Zariski's Main Theorem For Commutative Rings
Zariski (1949) reformulated his main theorem in terms of commutative algebra as a statement about local rings. Grothendieck (1961, Théorème 4.4.7) generalized Zariski's formulation as follows:
- If B is an algebra of finite type over a local Noetherian ring A, and n is a maximal ideal of B which is minimal among ideals of B whose inverse image in A is the maximal ideal m of A, then there is a finite A-algebra A′ with a maximal ideal m′ (whose inverse image in A is m) such that the localization Bn is isomorphic to the A-algebra A′m′.
If in addition A and B are integral and have the same field of fractions, and A is integrally closed, then this theorem implies that A and B are equal. This is essentially Zariski's formulation of his main theorem in terms of commutative rings.
Read more about this topic: Zariski's Main Theorem
Famous quotes containing the words main, theorem and/or rings:
“Parents need to recognize that the negative behavior accompanying certain stages is just a small part of the total child. It should not become the main focus or be pushed into the limelight.”
—Saf Lerman (20th century)
“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 (19131960)
“If a man do not erect in this age his own tomb ere he dies, he shall live no longer in monument than the bell rings and the widow weeps.”
—William Shakespeare (15641616)