Zariski's main theorem

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In algebraic geometry, a field in mathematics, Zariski's main theorem, or Zariski's connectedness theorem, is a theorem proved by Zariski (1943, 1949) which implies that fibers over normal points of birational projective morphisms of varieties are connected. The theorem can be stated in several ways which at first sight seem to be quite different. 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.

1 Zariski's main theorem for birational morphisms
2 Zariski's main theorem for quasifinite morphisms
3 Zariski's main theorem for commutative rings
4 See also
5 References

[edit] Zariski's main theorem for birational morphisms

Zariski's original version of his main theorem (Zariski 1943, p. 522) stated

"MAIN THEOREM: If W is an irreducible fundamental variety on V of a birational correspondence T between V and V′ and if T has no fundamental elements on V′ then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W."

This form of the theorem can be hard to understand because the language of algebraic geometry has changed. A version of his main theorem (Hartshorne 1977, corollary III.11.4) using current terminology states:

If f:X→Y is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of Y is connected.

(The inverse image of a non-normal point need not be connected: for example, X might be the normalization of a curve Y with a double point y; then the fiber over y has 2 points and is not connected.) This version easily implies the following result (Hartshorne 1977, Theorem V.5.2), which is closer to Zariski's original theorem:

If f:X→Y is a birational transformation of projective varieties with X normal, then the total transform of a fundamental point of f is connected and of dimension at least 1.

Another version, given by Grothendieck (1961, theorem 4.4.3) is

If f:X→Y is a quasiprojective morphism of Noetherian schemes, then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y.

[edit] Zariski's main theorem for quasifinite morphisms

Grothendieck (1966, Theorem 8.12.6) observed that Zariski's main theorem could easily be deduced from a more general theorem about the structure of quasifinite morphisms, and "Zariski's main theorem" is sometimes used to refer to this generalization. It is well known that quasi-compact open immersions and finite morphisms are quasi-finite. Zariski's main theorem for quasifinite morphisms, which is much harder than these facts is a kind of converse statement: if Y is a quasi-compact separated scheme and

f: X → Y

is a separated, quasi-finite, finitely presented morphism then there is a factorization into

X → Z → Y,

where the first map is an open immersion and the second one is finite.

The relation between this theorem about quasifinite morphisms and Zariski's original main theorem is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y, and Grothendieck's theorem about the structure of quasifinite morphisms can then be used to show that the fibers of f over normal points are connected.

[edit] 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, theorem 4.4.7) generalized Zariski's fomulation 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 B_n 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.