K-theory localization sequence
Localization is the algebraic act of inverting some elements of a ring — making fractions — and one wants to know how it changes the K-groups. The localization sequence is the bookkeeping that answers this: it is a long exact sequence that ties together the K-theory of the original ring, the K-theory of the localized ring, and a correction term measuring exactly what was lost or quotiented away in the process. It is the K-theoretic analogue of the long exact sequence of a pair in topology.
In the cleanest case, let R be a Dedekind domain (or a regular Noetherian ring) and S a multiplicative set, with localization R -> S^{-1}R. Quillen's localization theorem gives a long exact sequence ... -> K_n(R on torsion) -> K_n(R) -> K_n(S^{-1}R) -> K_{n-1}(R on torsion) -> ..., where the first term is the K-theory of the category of finitely generated torsion modules supported on the inverted primes. By dévissage that correction term decomposes as a sum of K-theories of the residue fields R/p over the primes p inverted by S.
The prototypical instance: for a Dedekind domain R with fraction field F, the sequence reads ... -> ⊕_p K_n(R/p) -> K_n(R) -> K_n(F) -> ⊕_p K_{n-1}(R/p) -> ..., the sum running over nonzero primes p. In degree zero this recovers the exact sequence relating the class group, the units, and the divisor map; in higher degrees it is the engine behind computations of K_*(Z) and K_* of rings of integers. The sequence requires the resolution and dévissage theorems and the localization theorem of Quillen, and it generalizes to schemes.
For R = Z, F = Q, the degree-(1,0) part of the sequence is 0 -> K_1(Z) -> K_1(Q) -> ⊕_p K_0(F_p) -> K_0(Z) -> K_0(Q) -> 0, which unwinds to Q^* = {±1} times the free group on the primes, the divisor map sending a rational to its order at each prime p.
The localization sequence for Z recovers the divisor map on Q^*.
The localization sequence is exact for G-theory (K-theory of all finitely generated modules) of any Noetherian ring, but for ordinary K-theory it requires regularity, since one needs finite projective dimension to compare projective modules before and after localization. This regularity hypothesis is exactly what negative K-theory measures the failure of.