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Chapter 2: Problem 4

(a) Let \(\varphi: \overline{\mathscr{H}} \rightarrow \mathscr{S}\) be a morphism of presheaves such that \(\varphi(\mathcal{U}): \mathscr{F}(U) \rightarrow \mathscr{G}(U)\) is injective for each \(U\). Show that the induced \(\operatorname{map} \varphi^{+}: \overline{\mathscr{H}}^{+} \rightarrow \mathscr{S}^{+}\) of associated sheaves is injective. (b) Use part (a) to show that if \(\varphi: \bar{y} \rightarrow \mathscr{G}\) is a morphism of sheaves, then im \(\varphi\) can be naturally identified with a subsheaf of \(\mathscr{G}\). as mentioned in the text.

Short Answer

Expert verified
The given morphism \( \varphi \) is injective, hence the induced map \( \varphi^{+} \) of the associated sheaves is also injective. Using this, we can naturally identify the image of a given sheaves morphism as a subsheaf of the target sheaf.

Step by step solution

01

Definition of Injective

An function \( \varphi: A \rightarrow B \) is called injective (or one-to-one), if every element of A is mapped to a unique element in B. In mathematical terms, this can be defined as for all \(x, y \in A\), \( \varphi(x) = \varphi(y) \) implies \( x = y \)
02

Proving \(\varphi^{+}\) is injective

Given presheaves \( \overline{\mathscr{H}} \) and \( \mathscr{S} \) and morphism \( \varphi: \overline{\mathscr{H}} \rightarrow \mathscr{S} \), it is known that \( \varphi(\mathcal{U}): \mathscr{F}(U) \rightarrow \mathscr{G}(U) \) is injective. The induced map \( \varphi^{+}: \overline{\mathscr{H}}^{+} \rightarrow \mathscr{S}^{+} \) is the map which connects the sheaves associated with the presheaves. To prove that \( \varphi^{+} \) is injective, it needs to be shown that for any \(x, y \in \overline{\mathscr{H}}^{+}\), if \( \varphi^{+}(x) = \varphi^{+}(y) \), then \( x = y \). As \( \varphi^{+} \) is an extension of \( \varphi \), we can infer that if \( \varphi \) is injective, then \( \varphi^{+} \) is also injective.
03

Using Part (a) to Identify the Subsheaf

Knowing that \( \varphi^{+} \) is injective, a morphism of sheaves \( \varphi: \bar{y} \rightarrow \mathscr{G} \) can be used to naturally identify a subsheaf of \( \mathscr{G} \). Specifically, this subsheaf will be the image of \( \varphi \), denoted as im \( \varphi \). This can be interpreted as the set of elements in \( \mathscr{G} \) that have a pre-image in \( \bar{y} \) under the morphism \( \varphi \).

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Key Concepts

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

Presheaves and Sheaves
In algebraic geometry, presheaves and sheaves are fundamental concepts used to manage data that "varies" across different spaces. A **presheaf** on a topological space assigns a set (or group, or ring) to each open set in a way that satisfies certain conditions. While presheaves can loosely connect data over open sets, they do not guarantee consistency over overlapping regions.
To refine this, we turn to **sheaves**. Sheaves add an extra layer of data consistency, ensuring that if some consistent data is given on overlapping open sets, it can be uniquely 'glued together' to form data on their union. This "gluing condition" is what differentiates a sheaf from a presheaf.
An example might clarify this: Imagine assigning functions to every open set of a space. A presheaf may just assign these functions without considering how they fit together. A sheaf ensures that functions assigned to overlapping open regions stitch together perfectly.
  • Presheaf: An assignment over open sets without strict consistency requirements.
  • Sheaf: Includes the additional gluing property, promoting consistency.
Exploring morphisms between these allows us to formally transfer such assignments from one sheaf to another, preserving this structure.
Morphism of Sheaves
A morphism between sheaves is a function that respects the structure these mathematical objects impose. Think of **morphisms** as the rules that guide how you map one sheaf (or presheaf) onto another while preserving their intrinsic properties. When we discuss morphisms of sheaves, we consider functions that allow compatible transformations between the data structures defined by sheaves so that the underlying relationships and dependencies are maintained.
In mathematical terms, if you have a morphism \(\varphi: \mathscr{F} \rightarrow \mathscr{G}\), this indicates a way to translate elements from sheaf \(\mathscr{F}\) to \(\mathscr{G}\) while respecting the rules that define sheaves. This becomes crucial when examining properties like injectivity for such mappings.
If an injective morphism exists, for any two sections (elements) in the source sheaf \(\mathscr{F}\), their images in \(\mathscr{G}\) remain distinct unless they were equal to begin with. This ensures that our transformation of data is unambiguous.
  • A morphism maintains relationships and dependencies between data structures.
  • Injective morphisms create distinct, consistent transformations.
Understanding these transforms aids in defining the structure and behavior of more complex geometric entities.
Injective Maps in Mathematics
Injective maps or injections are integral to understanding specific characteristics within algebraic structures. An **injective map** ensures that distinct elements in the domain map to distinct elements in the codomain.
Expressed mathematically, a map \(\varphi: A \rightarrow B\) is injective if, whenever \(\varphi(x) = \varphi(y)\), it follows that \(x = y\). In simpler terms: different inputs always lead to different outputs, making it a one-to-one function. This property is crucial when extending maps from presheaves to sheaves because it ensures that unique data remains unique when transferred.
In the context of sheaf theory:
  • An injective morphism of sheaves maintains clarity and detail when passing information between sheaves.
  • It helps in identifying subsheaves within a larger sheaf without losing data uniqueness.
Thus, injectivity is a linchpin in algebraic geometry, enabling the seamless extension of truths from simple structures to more complex sheaves by ensuring consistency and uniqueness of mappings.

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Most popular questions from this chapter

Properties of Morphisms of Finite Type. (a) A closed immersion is a morphism of finite type. (b) A quasi-compact open immersion (Ex. 3.2) is of finite type. (c) A composition of two morphisms of finite type is of finite type. (d) Morphisms of finite type are stable under base extension. (e) If \(X\) and \(Y\) are schemes of finite type over \(S,\) then \(X \times_{S} Y\) is of finite type over \(S.\) (f) If \(X \stackrel{f}{\rightarrow} Y \stackrel{g}{\rightarrow} Z\) are two morphisms, and if \(f\) is quasi-compact, and \(g^{\circ} f\) is of finite type, then \(f\) is of finite type. (g) If \(f: X \rightarrow Y\) is a morphism of finite type, and if \(Y\) is noetherian, then \(X\) is noetherian.

Support. Let \(\mathscr{F}\) be a sheaf on \(X\), and let \(s \in \mathscr{F}(U)\) be a section over an open set \(U\) The support of \(s\), denoted Supp s, is defined to be \(\left\\{P \in U | s_{P} \neq 0\right\\},\) where \(s_{P}\) denotes the germ of s in the stalk \(\overline{\mathscr{F}}_{p}\). Show that Supp s is a closed subset of \(U\). We define the support of \(\overline{\mathscr{F}}, \operatorname{Supp}, \overline{\mathscr{F}},\) to be \(\left\\{P \in X | \mathscr{F}_{P} \neq 0\right\\},\) It need not be a closed subset.

Let \(X\) be a variety of dimension \(n\) over \(k .\) Let \(\mathscr{E}\) be a locally free sheaf of \(\operatorname{rank}>n\) on \(X,\) and let \(V \subseteq \Gamma(X, \mathscr{E})\) be a vector space of global sections which generate \(\mathscr{E} .\) Then show that there is an element \(s \in V\), such that for each \(x \in X,\) we have \(s_{x} \notin \mathrm{m}_{x} \mathscr{E}_{x} .\) Conclude that there is a morphism \(\mathscr{O}_{x} \rightarrow \mathscr{E}\) giving rise to an exact sequence \\[ 0 \rightarrow \mathscr{O}_{X} \rightarrow \mathscr{E} \rightarrow \mathscr{E}^{\prime} \rightarrow 0 \\] where \(\mathscr{E}^{\prime}\) is also locally free. \([\text {Hint}:\) Use a method similar to the proof of Bertini's theorem \((8.18) .]\)

Examples of Valuation Rings. Let \(k\) be an algebraically closed field. (a) If \(K\) is a function field of dimension 1 over \(k(I, \$ 6),\) then every valuation ring of \(K / k\) (except for \(K\) itself) is discrete. Thus the set of all of them is just the abstract nonsingular curve \(C_{K}\) of \((\mathrm{I}, \$ 6)\) (b) If \(K / k\) is a function field of dimension two, there are several different kinds of valuations. Suppose that \(X\) is a complete nonsingular surface with function field \(K\) (1) If \(Y\) is an irreducible curve on \(X\), with generic point \(x_{1},\) then the local ring \(R=C_{x_{1}, x}\) is a discrete valuation ring of \(K k\) with center at the (nonclosed) point \(x_{1}\) on \(X\) (2) If \(f: X^{\prime} \rightarrow X\) is a birational morphism, and if \(Y^{\prime}\) is an irreducible curve in \(X^{\prime}\) whose image in \(X\) is a single closed point \(x_{0},\) then the local ring \(R\) of the generic point of \(Y^{\prime}\) on \(X^{\prime}\) is a discrete valuation ring of \(K k\) with center at the closed point \(x_{0}\) on \(X\) (3) Let \(r_{0} \in X\) be a closed point. Let \(f: X_{1} \rightarrow X\) be the blowing-up of \(x_{0}\) (I. \(\$ 4)\) and let \(E_{1}=f^{-1}\left(r_{0}\right)\) be the exceptional curve. Choose a closed point \(x_{1} \in E_{1},\) let \(f_{2}: X_{2} \rightarrow X_{1}\) be the blowing-up of \(x_{1},\) and let \(E_{2}=\) \(f_{2}^{-1}\left(x_{1}\right)\) be the exceptional curve. Repeat. In this manner we obtain a sequence of varieties \(X\), with closed points \(x_{i}\) chosen on them, and for each \(i,\) the local ring \(C_{1,1,1}, x_{1},\) dominates \(C_{x_{1}, x_{1}},\) Let \(R_{0}=\bigcup_{1=0}^{x} C_{x_{1}, x_{1}}\) Then \(R_{0}\) is a local ring, so it is dominated by some valuation ring \(R\) of \(K / k\) by \((\mathrm{I}, 6.1 \mathrm{A}) .\) Show that \(R\) is a valuation ring of \(K / k\). and that it has center \(x_{0}\) on \(X .\) When is \(R\) a discrete valuation ring? Note. We will see later (V.Ex. 5.6) that in fact the \(R_{0}\) of (3) is already a valuation ring itself, so \(R_{0}=R\). Furthermore, every valuation ring of \(K, k\) (except for \(K\) itself) is one of the three kinds just described.

Chow's Lemma. This result says that proper morphisms are fairly close to projective morphisms. Let \(X\) be proper over a noetherian scheme \(S\). Then there is a scheme \(X^{\prime}\) and a morphism \(g: X^{\prime} \rightarrow X\) such that \(X^{\prime}\) is projective over \(S,\) and there is an open dense subset \(L \subseteq X\) such that \(y\) induces an isomorphism of \(g^{-1}(U)\) to \(U\). Prove this result in the following steps. (a) Reduce to the case \(X\) irreducible. (b) Show that \(X\) can be covered by a finite number of open subsets \(U_{i}, i=1, \ldots, n\) each of which is quasi-projective over \(S\). Let \(U_{i} \rightarrow P_{i}\) be an open immersion of \(L_{i}\) into a scheme \(P_{i}\) which is projective over \(S\). (c) Let \(L=\bigcap U_{1},\) and consider the map \\[ f: U \rightarrow X \times_{s} P_{1} \times_{s} \cdots \times_{s} P_{n} \\] deduced from the given maps \(L \rightarrow X\) and \(U \rightarrow P_{1} .\) Let \(X^{\prime}\) be the closed image subscheme structure (Ex.3.11d) \(f(U)\). Let \(g: X^{\prime} \rightarrow X\) be the projection onto the first factor, and let \(h: X^{\prime} \rightarrow P=P_{1} \times_{s} \ldots \times_{s} P_{n}\) be the projection onto the product of the remaining factors. Show that \(h\) is a closed immersion. hence \(X\) ' is projective over \(S\). (d) Show that \(g^{-1}(L) \rightarrow l\) is an isomorphism. thus completing the proof.
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