direct products of sheaves preserve exactness

Question

In direct products exact in the category of quasi-coherent sheaves, Leonid Positselski gave a counterexample of direct products not preserving exactness. However, the products are taken in the category of quasi-coherent sheaves rather than the category of sheaves (of abelian groups). I want a counterexample of products not preserving exactness in the category of sheaves. Thanks in advance.

Answer 1

Let $X$ be the Hawaiian earring, let $A$ be the sheaf of continuous $\mathbb{R}$-valued functions on $X$, and let $B$ be the sheaf of continuous $S^1$-valued functions on $X$. The covering map $\pi:\mathbb{R}\to S^1$ gives an exact sequence $A\to B\to 0$, since every map to $S^1$ can locally be lifted along $\pi$. Now let $C=A^\mathbb{N}$ and $D=B^\mathbb{N}$ and consider the induced map $q:C\to D$ of sheaves; I claim $q$ is not surjective.

To prove this, let $v\in X$ be the point where all the circles meet and let $X_n$ be the $n$th circle. Consider the global section $s$ of $D$ whose $n$th coordinate is constant on $X_m$ for $m\neq n$ and maps $X_n$ homeomorphically to $S^1$. There is no neighborhood of $v$ on which every $s_n$ can be lifted to $\mathbb{R}$, since every neighborhood of $v$ contains some $X_n$ and $s_n$ cannot be lifted on all of $X_n$. Thus the germ of $s$ at $v$ is not in the image of $q$.