Is the constant sheaf $\mathbb{Q}$ injective?

Question

Let $X$ be a topological space, and let $\mathbb{Q}$ be the constant sheaf of abelian groups on $X$ associated to the group of rational numbers under addition. Is $\mathbb{Q}$ an injective object in $\mathfrak{Ab}(X)$, the category of sheaves of abelian groups on $X$? More generally, does the constant sheaf functor $\mathfrak{Ab}\to\mathfrak{Ab}(X)$ preserve injectives?

I suspect that the answer to both questions is no, else I would have seen this written somewhere, but I can't seem to prove my suspicion. Any enlightenment on this matter is much appreciated.

Answer 1

Of course the constant sheaf $\mathbb{Q}$ is not injective in general. Recall that for an injective sheaf $\mathscr{F}$, we have $H^1 ( X, \mathscr{F}) = 0$. However, for sufficiently nice spaces $X$ (say, paracompact), sheaf cohomology with coefficients in a constant sheaf $A$ is isomorphic to singular cohomology with coefficients in $A$; so in particular, by the universal coefficient theorem, $$H^n (X, \mathbb{Q}) \cong \mathrm{Hom} (H_n (X), \mathbb{Q})$$ where $H^n$ on the LHS is sheaf cohomology and $H_n$ on the RHS is singular homology. Thus for many spaces, the constant sheaf $\mathbb{Q}$ fails to be acyclic, let alone injective.

For example, $H^1 (S^1, \mathbb{Q}) \cong \mathbb{Q}$, so the constant sheaf $\mathbb{Q}$ on $S^1$ is not injective.

Answer 2

An injective sheaf is always flabby. Flabbyness of a constant sheaf imposes toplogical conditions for the ambient space. For example, a constant sheaf over a Haussdorff connected and locally connected space $X$ is never flabby unless $| X|=1$.

Answer 3

Let $k$ be a field. It can be shown that in the category of sheaves of $k$ vector spaces, a constant sheaf $M_X$ is injective if and only if it is flabby (see Sheaves on Manifolds, Kashiwara-Shapira, exercise II.10, p. 133). Now, if $X$ is an irreducible noetherian space, for exemple any irreducible complex algebraic variety endowed with the Zarisky topology, every constant sheaf is flabby, therefore injective.