How much information about a family is contained in its closed fibres?

Question

One general type of question I am unsure how to approach is "how much information can we deduce about a family given information about its closed fibres?" As a concrete example, say we have a projective variety $X$ over $\mathbb{C}$ and a family of coherent sheaf homomorphisms $\varphi:\mathcal{F}\rightarrow \mathcal{G}$ on $X$ parameterised by some connected $S$. Let's say we know these sheaves are well-behaved (for example, they have constant $h^0$'s), and let's say we know that at every closed $s\in S$ the fibre $\varphi_s: \mathcal{F}_s\rightarrow \mathcal{G}_s$ is an isomorphism; could we deduce that $\varphi$ itself is an isomorphism? Or what about if we know $H^0(\varphi_s): H^0(\mathcal{F}_s) \rightarrow H^0(\mathcal{G}_s)$ is an isomorphism, could we deduce $H^0(\varphi)$ is itself an isomorphism?

Answer 1

If $\varphi \colon \mathcal{F} \to \mathcal{G}$ is a morphism of coherent sheaves on $X \times S$ such that $\varphi_s \colon \mathcal{F}_s \to \mathcal{G}_s$ is an isomorphism for every closed $s \in S$, then indeed $\varphi$ is an isomorphism. Assume also that $\mathcal{G}$ is flat over $S$.

First, let $\mathcal{C}$ be the cokernel of $\varphi$. Then for every closed $s$ we have an exact sequence $$ \mathcal{F}_s \to \mathcal{G}_s \to \mathcal{C}_s \to 0 $$ (because the restriction functor is right exact). Your assumption implies $\mathcal{C}_s = 0$ for each $s$, and since $\mathcal{C}$ is coherent, it follows that $\mathcal{C} = 0$, so that $\varphi$ is surjective.

Now let $\mathcal{K}$ be the kernel of $\varphi$. Using flatness of $\mathcal{G}$ we obtain an exact sequence $$ 0 \to \mathcal{K}_s \to \mathcal{F}_s \to \mathcal{G}_s \to 0 $$ and again your assumption imples $\mathcal{K}_s = 0$, hence $\mathcal{K} = 0$, because it is coherent. Thus, $\varphi$ is an isomorphism.