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:
“Many women are reluctant to allow men to enter their domain. They don’t want men to acquire skills in what has traditionally been their area of competence and one of their main sources of self-esteem. So while they complain about the male’s unwillingness to share in domestic duties, they continually push the male out when he moves too confidently into what has previously been their exclusive world.”
—Bettina Arndt (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 (1913–1960)
“The next time the novelist rings the bell I will not stir though the meeting-house burn down.”
—Henry David Thoreau (1817–1862)