Zariski's main theorem
Encyclopedia
In algebraic geometry
, a field in mathematics
, Zariski's main theorem is a theorem proved by that 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 morphism
s of schemes
, which implies Zariski's original main theorem.
The name "Zariski's main theorem" comes from the fact that labeled it as the "MAIN THEOREM" of his paper, and does not imply that it was the most important theorem ever proved by Oscar Zariski
.
This form of the theorem can be hard to understand because the language of algebraic geometry has changed.
A version of his main theorem using current terminology states:
(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 , which is closer to Zariski's original theorem:
Another version, given by is
s, and "Zariski's main theorem" is sometimes used to refer to this generalization.
It is well known that 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
is a separated, quasi-finite, finitely presented morphism then there is a factorization into
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.
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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a field in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Zariski's main theorem is a theorem proved by that 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 morphism
Quasi-finite morphism
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is finite type and satisfies any of the following equivalent conditions:...
s of schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, which implies Zariski's original main theorem.
The name "Zariski's main theorem" comes from the fact that labeled it as the "MAIN THEOREM" of his paper, and does not imply that it was the most important theorem ever proved by Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
.
Zariski's main theorem for birational morphisms
Zariski's original version of his main theorem 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 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 , which is closer to Zariski's original theorem:
- If f:X→Y is a birational transformation of projective varieties with Y normal, then the total transform of a fundamental point of f is connected and of dimension at least 1.
Another version, given by 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.
Zariski's main theorem for quasifinite morphisms
observed that Zariski's main theorem could easily be deduced from a more general theorem about the structure of quasi-finite morphismQuasi-finite morphism
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is finite type and satisfies any of the following equivalent conditions:...
s, and "Zariski's main theorem" is sometimes used to refer to this generalization.
It is well known that 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.
Zariski's main theorem for commutative rings
reformulated his main theorem in terms of commutative algebra as a statement about local rings. 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.
Zariski's main theorem: topological form
A topological version of Zariski's main theorem says that if x is a (closed) point of a normal complex variety then there are arbitrarily small neighborhoods U of x such that the set of non-singular points of U is connected .Zariski's main theorem: power series form
A formal power series version of Zariski's main theorem says that if x is a point of a normal variety then the completion of the local ring at x is a normal integral domain .See also
- Deligne's connectedness theorem
- Fulton-Hansen connectedness theoremFulton-Hansen connectedness theoremIn mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1....
- Grothendieck's connectedness theoremGrothendieck's connectedness theoremIn mathematics, Grothendieck's connectedness theorem states thatif A is a complete local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec is -connected...