Thomason--Trobaugh Localization Theorem

The Thomason--Trobaugh theorem extends the localization sequence to K-theory of perfect complexes on general schemes, including singular and non-Noetherian ones. It is one of the foundational results of modern algebraic K-theory and was crucial for establishing K-theory as a well-behaved cohomology theory on algebraic varieties.

Statement

Theorem5.3Thomason--Trobaugh Localization

Let $X$ be a quasi-compact, quasi-separated scheme, $j: U \hookrightarrow X$ a quasi-compact open immersion, and $Z = X \setminus U$ the closed complement. Let $K(X \text{ on } Z)$ denote the K-theory spectrum of perfect complexes on $X$ acyclic on $U$ (i.e., with cohomology supported on $Z$ ). Then there is a homotopy fiber sequence of K-theory spectra:

$K(X \text{ on } Z) \to K(X) \xrightarrow{j^*} K(U)$

inducing a long exact sequence on homotopy groups:

$\cdots \to K_{n+1}(U) \xrightarrow{\partial} K_n(X \text{ on } Z) \to K_n(X) \xrightarrow{j^*} K_n(U) \to \cdots$

for all $n \in \mathbb{Z}$ (including negative K-groups).

Key innovations

RemarkPerfect complexes vs vector bundles

The crucial insight of Thomason--Trobaugh is that the localization sequence works for perfect complexes but not for vector bundles on singular schemes.

A perfect complex on $X$ is an object $\mathcal{E}^\bullet \in D^b(\mathcal{O}_X\text{-Mod})$ that is locally quasi-isomorphic to a bounded complex of free $\mathcal{O}_X$ -modules of finite rank. Key points:

On a regular scheme, every coherent sheaf has a finite resolution by vector bundles, so $\operatorname{Perf}(X) \simeq D^b(\mathcal{V}(X))$ and the K-theories agree.
On a singular scheme, $\operatorname{Perf}(X) \subsetneq D^b(\operatorname{Coh}(X))$ . For example, on $\operatorname{Spec} k[x]/(x^2)$ , the residue field module $k$ has infinite projective dimension and is not perfect.
The category $\operatorname{Perf}(X)$ is closed under the operations needed for localization: restriction to open subsets, cones, and shifts. Vector bundles are not closed under cones (a cone of a map of vector bundles is a perfect complex but not necessarily a vector bundle).

Definition5.9K-theory with support

For a closed subset $Z \subseteq X$ , define:

$\operatorname{Perf}_Z(X) = \{\mathcal{E}^\bullet \in \operatorname{Perf}(X) : \mathcal{E}^\bullet|_U \simeq 0\}$

the category of perfect complexes on $X$ that are acyclic on $U = X \setminus Z$ . The K-theory with support is:

$K_n(X \text{ on } Z) = K_n(\operatorname{Perf}_Z(X)).$

When $X$ is regular and $Z$ is a regular closed subscheme, $K_n(X \text{ on } Z) \cong K_n(Z)$ by devissage. For singular $Z$ , this is more subtle: $K_n(X \text{ on } Z)$ depends on the embedding $Z \hookrightarrow X$ , not just on $Z$ abstractly.

Proof ideas

Proof

The proof uses Waldhausen's machinery of categories with cofibrations and weak equivalences.

Step 1: Waldhausen K-theory. Define K-theory for a Waldhausen category $\mathcal{C}$ (a category with cofibrations and weak equivalences) using the $S_\bullet$ -construction: $K(\mathcal{C}) = \Omega |wS_\bullet \mathcal{C}|$ , the loop space of the geometric realization of the simplicial category of filtered objects.

Step 2: The localization setup. Consider three Waldhausen categories:

$\mathcal{A} = \operatorname{Perf}_Z(X)$ : perfect complexes acyclic on $U$ .
$\mathcal{B} = \operatorname{Perf}(X)$ : all perfect complexes on $X$ .
$\mathcal{C} = \operatorname{Perf}(U)$ : perfect complexes on $U$ .

The inclusion $\mathcal{A} \hookrightarrow \mathcal{B}$ and restriction $j^*: \mathcal{B} \to \mathcal{C}$ give functors. The key is to show this is a "short exact sequence" of Waldhausen categories.

Step 3: The approximation theorem. Show that $j^*: \operatorname{Perf}(X) \to \operatorname{Perf}(U)$ satisfies the hypotheses of Waldhausen's approximation theorem after taking appropriate completions. The crucial point: given a perfect complex $\mathcal{F}$ on $U$ , does it extend to a perfect complex on $X$ ?

Step 4: The Thomason--Trobaugh trick. The answer to extension is: not always for individual complexes, but every class $[\mathcal{F}] \in K_0(U)$ in the image of $j^*: K_0(X) \to K_0(U)$ does extend. More precisely, for every perfect complex $\mathcal{F}$ on $U$ , there exists a perfect complex $\mathcal{G}$ on $U$ such that $\mathcal{F} \oplus \mathcal{G}$ extends to $X$ . This "stable extension" property, combined with the Eilenberg swindle and the group completion, suffices.

Step 5: Fibration sequence. Using the localization theorem for Waldhausen categories (Waldhausen's fibration theorem), the sequence $K(\mathcal{A}) \to K(\mathcal{B}) \to K(\mathcal{C})$ is a homotopy fibration, giving the long exact sequence. $\square$

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Applications

ExampleBlowup formula

For the blowup $\pi: \tilde{X} \to X$ of a regular scheme $X$ along a regular center $Z$ of codimension $c \geq 2$ , 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 is proved using the Thomason--Trobaugh localization sequence for the open immersion $X \setminus Z \hookrightarrow \tilde{X}$ (complement of $E$ ) combined with the projective bundle formula for $K_n(E) = K_n(\mathbb{P}(\mathcal{N}))$ .

RemarkBass's fundamental theorem via Thomason--Trobaugh

The Thomason--Trobaugh theorem provides a clean proof of the fundamental theorem of K-theory. For $X = \operatorname{Spec} R$ , consider $\mathbb{A}^1_R = \operatorname{Spec} R[t]$ , $\mathbb{G}_m = \operatorname{Spec} R[t, t^{-1}]$ , and $\{0\} = \operatorname{Spec} R$ :

$\cdots \to K_{n+1}(R[t, t^{-1}]) \to K_n(R) \to K_n(R[t]) \to K_n(R[t, t^{-1}]) \to \cdots$

When $R$ is regular, homotopy invariance gives $K_n(R[t]) = K_n(R)$ , so the sequence splits into short exact sequences $0 \to K_n(R) \to K_n(R[t, t^{-1}]) \to K_{n-1}(R) \to 0$ , recovering the Bass--Heller--Swan decomposition.

For non-regular $R$ , the nil-groups appear as the failure of the Cartan map $K_n(R) \to K_n(R[t])$ to be an isomorphism.