Zariski's Main Theorem - Zariski's Main Theorem For Commutative Rings

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 Am.

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:

    Women are taught that their main goal in life is to serve others—first men, and later, children. This prescription leads to enormous problems, for it is supposed to be carried out as if women did not have needs of their own, as if one could serve others without simultaneously attending to one’s own interests and desires. Carried to its “perfection,” it produces the martyr syndrome or the smothering wife and mother.
    Jean Baker Miller (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)

    Ye say they all have passed away,
    That noble race and brave;
    That their light canoes have vanished
    From off the crested wave;
    That, mid the forests where they roamed,
    There rings no hunters’ shout;
    But their name is on your waters,
    Ye may not wash it out.
    Lydia Huntley Sigourney (1791–1865)