K-theory of schemes with support.

Question

Hi, I'm having a little trouble with a theorem i found in K-theory of schemes. The theorem is the folowing, if $X$ is a regular noetherian scheme, and $Y$ a closed subscheme of $X$ of codimension $m$, then we get an exact sequence $$0\to F^{m+1}K_0^Y(X)\to K_0^Y(X)\to \bigoplus_{x\in Y\cap X^{(m)}}K_0^{x}(\mathcal O_X,x)$$ where $X^{(m)}$ is the set of points of $X$ of codimension $m$ in $X$ (strictly speaking the points whose closure is of codimension $m$). The proposition seems natural enough, but the proof doesnt go as i would have prooved this result, and relies on an isomorphism, which is stated without anymore precision (i dont know if this is supposed to be obvious, and i dont really find it obvious). The isomorphism in question is the following $$\lim_{Z\subset Y, \text{codim} Z \geq m+1} K_0^{Y- Z}(X-Z) \simeq \bigoplus_{x\in Y\cap X^{(m)}}K_0^{x}(\mathcal O_{X,x})$$ Where the lim, on the left side, means direct limit. And this is this last isomorphism which doesnt seem self evident. Could anyone enlighten me about how to prove this last statement. Thank you.

Answer 1

Since $X$ is regular, your desired isomorphism reduces to

$$lim G_0(Y-Z) \approx \oplus G_0(k(y))$$

where the lim is over points of $Y$ with codimension at least 1, and the sum is over points of $Y$ of codimension 0.

(Here $G_0$ is the $K$-theory of coherent sheaves, sometimes designated $K_0'$.)

If you let ${\cal M}_0$ be the category of coherent $Y$-modules and ${\cal M}_1$ the subcategory of modules with codimension of support at least 1, then Quillen's localization theorem gives you an exact sequence

$$K_0({\cal M}_1)\rightarrow K_0({\cal M}_0) \rightarrow K_0({\cal M}_0/{\cal M}_1)\rightarrow 0$$

The cokernel of the first map is $lim G_0(Y-Z)$, so it suffices to show that

$$K_0({\cal M}_0/{\cal M}_1)\approx \oplus G_0(k(y))$$

This in turn follows from devissage as in the proof of Theorem 5.4 on page 131 of Quillen's paper.