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Chapter 2: Problem 3

(a) Show that a morphism \(f: X \rightarrow Y\) is of finite type if and only if it is locally of finite type and quasi-compact. (b) Conclude from this that \(f\) is of finite type if and only if for every open affine subset \(V=\operatorname{Spec} B\) of \(Y, f^{-1}(V)\) can be covered by a finite number of open affines \(U_{i}=\operatorname{Spec} A_{i},\) where each \(A_{i}\) is a finitely generated \(B\) -algebra. (c) Show also if \(f\) is of finite type, then for every open affine subset \(V=\operatorname{Spec} B \subseteq\) \(Y,\) and for every open affine subset \(U=\operatorname{Spec} A \subseteq f^{-1}(V), A\) is a finitely gener- ated \(B\) -algebra.

Short Answer

Expert verified
The solution to the exercise involves using the concept of finite type in algebraic geometry, relating it to the concepts of locally of finite type and quasi-compactness, and applying it to the concepts of open affine subsets, their pre-images, and finitely generated algebras. The key tool in the proof is the definition of finite type.

Step by step solution

01

Showing a morphism is of finite type

Let's start by showing that if \(f\) is locally of finite type and quasi-compact then it's of finite type. Confirm that a morphism \(f: X \rightarrow Y\) is locally of finite type, or in other words, for every point \(x\) in \(X\), there's an open neighborhood \(U\) of \(x\) in \(X\) and an open neighborhood \(V\) of \(f(x)\) in \(Y\) such that the restricted morphism \(f|_{U} : U \rightarrow V\) is of finite type. This means that there exist \(f^{V}\) finitely many elements in the image of \(f\) such that every point \(y\) in \(Y\) has an open neighborhood \(W\) contained in one of these elements. In other terms, any morphism \(f: X \rightarrow Y\) is of finite type if every \(f^{-1}(V)\) can be covered by finitely many \(U_{i}\) such that for each \(i\) the morphism \(f|_{U_{i}} : U_{i} \rightarrow V\) is of finite type.
02

Infinite Type and Covering

Now that we established \(f\) is of finite type, it can be used to argue the claim in (b). Begin by taking any open affine subset \(V = \operatorname{Spec} B\) of \(Y\). Because \(f\) is of finite type, we know that \(f^{-1}(V)\) can be covered by \(U_{i} = \operatorname{Spec} A_{i}\), where each \(A_{i}\) is a finitely generated \(B\)-algebra. Such \(A_{i}\)'s exist because \(f\) being locally of finite type implies the restricted morphism \(f|_{U_{i}} : U_{i} \rightarrow V\) is of finite type for each \(i\), and this is equivalent to \(A_i\) being a finitely generated \(B\)-algebra. The other direction has the same method of proof, namely that each such \(A_i\) is a finitely generated \(B\)-algebra implies that \(f\) is locally of finite type, and there existing finitely many \(U_i\) covering \(f^{-1}(V)\) means \(f\) is quasi-compact, so \(f\) is of finite type.
03

Showing A is a Finitely Generated B-Algebra

Now, to show the last part of the exercise, we again start by any open affine subset \(V = \operatorname{Spec} B \subseteq Y\), and any open affine subset \(U = \operatorname{Spec} A \subseteq f^{-1}(V)\). The proof follows directly from the definition of finite type, as in the last step. The restricted morphism \(f|_{U} : U \rightarrow V\) being of finite type because \(f\) is of finite type implies \(A\) is a finitely generated \(B\)-algebra.

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Key Concepts

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

locally of finite type
When we talk about a morphism being "locally of finite type," we mean that over small enough regions, the morphism behaves like a finite type morphism. In more technical terms, given a morphism \(f: X \rightarrow Y\), at any point \(x\) in \(X\), there is an open neighborhood \(U\) around \(x\) and an open neighborhood \(V\) around \(f(x)\) such that the restricted morphism \(f|_{U}: U \rightarrow V\) is of finite type. Breaking it down:
  • An open neighborhood can be thought of as a small, manageable subset around a point.
  • Being of finite type means that these maps are constructed from a finite amount of data.
This concept is crucial in algebraic geometry as it allows us to study complex structures by breaking them down into these manageable pieces.
By proving that a morphism is locally of finite type, we can assure that on suitably small scales, our algebraic varieties are tractable and well-behaved, setting the stage for more comprehensive analyses of their global properties.
quasi-compact
The term "quasi-compact" in algebraic geometry relates closely to how spaces can be covered using only a finite number of smaller, simpler pieces. If a space is quasi-compact, it can be covered by finitely many open affines.
To understand it better:
  • "Quasi" in this context often refers to an affinity with compactness, which implies finite coverage.
  • Open affines are simpler building blocks, commonly expressed as \(\operatorname{Spec} A\), representing basic algebraic structures.
In practice, when a morphism \(f: X \rightarrow Y\) is quasi-compact, it means for any affine open subset \(V = \operatorname{Spec} B\) in \(Y\), the preimage \(f^{-1}(V)\) can be finitely covered by affines \(U_i = \operatorname{Spec} A_i\).
Understanding quasi-compactness is important because it helps manage infinite possibilities, allowing us to work within a finite framework, which is crucial for computational and theoretical approaches in algebraic geometry.
finitely generated algebra
In algebraic terms, an algebra being "finitely generated" means it can be built using finitely many building blocks. Consider an algebra \(A\), which is formed using this finite set over a base ring \(B\). The algebra consists of all possible combinations of these blocks in terms of addition and multiplication.
Let's break it down:
  • "Finitely generated" implies that there's a finite number of generators from which the entire algebra can be described.
  • This is akin to having a basic set of instruments in a band to create a variety of music. From these few elements, various combinations form the entire piece.
In the context of a morphism being of finite type, having \(A\) as a finitely generated \(B\)-algebra (where \(U = \operatorname{Spec} A\) and \(V = \operatorname{Spec} B\)) implies that the morphism has a finite description in terms of these generators.
This concept is essential as it ensures that we can tackle problems with finite, rather than infinite, descriptions, contributing to the feasibility of algebraic operations and solutions in practical scenarios.

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Most popular questions from this chapter

If \(X\) is a scheme of finite type over a field, show that the closed points of \(X\) are dense. Give an example to show that this is not true for arbitrary schemes.

Show that a morphism \(f: X \rightarrow Y\) is finite if and only if for erery ' open affine subset \(V=\operatorname{Spec} B\) of \(Y, f^{-1}(V)\) is affine, equal to Spec \(A,\) where \(A\) is a finite \(B\) -module.

Show that a sheaf of \(0 x\) -modules \(\mathscr{F}\) on a scheme \(X\) is quasi- coherent if and only if every point of \(X\) has a neighborhood \(U,\) such that \(\left.\mathscr{F}\right|_{U}\) is isomorphic to a cokernel of a morphism of free sheaves on \(U .\) If \(X\) is noetherian, then \(\mathscr{F}\) is coherent if and only if it is locally a cokernel of a morphism of free sheaves of finite rank. (These properties were originally the definition of quasi-coherent and coherent sheaves.)

Shyscraper Sheares. Let \(X\) be a topological space. let \(P\) be a point, and let \(A\) be an abelian group. Define a sheaf \(i_{P}(A)\) on \(X\) as follows: \(i_{p}(A)(\mathcal{L})=A\) if \(P \in L, 0\) otherwise. Verify that the stalk of \(i_{P}(A)\) is \(A\) at every point \(Q \in\left\\{P \text { ; }^{-} \text {, and } 0\right.\) elsewhere, where \(\\{P \text { ; - denotes the closure of the set consisting of the point } P\) Hence the name "skyscraper sheaf." Show that this sheaf could also be described \(\operatorname{as} i_{*}(A),\) where \(A\) denotes the constant sheaf \(A\) on the closed subspace \(\\{P\\}^{-},\) and \(i:\\{P\\}^{-} \rightarrow X\) is the inclusion.

Schemes Orer R. For any scheme \(X_{0}\) over \(\mathbf{R}\), let \(X=X_{0} \times_{\mathbf{R}} \mathbf{C}\). Let \(\alpha: \mathbf{C} \rightarrow \mathbf{C}\) be complex conjugation, and let \(\sigma: X \rightarrow X\) be the automorphism obtained by keeping \(X_{0}\) fixed and applying \(\alpha\) to \(\mathbf{C}\). Then \(X\) is a scheme over \(\mathbf{C}\), and \(\sigma\) is a semi- linear automorphism, in the sense that we have a commutative diagramsince \(\sigma^{2}=\) id. we call \(\sigma\) an involution. (a) Now let \(X\) be a separated scheme of finite type over \(\mathbf{C}\), let \(\sigma\) be a semilinear involution on \(X,\) and assume that for any two points \(\mathrm{r}_{1}, \mathrm{r}_{2} \in X,\) there is an open affine subset containing both of them. (This last condition is satisfied for example if \(X\) is quasi-projective.) Show that there is a unique separated scheme \(X_{0}\) of finite type over \(\mathbf{R},\) such that \(X_{0} \times_{\mathbf{R}} \mathbf{C} \cong X,\) and such that this isomorphism identifies the given involution of \(X\) with the one on \(X_{0} \times_{\mathbf{R}} \mathbf{C}\) described above. For the following statements, \(X_{0}\) will denote a separated scheme of finite type over \(\mathbf{R}\), and \(X, \sigma\) will denote the corresponding scheme with involution over \(\mathbf{C}\) (b) Show that \(X_{0}\) is affine if and only if \(X\) is. (c) If \(X_{0}, Y_{0}\) are two such schemes over \(\mathbf{R}\), then to give a morphism \(f_{0}: X_{0} \rightarrow Y_{0}\) is equivalent to giving a morphism \(f: X \rightarrow Y\) which commutes with the involutions, i.e., \(f \quad \sigma_{X}=\sigma_{Y} \quad f\) (d) If \(X \geqq \mathbf{A}_{\mathbf{C}}^{1},\) then \(X_{0} \cong \mathbf{A}_{\mathbf{R}}^{1}\) (e) If \(X \cong \mathbf{P}_{\mathbf{C}}^{1}\). then either \(X_{0} \cong \mathbf{P}_{\mathbf{R}}^{1},\) or \(X_{0}\) is isomorphic to the conic in \(\mathbf{P}_{\mathbf{R}}^{2}\) given by the homogencous equation \(x_{0}^{2}+1_{1}^{2}+r_{2}^{2}=0\)
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