Product of skyscraper sheaf vs constant sheaf

Solution 1:

Your issue is the claim that the stalk of $\mathcal{A}$ at any point is simply $A.$ Let's look at an example: take $X = \Bbb{R}$ and $A = \Bbb{Z}$ for concreteness, and let's look at the stalk of $\mathcal{A}$ at $0.$

For any sheaf $\mathcal{F}$ on any space $X,$ we have an explicit description of the stalk of $\mathcal{F}$ at a point $x$: $$ \mathcal{F}_x \cong \{\sigma\in \mathcal{F}(U)\mid U\ni x\}/\sim, $$ where $\sigma\in \mathcal{F}(U)$ and $\tau\in\mathcal{F}(V)$ are equivalent if there exists some open $W\ni x$ such that $\left.\sigma\right|_W = \left.\tau\right|_W.$

So, for our particular $\mathcal{A},$ an element of $\mathcal{A}_0$ is represented by some $(\sigma_x)_{x\in U}$ with $U\ni 0$, where each $\sigma_x\in\Bbb{Z}.$ There is a morphism of groups \begin{align*} e : \mathcal{A}_0&\to \Bbb{Z}\\ [(\sigma_x)]&\mapsto\sigma_0. \end{align*} The kernel of this map consists of $[(\sigma_x)]$ such that $\sigma_0 = 0.$ However, I claim that $e$ is not injective. In particular, if we let $U = (-\epsilon,\epsilon)$ for some $\epsilon > 0$ and set $$ \tau_x := \begin{cases} 0,&x = 0,\\ 1,&x\neq 0. \end{cases} $$

If $\tau = (\tau_x)_{x\in U},$ then $[\tau]\in\ker e,$ but $\tau\not\sim 0$: we have $\left.\tau\right|_V = (\tau_x)_{x\in V}\neq (0)_{x\in V}$ for any open $V\ni 0,$ since any open neighborhood of $0$ contains nonzero real numbers.

In fact, notice that \begin{align*} \mathcal{A}(U)&\to\{f : U\to A\}\\ s = (s_x)_{x\in U}&\mapsto\left.\begin{cases}f_s : &U&\to A\\ &x&\mapsto s_x\end{cases}\right\} \end{align*} is an isomorphism for any $U.$ That is, your sheaf is the sheaf of functions into $A.$ One common interpretation of the elements of the stalk of a sheaf of functions is as "germs of those functions." With this interpretation, it might be clearer that the stalk of $\mathcal{A}$ at a point $x$ is not simply $A,$ since a germ of a function $f$ at a point $x$ is not determined only by the value of $f$ at $x,$ but also by "infinitesimal data" of $f$ near $x.$