Direct and inverse limits of sheaves

Question

Is the direct limit of sheaves a sheaf? Is the inverse limit of sheaves a sheaf? I guess another way of saying it is whether sheafification commutes with direct limit or inverse limit. While we are at it, does sheafification commutes with direct sum or product?

Answer 1

Keenan Kidwell has answered your main question but I think I can add a little exposition earned from doing all the $\textrm{Ch II}\S1$ exercises in Hartshorne.

A finite (presheaf) product or coproduct of sheaves is again a sheaf, since $\mathscr{F} \oplus_{\mathsf{PSh}} \mathscr{G} \cong \mathscr{F} \times_{\mathsf{PSh}} \mathscr{G}$. Note that in general even finite colimits of sheaves have to be taken in $\mathsf{PSh}$ and then sheafified, e.g. cokernels.

However even a countably infinite family of sheaves can have the property that the direct sum(as presheaves) is not a sheaf! Take our space $\mathbb{N}_{\geq 0}$ with the discrete topology, and let $\mathscr{F}_i$ for $i \geq 0$ be the skyscraper sheaf with stalk $\mathscr{F}_{i,i}=\mathbb{Z}$ and zero elsewhere. Now we see that $\oplus_{i,\mathsf{PSh}} \mathscr{F}_i$ isn't a sheaf! Take the open sets $\{j\}_{j \geq 0}$ and the compatible family of sections $i_j(j)$ for $j \in \mathscr{F}_j(\{j\}) \cong \mathbb{Z}$ and where $i_j : \mathscr{F}_j \hookrightarrow \oplus_{i,\mathsf{PSh}} \mathscr{F}_{i}$.

EDIT: It occurs to me here that it's easier to show in general that if $\{ U_i \}$ is a family of disjoint open sets then $\mathscr{F}(\cup U_i) = \prod_i \mathscr{F}(U_i)$ if $\mathscr{F}$ is a sheaf. This immediately shows that this direct sum is not a sheaf.

It's a little dense, but if you do your part to break up that paragraph you'll see this family has no gluing, precisely because direct sums only have finitely many nonzero terms.

Answer 2

It doesn't really make sense to ask "is the direct limit of sheaves a sheaf?" I assume you mean to ask whether or not the natural choice for a presheaf direct limit of a directed system of sheaves a sheaf. The answer is no in general. You have to sheafify. That is, given a directed system of sheaves $(\mathscr{F}_i,\varphi_{ij})$, the presheaf defined by $U\mapsto\varinjlim_i\mathscr{F}_i(U)$, taken with respect to the maps $\varphi_{ij}(U)$, is usually only a presheaf. The associated sheaf is what we call $\varinjlim_i\mathscr{F}$, and you can verify that it has the properties to be the (i.e. categorical) direct limit of the $\mathscr{F}_i$. I believe one case where the direct limit presheaf is already a sheaf is when $X$ is a Noetherian topological space (i.e. every subset is quasi-compact).

For an inverse system of sheaves $(\mathscr{F}_i,\varphi_{ij})$, the presheaf $U\mapsto\varprojlim_i\mathscr{F}_i(U)$ actually is a sheaf, $\varprojlim_i\mathscr{F}_i$, and it is the categorical inverse limit of the $\mathscr{F}_i$.

In general, for colimits of diagrams of sheaves, one takes the natural presheaf (take colimits open set by open set), and then has to sheafify, while for limits, the natural presheaf is already a sheaf.

Answer 3

If your space is a noetherian topological space, then the presheaf direct limit of a directed system of sheaves is a sheaf. This follows from Theorem 3.1.10 (page 162) of Godement's Topologie algebrique et theorie des faisceaux, Hermann, Paris (1998). The statement of Theorem 3.1.10 in Godement's book is as follows: suppose $\mathcal{F}=\varinjlim_{\lambda}\mathcal{F}_\lambda$ is a direct limit of sheaves of sets on a noetherian space $X$. Then the canonical map: $$\varinjlim_\lambda\mathcal{F}_\lambda(X)\rightarrow\mathcal{F}(X)$$ is bijective. (Actually, in the statement of Theorem 3.1.10, $X$ is assumed to be compact, but following the proof, the theorem is also proved for noetherian spaces.)

As every subspace of a noetherian space is noetherian, and since passing to the sheaf associated with a presheaf commutes with restricting to open subsets, from Godement's Theorem 3.1.10 it follows that for every open set $U\subseteq X$, the canonical map: $$\varinjlim_\lambda\mathcal{F}_\lambda(U)\rightarrow\mathcal{F}(U)$$ is bijective.

This is for sheaves of sets but you can also deduce it for sheaves with values in other categories. For example, for sheaves of abelian groups, since any group homomorphism that is bijective as a map of underlying sets, is an isomorphism.