A NOTE ON THE FIBER DIMENSION THEOREM

The aim of this work is to prove a version of the Fiber Dimension Theorem, emphasizing the case of non-closed points. Resumo El objetivo de este trabajo es probar una versión del Teorema de la Dimensión de la Fibra, enfatizando el caso de puntos no cerrados.


INTRODUCTION
The fiber dimension theorem is an important mathematical tool in the everyday life of an algebraic geometer.For example, we can calculate the dimension of the space of matrices whose rank is less than or equal to k; the dimension of the incidence variety Γ = {(π, , p) | p ∈ ⊂ π} where π is a plane, is a line and p is a point in some projective space as indicated in the figure below (see Harris [8] for other applications).
We can also use this theorem to know for example if a nonsingular hypersurface contains a finite number of lines, planes, etc., in particular, we can show that a nonsingular cubic surface in the three-dimensional projective space contains exactly 27 lines (see Reid [12], Beauville [3], Crauder-Miranda [4]).
The aim of this article is to prove the following theorem.
1.1.Theorem.(Fiber dimension theorem) Let X and Y be integral affine schemes of finite type over a field K, f : X −→ Y be a dominant morphism.Then there exists a nonempty open subset U ⊆Y such that dim f −1 (y) = r for all y ∈ U (r = dim X − dim Y ).
Indeed, the theorem above can be read from Eisenbud's compendium, Corollary 14.5 (see p. 310 in [6]).But, the strategy that we will use to prove the theorem 1.1 above, was suggested by exercise 3.22 in Hartshorne's book [10] at page 95.In fact, the exercise 3.22 in Hartshorne's book tell us that it is enough to prove the following theorem.
1.2.Theorem.(see Theorem 3 in chapter I, section 8 of [11]) Let X and Y be affine schemes and f : It is important to notice that the proof of Theorem 1.1 is not just an application of Theorem 1.2, due to the existence of non-closed points, as is the case of affine schemes with the Zariski topology (see Section 3).In fact, in this work we are not going to prove Theorem 1.2.We focus our attention to the proofs of the results that we will use to conclude Theorem 1.1 in the case of non-closed points, showing that, we need to be more careful with these ones.
For those who are not familiar with the nature of algebraic varieties, in particular with schemes of points, where we can find non-closed points, fat points, etc., we recommend the beautiful exposition in Harris-Eisenbud [9], where certain fat points appear as the limits of a finite number of simple points and other very interesting results.Also in [7] you can find some results on fat points in the projective plane made by Harbourne.
We refer the reader to the study of quadruplets of points in the projective plane made by Avritzer-Vainsencher in [2] and conical sextuplets made by Rojas-Vainsencher in [13], where they give an explicit descriptions of the varieties that parametrize these subschemes of degree 4 (respectively 6) contained in a conic.

Affine varieties and Zariski's topology
By the affine space over a field C, we mean simply the vector space C n ; this is usually denoted by A n (the main distinction between affine space and the vector space C n is that the origin plays no special role in affine space).An affine variety X in A n is simply the common zero locus of a collection of polynomials in C[x 1 , ..., x n ].
More precisely, let f 1 , ..., f k ∈ C[x 1 , ..., x n ] be polynomials.Then is the affine variety in A n determined by the polynomials f 1 , ..., f k .
Thus, if we consider f = 0 ∈ C[x 1 , ..., x n ] then we have that The Zariski Topology on A n , is simply the topology where the closed sets are given by Z(I) for some ideal

The Zariski topology on Spec(A)
Let A be a commutative ring with unity and be the prime spectrum of A.

Remarks.
1.If the ring A is an integral domain then we will denote by 0 ∈ Spec(A) the prime ideal h0i, where 0 ∈ A is the zero element of the ring A.

We will use the notation
Note that, A n can be considered as a subset of where to each point (a 1 , ..., a n ) ∈ A n we associate the maximal ideal

Thus the above identification implies that
) possesses exactly one more point than A 1 , namely x = 0. Nevertheless, in the case n > 1, there are an infinite number of points, or more precisely, of prime ideals.
Then we can make the following question: ) such that under the identification in (3.2) the points of A n still remain closed?
Yes, it is enough to observe that for any commutative ring with unity A, the family of sets defines a topology on Spec(A), called the Zariski topology of Spec(A) (see problem 15, p. 12 in Atiyah-MacDonald [1]).
In fact, we have the following bijection C , we would like to known: is x = 0 a closed point of A 1  C or not?
Note that, {0} ⊂ V (0) since 0 ∈ V (0).On the other hand Therefore, if 0 ∈ V (I) then we have that I ⊆ 0, that is, C .In fact, 0 is the generic point of A 1 C .More generally, if X is a topological space and x ∈ X is a point of X such that {x} = X, then x is called a generic point of X (see p. 74 in Hartshorne [10]).

Remarks.
1.The consequence of 0 to be the generic point of A 1  C follows from the fact that C[x] is an integral domain, as the following result guarantees.
Let A be an integral domain then 0 ∈ Spec(A) is the generic point of Spec(A).2. Using a similar reasoning for x ∈ Spec(A) in place of 0 ∈ Spec(C[x]) above, we conclude that {x} = V (x).From this, we can deduce that, x is a generic point of Spec(A) if and only if x = N A .N A denote the nilradical of the ring A as in (4.1).

The sheaf of regular functions on Spec(A)
Let X be any topological space.A presheaf F on X assigns to each open set U in X a set, denoted F(U ), and to every pair of nested open sets A presheaf F on X is a sheaf if for each open covering U = ∪ a∈I U a of an open set U ⊂ X and each collection of elements f a ∈ F(U a ) for each a ∈ I having the property that 3.3.Remark.If each F(U ) is a group, ring, etc. and each restriction map r V,U is an homomorphism of groups, rings, etc. then F is called a presheaf of groups, rings, etc.
The concept of B-sheaf (which will be soon introduced) will allow us to define a sheaf from certain special types of bases for the topological space X.Thus it is not necessary to define a sheaf in all the open sets of the topology of X.
Given a base B for the open sets of a topological space X, we say that a collection of sets, groups, rings, etc.
And it can be shown that.
3.4.Proposition.Every B-sheaf on X extends uniquely to a sheaf on X.
Coming back to the topological space Spec(A) defined in (3.1) we have that We define the sheaf of regular functions O Spec(A) on Spec(A) as follows where A f denoted the localization of the ring A at f ∈ A.
Note that, Spec(A) g ⊂ Spec(A) f if and only if some power of g is a multiple of f then we define the restriction map where g N = µf.
In fact, if B is the collection of distinguished open sets Spec(A) f of Spec(A) then O Spec(A) is a B-sheaf on Spec(A) (see proposition I-18, p. 19 in Harris-Eisenbud [9]).

Affine schemes
An affine scheme is a pair (Spec(A), O Spec(A) ) where Spec(A) is the prime spectrum of the commutative ring A considered as a topological space with the Zariski topology and O Spec(A) is the sheaf of regular functions over Spec(A). 4.1.Remarks.
1. Let (Spec(A), O Spec(A) ) be an affine scheme, then the ring A is called the coordinate ring of this affine scheme.

The closed set V (I) ⊂ A n
C can be identified with the affine scheme where In the next section, we will define morphisms between affine schemes.

Morphisms between affine schemes
Let us take X = Spec(A) and Y = Spec(B) with A and B commutative rings with unity.Note that every homomorphism of rings ϕ : B −→ A determines a continuous function ϕ * : X −→ Y , defined by ϕ * (x) = ϕ −1 (x).
We will say that f : X −→ Y is a morphism between affine schemes if it is obtained from a ring homomorphism, that is, there exist ϕ : B −→ A us above such that f = ϕ * .
4.2.Remark.More precisely a morphism between two affine schemes is a pair (f, f # ), where f : Spec(A) −→ Spec(B) is continuous function and open and it is called the direct image of O Spec(A) by f ), is a family of ring homomorphisms {f # Spec(B)g } g∈B commuting with the restriction maps given by where we assume that f = ϕ * for some ring homomorphism ϕ : B −→ A and we use the basic fact that f −1 (Spec(B) g ) = Spec(A) ϕ(g) for all g ∈ B.
In fact, we have the following equivalence of categories Thus we have that Hom Rings (B, A) ∼ = Hom AffineSchemes (X, Y ).That is, each morphism between affine schemes is determined by a unique homomorphism of rings (see Theorem I-40 in Harris-Eisenbud [9]).
Let us see some examples.

Examples.
1. Let A be a ring, I ⊂ A be an ideal and π : A −→ A/I the canonical homomorphism.Then π induces the morphism 2. There exists a morphism f : The answer will be affirmative if we can find ϕ : Finally, let us remember that a function f :

The notion of dimension
In this section, we consider the notion of dimension from an algebraic and topological point of view.

• Algebraic The Krull dimension
Let A be a commutative ring and Spec(A) the prime spectrum of A. Consider all the chains of the form

The transcendence degree
The concept of dimension also has a close relation with the maximum number of algebraically independent variables in a field.More explicitly, let us consider a finitely generated K-algebra A (K is a field), that is, there exists a surjective ring homomorphism ϕ Note that, if A is an integral domain then F rac(A), the field of fractions of the ring A, is an extension field of K.And we can ask for the maximum number of algebraically independent elements of F rac(A) over K, that is, the transcendence degree of F rac(A) over K.

• Topological
Let X be a topological space and Y ⊆ X be a subset of X.Consider Y equipped with the natural topology induced from that of X. Y is called irreducible if Y is not union of two proper closed subsets of Y , see Hartshorne [10] and Atiyah-MacDonald [1].

Consider all the chains of the form
5.1.Remark.Note that, if X is a noetherian topological space (see Hartshorne [10], Atiyah-MacDonald [1]), then X admits a decomposition as follows where each X i is an irreducible closed subset of X, such that, for all i 6 = j ∈ {1, ..., m} X i 6 ⊆ X j .Each one of these X i is called an irreducible component of X and in this case it is verified that, dim X = max ½ dim X i | i ∈ {1, ..., m} ¾ .(5.1) Thus, in order to compute dimensions of topological spaces, it will be enough to consider the case of irreducible ones.
Let us see the following example.C .Note that hx 1 x 2 , x 1 x 3 i = hx 1 i ∩ hx 2 , x 3 i .
Therefore X = V (x 1 ) ∪ V (x 2 , x 3 ).Thus X consists of two irreducible components, namely, the line V (x 2 , x 3 ) and the plane V (x 1 ).And we conclude that dim X = 2.
It is important to note that these three approaches to the notion of dimension agree, as the following theorem guarantees.
5.3.Theorem.Let X = Spec(A) with A a finitely generated K-algebra without divisors of zero then dim Spec(A) = dim Krull A = trdeg K K(X) where K(X) = F rac(A) is called the field of rationals functions of X.
4 in Hartshorne[10])Coming back to the case of A 1 F(U ) for open sets U ∈ B and maps r V,U : F(V ) −→ F(U ) for V ⊂ U form a B-sheaf if they satisfy the sheaf axiom with respect to inclusions of basic open sets in basic open sets and coverings of basic open sets by basic open sets.(The condition that f a ∈ F(U a ) for each a ∈ I, U a ∈ B having the property r U a ,U a ∩U b (f a ) = r U b ,U a ∩U b (f b ) ∀ a, b ∈ I; U a , U b ∈ B must be replaced by the following condition form a base for the Zariski topology on Spec(A).The open sets Spec(A) f are called distinguished (or basic) open sets of Spec(A).