Thomason–Trobaugh localization theorem

Remarks

The key geometric fact is that a perfect complex on $U$ extends to a perfect complex on $X$ after adding a suitable complex as a direct summand. Its class in $K_0(U)$ is the obstruction to extending it directly. With this, Waldhausen's fibration theorem gives the sequence in degrees $n \ge 1$; for connective K-theory the map $K_0(X) \to K_0(U)$ need not be surjective. The nonconnective spectra repair this, and the cokernel of $K_0(X) \to K_0(U)$ is accounted for by negative K-groups of $X$ on $Z$.

Example: if $X$ is a separated regular Noetherian scheme, every coherent sheaf has a finite resolution by vector bundles, and $K(X \text{ on } Z)$ is equivalent to the G-theory spectrum of $Z$. The negative groups vanish, and the theorem gives Quillen's sequence $\cdots \to G_n(Z) \to K_n(X) \to K_n(U) \to G_{n-1}(Z) \to \cdots$.

Going deeper

For a student

The localization theorem is the basis of Thomason–Trobaugh's proof that K-theory satisfies Mayer–Vietoris sequences for Zariski open covers, and of their excision results. It replaces Quillen's localization theorem, which concerns coherent sheaves and so G-theory, by a statement about K-theory itself, valid for all quasi-compact quasi-separated schemes. Many later descent results for K-theory, in the Nisnevich and other topologies, start from it.