Unibranch local ring

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In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of Ared is also a local ring.[citation needed] A unibranch local ring is said to be geometrically unibranch if the residue field of B is a purely inseparable extension of the residue field of Ared. A complex variety X is called topologically unibranch at a point x if for all complements Y of closed algebraic subsets of X there is a fundamental system of neighborhoods (in the classical topology) of x whose intersection with Y is connected.

In particular, a normal ring is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:

Theorem (EGA, 4.3.7) Let X and Y be two integral locally noetherian schemes and f \colon X \to Y a proper dominant morphism. Denote their function fields by K(X) and K(Y), respectively. Suppose that the algebraic closure of K(Y) in K(X) has separable degree n and that y \in Y is unibranch. Then the fiber f^{-1}(y) has at most n connected components. In particular, if f is birational, then the fibers of unibranch points are connected.

In EGA, the theorem is obtained as a corollary of Zariski's main theorem.

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