Localization of A Ring - Applications

Applications

Two classes of localizations occur commonly in commutative algebra and algebraic geometry and are used to construct the rings of functions on open subsets in Zariski topology of the spectrum of a ring, Spec(R).

• The set S consists of all powers of a given element r. The localization corresponds to restriction to the Zariski open subset U_r ⊂ Spec(R) where the function r is non-zero (the sets of this form are called principal Zariski open sets). For example, if R = K is the polynomial ring and r = X then the localization produces the ring of Laurent polynomials K. In this case, localization corresponds to the embedding U ⊂ A1, where A1 is the affine line and U is its Zariski open subset which is the complement of 0.

• The set S is the complement of a given prime ideal P in R. The primality of P implies that S is a multiplicatively closed set. In this case, one also speaks of the "localization at P". Localization corresponds to restriction to arbitrary small open neighborhoods of the irreducible Zariski closed subset V(P) defined by the prime ideal P in Spec(R).