraordinary cohomology theory assigning to objects in certain categories (such as the category of rings and the category of topological spaces) algebraic invariants that can then be used to study the objects with the methods of algebra.
Consider the case of cohomology theory for topological spaces. In the category of spaces, we consider spaces to be "the same" if they are homeomorphic. This turns out to be a difficult and often too restrictive definition of sameness, and for many situations it is sufficient if the spaces are homotopic or quasi-isomorphic. Additionally, the invariants provided by cohomology theories (and homotopy theories) allow the distinguishing of many spaces in cases where it is difficult to prove directly that there does not exist a homoemorphism between two spaces, because in this regard, algebraic objects are easier to work with given current knowledge.
Consider Morita theory for rings. Rather than directly compare rings, which is in general difficult, we can generate the category of modules of that ring (which is the category of representations of the ring as an abelian group). Morita theory considers rings to be the same if they have the same collection of representations, defining equivalence as category-theoretic equivalence of the categories of representations. Such Morita equivalent rings share many ring-theoretic properties such as the properties of being simple, primitive, artinian, and noetherian. Algebraic K-theory compares rings in a similar but less precise way by extracting algebraic invariants from the category of projective modules over the ring, such as the Grothendieck group for the functor K_0.
This approach can be extended via homotopy theory to define an infinite family of invariants. To define the higher K-groups, first a simplicial set is formed from the nerve construction on the category of projective R-modules. The homotopy theory of simplicial sets can now be applied giving a family of functors K_n for all positive n by turning the simplicial set into a space (geometric realization) and applying homotopy theory for spaces. In some sense, Algebraic K-theory measures the topological and algebraic properties of the category of the representations of a ring, which can then give useful information about the ring itself.
These constructions for K-theory work for any object that admits a nice category like the category of projective R-modules over a ring R, such as bundles over topological spaces (see Swan's theorem) and C*-algebras. Such categories are called exact categories, defined by Quillen for the foundations of K-theory.
I suggest then moving some of the material on exact categories and k-theory from the algebraic k-theory article to this article, since this is where the generality belongs. -- Marc 74.139.223.145 (talk) 01:19, 19 January 2008 (UTC)