# Alternative Construction of Sheaf from Sheaf on a Base

In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too.

Suppose $\mathscr F$ is a sheaf on bases of a topological space $X$. For an open set $U\subset X$ where $U$ is open, define

$\bar{\mathscr F}(U)=lim_{B\subset U}\mathscr F(B)$, where the $B$s are basic open sets.

This seems to satisfy all the sheaf conditions since limits commute with products and kernels and hence the equalizer diagram for sheaves is satisfied. But I must have overlooked something since I haven't seen this anywhere else. Could you tell me if I'm right?

• What kind of limit is that? – Lord Shark the Unknown Jul 28 '18 at 9:02
• An inverse limit. – Jehu314 Jul 28 '18 at 9:17
• An inverse limit? Over what ordered set/category? – Lord Shark the Unknown Jul 28 '18 at 9:18
• Thus if you have been able to show that $\overline{\mathscr{F}}$ is a sheaf, you're done! They're isomorphic on the level of basic opens so they're isomorphic as sheaves. – Samir Canning Jul 30 '18 at 15:52
• For reference, this is Proposition I-12 in Eisenbud-Harris, "Geometry of Schemes". – DKS Nov 3 '18 at 17:53

Let $$\mathcal{K}$$ be a category. In paragraph (3.2.1) (p. 26), given a sheaf $$\mathscr F$$ on a base $$\mathscr B$$ of open sets, he defines a sheaf $$\mathscr F'$$ by $$\mathscr{F}'(U) = \varprojlim_V \mathscr{F}(V)$$ where $$V$$ ranges over the ordered set of sets $$V \in \mathscr{B}$$ with $$V \subseteq U$$.
Assume still that $$\mathcal{K}$$ admits projective limits. Then the category of sheaves on $$X$$ with values in $$\mathcal{K}$$ also admits projective limits;...the verification of axiom (F) [the sheaf axiom] results from the transitivity of projective limits...