Why we want exact sequences at all
An invariant you cannot compute is decoration. The reason homological algebra is powerful is that its invariants sit in exact sequences: knowing two of three terms, plus the connecting map, often pins down the third. K-theory was engineered to behave the same way. The slogan is that K_0 and K_1 are the bottom two pieces of a single long exact sequence, joined by a connecting homomorphism ∂: K_1 → K_0 that plays the role of the boundary map.
The localization sequence at work
The single most useful exact sequence relates a ring, a localization, and what you killed off. For a Dedekind domain R with fraction field F, inverting a maximal ideal — or all of them at once — produces the localization sequence. Worked out at the bottom, it reads as a short exact sequence that literally reconstructs the ideal class group.
R a Dedekind domain, F = Frac(R), primes p ranging over Max(R).
Localization sequence (low-degree piece):
K_1(R) -> K_1(F) --d--> (+)_p K_0(R/p) -> K_0(R) -> K_0(F) -> 0
Unpack each term:
K_1(F) = F^x (units of the field)
K_0(R/p) = Z (each residue field, one Z per prime)
K_0(F) = Z (F is a field)
The map d: F^x -> (+)_p Z sends a unit to its tuple of valuations:
d(x) = ( v_p(x) )_p = the divisor of x.
Now take cokernels. The free group on primes (+)_p Z is the group of
fractional ideals; modding by the image of d (principal divisors) gives
ker( K_0(R) -> Z ) = Coker(d) = Cl(R) = ideal class group.
So K_0(R) ~= Z (+) Cl(R), i.e. reduced K_0 = Cl(R).Stare at that map ∂(x) = (v_p(x))_p. It is the assignment of a principal ideal to its prime factorization — the same data as classical prime factorization, now reborn as a K-theory connecting homomorphism. The class group, an object you first met as a measure of failure of unique factorization, drops out of a snake-lemma-style boundary map. This is the moment K-theory stops feeling abstract.
Functoriality and exact categories
All of this works because K_n are functors. A ring map R → S induces K_n(R) → K_n(S); a finite flat map even induces a wrong-way transfer S → R on K-theory. The natural input is not a ring but an exact category — a category of modules together with a chosen class of short exact sequences — and K-theory is a functor out of exact categories. This is why K_0 of projectives and G_0 of all modules are both defined, and why the localization sequence above is really a statement about a quotient of exact categories.
There is also an idempotent completion subtlety: K_0 of an exact category equals K_0 of its idempotent completion only after you split every projector. Glossing over it is the classic source of off-by-a-summand errors. Keep the categorical framing in your pocket; it is what lets the same theorems serve algebra, geometry, and topology at once.