Let $\mathbb{G}_m$ the multiplicative group, with coordinate ring $\mathbb{C}[x^{\pm 1}]$, and considered as a sheaf of abelian groups over $\mathrm{Spec}\,\mathbb{C}$ in the Zariski topology. Let $X$ be another complex scheme (I am interested in $X$ being a smooth projective variety, but I do not think it matters anyway). Denote by $p$ the structure morphism of $X$ down to $\mathrm{Spec}\,\mathbb{C}$.
There are two (seemingly different) ways of pulling $\mathbb{G}_m$ back to $X$:
- As sheaves of abelian groups, $p^{-1}\mathbb{G}_m$ is the sheafification of the presheaf $$(p^{-1}\mathbb{G}_m)^\mathrm{pre}(U) = \varinjlim_{V \supseteq p(U)}\,\mathbb{G}_m(V)$$ Since $p(U)$ is the whole of $\mathrm{Spec}\,\mathbb{C}$ for any nonempty $U$, we have $(p^{-1}\mathbb{G}_m)^\mathrm{pre}(U) = \mathbb{G}_m(\mathrm{Spec}\,\mathbb{C}) = \mathbb{C}^\times$, so $p^{-1}\mathbb{G}_m$ is the locally constant sheaf $\underline{\mathbb{C}^\times}$.
- As schemes, the pullback $X \times_{\mathrm{Spec}\,\mathbb{C}}\mathbb{G}_m$ is an abelian group scheme over $X$ whose sheaf of sections is $\mathbb{G}_m(\mathcal{O}_X)$ —that is, the sheaf of functions from $X$ to $\mathbb{G}_m$.
I have always thought that the first construction gave the result of the second, but I seem to have argued otherwise here. Have I made a mistake? Or am I right but the pullback as schemes is some other functor (maybe the exceptional inverse image $p^!$)?
I think this sounds right. While this works when talking about sheaves on topological spaces, we have a problem here because sheaf represented by a scheme "forgets" the scheme structure.
In fact, the category of sheaves on $\text{Spec} \mathbb{C}$ is equivalent (even isomorphic, I think) to the category of sets! So the sheaf on $\text{Spec} \mathbb{C}$ represented by $\mathbb{G}_m$ really is just the (discrete) set of elements of $\mathbb{C}^\times$, and so its pullback really should be a locally constant sheaf.