Quillen's Localization Theorem

Quillen's localization theorem is the foundational result that produces long exact sequences in algebraic K-theory from localization of abelian categories. It is the engine behind virtually all computations of higher K-groups of rings and schemes.

Statement for schemes

Theorem3.5Quillen's Localization for Schemes

Let $X$ be a Noetherian scheme, $Z \hookrightarrow X$ a closed subscheme with open complement $j: U = X \setminus Z \hookrightarrow X$ . Then there is a long exact sequence of G-theory groups:

$\cdots \to G_{n+1}(U) \xrightarrow{\partial} G_n(Z) \xrightarrow{i_*} G_n(X) \xrightarrow{j^*} G_n(U) \xrightarrow{\partial} G_{n-1}(Z) \to \cdots \to G_0(Z) \to G_0(X) \to G_0(U) \to 0$

where:

$i_*$ is induced by pushforward $i_*: \operatorname{Coh}(Z) \to \operatorname{Coh}(X)$
$j^*$ is induced by restriction $j^*: \operatorname{Coh}(X) \to \operatorname{Coh}(U)$
$\partial$ is the connecting homomorphism

Proof outline

Proof

The proof applies the localization theorem for abelian categories to $\mathcal{A} = \operatorname{Coh}(X)$ with Serre subcategory $\mathcal{B} = \operatorname{Coh}_Z(X)$ (coherent sheaves supported on $Z$ ).

Step 1: Identify the Serre subcategory. The category $\operatorname{Coh}_Z(X)$ of coherent sheaves on $X$ supported set-theoretically on $Z$ is a Serre subcategory of $\operatorname{Coh}(X)$ : if $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is exact with $\mathcal{F}$ supported on $Z$ , then so are $\mathcal{F}'$ and $\mathcal{F}''$ , and conversely.

Step 2: Identify the quotient category. The Serre quotient $\operatorname{Coh}(X) / \operatorname{Coh}_Z(X)$ is equivalent to $\operatorname{Coh}(U)$ . This is because a morphism $\mathcal{F} \to \mathcal{G}$ in $\operatorname{Coh}(X)$ becomes an isomorphism in the quotient iff its kernel and cokernel are supported on $Z$ , which is equivalent to the restriction $\mathcal{F}|_U \to \mathcal{G}|_U$ being an isomorphism.

Step 3: Apply devissage. The K-groups of $\operatorname{Coh}_Z(X)$ are identified with $G_n(Z)$ via devissage. If $Z$ is reduced, this is immediate from the equivalence $\operatorname{Coh}_Z(X) \simeq \operatorname{Coh}(Z)$ (using pushforward along the closed immersion). If $Z$ is not reduced, one uses the filtration by powers of the ideal sheaf $\mathcal{I}_Z$ and devissage to reduce to the reduced case.

Step 4: The long exact sequence. The localization theorem for the Serre subcategory $\operatorname{Coh}_Z(X) \subseteq \operatorname{Coh}(X)$ with quotient $\operatorname{Coh}(U)$ gives:

$\cdots \to K_{n+1}(\operatorname{Coh}(U)) \to K_n(\operatorname{Coh}_Z(X)) \to K_n(\operatorname{Coh}(X)) \to K_n(\operatorname{Coh}(U)) \to \cdots$

Substituting $G_n(X) = K_n(\operatorname{Coh}(X))$ etc., we obtain the stated sequence.

Step 5: The Q-construction proof. At the level of Q-constructions, the sequence arises from the fibration

$BQ\operatorname{Coh}_Z(X) \to BQ\operatorname{Coh}(X) \to BQ\operatorname{Coh}(U)$

where the second map is a "Quillen fibration" (satisfies the hypotheses of Theorem B). The fiber is identified with $BQ\operatorname{Coh}_Z(X)$ up to homotopy, giving the fibration sequence whose long exact sequence in homotopy groups is the localization sequence. $\square$

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Applications

ExampleG-theory of projective space

Using iterated localization for $\mathbb{P}^n_k$ with the filtration by linear subspaces:

$\mathbb{P}^0 \subset \mathbb{P}^1 \subset \cdots \subset \mathbb{P}^n$

At each step, $\mathbb{P}^i \setminus \mathbb{P}^{i-1} \cong \mathbb{A}^i$ . The localization sequence and homotopy invariance ( $G_n(\mathbb{A}^i) = K_n(k)$ ) give:

$G_n(\mathbb{P}^n_k) \cong K_n(k)^{n+1}$

for all $n \geq 0$ . For the K-theory of vector bundles, the more refined computation gives:

$K_0(\mathbb{P}^n) \cong \mathbb{Z}^{n+1}$

generated by $[\mathcal{O}], [\mathcal{O}(1)], \ldots, [\mathcal{O}(n)]$ .

ExampleK-theory of blowups

For the blowup $\tilde{X} = \operatorname{Bl}_Z(X)$ of a smooth variety $X$ along a smooth center $Z$ of codimension $c$ , with exceptional divisor $E \cong \mathbb{P}(\mathcal{N}_{Z/X})$ :

$K_n(\tilde{X}) \cong K_n(X) \oplus \bigoplus_{i=1}^{c-1} K_n(Z)$

This blowup formula is proved using the localization sequence for $E \hookrightarrow \tilde{X}$ and $Z \hookrightarrow X$ , combined with the projective bundle formula for $E$ .

RemarkFrom G-theory to K-theory

The localization sequence is stated for G-theory. For K-theory of vector bundles, the analogous statement requires the scheme to be regular, or one must work with Thomason--Trobaugh K-theory (derived categories):

For a regular scheme $X$ with closed regular subscheme $Z$ :

$\cdots \to K_{n+1}(U) \to K_n(Z) \to K_n(X) \to K_n(U) \to \cdots$

For singular schemes, the localization sequence in K-theory (of perfect complexes) was established by Thomason--Trobaugh using Waldhausen's framework, replacing exact categories with categories with cofibrations and weak equivalences.