# II.1.10

215,872pages on
this wiki

### Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

This problem has Hartshorne Height 1.

### Sketch

• Fix a sheaf $G$ and a system of compatible morphisms $F_i \to G$.

For every $U$ you have the desired factorization on the level of presheaves $F_i(U) \to \varinjlim F_i(U) \to G(U)$

• Show this is compatible with the restriction morphisms, i.e. for $V\subset U$ and any $F_i \to F_j$ in the directed system (possibly the identity) then
$F_i(U) \to F_j(V) \to \varinjlim F_i(V) \to G(V)$

is the same as

$F_i(U) \to \varinjlim F_i(U) \to G(U) \to G(V)$

(use the universal property of direct limit applied to the map $F_i(U) \to G(V)$).

• Use Proposition-Definition II.1.2