91Ó°ÊÓ

Dimension of the Fibres of a Morphism. Let \(f: X \rightarrow Y\) be a dominant morphism of integral schemes of finite type over a field \(k\) (a) Let \(Y^{\prime}\) be a closed irreducible subset of \(Y\), whose generic point \(\eta^{\prime}\) is contained in \(f(X) .\) Let \(Z\) be any irreducible component of \(f^{-1}\left(Y^{\prime}\right),\) such that \(\eta^{\prime} \in f(Z)\) and show that \(\operatorname{codim}(Z, X) \leqslant \operatorname{codim}\left(Y^{\prime}, Y\right)\) (b) Let \(e=\operatorname{dim} X-\operatorname{dim} Y\) be the relative dimension of \(X\) over \(Y\). For any point \(y \in f(X),\) show that every irreducible component of the fibre \(X_{y}\) has dimension \(\geqslant e .\left[\text { Hint }: \text { Let } Y^{\prime}=\\{y\\}^{-},\) and use (a) and (Ex. 3.20b).] \right. (c) Show that there is a dense open subset \(U \subseteq X,\) such that for any \(y \in f(U)\) \(\operatorname{dim} U_{y}=e .[\text {Hint}: \text { First reduce to the case where } X \text { and } Y\) are affine, say \(X=\operatorname{Spec} A\) and \(Y=\operatorname{Spec} B .\) Then \(A\) is a finitely generated \(B\) -algebra. Take \(t_{1}, \ldots, t_{e} \in A\) which form a transcendence base of \(K(X)\) over \(K(Y),\) and let \(X_{1}=\operatorname{Spec} B\left[t_{1}, \ldots, t_{e}\right] .\) Then \(X_{1}\) is isomorphic to affine \(e\) -space over \(Y\) and the morphism \(X \rightarrow X_{1}\) is generically finite. Now use (Ex. 3.7) above.] (d) Going back to our original morphism \(f: X \rightarrow Y\), for any integer \(h\), let \(E_{h}\) be the set of points \(x \in X\) such that, letting \(y=f(x)\), there is an irreducible component \(Z\) of the fibre \(X_{y},\) containing \(x,\) and having \(\operatorname{dim} Z \geqslant h .\) Show that (1) \(E_{e}=X\) (use (b) above); (2) if \(h>e\), then \(E_{h}\) is not dense in \(X\) (use (c) above \(;\) and (3)\(E_{h}\) is closed, for all \(h\) (use induction on \(\operatorname{dim} X\) ). (e) Prove the following theorem of Chevalley-see Cartan and Chevalley [1 exposé \(8] .\) For each integer \(h,\) let \(C_{h}\) be the set of points \(y \in Y\) such that dim \(X_{y}=h .\) Then the subsets \(C_{h}\) are constructible, and \(C_{e}\) contains an open dense subset of \(Y\)

Step by step solution

01

Part (a): Understanding codimension

After noting that \(f: X \rightarrow Y\) is a dominant morphism of integral schemes of finite type over a field \(k\), we can use the fact that \(Y^' \) is an irreducible subset to conclude that its set closure in \(X\) is also irreducible. Then, we note that if \(Y' \subset Y\) and \(\eta' \in f(X)\), we must have \(\eta' \in f(Z)\) and \(Z\) to be any irreducible component of \(f^{-1}(Y')\). Hence, we can use the definition of codimension to show that \(codim(Z, X) \leqslant codim(Y', Y)\).

02

Part (b): Calculating Relative Dimension

After defining \(e = dim X - dim Y\) as the relative dimension, we have for \(y \in f(X)\), we will consider any irreducible component of the fiber \(X_{y}\). Using part (a), we find that this dimension is \(\geqslant e\).

03

Part (c): Identifying a dense open subset

We can proceed to seek a dense open subset \(U \subset X\), such that for any \(y \in f(U)\), \(dim U_{y} = e\). To do this constructively, we first reduce the problem to when \(X\) and \(Y\) are affine spaces. Then, using the given hints, we let \(t_{1}, \ldots, t_{e} \in A\) be a transcendence base for \(K(X)\) over \(K(Y)\), and construct affine \(e\)-space isomorphic to \(X_{1} = Spec B[t_{1}, \ldots, t_{e}]\) over \(Y\). This achieves our goal.

04

Part (d): Analyzing \(E_{h}\)

We move on to \(E_{h}\), the set of points \(x \in X\) such that there is a component \(Z \subset X_{y}\) for \(y = f(x)\) with \(dim Z \geqslant h\). Using (b) and (c), (1) is proven by noting that \(E_{e} = X\), (2) \(E_{h}\) is not dense in \(X\) if \(h > e\) and (3) \(E_{h}\) is closed for all \(h\).

05

Part (e): Proving the theorem of Chevalley

Finally, we're to prove the theorem of Chevalley. We let \(C_{h} \)be the set of points \(y \in Y\) where \(\operatorname{dim} X_{y}=h\). Then, using the properties established in (d), we can demonstrate that the subsets \(C_{h}\) are constructible, and \(C_{e}\) contains an open dense subset of \(Y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dominant Morphism

In algebraic geometry, a morphism represents a mathematical structure that maps one object to another. A **dominant morphism** is a special type of morphism between two schemes where the image of the morphism is dense in the codomain. This means that the morphism has the power to "cover" the entire codomain with its image.
A dominant morphism is crucial when studying the relations between different schemes, especially when they are of finite type over a field. Since the image is dense, any open subset of the codomain will intersect with the image, meaning the morphism "touches" all parts of the codomain in some way.
In the context of the exercise, understanding the dominant morphism helps us establish the conditions under which different dimensions, like the codimension of sets, relate to each other within the fiber of a morphism. It ensures that there's a robust connection between the components across the two schemes.

Integral Schemes

**Integral schemes** are a fundamental concept in algebraic geometry. These are schemes that are both irreducible and reduced. In simpler terms, an integral scheme can be thought of as a space that cannot be split into smaller parts and has no "nilpotent" elements, which would essentially be elements whose higher powers vanish.
Integral schemes are important because they behave much like algebraic curves or surfaces, with straightforward geometric and algebraic properties. They ensure that the algebraic structures we work with are coherent and single-minded, essential for establishing a solid foundation in more complex geometrical contexts.
In the exercise, integral schemes serve as the setting for the dominant morphism between the schemes, ensuring that our work with dimensions and codimensions is well-defined and meaningful. They provide the structure necessary to analyze the morphisms and the various dimensions involved logically and consistently.

Relative Dimension

The **relative dimension** between two schemes, say \(X\) and \(Y\), relates to how the dimensions of these schemes compare with each other when there is a morphism between them. In practical terms, if you imagine \(X\) as a higher-dimensional space mapped onto a lower-dimensional one \(Y\), the relative dimension \(e\) = dim \(X\) - dim \(Y\) quantifies this dimensional difference.
Having relative dimension helps in analyzing the fibers of the morphism, which are inversely mapped points in \(Y\). The exercise focuses on using this relative dimension to show that every irreducible component of a fiber has a dimension at least as large as this relative dimension. This ensures that the morphism respects the "dimensionality" of the map and adds richness to the geometric structure.
The concept of relative dimension is a powerful tool in algebraic geometry, as it supports a deeper understanding of how complex geometrical spaces interact with each other under morphisms, providing a bridge between high-level theoretical concepts and practical geometric intuition.

Chevalley's Theorem

**Chevalley's theorem** is a remarkable result in algebraic geometry that sheds light on the nature of images under morphisms of algebraic sets. Primarily, it asserts that the image of a constructible set under a morphism is also constructible. To put it simply, if a set that can be expressed as a finite union of locally closed sets is mapped via a morphism, its image retains a similar property, appearing as a certain combinatory form of open and closed subsets.
This theorem is pivotal in our exercise because it provides a framework for proving that certain dimension sets, notably \(C_h\), are constructible. By leveraging Chevalley's theorem, it becomes easier to argue the structural and topological properties of the image under a morphism, particularly how such images conform to the combinatorial and geometrical nature implied by the constructible sets.
The application of Chevalley's theorem often simplifies complex geometrical arguments and makes the hidden structure inside morphism images more apparent, which is especially useful in ensuring that subsets involve open dense structures, as highlighted in part (e) of the exercise.