Commutative algebra | Algebraic geometry
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. 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 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. (Wikipedia).
From playlist UNSW Medicine
23C3: Hacking fingerprint recognition systems (de)
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UNSW Online courses: https://studyonline.unsw.edu.au/ Studying Online with UNSW is like studying at university in all respects other than you get to choose when you study and where you study.
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Quantum Computing Concepts - Quantum Bits
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From playlist Quantum Computing Concepts
Learn what it takes to teach online and face-to-face students together at UNSW, and about the benefits a hybrid approach can offer. Find out more at unsw.to/hybridlearning
From playlist Hybrid Learning & Teaching
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From playlist Commutative algebra
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From playlist Commutative algebra
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From playlist Algebraic geometry II: Schemes
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From playlist Commutative algebra
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From playlist HIM Lectures: Junior Trimester Program "Topology"
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