Unibranch local ring

In algebraic geometry, a local ring A is said to be unibranch if the reduced ring A_red (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of A_red 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 A_red. 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, III.4.3.7) harv error: no target: CITEREFEGA (help) Let X and Y be two integral locally noetherian schemes and ${\displaystyle f\colon X\to Y}$ 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 ${\displaystyle y\in Y}$ is unibranch. Then the fiber ${\displaystyle f^{-1}(y)}$ 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.

References[]

• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.