A sheaf \(\mcf\) on a topological space \(X\) is flasque if for every inclusion \(V \subseteq U\) of open sets the restriction map \(\mcf(U) \to \mcf(V)\) is surjective.
Show that a constant sheaf on an irreducible topological space is flasque.
If \[ 0 \to \mcf' \to \mcf \to \mcf'' \to 0 \] is an exact sequence of sheaves and \(\mcf'\) is flasque, then for any open set \(U\) the sequence \[ 0 \to \mcf'(U) \to \mcf(U) \to \mcf''(U) \to 0 \] of abelian groups is also exact.
If \[ 0 \to \mcf' \to \mcf \to \mcf'' \to 0 \] is an exact sequence of sheaves and \(\mcf'\) and \(\mcf\) are flasque, then \(\mcf''\) is flasque.
If \(f: X \to Y\) is a continuous map and \(\mcf\) is a flasque sheaf on \(X\), then \(f_* \mcf\) is a flasque sheaf on \(Y\).
Let \(\mcf\) be any sheaf on \(X\). Define a new sheaf \(\mcg\), called the sheaf of discontinuous sections of \(\mcf\), as follows: for each open set \(U \subseteq X\), let \(\mcg(U)\) be the set of maps \(s: U \to \Union_{P \in U} \mcf_P\) such that \(s(P) \in \mcf_P\) for each \(P \in U\). Show that \(\mcg\) is a flasque sheaf and that there is a natural injective morphism \(\mcf \to \mcg\).
Solution
1. A constant sheaf on an irreducible topological space is flasque.
Let \(A\) be an abelian group and let \(\underline A\) denote the constant sheaf on an irreducible space \(X\).
Every nonempty open subset \(U\subseteq X\) is irreducible. A section of \(\underline A\) over \(U\) is a locally constant map \[ U\longrightarrow A \] with \(A\) discrete. Such a map must be constant: if it took two distinct values, the inverse images of those values would give disjoint nonempty open subsets whose union disconnects the image partition, contradicting irreducibility.
Hence \[ \underline A(U)\cong A \] for every nonempty open \(U\), and restriction between two nonempty opens is the identity on \(A\). Restriction to the empty set is also surjective. Thus every restriction map is surjective, so \(\underline A\) is flasque.
2. Suppose\[ 0\longrightarrow\mathcal F' \xrightarrow{\iota} \mathcal F \xrightarrow{q} \mathcal F'' \longrightarrow0 \]is exact and \(\mathcal F'\) is flasque. Then for every open \(U\subseteq X\), the map\[ q(U):\mathcal F(U)\longrightarrow\mathcal F''(U) \]is surjective.
Fix \[ s''\in\mathcal F''(U). \] If \(U=\varnothing\) there is nothing to prove, so assume \(U\ne\varnothing\).
Consider pairs \((V,s)\) where \(V\subseteq U\) is open and \[ s\in\mathcal F(V) \] satisfies \[ q(s)=s''|_V. \] Order these pairs by extension: \[ (V,s)\le (W,t) \] if \(V\subseteq W\) and \(t|_V=s\).
The collection is nonempty. Indeed, surjectivity of the map of stalks \[ \mathcal F_P\to\mathcal F''_P \] at any \(P\in U\) gives a germ lifting \(s''_P\); represent that germ by a local section and shrink the neighborhood until its image equals \(s''\) there.
Every chain has an upper bound: the domains form a nested family of opens, and the sections agree on inclusions by definition of the order, so the sheaf gluing axiom produces a section on their union. Zorn’s lemma therefore gives a maximal pair \[ (V,s). \]
We claim \(V=U\). Suppose not and choose \[ P\in U\setminus V. \] As above, choose an open neighborhood \(W\subseteq U\) of \(P\) and \[ t\in\mathcal F(W) \] with \[ q(t)=s''|_W. \]
On \(V\cap W\), \[ q(t-s)=0. \] Exactness identifies the kernel sheaf of \(q\) with \(\mathcal F'\), so there is \[ a\in\mathcal F'(V\cap W) \] whose image under \(\iota\) is \[ t|_{V\cap W}-s|_{V\cap W}. \]
Because \(\mathcal F'\) is flasque, extend \(a\) to \[ \widetilde a\in\mathcal F'(W). \] Replace \(t\) by \[ t'=t-\iota(\widetilde a). \] Then \(q(t')=s''|_W\) and \[ t'|_{V\cap W}=s|_{V\cap W}. \] Hence \(s\) and \(t'\) glue to a section on \[ V\cup W, \] still lifting \(s''\). Since \(P\in W\setminus V\), this strictly enlarges the maximal pair, contradiction.
Therefore \(V=U\), so \(s''\) has a global lift in \(\mathcal F(U)\).
3. Under the hypotheses of 2, for every open \(U\) the sequence\[ \boxed{ 0\longrightarrow\mathcal F'(U) \longrightarrow\mathcal F(U) \longrightarrow\mathcal F''(U) \longrightarrow0 } \]is exact.
The global-section functor is always left exact, so exactness holds at the first two terms. Step 2 supplies the missing surjectivity onto \(\mathcal F''(U)\).
4. If \(\mathcal F'\) and \(\mathcal F\) are flasque in an exact sequence\[ 0\to\mathcal F'\to\mathcal F\to\mathcal F''\to0, \]then \(\mathcal F''\) is flasque.
Let \(V\subseteq U\) be open and take \[ s''\in\mathcal F''(V). \] By 3 applied to the open \(V\), lift \(s''\) to \[ s\in\mathcal F(V). \] Since \(\mathcal F\) is flasque, extend \(s\) to \[ \widetilde s\in\mathcal F(U). \] Its image \[ q(\widetilde s)\in\mathcal F''(U) \] restricts to \(s''\). Therefore \[ \mathcal F''(U)\to\mathcal F''(V) \] is surjective for every inclusion \(V\subseteq U\), so \(\mathcal F''\) is flasque.
5. If \(f:X\to Y\) is continuous and \(\mathcal F\) is flasque on \(X\), then \(f_*\mathcal F\) is flasque on \(Y\).
For an open \(U\subseteq Y\), \[ (f_*\mathcal F)(U)=\mathcal F(f^{-1}(U)). \] If \(V\subseteq U\), then \[ f^{-1}(V)\subseteq f^{-1}(U). \] The restriction map \[ (f_*\mathcal F)(U) =\mathcal F(f^{-1}(U)) \longrightarrow \mathcal F(f^{-1}(V)) =(f_*\mathcal F)(V) \] is therefore surjective by flasqueness of \(\mathcal F\).
6. The presheaf \(\mathcal G\) of discontinuous sections is a sheaf.
For an open \(U\subseteq X\), \[ \mathcal G(U) =\prod_{P\in U}\mathcal F_P \] as an abelian group: a section is simply a choice \[ s(P)\in\mathcal F_P \] for every \(P\in U\). Restriction is literal restriction of the underlying function.
If such functions agree on overlaps of an open cover, they glue uniquely pointwise to a function on the union. Thus the sheaf axioms hold.
7. The sheaf \(\mathcal G\) is flasque.
Let \(V\subseteq U\) and \[ s\in\mathcal G(V). \] Define \[ \widetilde s(P)= \begin{cases} s(P),&P\in V,\\ 0\in\mathcal F_P,&P\in U\setminus V. \end{cases} \] This is an element of \(\mathcal G(U)\) and restricts to \(s\). Hence every restriction map \[ \mathcal G(U)\to\mathcal G(V) \] is surjective.
8. There is a natural injective sheaf morphism\[ \boxed{ \eta:\mathcal F\hookrightarrow\mathcal G, \qquad s\longmapsto(P\mapsto s_P). } \]
For each open \(U\), define \[ \eta_U(s)(P)=s_P. \] Taking germs commutes with restriction, so the \(\eta_U\) define a morphism of sheaves.
Suppose \[ \eta_U(s)=0. \] Then \[ s_P=0 \] for every \(P\in U\). For each \(P\), equality of the germ with zero gives an open neighborhood \(P\in U_P\subseteq U\) on which \[ s|_{U_P}=0. \] The \(U_P\) cover \(U\), so the uniqueness axiom for \(\mathcal F\) gives \[ s=0. \] Thus every \(\eta_U\) is injective.
9. Q.E.D.
Step 1 proves part (a), 3 proves part (b), 4 proves part (c), 5 proves part (d), and <1>6–<1>8 prove part (e).