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 In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...

, and the

integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' ...

''B'' of ''A''_red is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of ''B'' is a

purely inseparable extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...

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 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^(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

{{math-stub Algebraic geometry Commutative algebra