# 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

$$\bar{\mathscr F}(U)=\varprojlim_{B\subset U}\mathscr F(B)$$, 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? Jul 28, 2018 at 9:02
• An inverse limit. Jul 28, 2018 at 9:17
• The sheaf obtained from a sheaf on the base is unique and a morphism of sheaves (so in particular an isomorphism of sheaves) is determined by a morphism (resp. isomorphism) on the base. Can you write down a map on the between your object and Vakil's on the level of basic opens? Alternatively, you can look up what a cofinal system is and how inverse limits over cofinal systems work. Jul 28, 2018 at 17:54
• 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. Jul 30, 2018 at 15:52
• For reference, this is Proposition I-12 in Eisenbud-Harris, "Geometry of Schemes".
– DKS
Nov 3, 2018 at 17:53

Such a construction works: here's a reference (title: "Construction of a sheaf from the data on a basis of open sets"). Here (pp. 26) is the one pointed out by @DKS in the comments.

Edit (two years later!): A conceptual way to look at this construction is that the formula $$\begin{equation}\tag{1}\mathscr{F}(U)\cong\underset{\overset{\longleftarrow}{B\subset U}}{\lim}(\mathscr{F}(B))\end{equation}$$ is precisely the usual formula computing right Kan extensions. Let $$\mathsf{Open}(X)$$ be the category whose

• Objects are the open subsets of $$X$$;
• Morphisms are inclusions.

Given a basis $$\mathcal{B}$$ of $$X$$, write $$\mathsf{Open}(\mathcal{B})$$ for the full subcategory of $$\mathsf{Open}(X)$$ spanned by the opens of $$X$$ in $$\mathcal{B}$$. Then we have a canonical inclusion $$i\colon\mathsf{Open}(\mathcal{B})\hookrightarrow\mathsf{Open}(X),$$ and the presheaf associated to a presheaf $$\mathscr{F}\colon\mathsf{Open}(\mathcal{B})^\mathsf{op}\to\mathcal{A}$$ on the basis $$\mathcal{B}$$ taking values on a category $$\mathcal{A}$$ is precisely the right Kan extension of $$\mathscr{F}$$ along $$i$$: You can go from this to Equation $$(1)$$ by computing $$\mathsf{Ran}_i(\mathscr{F})$$ as the limit in Eq. 6.2.3 of Riehl's book:

which in our context takes the form $$\mathscr{F}(U)\cong\lim\left((\underline{U}\downarrow i)^\mathsf{op}\twoheadrightarrow\mathsf{Open}(\mathcal{B})^\mathsf{op}\overset{\mathscr{F}}{\longrightarrow}\mathcal{A}\right).$$ But $$(\underline{U}\downarrow i)^\mathsf{op}$$ is precisely the category whose objects are inclusions of the form $$B\hookrightarrow U$$ for $$B\in\mathcal{B}$$, so projecting from this to $$\mathsf{Open}(\mathcal{B})^\mathsf{op}$$ and then applying $$\mathscr{F}$$ recovers Eq. $$(1)$$.

For sheaves, you first take the right Kan extension of $$\mathscr{F}$$ on $$\mathcal{B}$$ as above and then sheafify. If $$\mathscr{F}$$ is already a sheaf on $$\mathcal{B}$$, then so is the resulting sheaf on $$X$$ (Görtz–Wedhorn, Prop. 2.20).

Yes, your construction works and indeed this is how Grothendieck defines the sheaf induced by a sheaf on a base in EGA I.

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 $$V \subseteq U$$.

He remarks in paragraph (3.2.6):

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...