# The localisation long exact sequence in K-theory over an arbitrary base

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?))

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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

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 $K(Z)$.

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 $$K(X)\to K(X[[t]])\to K(X((t)))$$

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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!

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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.

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«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