Local-Global Ext sequence

Question

Let $F,G$ be two sheaves of $\mathcal{O}_X$-modules, where $X$ is a scheme. The local-global Ext exact sequence starts like this:

$$0\to H^1(\mathcal{Hom}(F,G))\to Ext(F,G)\to H^0(\mathcal{Ext}(F,G))\to \dots$$

I want to understand the maps thinking of $Ext$ as equivalence classes of short exact sequences. However I am having difficulty in finding an explicit construction for the first map (and I also have doubts about the second). Can someone help?

Answer 1

Fix an affine open cover $\{U_i\}$ of $X$, $U_i = \mathrm{Spec}(R_i)$. Let $F_i, G_i$ denote the restrictions of $F,G$ to those open sets. Write $U_i \cap U_j = U_{ij} = \mathrm{Spec}(R_{ij})$, and so on.

An $H_1$ class is represented by, for each $i,j$, an $R_i$-module map $\phi_{ij} : F_{ij} \to G_{ij}$, such that $\phi_{ij} + \phi_{jk} = \phi_{ik}$ on $U_{ijk}$. (For notational purposes, define $\phi_{ji} = -\phi_{ij}$.)

I claim that this builds an extension of sheaves,

$$0 \to G \to S \to F \to 0.$$

Here's a key observation: this extension should give the zero class in $H^0(\mathcal{E}xt^1(F,G))$ -- by exactness of the local-to-global sequence. This means that the extension should be locally trivial. So, on $U_i$, we have to put $S_i = G_i \oplus F_i$, with the obvious inclusion and surjection.

But, we glue these extensions together using the $\phi_{ij}$'s: glue $S_i$ to $S_j$ (only over $U_{ij}$) by

$$(g_i,f_i) \mapsto (g_i + \phi_{ij}(f_i), f_i).$$

Check that:

• This is an isomorphism from $S_i$ to $S_j$ (only defined over $U_{ij}$).
• On triple overlaps, the composite isomorphism $S_i \to S_j \to S_k$ equals the direct one $S_i \to S_k$.
• The isomorphism commutes with the inclusion $g_i \mapsto (g_i,0)$.
• It commutes with the projection $(g_i,f_i) \mapsto f_i$.

Thus we have glued together the collection of short exact sequences

$$0 \to G_i \to S_i \to F_i \to 0.$$

Finally, note that this construction used a representative for an $H^1$ class. But if we alter the $\phi_{ij}$'s by $\phi_{ij} + (\epsilon_i - \epsilon_j)$ for some arbitrary collection of $R_i$-module maps $\epsilon_i : F_i \to G_i$, then the resulting extensions are isomorphic, $S_i^{\text{(old)}} \to S_i^{\text{(new)}}$ by

$$(g_i,f_i) \to (g_i + \epsilon_i(f_i),f_i).$$

Clearly this is an isomorphism of extensions (respects the inclusion and projection).

PS: The second map is maybe easier to understand. An element of $H^0(\mathcal{E}xt^1(F,G))$ is a choice of local extensions on each $U_i$, which agree on overlaps in the weaker sense that the two short exact sequences happen to be isomorphic on each $U_{ij}$. However, these isomorphisms can be completely unrelated to one another! They don't have to be compatible, gluing to a global extension of sheaves (though it's fine if they do, such as if they come from an element of $\mathrm{Ext}^1(F,G)$ -- an honest global extension.)