Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory

... Kn+1(Dx) --> Kn(k) --> Kn(D) --> Kn(Dx) ...

I want to know, what happens if we replace the base k by a more general scheme?

(I am particularly interested in the map K2(Dx) --> K1(k) (which must be the tame symbol right?))

share|improve this question
    
Is your question about the punctured disc over an arbitrary affine scheme? Or, are you asking about localization for an arbitrary open subscheme of an arbitrary scheme? –  Benjamin Antieau Nov 15 '09 at 16:27
    
I am asking about the punctured disk over an arbitrary (affine) scheme. –  Peter McNamara Nov 16 '09 at 4:20

3 Answers 3

I'm not sure that what I have to say really addresses the heart of your question, but it seems at least related.

Background

The general Localization Theorem (7.4 of Thomason-Trobaugh) states the following. Suppose $X$ a quasiseparated, quasicompact scheme, suppose $U$ a Zariski open in $X$ such that $U$ is also quasiseparated and quasicompact, and suppose $Z$ the closed complement. Then the following sequence of spectra is a fiber sequence: $$K^B(X\textrm{ on }Z)\to K^B(X)\to K^B(U).$$ Here $K^B$ refers to the Bass nonconnective delooping of algebraic $K$-theory. One thus gets a long exact sequence $$\cdots\to K_n^B(X\textrm{ on }Z)\to K_n^B(X)\to K_n^B(U)\to K_{n-1}^B(X\textrm{ on }Z)\to\cdots$$ (If one tries to work only with the connective version, then the exact sequence ends awkwardly, since $K_0(X)\to K_0(U)$ is not in general surjective; indeed, the obstruction to lifting $K_0$-classes from $U$ to $X$ is precisely $K_{-1}(Z)$ by Bass's fundamental theorem.)

The term $K^B(X\textrm{ on }Z)$ is the Bass delooping of the $K$-theory of the ∞-category of perfect complexes of quasicoherent $\mathcal{O}$-modules that are acyclic on $U$. Identifying this fiber term with $K^B(Z)$ is generally a delicate matter. Let me summarize one situation in which it can be done.

Suppose that $X$ admits an ample family of line bundles [Thomason-Trobaugh 2.1.1, SGA VI Exp. II 2.2.3], and suppose that $Z$ admits a subscheme structure such that the inclusion $Z\to X$ is a regular immersion (so that the relative cotangent complex $\mathbf{L}_{X|Z}$ is $I/I^2[1]$, where $I$ is the ideal of definition), and $Z$ is of codimension $k$ in $d$ in $X$. Then the spectrum $K^B(X\textrm{ on }Z)$ coincides with a nonconnective delooping of the Quillen $K$-theory of the exact category of pseudocoherent $\mathcal{O}_X$-modules of Tor-dimension $\leq k$ supported on $Z$. If now $Z$ and $X$ are regular noetherian schemes, then a dévissage argument now permits us to identify $K^B(X\textrm{ on }Z)$ with $K(Z)$.

Your case

Now I'm assuming that $K(D)$ refers just to the $K$-theory of the ring $k[[t]]$ (and not, for instance, the $K$-theory of the formal scheme $\mathrm{Spf}(k[[t]])$), then the discussion above applies to give you your desired localization sequence $$K^B(X)\to K^B(X[[t]])\to K^B(X((t)))$$ for any scheme $X$ admitting an ample family of line bundles. If in particular $X$ is regular, then the negative $K$-theory vanishes, and we have a localization sequence $$K(X)\to K(X[[t]])\to K(X((t)))$$

share|improve this answer

I do not have the reference with me right now, but I think the localization sequence for K-theory over general base was handled in:

R. W. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, "The Grothendieck Festschrift", (1990) 247--435.

There is a link with Google book but it was missing the relevant pages!

share|improve this answer
3  
That paper is one of my favorite stories. Trobaugh, despite being dead, told Thomason how to prove the main theorem in a dream. ams.org/notices/199608/comm-thomason.pdf –  Graham Leuschke Dec 29 '09 at 20:14

This is not a direct answer to the original question, but is what I am interested in.

I found the following in 12.14(iii) of Brylinski and Deligne's paper "Central Extensions of Reductive Groups by K_2". I'll quote the relevant paragraph and comment afterwards.

Suppose that V is henselian and essentially of finite type over a field. For j (resp i) the inclusion of G (resp G_s) in G_V, Quillen resolution gives a short exact sequence of sheaves on G_V. $$0 \to K_2 \to j_\*K_2 \to i_\*K_1(D) \to 0$$

The K's are sheafified K-theory on the big Zariski site. G is the generic fibre of a smooth group scheme G_V, with special fibre G_s.

What I don't know is what "essentially of finite type over a field" means, nor how this exact sequence arises.

share|improve this answer
    
«Algebra of essentially of finite type» tends to mean «a localization of an algebra of finite type» –  Mariano Suárez-Alvarez Nov 17 '09 at 15:26
    
the j and the i in the short exact sequence I have should both be accompanied by a lower star that I don't know how to edit to make appear. –  Peter McNamara Dec 30 '09 at 23:45
    
I've fixed the short exact sequence for you: you can always use the TeX's double-dollar sign to write math. –  Mariano Suárez-Alvarez Jan 12 '10 at 20:52

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.