holo/TheRisingSea-Solutions
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Up to Ex 2.3.D.

Browse Source
This commit is contained in:
Interitio
2025-08-29 23:30:00 +10:00
parent 62ec5dafe4
commit dcd73a79c1
1 changed files with 224 additions and 10 deletions
Download Patch File Download Diff File Expand all files Collapse all files

234
chapter2.tex
View File

@@ -55,28 +55,31 @@
and let $\eta \colon A \implies B$ be a natural transformation between them.
Then $\eta$ induces a map on the colimits, if they exist.
\[
\colim_{i \in I} B(i) \longrightarrow \colim_{i \in I} A(i)
\colim_{i \in I} A(i) \longrightarrow \colim_{i \in I} B(i)
\]
\end{Lemma}
\begin{proof}
Let $Z$ be any co-cone over the diagram $B$,
written as a natural transformation as in the previous lemma.
written as a natural transformation $\varepsilon$ as in the previous lemma.
\[
\begin{tikzcd}[row sep=large]
& * \ar[rd, "Z"] \\
I \ar[ru]
\ar[rr, bend left, "A"{anchor=center, fill=white, name=A}]
\ar[rr, bend right, "B"'{name=B}]
\ar[from=B.north-|A, shorten=4pt, to=A, Rightarrow, "\eta"]
\ar[from=A, ur, Rightarrow, shorten=2pt, "\varepsilon"]
\ar[rr, bend left, "B"{anchor=center, fill=white, name=B}]
\ar[rr, bend right, "A"'{name=A}]
\ar[from=A.north-|B, shorten=4pt, to=B, Rightarrow, "\eta"]
\ar[from=B, ur, Rightarrow, shorten=2pt, "\varepsilon"]
& & C\\
\end{tikzcd}
\]
The composition $\varepsilon \circ \eta$ then describes $Z$
as a co-cone over the diagram $B$.
If we choose $Z$ to be the colimit over $A$,
and assume the colimit of $B$ also exists,
then we get the map desired by the lemma.
as a co-cone over the diagram $A$ as well.
If the diagram $A$ has a colimit, then it must admit a map to
this cocone $Z$ by universal property.
Thus, if $Z$ is assumed to be the colimit of $B$,
we have induced a map from the colimit of $A$ to the colimit of $B$
as desired.
\end{proof}
\begin{qanda}
@@ -647,7 +650,218 @@
Thus, $\cHom(\cF, \cG)$ is a presheaf satisfying both identity and gluing, and is thus a sheaf,
as desired.
% Exercise 2.3.D
\question
\begin{enumerate}
\item Let $\cF$ be a sheaf of sets on $X$.
Show that $\cHom(\underline{\{p\}}, \cF) \isom \cF$,
where $\underline{\{p\}}$ is the constant sheaf with values in $\{p\}$.
\item Let $\cF$ be a sheaf of abelian groups on $X$.
Show that $\cHom_{Ab_X}(\underline{\bZ}, \cF) \isom \cF$
in the category of sheaves of abelian groups.
\item Let $\cF$ be an $\cO_X$-module.
Show that $\cHom_{Mod_{\cO_X}}(\cO_X, \cF) \isom \cF$
in the category of $\cO_X$-modules.
\end{enumerate}
\answer
\begin{enumerate}
\item We first consider the sheaf $\constset{p}$.
Take any $U$ open in $X$, then the sections $\constset{p}(U)$
are the functions $f\colon U \to \{p\}$ such that $f^{-1}(p)$ is open.
That is, $\constset{p}(U)$ is the single-element set $*$.
The restrictions are then the trivial map $* \to *$.
Now, consider $\cHom(\constset{p}, \cF)$.
The sections of this sheaf hom over $U$ are the natural transformations
$\constset{p}|_U \to \cF|_U$.
However, such a natural transformation is precisely the choice of
an element of $\cF(U)$.
Note that although the natural transformation
describes a map $* \to \cF(V)$ for every open set $V \subseteq U$,
the natural transformation diagram ensures that
the chosen section over $V$ is the restriction of the
section over $U$, so the image of the $U$ component
does fully determine the transformation.
\[
\begin{tikzcd}
* \ar[r] \ar[d] & \cF(U) \ar[d] \\
* \ar[r] & \cF(V)
\end{tikzcd}
\]
This then defines a natural transformation $\cHom(\constset{p}, \cF) \Rightarrow \cF$,
which is isomorphic on components and is thus a natural isomorphism
as desired.
\item
Next we consider $\ul{\bZ}$, the constant sheaf on $\bZ$.
For $U$ open in $X$, the sections over $U$ are functions
$\func{f}{U}{\bZ}$ such that $f^{-1}(n)$ is open in $U$ for each $n \in \bZ$.
The group structure on $\ul{\bZ}(U)$ is given by pointwise addition,
that is, $(f+g)(x) = f(x) + g(x) \in \bZ$.
This is well-defined since for any $n \in \bZ$,
\[
(f+g)^{-1}(n) = \bigcup_{a+b = n} f^{-1}(a) \cap g^{-1}(b).
\]
Unlike in the previous part, $\ul{\bZ}(U)$ now has more than a single element.
However, we shall show that each $\eta \in \Hom(\ul{\bZ}|_U, \cF|_U)$
is still fully determined by where a single function in $\ul{\bZ}(U)$ is sent.
For $n \in \bZ$, let $\ul{n}^V\colon V \to \bZ$ denote the function in $\ul{\bZ}(V)$
sending every point to $n$. Let $\sigma = \eta_U(\ul{1}^U)$.
Now, take any $f \in \ul{\bZ}(U)$.
Then for each $n$, by definition, $U_n \ceq f^{-1}(n)$ is open in $U$.
Restricting, we have the following two diagrams.
\[
\begin{tikzcd}
f \ar[r, mapsto] \ar[d, mapsto] & \eta_U(f) \ar[d, mapsto] \\
\ul{n}^{U_n} \ar[r, mapsto] & \eta_U(f)|_{U_n}
\end{tikzcd}
\qquad
\begin{tikzcd}
\ul{1}^U \ar[r, mapsto] \ar[d, mapsto] & \sigma \ar[d, mapsto] \\
\ul{1}^{U_n} \ar[r, mapsto] & \sigma|_{U_n}
\end{tikzcd}
\]
Since $\eta_{U_n}$ is a group homomorphism, we have
\[
\eta_U(f)|_{U_n} = \eta_{U_n}(\ul{n}^{U_n}) = n \cdot \eta_{U_n}(\ul{1}^{U_n})
= n \cdot \sigma|_{U_n}
\]
Thus the restrictions of $\eta_U(f)$ to the cover $U_n$ of $U$
are fully determined by the values of $\sigma|_{U_n}$.
Since $\cF$ is a sheaf, we then have that $\eta_U(f)$ itself is fully determined by
$\sigma$, and thus we have a correspondence between $\cHom(\ul{\bZ}|_U, \cF|_U)$
and $\cF(U)$ as desired.
\end{enumerate}
% Question 2.3.E
\question
If $\phi\colon \cF \to \cG$ is a morphism of presheaves,
the presheaf kernel $\ker_{pre}(\phi)$ is defined by
$(\ker_{pre}\phi)(U) \ceq \ker \phi(U)$
Show that the presheaf kernel defined in this way is a presheaf.
% Question 2.3.F
\question
Show that the presheaf cokernel satisfies the universal property of cokernel.
% Question 2.3.G
\question
Show that $\cF \mapsto \cF(U)$
is an exact functor $Ab^{pre}_X \to Ab$.
% Question 2.3.H
\question
Show that a sequence of presheaves
\[
0 \to \cF_1 \to \cF_2 \to \dots \cdot \cF_n \to 0.
\]
is exact if and only if
\[
0 \to \cF_1(U) \to \cF_2(U) \to \dots \to \cF_n(U) \to 0.
\]
is exact for all $U$.
% Question 2.3.I
\question
Assume $\phi\colon \cF \to \cG$ is a morphism of sheaves.
Show $\ker_{pre}\phi$ is a sheaf,
and satisfies the universal property of kernels in the category of sheaves.
% Question 2.3.J
\question
Take $X$ to be $\cC$ with the standard topology,
and let $\cO_X$ be the sheaf of holomorphic functions.
Let $\cF$ be the presheaf of functions on $X$ admitting a holomorphic logarithm.
Show that the follow sequence of preshaves is exact
\[
0 \to \ul{\bZ} \to \cO_X \to \cF \to 0.
\]
Here $\ul{\bZ} \to \cO_X$ is the natural inclusion, and $\cO_X \to \cF$
is given by $f \mapsto \exp(2\pi i f)$.
Show that even though $\cF$ is a presheaf cokernel of a morphism of sheaves,
$\cF$ is not a sheaf itself.
% Question 2.4.A
\question
Prove that a section of a sheaf of sets is determined by its germs, i.e.,
the natural map
\[
\cF(U) \to \prod_{p\in U} \cF_p
\]
is injective.
% Question 2.4.B
\question
Call a family of germs over an open set $U$ \emph{compatible}
if they are locally the germs of some cover of $U$.
Formally, say that $(s_p)_{p \in U}$ is a compatible family of germs
if for each $p \in U$,
there is some neighbourhood $U_p \ni p$
where for some section $t \in \cF(U_p)$ over that neighbourhood,
$t_q = s_q$ for all $q \in U_p$.
% Question 2.4.C
\question
If $\phi_1$ and $\phi_2$ are morphisms from a presheaf of sets $\cF$
to a sheaf of sets $\cG$ that induce the same maps on each stalk,
show that $\phi_1 = \phi_2$.
% Question 2.4.D
\question
Show that a morphism of sheaves of sets is an isomorphism
if and only if it induces an isomorphism of all stalks.
% Question 2.4.E
\question
\begin{enumerate}
\item Show that $\cF(U) \to \prod_{p \in U} \cF_p$
need not be injective if $\cF$ is not a sheaf.
\item Show that morphisms are not determined by stalks
for general presheaves.
\item Show that isomorphisms are not determined by stalks
for general presheaves.
\end{enumerate}
% Question 2.4.F
\question
Show that sheafification is unique up to unique isomorphism,
assuming it exists. Show that $\cF$ is a sheaf,
then the sheafification is $id\colon \cF \to \cF$.
% Question 2.4.G
\question
Show that sheafification is a functor from presheaves on $X$ to sheaves on $X$.
% Question 2.4.H
\question
Show that $\cF^{sh}$ forms a sheaf.
% Question 2.4.I
\question
Describe a natural map of presheaves $sh\colon \cF \to \cF^{sh}$
% Question 2.4.K
\question
Show that the sheafification functor is left-adjoint
to the forgetful functor from sheaves on $X$ to presheaves on $X$.
% Question 2.4.L
\question
Show $\cF \to \cF^{sh}$ induces an isomorphism of stalks.
% Question 2.4.M
\question
Suppose $\phi \colon \cF \to \cG$ is a morphism of sheaves of sets on $X$.
Show that the following are equivalent.
\begin{enumerate}
\item $\phi$ is a monomorphism in the category of sheaves.
\item $\phi$ is injective on the level of stalks, i.e.
$\phi_p\colon \cF_p \to \cG_p$ injective
\item $\phi$ is injective on the level open sets.
\end{enumerate}
\end{qanda}