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2021_Algebra_I.tex
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@ -2285,16 +2285,16 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic
\begin{definition}[Sheaf]
Let $X$ be any topological space.
A \vocab{presheaf} $\cG$ of sets (or rings, (abelian) groups) on $X$ associates a set (or rings, or (abelian) group) $\cG(U)$ to every open subset $U$ of $X$, and a map (or ring or group homomorphism) $\cG(U) \xrightarrow{r_{U,V}} \cG(V)$ to every inclusion $V \subseteq U$ of open subsets of $X$ such that $r_{U,W} = r_{V,W} r_{U,V}$ for inclusions $U \subseteq V \subseteq W$ of open subsets.
A \vocab{presheaf} $\mathcal{G}$ of sets (or rings, (abelian) groups) on $X$ associates a set (or rings, or (abelian) group) $\mathcal{G}(U)$ to every open subset $U$ of $X$, and a map (or ring or group homomorphism) $\mathcal{G}(U) \xrightarrow{r_{U,V}} \mathcal{G}(V)$ to every inclusion $V \subseteq U$ of open subsets of $X$ such that $r_{U,W} = r_{V,W} r_{U,V}$ for inclusions $U \subseteq V \subseteq W$ of open subsets.
Elements of $\cG(U)$ are often called \vocab{sections} of $\cG$ on $U$ or \vocab{global sections} when $U = X$.
Elements of $\mathcal{G}(U)$ are often called \vocab{sections} of $\mathcal{G}$ on $U$ or \vocab{global sections} when $U = X$.
Let $U \subseteq X$ be open and $U = \bigcup_{i \in I} U_i$ an open covering.
A family $(f_i)_{i \in I} \in \prod_{i \in I} \cG(U_i)$ is called \vocab[Sections!compatible]{compatible} if $r_{U_i, U_i \cap U_j}(f_i) = r_{U_j, U_i \cap U_j}(f_j)$ for all $i,j \in I$.
A family $(f_i)_{i \in I} \in \prod_{i \in I} \mathcal{G}(U_i)$ is called \vocab[Sections!compatible]{compatible} if $r_{U_i, U_i \cap U_j}(f_i) = r_{U_j, U_i \cap U_j}(f_j)$ for all $i,j \in I$.
Consider the map
\begin{align}
\phi_{U, (U_i)_{i \in I}}: \cG(U) &\longrightarrow \{(f_i)_{i \in I} \in \prod_{i \in I} \cG(U_i) | r_{U_i, U_i \cap U_j}(f_i) = r_{U_j, U_i \cap U_j}(f_j) \text{ for } i,j \in I \} \\
\phi_{U, (U_i)_{i \in I}}: \mathcal{G}(U) &\longrightarrow \{(f_i)_{i \in I} \in \prod_{i \in I} \mathcal{G}(U_i) | r_{U_i, U_i \cap U_j}(f_i) = r_{U_j, U_i \cap U_j}(f_j) \text{ for } i,j \in I \} \\
f &\longmapsto (r_{U, U_i}( f))_{i \in I}
\end{align}
@ -2306,13 +2306,13 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic
The bijectivity of the $\phi_{U, (U_i)_{i \in I}}$ is called the \vocab{sheaf axiom}.
\end{definition}
\begin{dtrivial}
A presheaf is a contravariant functor $\cG : \mathcal{O}(X) \to C$ where $\mathcal{O}(X)$ denotes the category of open subsets of $X$ with inclusions as morphisms and $C$ is the category of sets, rings or (abelian) groups.
A presheaf is a contravariant functor $\mathcal{G} : \mathcal{O}(X) \to C$ where $\mathcal{O}(X)$ denotes the category of open subsets of $X$ with inclusions as morphisms and $C$ is the category of sets, rings or (abelian) groups.
\end{dtrivial}
\begin{definition}
A subsheaf $\cG'$ is defined by subsets (resp. subrings or subgroups) $\cG'(U) \subseteq \cG(U)$ for all open $U \subseteq X$ such that the sheaf axioms still hold.
A subsheaf $\mathcal{G}'$ is defined by subsets (resp. subrings or subgroups) $\mathcal{G}'(U) \subseteq \mathcal{G}(U)$ for all open $U \subseteq X$ such that the sheaf axioms still hold.
\end{definition}
\begin{remark}
If $\cG$ is a sheaf on $X$ and $\Omega \subseteq X$ open, then $\cG\defon{\Omega}(U) \coloneqq \cG(U)$ for open $U \subseteq \Omega$ and $r_{U,V}^{(\cG\defon{\Omega})}(f) \coloneqq r_{U,V}^{(\cG)}(f)$ is a sheaf of the same kind as $\cG$ on $\Omega$.
If $\mathcal{G}$ is a sheaf on $X$ and $\Omega \subseteq X$ open, then $\mathcal{G}\defon{\Omega}(U) \coloneqq \mathcal{G}(U)$ for open $U \subseteq \Omega$ and $r_{U,V}^{(\mathcal{G}\defon{\Omega})}(f) \coloneqq r_{U,V}^{(\mathcal{G})}(f)$ is a sheaf of the same kind as $\mathcal{G}$ on $\Omega$.
\end{remark}
\begin{remark}
The notion of restriction of a sheaf to a closed subset, or of preimages under general continuous maps, can be defined but this is a bit harder.
@ -2321,7 +2321,7 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic
It is often convenient to write $f \defon{V}$ instead of $r_{U,V}(f)$.
\end{notation}
\begin{remark}
Applying the \vocab{sheaf axiom} to the empty covering of $U = \emptyset$, one finds that $\cG(\emptyset) = \{0\} $.
Applying the \vocab{sheaf axiom} to the empty covering of $U = \emptyset$, one finds that $\mathcal{G}(\emptyset) = \{0\} $.
\end{remark}
@ -2335,7 +2335,7 @@ If $P_e $ with $e < d$ was $\neq 0$, it could not be a multiple of $P$ contradic
Similarly, a ring structure on $G$ can be used to define the structure of a sheaf of rings on $\fG$.
\end{example}
\begin{example}
If in the previous example $G$ carries a topology and $\cG(U) \subseteq \fG(U)$ is the subset (subring, subgroup) of continuous functions $U \xrightarrow{f} G$, then $\cG$ is a subsheaf of $\fG$, called the sheaf of continuous $G$-valued functions on (open subsets of) $X$.
If in the previous example $G$ carries a topology and $\mathcal{G}(U) \subseteq \fG(U)$ is the subset (subring, subgroup) of continuous functions $U \xrightarrow{f} G$, then $\mathcal{G}$ is a subsheaf of $\fG$, called the sheaf of continuous $G$-valued functions on (open subsets of) $X$.
\end{example}
\begin{example}
@ -2701,44 +2701,44 @@ If $X$ is a set, then $\cB \subseteq \mathcal{P}(X)$ is a base for some topology
\subsection{Stalks of sheaves}
\begin{definition}[Stalk]
Let $\cG$ be a presheaf of sets on the topological space $X$, and let $x \in X$.
The \vocab{stalk} (\vocab[Stalk]{Halm}) of $\cG$ at $x$ is the set of equivalence classes of pairs $(U, \gamma)$, where $U$ is an open neighbourhood of $x$ and $\gamma \in \cG(U)$
Let $\mathcal{G}$ be a presheaf of sets on the topological space $X$, and let $x \in X$.
The \vocab{stalk} (\vocab[Stalk]{Halm}) of $\mathcal{G}$ at $x$ is the set of equivalence classes of pairs $(U, \gamma)$, where $U$ is an open neighbourhood of $x$ and $\gamma \in \mathcal{G}(U)$
and the equivalence relation $\sim $ is defined as follows:
$( U , \gamma) \sim (V, \delta)$ iff there exists an open neighbourhood $W \subseteq U \cap V$ of $x$ such that $\gamma \defon{W} = \delta \defon{W}$.
If $\cG$ is a presheaf of groups, one can define a groups structure on $\cG_x$ by
If $\mathcal{G}$ is a presheaf of groups, one can define a groups structure on $\mathcal{G}_x$ by
\[
((U, \gamma) / \sim ) \cdot \left( (V,\delta) / \sim \right) = (U \cap V, \gamma \defon{U \cap V} \cdot \delta\defon{U \cap V}) / \sim
\]
If $\cG$ is a presheaf of rings, one can similarly define a ring structure on $\cG_x$.
If $\mathcal{G}$ is a presheaf of rings, one can similarly define a ring structure on $\mathcal{G}_x$.
If $U$ is an open neighbourhood of $x \in X$, then we have a map (resp. homomorphism)
\begin{align}
\cdot_x : \cG(U) &\longrightarrow \cG_x \\
\cdot_x : \mathcal{G}(U) &\longrightarrow \mathcal{G}_x \\
\gamma &\longmapsto \gamma_x \coloneqq (U, \gamma) / \sim
\end{align}
\end{definition}
\begin{fact}
Let $\gamma,\delta \in \cG(U)$. If $\cG$ is a sheaf\footnote{or, more generally, a separated presheaf} and if for all $x \in U$, we have $\gamma_x = \delta_x$, then $\gamma = \delta$.
Let $\gamma,\delta \in \mathcal{G}(U)$. If $\mathcal{G}$ is a sheaf\footnote{or, more generally, a separated presheaf} and if for all $x \in U$, we have $\gamma_x = \delta_x$, then $\gamma = \delta$.
In the case of a sheaf, the image of the injective map $\cG(U) \xrightarrow{\gamma \mapsto (\gamma_x)_{x \in U}} \prod_{x \in U} \cG_x$
is the set of all $(g_x)_{x \in U} \in \prod_{x \in U} \cG_x $ satisfying the following \vocab{coherence condition}:
For every $x \in U$, there are an open neighbourhood $W_x \subseteq U$ of $x$ and $g^{(x)} \in \cG(W_x)$ with $g_y^{(x)} = g_y$ for all $y \in W_x$.
In the case of a sheaf, the image of the injective map $\mathcal{G}(U) \xrightarrow{\gamma \mapsto (\gamma_x)_{x \in U}} \prod_{x \in U} \mathcal{G}_x$
is the set of all $(g_x)_{x \in U} \in \prod_{x \in U} \mathcal{G}_x $ satisfying the following \vocab{coherence condition}:
For every $x \in U$, there are an open neighbourhood $W_x \subseteq U$ of $x$ and $g^{(x)} \in \mathcal{G}(W_x)$ with $g_y^{(x)} = g_y$ for all $y \in W_x$.
\end{fact}
\begin{proof}
Because of $\gamma_x = \delta_x$, there is $x \in W_x \subseteq U$ open such that $\gamma\defon{W_x} = \delta\defon{W_x}$. As the $W_x$ cover $U$, $\gamma = \delta$ by the sheaf axiom.
\end{proof}
\begin{definition}
Let $\cG$ be a sheaf of functions.
Let $\mathcal{G}$ be a sheaf of functions.
Then $\gamma_x$ is called the \vocab{germ} of the function $\gamma$ at $x$.
The \vocab[Germ!value at $x$]{value at $x$ } of $g = (U, \gamma) / \sim \in \cG_x$ defined as $g(x) \coloneqq \gamma(x)$, which is independent of the choice of the representative $\gamma$.
The \vocab[Germ!value at $x$]{value at $x$ } of $g = (U, \gamma) / \sim \in \mathcal{G}_x$ defined as $g(x) \coloneqq \gamma(x)$, which is independent of the choice of the representative $\gamma$.
\end{definition}
\begin{remark}
If $\cG$ is a sheaf of $C^{\infty}$-functions (resp. holomorphic functions), then $\cG_x$ is called the ring of germs of $C^\infty$-functions (resp. of holomorphic functions) at $x$.
If $\mathcal{G}$ is a sheaf of $C^{\infty}$-functions (resp. holomorphic functions), then $\mathcal{G}_x$ is called the ring of germs of $C^\infty$-functions (resp. of holomorphic functions) at $x$.
\end{remark}
\subsubsection{The local ring of an affine variety}