From dcd73a79c1ae8c4d78bc8c895bcdb625ae423414 Mon Sep 17 00:00:00 2001 From: Interitio Date: Fri, 29 Aug 2025 23:30:00 +1000 Subject: [PATCH] Up to Ex 2.3.D. --- chapter2.tex | 234 ++++++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 224 insertions(+), 10 deletions(-) diff --git a/chapter2.tex b/chapter2.tex index 760a014..85ade83 100644 --- a/chapter2.tex +++ b/chapter2.tex @@ -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}