“Unibranch local ring”的意思、由来-开放百科全书

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Unibranch local ring

释义

References

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.{{fact|date=August 2012}} 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 {{harv|EGA|loc=III.4.3.7}} Let X and Y be two integral locally noetherian schemes and 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 is unibranch. Then the fiber 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

{{EGA |book=III-1}}

2 : Algebraic geometry|Commutative algebra

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