I'm thinking about some arguments from Glen Bredon's 'Sheaf Theory' and have a lot of problems with understanding how they work. The first question deals with explanations from page 26:
We denote by $X$ the topological space, and $\mathcal{A}$ a locally constant sheaf on $X$. For $U \subset X$. Let $S^p(U;\mathcal{A})$ be the group of singular $p$-chains of $U$ with values in $\mathcal{A}$. That is, an element $f \in S^p(U;\mathcal{A})$ is a function that assigns to each singular $p$-simplex $\sigma: \Delta_p \to U $ of $U$, a cross section $f(\sigma) \in \Gamma(\sigma^*(\mathcal{A}))$, where $\Delta_p$ denotes the standard $p$-simplex.
Let $\mathcal{U} := \{U_{\alpha} \}$ be a covering of $X$ by open sets, and let $S^p(\mathcal{U};\mathcal{A})$ be the group of singular cochains based on ll-small singular simplices. Then Bredon uses an argument I do not understand. He says
'Then a subdivision argument shows that the surjection
$$j_{\mathcal{U}}: S^*(U;\mathcal{A}) \to S^*(\mathcal{U};\mathcal{A})$$
induces a cohomology isomorphism.'
Question #1: what is this 'subdivision argument' and why does it imply that $j_{\mathcal{U}}$ induces an isomorphism on cohomology?
Question #2: On page 180, Bredon remarks moreover another important property of the group of singular $p$-chains and their associated sheaf: assume that $L$ is a principal ideal domain, which will be taken as the base ring. Let $\mathcal{A}:= L$ be the locally constant sheaf with values in $L$ and let $\mathfrak{S}^* = \mathfrak{S}^*(X; L) $ the sheafification of presheaf $ U \to S^*(U;\mathcal{A})$. Therefore the sheaf $\mathfrak{S}^*$ can be naturally considered as $L$-module.
Bredon claims in the middle of page 180 that the stalks of $\mathfrak{S}^*$ are torsion-free. Why is that true?
Since the stalks of the sheaf $\mathfrak{S}^p(X; L)$ and the presheaf associating $ U \to S^p(U;L)$ are equal, it is sufficient to think about the direct limit of the $S^p(U; L)$, where the $U$ form an open system of neighborhoods around $x \in X$.
For sufficiently small open neighbourhood $x \in U \subset X$, the $S^p(U; L)$ is a free $L$-module, so torsion free. It's direct limit, which is by definition the stalk at $x \in X$ is in general not free. Is it nevertheless torsion free?