Alggeom proofs

Signed-off-by: zeramorphic <[email protected]>

iii/alggeom/02_sheaves.tex

@@ -44,7 +44,7 @@ \subsection{Sheaves}
     A \emph{sheaf} on \( X \) is a presheaf \( \mathcal F \) on \( X \) such that
     \begin{enumerate}
         \item if \( U \subseteq X \) is open and \( \qty{U_i} \) is an open cover of \( U \), then for \( s \in \mathcal F(U) \), if \( \res_{U_i}^U s = 0 \) for all \( i \), then \( s = 0 \); and
-        \item if \( U, \qty{U_i} \) are as in (i), given \( s_i \in \mathcal F(U_i) \) such that \( \res^{U_i}_{U_i \cap U_j} s_i = \res^{U_j}_{U_i \cap U_j} s_j \) for all \( i, j \), then there exists \( s \in \mathcal F(U) \) such that \( \res^U_{U_i} s = s_i \). 
+        \item if \( U, \qty{U_i} \) are as in (i), given \( s_i \in \mathcal F(U_i) \) such that \( \res^{U_i}_{U_i \cap U_j} s_i = \res^{U_j}_{U_i \cap U_j} s_j \) for all \( i, j \), then there exists \( s \in \mathcal F(U) \) such that \( \res^U_{U_i} s = s_i \).
     \end{enumerate}
 \end{definition}
 \begin{remark}
@@ -71,7 +71,7 @@ \subsection{Sheaves}
         Let
         \[ \mathcal O_V(U) = \qty{f \in k(V) \mid \forall p \in U,\, f \text{ regular at } p} \]
         where a function \( f \) is regular at \( p \) precisely if it can be represented as a quotient \( \frac{g}{h} \) in a neighbourhood of \( p \) on which \( h \) is nonzero.
-        This is called the \emph{structure sheaf} of \( V \); it is a sheaf since regularity is a local condition. 
+        This is called the \emph{structure sheaf} of \( V \); it is a sheaf since regularity is a local condition.
     \end{enumerate}
 \end{example}
 
@@ -162,19 +162,61 @@ \subsection{Sheafification}
 \end{proposition}
 \begin{proof}
     Let \( \mathcal F \) be a presheaf on \( X \).
-    % Define \( \mathcal F^{\mathrm{sh}}(U) \) to be the set of functions
-    % \[ f : U \to \coprod_{p \in U} \mathcal F_p \]
-    % such that blah
+    Define \( \mathcal F^{\mathrm{sh}}(U) \) to be the set of dependent functions
+    \[ f : \prod_{p \in U} \mathcal F_p \]
+    such that each \( p \in U \) has a section \( s \in \mathcal F(V_p) \) over some neighbourhood \( V_p \subseteq U \), such that for each \( q \in V_p \), we have \( (V_p, s) = f(q) \) in \( \mathcal F_q \).
+    This condition makes \( f \) look locally like some element of the original presheaf.
 
-    \[ \mathcal F^{\mathrm{sh}}(U) = \qty{f : U \to \coprod_{p \in U} \mathcal F_p \midd f(p) \in \mathcal F_p,\, \forall p \in U,\, \exists V_p \subseteq U,\, p \in V_p,\, s \in \mathcal F(V_p) \text{ such that } \forall q \in V_p,\, (V_p, s) = f(q) \in \mathcal F_q} \]
-    % locally, this function looks like some element of the original presheaf
-    One can check that this is a sheaf, and that it satisfies the required universal property.
-    % exercise.
+    The action of the restriction map \( \res^V_U \) on \( f : \prod_{p \in V} \mathcal F_p \) yields \( g : \prod_{p \in U} \mathcal F_p \) given by \( g(p) = f(p) \).
+    This is clearly functorial, so \( \mathcal F^{\mathrm{sh}} \) is a presheaf.
+    We define \( \mathrm{sh} : \mathcal F \to \mathcal F^{\mathrm{sh}} \) by
+    \[ \mathrm{sh}_U(s)(p) = (U, s) \]
+    This is a morphism of presheaves as the naturality square
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhjYWwgRihWKSJdLFsxLDAsIlxcbWF0aGNhbCBGKFUpIl0sWzEsMSwiXFxtYXRoY2FsIEZee1xcbWF0aHJte3NofX0oVSkiXSxbMCwxLCJcXG1hdGhjYWwgRl57XFxtYXRocm17c2h9fShWKSJdLFswLDEsIlxccmVzXlZfVSJdLFsxLDIsIlxcbWF0aHJte3NofV9VIl0sWzAsMywiXFxtYXRocm17c2h9X1YiLDJdLFszLDIsIlxccmVzX1VeViIsMl1d
+\[\begin{tikzcd}
+	{\mathcal F(V)} & {\mathcal F(U)} \\
+	{\mathcal F^{\mathrm{sh}}(V)} & {\mathcal F^{\mathrm{sh}}(U)}
+	\arrow["{\res^V_U}", from=1-1, to=1-2]
+	\arrow["{\mathrm{sh}_U}", from=1-2, to=2-2]
+	\arrow["{\mathrm{sh}_V}"', from=1-1, to=2-1]
+	\arrow["{\res_U^V}"', from=2-1, to=2-2]
+\end{tikzcd}\]
+    commutes:
+    \[ \mathrm{sh}_U(\res^V_U(s))(p) = (U, \res^V_U(s)) \underset{\text{in } \mathcal F_p}{=} (V, s) = (\mathrm{sh}_V(s))(p) = (\res^V_U(\mathrm{sh}_V(s)))(p) \]
+
+    Suppose \( U \subseteq X \) is open, \( \qty{U_i}_{i \in I} \) is an open cover of \( U \), and \( f : \prod_{p \in U} \mathcal F_p \) is such that \( \res^U_{U_i} f = 0 \) for each \( i \in I \).
+    Then for \( p \in U_i \), \( f(p) = f_i(p) = 0 \), so \( f = 0 \) as required.
+
+    Now suppose \( U, \qty{U_i} \) are as above, and \( f_i : \prod_{p \in U_i} \mathcal F_p \) are in \( \mathcal F^{\mathrm{sh}}(U_i) \), and \( f_i, f_j \) agree on \( U_i \cap U_j \).
+    Define \( f(p) = f_i(p) \) for any \( U_i \) containing \( p \); this is well-defined as \( f_i(p) = f_j(p) \) for \( p \in U_i \cap U_j \).
+    For each \( p \in U \), we can choose a section \( s \in \mathcal F(V_p) \) with \( p \in V_p \subseteq U_i \), such that for each \( q \in V_p \) we have \( (V_p, s) = f(q) \) in \( \mathcal F_q \), so \( f \in \mathcal F^{\mathrm{sh}}(U) \).
+    Thus \( \mathcal F^{\mathrm{sh}} \) is a sheaf.
+
+    We show \( \mathcal F^{\mathrm{sh}} \) satisfies the required universal property.
+    Let \( \varphi : \mathcal F \to \mathcal G \) be a morphism of presheaves, where \( \mathcal G \) is a sheaf.
+    We now define the action of the map \( \psi : \mathcal F^{\mathrm{sh}} \to \mathcal G \) on \( f \in \mathcal F^{\mathrm{sh}}(U) \).
+    For each point \( p \in U \), let \( s_p \in \mathcal F(V_p) \) be a section over some neighbourhood \( V_p \subseteq U \) such that for all \( q \in V_p \), we have \( (V_p, s_p) = f(q) \) in \( \mathcal F_q \).
+    Then consider the collection \( \varphi(s_p) \in \mathcal G(V_p) \).
+    As the \( s_p \) are locally compatible, the \( \varphi(s_p) \) can be glued together to form \( \psi(f) \in \mathcal G(U) \).
+
+    If \( U \subseteq V \) and \( f \in \mathcal F^{\mathrm{sh}}(V) \), \( \res_U^V \psi(f) \) coincides locally with \( \psi(\res_U^V f) \) by construction, so they agree by the first sheaf axiom.
+    Thus \( \psi : \mathcal F^{\mathrm{sh}} \to \mathcal G \) is a morphism of (pre)sheaves.
+
+    We now show \( \psi \circ \mathrm{sh} = \varphi \).
+    Let \( U \subseteq X \) be open, and consider \( s \in \mathcal F(U) \).
+    Then \( \psi_U(\mathrm{sh}_U(s)) = \psi_U((U, s)_{p \in U}) \).
+    Gluing together the equal germs \( (U, \varphi_U(s)) \) gives \( \psi_U((U, s)_{p \in U}) = \varphi_U(s) \), as required.
+    For uniqueness, note that \( \mathrm{sh} : \mathcal F \to \mathcal F^{\mathrm{sh}} \) is injective.
 \end{proof}
 \begin{corollary}
     The stalks of \( \mathcal F \) and \( \mathcal F^{\mathrm{sh}} \) coincide.
 \end{corollary}
-% TODO prove this
+\begin{proof}
+    Suppose \( (U, f) \) is a germ of \( \mathcal F^{\mathrm{sh}} \) at \( p \in X \).
+    Then \( f(p) \in \mathcal F_p \) is a germ of \( \mathcal F \) at \( p \).
+    If \( (U, s) \in \mathcal F_p \), we can produce the germ \( (U, (U, s)_{p \in U}) \) of \( \mathcal F^{\mathrm{sh}} \) at \( p \in X \).
+    These are inverse operations, and hence give a bijection of stalks.
+\end{proof}
 % exercise: find a nonzero presheaf F with F^sh = 0. (ES1 Q10)
 
 \subsection{Kernels and cokernels}
@@ -189,7 +231,17 @@ \subsection{Kernels and cokernels}
 \begin{proposition}
     The presheaf kernel for a morphism of sheaves is a sheaf.
 \end{proposition}
-% exercise
+\begin{proof}
+    Let \( \varphi : \mathcal F \to \mathcal G \) be a morphism of sheaves, let \( U \subseteq X \) be open, and let \( \qty{U_i}_{i \in I} \) be an open cover of \( U \).
+    Let \( f \in (\ker \varphi)(U) \) be such that \( \res^U_{U_i} f = 0 \) for each \( f \).
+    Then as \( f \in \mathcal F(U) \), we can use the fact that \( \mathcal F \) is a sheaf to conclude \( f = 0 \).
+
+    Now suppose \( f_i \in (\ker \varphi)(U_i) \) agree on their intersections.
+    Then they can be glued as elements of \( \mathcal F(U_i) \) into \( f \in \mathcal F(U) \).
+    As \( \varphi_{U_i}(f_i) = 0 \) for each \( i \in I \),
+    \[ 0 = \varphi_{U_i}(\res_{U_i}^U f) = \res_{U_i}^U \varphi_U(f) \]
+    So as \( \mathcal G \) is a sheaf, \( \varphi_U(f) = 0 \) in \( \mathcal G(U) \).
+\end{proof}
 However, the presheaf cokernel of a morphism of sheaves is not in general a sheaf.
 \begin{example}
     Consider \( X = \mathbb C \) with the Euclidean topology, and let \( \mathcal O_X \) be the sheaf of holomorphic functions on \( X \) under addition.

iii/cat/03_adjunctions.tex

@@ -17,20 +17,16 @@ \subsection{Definition and examples}
         \[ \mathbf{Set}(UX, Y) \leftrightarrow \mathbf{Top}(X, IY) \]
         \item Consider the functor \( \ob : \mathbf{Cat} \to \mathbf{Set} \) which maps each category to each set of objects.
         It has a left adjoint \( D \) which turns each set \( X \) into a discrete category in which the objects are elements of \( X \), and the only morphisms are identities.
-        % fix
-        % \[ \mathbf{Set}(\pi_0 \mathcal C, X) \leftrightarrow \mathbf{Cat}(\mathcal C, DX) \]
         It also has a right adjoint \( I \) which turns each set \( X \) into an indiscrete category in which the objects are elements of \( X \), and there is exactly one morphism between any two elements of \( X \).
-        % fix
-        % \[ \mathbf{Set}(\pi_0 \mathcal C, X) \leftrightarrow \mathbf{Cat}(\mathcal C, DX) \]
         In addition, \( D : \mathbf{Set} \to \mathbf{Cat} \) has a left adjoint \( \pi_0 : \mathbf{Cat} \to \mathbf{Set} \), where \( \pi_0 \mathcal C \) is the set of connected components of \( \ob \mathcal C \) under the graph induced by its morphisms.
-        \[ \mathbf{Set}(\pi_0 \mathcal C, X) \leftrightarrow \mathbf{Cat}(\mathcal C, DX) \]
+        \[ \mathbf{Set}(\pi_0 \mathcal C, X) \leftrightarrow \mathbf{Cat}(\mathcal C, DX);\quad \mathbf{Cat}(D X, \mathcal C) \leftrightarrow \mathbf{Set}(X, \ob \mathcal C);\quad \mathbf{Set}(\ob \mathcal C, X) \leftrightarrow \mathbf{Cat}(\mathcal C, IX) \]
         Thus we have a chain
         \[ \pi_0 \dashv D \dashv \ob \dashv I \]
         \item For any set \( A \), we have a functor \( (-) \times A : \mathbf{Set} \to \mathbf{Set} \).
         This functor has a right adjoint, which is the functor \( \mathbf{Set}(A, -) : \mathbf{Set} \to \mathbf{Set} \).
         \[ \mathbf{Set}(B \times A, C) \leftrightarrow \mathbf{Set}(B, \mathbf{Set}(A, C)) \]
         Applying this bijection is sometimes called \emph{currying} or \emph{\( \lambda \)-conversion}.
-        We say that a category \( \mathcal C \) with binary products is \emph{cartesian closed} if \( (-) \times A \mathcal C \to \mathcal C \) has a right adjoint, written \( [A, -] \) or \( (-)^A \), for each \( A \).
+        We say that a category \( \mathcal C \) with binary products is \emph{cartesian closed} if \( (-) \times A : \mathcal C \to \mathcal C \) has a right adjoint, written \( [A, -] \) or \( (-)^A \), for each \( A \).
         For example, \( \mathbf{Cat} \) is cartesian closed, where \( \mathcal D^{\mathcal C} = [\mathcal C, \mathcal D] \) is the functor category that this notation already refers to.
     \end{enumerate}
 \end{example}