In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski, states that there is only one branch at any point of a normal variety. Informally, the reason it is true is that any branch locus is a singularity of codimension 1, while singularities of normal varieties all have codimension at least 2.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, as there are several ways to make the informal notion of having only one branch precise. In particular the name "Zariski's main theorem" is also used for a closely related theorem of Grothendieck that describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem.
The name "Zariski's main theorem" comes from the fact that it was labeled as the "MAIN THEOREM" in Zariski (1943).
Read more about Zariski's Main Theorem: Zariski's Main Theorem For Birational Morphisms, Zariski's Main Theorem For Quasifinite Morphisms, Zariski's Main Theorem For Commutative Rings, Zariski's Main Theorem: Topological Form, Zariski's Main Theorem: Power Series Form
Famous quotes containing the words main and/or theorem:
“Sinclair Lewis is the perfect example of the false sense of time of the newspaper world.... [ellipsis in source] He was always dominated by an artificial time when he wrote Main Street.... He did not create actual human beings at any time. That is what makes it newspaper. Sinclair Lewis is the typical newspaperman and everything he says is newspaper. The difference between a thinker and a newspaperman is that a thinker enters right into things, a newspaperman is superficial.”
—Gertrude Stein (1874–1946)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)