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Chapter 2: Problem 1

Show that a morphism \(f: X \rightarrow Y\) is locally 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 open affine subsets \(U_{j}=\operatorname{Spec} A_{j},\) where each \(A_{j}\) is a finitely generated \(B\) -algebra.

Short Answer

Expert verified
The morphism \(f: X \rightarrow Y\) is locally of finite type if and only if for every open affine subset \(V=\operatorname{Spec} B\) of \(Y\), the inverse image \(f^{-1}(V)\) can be covered by open affine subsets \(U_j=\operatorname{Spec} A_j\), where each \(A_j\) is a finitely generated \(B\)-algebra.

Step by step solution

01

Assumption and Initial Consideration

Assume that the morphism \(f: X \rightarrow Y\) is locally of finite type. This implies that for every point \(x \in X\), there exists an open neighborhood \(U\) of \(x\) and open neighborhood \(V\) of \(f(x)\) in \(Y\) such that \(f(U)\subset V\), and \(U\) can be covered by open affine subsets where the corresponding ring map is of finite type.
02

Subset Selection and Morphism Inversion

Consider the open affine subset \(V=\operatorname{Spec} B\) of \(Y\), with \(U_j=\operatorname{Spec} A_j\) being the open affine subsets of \(f^{-1}(V)\). As \(f\) is locally of finite type, for any \(x \in f^{-1}(V)\), we can find such a \(U_j\) containing \(x\). The induced morphism \(f|_{U_j} : U_j \rightarrow V\) correspond to a ring map \(B \rightarrow A_j\). We denote this map by \( \phi_j: B \rightarrow A_j \). According to the assumption, this map \( \phi_j \) is of finite type.
03

Application of Definition

Applying the definition of a map being of finite type, each \(A_j\) is a finitely generated \(B\)-algebra. Consequently, \(f^{-1}(V)\) can be covered by open affine subsets \(U_j=\operatorname{Spec} A_j\), where each \(A_j\) is a finitely generated \(B\)-algebra.
04

Reversal of Assumption

Now, consider the reverse: Let \(f^{-1}(V)\) be able to be covered by open affine subsets \(U_j=\operatorname{Spec} A_j\), where each \(A_j\) is a finitely generated \(B\)-algebra. We can choose an open neighborhood \(V\) of \(y=f(x)\), so that \(f^{-1}(V)\) is covered by these \(U_j\)'s. Therefore, the morphism \(f\) is still locally of finite type.

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Key Concepts

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

Spec (Spectrum of a Ring)
The concept of Spec, or the Spectrum of a ring, is a foundational idea in algebraic geometry. It converts the study of a commutative ring into a geometric problem by considering all prime ideals of the ring as points in a geometric space. This space is equipped with a natural topology, known as the Zariski topology, where the closed sets are defined by vanishing of sets of polynomials.

When we say Spec of a ring B, denoted as \(\operatorname{Spec} B\), we refer to the set of all prime ideals of B structured in a geometric manner. Each point in this space represents a prime ideal, and the open sets in this space correspond to complements of \(V(I)\), where \(I\) is an ideal of B.

An important property of Spec is that open affine subsets, like the \(U_j=\operatorname{Spec} A_j\) mentioned in our problem, form a base for the Zariski topology. This allows us to 'cover' the space by considering all such open subsets, which is analogous to covering a Venn diagram with overlapping circles, each representing an open affine subset.
Algebraic Geometry
Algebraic geometry is the study where geometry and algebra intersect, focusing on the properties of spaces defined by solutions to polynomial equations. These spaces, known as algebraic varieties, can be extremely complex, and algebraic geometry provides the tools to understand their structure and relationships.

In the case of our problem, algebraic geometry provides the framework to study the morphism \(f: X \rightarrow Y\) by examining the behavior of prime ideals and their corresponding geometric spaces, or spectra. For example, knowing that \(f^{-1}(V)\) can be covered by open affine subsets \(U_j\) is key in understanding the local behavior of the map \(f\), which is akin to observing a complex shape through small, manageable patches.
Finitely Generated Algebra
A finitely generated algebra over a ring B is an algebra for which there exists a finite set of generators. In simple terms, this means that every element of the algebra can be expressed as a combination, involving addition and multiplication, of elements from this finite set and elements of B.

In our exercise, each \(A_j\) is a finitely generated \(B\)-algebra. This concept is important because it implies that we can capture all the algebraic properties of \(A_j\) with just a few 'building blocks' (the generators). This finite nature allows for a more tractable study of the algebraic properties of \(A_j\), making complex problems more accessible - much like how building an intricate LEGO model is possible through the combination of a finite set of distinct brick types.
Ring Homomorphism
A ring homomorphism is a map between two rings that respects the ring operations; that is, it preserves addition and multiplication. Formally, a function \(\phi: B \rightarrow A\) is a ring homomorphism if for all \(b_1, b_2\) in B, \(\phi(b_1 + b_2) = \phi(b_1) + \phi(b_2)\) and \(\phi(b_1 \cdot b_2) = \phi(b_1) \cdot \phi(b_2)\).

In the exercise, \(\phi_j: B \rightarrow A_j\) corresponds to such a map, induced by the local behavior of the morphism \(f\) when restricted to open affine subsets. Understanding ring homomorphisms is essential as they serve as the 'bridge' between algebraic structures, allowing us to translate properties and operations from one ring to another. This 'bridging' is critical when we deal with the local-global principle, moving from understanding local properties of a morphism, such as being of finite type, to the global perspective on the morphism鈥檚 behavior over the whole space.

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Most popular questions from this chapter

Extending a Sheaf by Zero. Let \(X\) be a topological space, let \(Z\) be a closed subset. let \(i: Z \rightarrow X\) be the inclusion, let \(U=X-Z\) be the complementary open subset and let \(j: U \rightarrow X\) be its inclusion. (a) Let \(\mathscr{J}\) be a sheaf on \(Z\). Show that the stalk \(\left(i_{*}, \overline{\mathscr{H}}\right)_{p}\) of the direct image sheaf on \(X\) is \(\mathscr{F}_{P}\) if \(P \in Z, 0\) if \(P \notin Z\). Hence we call \(i_{*}\). \(\bar{y}\) the sheaf obtained by extending of \(i_{*} \overline{\mathscr{H}},\) and say "consider \(\mathscr{F}\) as a sheaf on \(X\)," when we mean "consider \(i_{*}\). (b) Now let \(\overline{\mathscr{H}}\) be a sheaf on \(U\). Let \(j\), \((\overrightarrow{\mathscr{H}})\) be the sheaf on \(X\) associated to the presheaf \(V \mapsto \mathscr{F}(V)\) if \(V \subseteq U, V \mapsto 0\) otherwise. Show that the stalk \((j,(\mathscr{F}))_{P}\) is equal to \(\overline{\mathscr{I}}_{p}\) if \(P \in U, 0\) if \(P \notin U\), and show that \(j\), \(\overline{\mathscr{H}}\) is the only sheafon \(X\) which has this property, and whose restriction to \(U\) is \(\mathscr{F}\). We call \(j\). F. Fhe sheaf obtained by extending \(\mathscr{F}\) by zero outside \(U\) (c) Now let \(\mathscr{F}\) be a sheaf on \(X\). Show that there is an exact sequence of sheaves on \(X\) $$0 \rightarrow j \cdot\left(\left.\overline{\mathscr{H}}\right|_{c}\right) \rightarrow \overline{\mathscr{H}} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{Z}\right) \rightarrow 0$$

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 expos茅 \(8] .\) 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 \(Y\)

Let \(R\) be a discrete valuation ring with quotient field \(K\), and let \(X=\operatorname{Spec} R\). (a) To give an \(\mathscr{O}_{x}\) -module is equivalent to giving an \(R\) -module \(M,\) a \(K\) -vector space \(L,\) and a homomorphism \(\rho: M \otimes_{R} K \rightarrow L\) (b) That \(\mathscr{O}_{x}\) -module is quasi-coherent if and only if \(\rho\) is an isomorphism.

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.

The Grothendieck Group of a Nonsingular Curce. Let \(X\) be a nonsingular curve over an algebraically closed field \(k .\) We will show that \(K(X) \cong \operatorname{Pic} X \oplus \mathbf{Z},\) in several steps. (a) For any divisor \(D=\sum n_{i} P_{i}\) on \(X\), let \(\psi(D)=\sum n_{i \hat{i}}\) ' \(\left(k\left(P_{i}\right)\right) \in K(X),\) where \(k\left(P_{i}\right)\) is the skyscraper sheaf \(k\) at \(P_{t}\) and 0 elsewhere. If \(D\) is an effective divisor, let \(\mathrm{C}_{D}\) be the structure sheaf of the associated subscheme of codimension \(1,\) and show that \(\psi(D)=\dot{\gamma}\left(C_{D}\right) .\) Then use (6.18) to show that for any \(D, \psi(D)\) depends only on the linear equivalence class of \(D,\) so \(\psi\) defines a homomorphism \(\psi: \mathrm{Cl} X \rightarrow K(X)\) (b) For any coherent sheaf \(\mathscr{F}\) on \(X\), show that there exist locally free sheaves \(\delta_{0}\) and \(\mathscr{E}_{1}\) and an exact sequence \(0 \rightarrow \mathscr{E}_{1} \rightarrow \mathscr{B}_{0} \rightarrow \mathscr{H} \rightarrow 0 .\) Let \(r_{0}=\operatorname{rank} \delta_{0}\) \(r_{1}=\operatorname{rank} \delta_{1},\) and define det \(\tilde{\mathscr{H}}=\left(\bigwedge^{r_{0}} \mathscr{E}_{0}\right) \otimes\left(\bigwedge^{r_{1}} \delta_{1}\right)^{-1} \in \operatorname{Pic} X .\) Here \(\wedge \mathrm{de}\) notes the exterior power (Ex. 5.16). Show that det \(\mathscr{H}\) is independent of the resolution chosen, and that it gives a homomorphism det: \(K(X) \rightarrow\) Pic \(X\) Finally show that if \(D\) is a divisor, then \(\operatorname{det}(\psi(D))=\mathscr{L}(D)\) (c) If \(\mathscr{F}\) is any coherent sheaf of rank \(r,\) show that there is a divisor \(D\) on \(X\) and an exact sequence \(0 \rightarrow \mathscr{P}(D)^{\oplus r} \rightarrow \mathscr{H} \rightarrow \mathscr{I} \rightarrow 0\), where \(\mathscr{J}\) is a torsion sheaf. Con- clude that if \(\mathscr{F}\) is a sheaf of rank \(r,\) then \(\gamma(\mathscr{F})-r \gamma\left(\mathcal{O}_{X}\right) \in \operatorname{Im} \psi\) (d) Using the maps \(\psi,\) det, rank,and \(1 \mapsto \gamma\left(C_{X}\right)\) from \(\mathbf{Z} \rightarrow K(X),\) show that \(K(X) \cong\) Pic \(X \oplus \mathbf{Z}\)
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