# Difference between revisions of "Math 764 -- Algebraic Geometry II -- Homeworks"

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## Homeworks

I tried to convert the homeworks into the wiki format with pandoc. This does not always work as expected; in case of doubt, check the pdf files.

### Homework 1

Due Friday, February 3rd

In all these problems, we fix a topological space $X$; all sheaves and presheaves are sheaves on $X$.

1. Example: Let $X$ be the unit circle, and let ${\mathcal{F}}$ be the sheaf of $C^\infty$-functions on $X$. Find the (sheaf) image and the kernel of the morphism $\frac{d}{dt}:{\mathcal{F}}\to{\mathcal{F}}.$ Here $t\in{\mathbb{R}}/2\pi{\mathbb{Z}}$ is the polar coordinate on the circle.
2. Sheaf operations: Let ${\mathcal{F}}$ and ${\mathcal{G}}$ be sheaves of sets. Recall that a morphism $\phi:{\mathcal{F}}\to {\mathcal{G}}$ is a (categorical) monomorphism if and only if for any sheaf ${\mathcal{F}}'$ and any two morphisms $\psi_1,\psi_2:{\mathcal{F}}'\to {\mathcal{F}}$, the equality $\phi\circ\psi_1=\phi\circ\psi_2$ implies $\psi_1=\psi_2$. Show that $\phi$ is a monomorphism if and only if it induces injective maps on all stalks.
3. Let ${\mathcal{F}}$ and ${\mathcal{G}}$ be sheaves of sets. Recall that a morphism $\phi:{\mathcal{F}}\to{\mathcal{G}}$ is a (categorical) epimorphism if and only if for any sheaf ${\mathcal{G}}'$ and any two morphisms $\psi_1,\psi_2:{\mathcal{G}}\to{\mathcal{G}}'$, the equality $\psi_1\circ\phi=\psi_2\circ\phi$ implies $\psi_1=\psi_2$. Show that $\phi$ is a epimorphism if and only if it induces surjective maps on all stalks.
4. Show that any morphism of sheaves can be written as a composition of an epimorphism and a monomorphism. (You should know what order of composition I mean here.)
5. Let ${\mathcal{F}}$ be a sheaf, and let ${\mathcal{G}}\subset{\mathcal{F}}$ be a sub-presheaf of ${\mathcal{F}}$ (thus, for every open set $U\subset X$, ${\mathcal{G}}(U)$ is a subset of ${\mathcal{F}}(U)$ and the restriction maps for ${\mathcal{F}}$ and ${\mathcal{G}}$ agree). Show that the sheafification $\tilde{\mathcal{G}}$ of ${\mathcal{G}}$ is naturally identified with a subsheaf of ${\mathcal{F}}$.
6. Let ${\mathcal{F}}_i$ be a family of sheaves of abelian groups on $X$ indexed by a set $I$ (not necessarily finite). Show that the direct sum and direct product of this family exists in the category of sheaves of abelian groups. (E.g., a direct sum would be a sheaf of abelian groups ${\mathcal{F}}$ together with a universal family of homomorphisms ${\mathcal{F}}_i\to {\mathcal{F}}$.) Do these operations agree with (a) taking stalks at a point $x\in X$ (b) taking sections over an open subset $U\subset X$?
7. Locally constant sheaves:

Definition. A sheaf ${\mathcal{F}}$ is constant over an open set $U\subset X$ if there is a subset $S\subset F(U)$ such that the map ${\mathcal{F}}(U)\to{\mathcal{F}}_x:s\mapsto s_x$ (the germ of $s$ at $x$) gives a bijection between $S$ and ${\mathcal{F}}_x$ for all $x\in U$.

${\mathcal{F}}$ is locally constant (on $X$) if every point of $X$ has a neighborhood on which ${\mathcal{F}}$ is constant.

Recall that a covering space $\pi:Y\to X$ is a continuous map of topological spaces such that every $x\in X$ has a neighborhood $U\ni x$ whose preimage $\pi^{-1}(U)\subset U$ is homeomorphic to $U\times Z$ for some discrete topological space $Z$. ($Z$ may depend on $x$; also, the homeomorphism is required to respect the projection to $U$.)

Show that if $\pi:Y\to X$ is a covering space, its sheaf of sections ${\mathcal{F}}$ is locally constant. Moreover, prove that this correspondence is an equivalence between the category of covering spaces and the category of locally constant sheaves. (If $X$ is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of $X$.)

8. Sheafification: (This problem may be hard, but it is still a good idea to try it) Prove or disprove the following statement (contained in the lecture notes). Let ${\mathcal{F}}$ be a presheaf on $X$, and let $\tilde{\mathcal{F}}$ be its sheafification. Then every section $s\in\tilde{\mathcal{F}}(U)$ can be represented as (the equivalence class of) the following gluing data: an open cover $U=\bigcup U_i$ and a family of sections $s_i\in{\mathcal{F}}(U_i)$ such that $s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}$.