Unramified

Unramified is a term used in several branches of mathematics to describe a lack of ramification in a relation between objects. In algebraic number theory, a finite extension L/K of number fields is studied prime by prime. For a prime p of K, write pO_L = ∏ P_i^{e_i}. The prime p is unramified in L/K if every ramification index e_i equals 1; equivalently no prime above p is ramified. Equivalently, p does not divide the discriminant D_{L/K}. Unramified primes reflect that the residue field extensions have certain sizes and that the local extension L_{P_i}/K_p is unramified in the local sense. In local fields, an extension L_p/K_p is unramified if its ramification index is 1.

In algebraic geometry, a morphism f: X → S is unramified at a point x if it is locally of finite type and the sheaf of relative differentials Ω_{X/S} vanishes at x; equivalently the fiber over the image is discrete and the diagonal morphism X → X ×_S X is an open immersion. Intuitively, unramified morphisms have no infinitesimal thickenings in the fibers. If, in addition, f is flat, the morphism is étale, a stronger, particularly well-behaved class of maps.

In practice, unramified extensions are those with good reduction at the corresponding primes, and the notion specializes to separable field extensions where no ramification occurs. Across contexts, unramified serves as a counterpart to ramified, where ramification measures the departure from simple, smooth behavior.