Localization: inverting to zoom in
The field of fractions inverts every nonzero element of a domain. Localization is the controlled version: invert only a chosen multiplicative set. To study a ring near a prime P, invert everything *outside* P — the set S = R∖P — getting the localization at P, written R_P. Its elements are fractions a/s with s ∉ P.
The magic: R_P is always a local ring — it has a unique maximal ideal PR_P. Everything outside P became a unit, so P is the only obstruction left. Localization is exact and commutes with quotients and finite intersections, which makes it the standard tool for reducing a global question to one prime at a time.
R = Z, P = (5). Then R_P = Z_(5) = { a/b : b not divisible by 5 }.
Units: a/b with 5 not dividing a (and not b).
Only non-unit prime: (5)Z_(5). So Z_(5) is local, max ideal (5).
e.g. 1/3, 7/2, 100/3 are all in Z_(5); but 1/5 is NOT.
Primes of R_P <-> primes of R contained in P:
Spec Z_(5) = { (0), (5)Z_(5) } -- just the chain (0) c (5).
Localization 'deletes' every prime not below P and keeps the rest.
This is exactly zooming the picture in onto the point P.Discrete valuation rings: the smooth points of curves
Localize a nice curve at a point and you get a discrete valuation ring (DVR): a local PID that is not a field. Equivalently, a domain with a valuation v: K^× → Z measuring the order of vanishing of a function at that point. There is a single uniformizer π generating the maximal ideal, and every nonzero element is uniquely a unit times a power of π.
Many equivalent definitions of a DVR (R local, domain, NOT a field):
(a) R is a PID with a unique nonzero prime ideal;
(b) the maximal ideal m is principal, m = (pi), and n m^n = 0;
(c) R is integrally closed, Noetherian, with Krull dimension 1;
(d) there is a discrete valuation v on Frac(R) with R = {x : v(x) >= 0}.
Prototype: R = k[x]_(x) = rational functions f/g with g(0) != 0.
Uniformizer pi = x. v(f) = order of vanishing of f at 0.
v(x^3 * unit) = 3, v((x^2+x)/(1+x)) = v(x(x+1)/(1+x)) = 1.
Ideals are exactly (x^n), a single chain (1) > (x) > (x^2) > ...
A Dedekind domain is precisely a domain that is a DVR at every nonzero prime.Krull dimension: counting chains of primes
Everything assembles into one number. The Krull dimension of R is the supremum of lengths d of strict chains of primes P_0 ⊊ P_1 ⊊ … ⊊ P_d. The height of a prime P is the dimension of the local ring R_P — the longest chain *below* P. This is the purely algebraic definition of dimension, and Noether normalization made it agree with the geometric one: dim k[x_1,…,x_n] = n.
dim Z = 1: longest chain is (0) c (p), length 1. dim k = 0: a field has only the prime (0). length 0. dim k[x] = 1: (0) c (x - a). dim k[x,y] = 2: (0) c (x) c (x, y). Three primes, chain length 2. In general dim k[x_1,...,x_n] = n (a clean Noether-normalization corollary). Krull's principal ideal (Hauptidealsatz): in a Noetherian ring, a minimal prime over a SINGLE nonzero element f has height <= 1. => cutting by one equation drops dimension by at most one. Geometric reading: a hypersurface V(f) in n-space has dimension >= n-1. Iterating: V(f_1,...,f_c) has every component of dimension >= n - c.
The capstone is Krull's principal ideal theorem and its converse: in a Noetherian local ring, the height of a prime equals the *minimum* number of elements needed to generate an ideal with that prime as a minimal prime. When that minimum equals the dimension — generators as few as the dimension allows — the local ring is a regular local ring, the algebraic signature of a smooth point. From finiteness (Hilbert) to geometry (Nullstellensatz) to dimension (Krull): the same chains of primes run through all of it.