Quillen's localization theorem

Remarks

Quillen proved the theorem with his Theorem B, applied to the functor $Q\mathcal{A} \to Q(\mathcal{A}/\mathcal{B})$: he showed that its homotopy fibres are equivalent to $BQ\mathcal{B}$.

Example: let $R$ be a Dedekind domain with fraction field $F$. Take for $\mathcal{A}$ the finitely generated $R$-modules and for $\mathcal{B}$ the torsion ones; then $M \mapsto M \otimes_R F$ identifies $\mathcal{A}/\mathcal{B}$ with the finite-dimensional $F$-vector spaces. Every finitely generated torsion module has a finite filtration with quotients $R/\mathfrak{m}$, so dévissage gives $K_n(\mathcal{B}) \cong \bigoplus_{\mathfrak{m}} K_n(R/\mathfrak{m})$, the sum over maximal ideals. Since $R$ is regular, the resolution theorem gives $K_n(\mathcal{A}) = K_n(R)$. The result is the localization sequence $$\cdots \to \bigoplus_{\mathfrak{m}} K_n(R/\mathfrak{m}) \to K_n(R) \to K_n(F) \to \bigoplus_{\mathfrak{m}} K_{n-1}(R/\mathfrak{m}) \to \cdots.$$

Going deeper

For a student

The main use is in G-theory: for a Noetherian scheme $X$, a closed subscheme $Z$ and the open complement $U$, it gives the sequence $\cdots \to G_n(Z) \to G_n(X) \to G_n(U) \to G_{n-1}(Z) \to \cdots$. Together with dévissage and the resolution theorem it computes K-groups of Dedekind domains from those of their fraction fields and residue fields. It is also a key step in Quillen's proof of the fundamental theorem for polynomial and Laurent polynomial rings. For K-theory of vector bundles instead of coherent sheaves, the right analogue is the Thomason–Trobaugh localization theorem.