Talk:K-theory

Strange omissions

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)

No definitions redux

One Marc, on Jan 19th of 2008, correctly noted that this article contains no definitions. As a matter of fact, a worrying fraction of the articles on algebraic geometry and algebraic topology seem to consist more of Grothendieck/Serre fanboyism than of actual information. When not indulging in genius-worship, their time is spent on wildly informal descriptions of the various (supposedly very impressive) "connections" between a dozen concepts and theorems which are nowhere defined nor stated. Seriously, it all reads like sci.math crank posts.

No definitions

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)

I added a Lead rewrite tag since this issue has not been resolved. Linking to the algebraic K-theory and topological K-theory articles does not help since those articles refer back to this one for their definitions.--RDBury (talk) 15:14, 12 July 2010 (UTC)

Reference

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)

Is the definition of Serre's Conjecture wrong?

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)

Possibly repeated entry

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)

To do

• 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)

Suggestions concerning various K-theory articles (mostly for the introductions)

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)

• A completely different problem is that in 1957 Grothendieck invented and called "K-theory", K(X), an object which is nowadays called K₀(X). The Grothendieck group construction is relatively elementary and very versatile. There were and there are useful theorems (like the Grothendieck-Riemann-Roch theorem, statements about representation rings of semisimple groups, and surely many others) which only use that K₀ piece. Sometimes they even follow Grothendieck's usage and refer to this piece as a "K-theory" without further qualifications. I would say K₀ has more fame and applications than higher K-groups. Thus I think this broad article should concentrate on K₀ first (already true), and mention that occasionally, and historically, the words "K-theory" mean only this part (not yet clear from the text).
• There must be included an explanation of why mathematicians started to suspect that Grothendieck's K(X) is actually K₀(X). Perhaps the details should be left to Algebraic K-theory and Topological K-theory, but a nontechnical overview here would be nice. This is necessary to motivate the split between the algebraic and topological parts. The Grothendieck group K₀ may be applied to any commutative monoid, but it's impossible to define even K₁ without more information on where does that specific monoid come from. So it's only higher K-theory that clearly separates algebraic K-theory from topological K-theory.

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)

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)

• So I think a section explicitly dedicated to higher K-groups is needed, which would explain why people expect them and concede the point that they're more complicated and harder to interpret. On the other hand, the more specialized articles on algebraic and topological K-theory should probably acknowledge that they are mostly about higher K-groups.
• That being said, the topological K-theory has a unique feature: due to the miracle of Bott periodicity and the philosophy of cohomology theories the passage from K⁰ to Kⁿ is a piece of cake. This fact should be highlighted in that hypothetical section on higher K-groups here. Maybe it also deserves an explicit formulation in the specialized article.
• Further remark on the language: in the field of algebraic K-theory, the historical motivation was to define K_i(R) for a ring R. There was no satisfactory solution until the work of Quillen, who used homotopy theory. The study of the particular homotopical constructions became a part of the field. For a very long time the only definition of K_i was "apply Quillen's construction, the result is what's called higher algebraic K-theory of a ring". Which is not how mathematics usually works: usually there are abstract definitions and concrete constructions, but in that case the construction was the only ingredient that we got. This should be acknowledged in this or that article. Probably this is the reason why people complain about the lack of clear definitions on K-theory pages -- sometimes there are none. The universal description that we now have ('universal additive Morita-invariant functor on small stable infinity-categories') is too technical to substitute, even for many mathematicians, including me.
• As the detailed investigation of various homotopical constructions became accepted as a part of algebraic K-theory, the rings slowly started to disappear from a field that arose in order to study them. After Waldhausen's work one could form, say, algebraic K-theory of a category of topological spaces -- no rings involved. Thus I think now words 'algebraic K-theory' don't mean that a ring is hiding somewhere (to confirm, look at the publications in the journal K-theory or a corresponding arXiv section; that's also a sense in which algebraic K-theory is used in some fancy modern theories). But historically the ring was always present somewhere. I'm not sure, but my impression is that sometimes number theorists say 'algebraic K-theory' but they are still only interested in algebraic K-theory of rings of integers of certain fields. At least, the current article algebraic K-theory (including its extensive section on history) seems to be written from the position that rings are primary objects of interest for the field, and algebraic K-theory of other objects is a matter of curiousity rather than the focus. This position may very well be the correct one, I'm not qualified to judge. But perhaps an explicit pronouncement on this question would be welcome.

Sorry for a very long rant! Maybe it would've been quicker to implement some of the suggestions than to write them all down. I think that for the articles on these kinds of topics there is always a coordination problem: the area is not small enough to fit in one article, but the subdivision is not immediately clear. For a sad example, have you tried to sort through a mess of multiple articles related to connections? Perhaps this centralized plan on what should the introductions say about their respective subtopics would be useful. Feel free to write any objections to this overview, if you wish. Dpirozhkov (talk) 00:13, 8 December 2017 (UTC)

@Jakob.scholbach: @Dpirozhkov: So, I'm quite late but reading the above reminded me of Spectral sequence#Further examples. Perhaps having the analogous list for K-theory might be a way to implement the above suggestion? -- Taku (talk) 02:04, 28 December 2017 (UTC)