Skyscraper sheaves

This article is part of my migration effort, moving some of my articles over from the excellent Functor Network.

Let $X$ be a topological space, let $p$ be a point of $X$, and let $S$ be any set. In the notation of Vakil, define the skyscraper sheaf supported at $p$ by the formula $$\begin{equation*} (i_{p,*}S)(U) = \begin{cases} S & \text{if $p \in U$;} \\ 1 & \text{otherwise.} \end{cases} \end{equation*}$$ Here $1$ is the singleton set. We can also define such a sheaf in other categories (abelian groups, rings, etc.), replacing the singleton set by the appropriate terminal object.

If $V \subseteq U$ is a containement of open sets, then the restriction $\rho_{U,V}$ is given as follows:

• if $p \in V$, then $\rho_{U,V}$ is the identity on $S$;
• otherwise, $\rho_{U,V}$ is the unique map to $1$.

For $W \subseteq V \subseteq U$, we have $\rho_{U,W} = \rho_{V,W} \circ \rho_{U,V}$, because if $p \notin W$ both sides are the unique map to $1$, and otherwise both sides are the identity map on $S$. Therefore we have a presheaf of sets on $X$ (and this could also be a presheaf of other objects as well, as long as there is a terminal object).

Suppose $\{U_i\}_{i \in I}$ is an open cover of some open set $U$ in $X$, and suppose $\{s_i \in (i_{p,*}S)(U_i)\}_{i \in I}$ is a collection of sections such that, for any $i,j \in I$, we have $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$. If $p \notin U$, then the unique element of $(i_{p,*}S)(U)$ is evidently the unique gluing of the sections. If $p \in U$, then there is some $i_0 \in I$ such that $p \in U_i$; let $s=s_{i_0}$ be the gluing of the sections. Because restriction to $U_{i_0}$ is the identity, the gluing is clearly unique. We want to show that for any $i \in I$, we have $s|_{U_i} = s_i$. If $p \notin U_i$, then it’s obviously true. If $p \in U_i$, then $p \in U_i \cap U_{i_0}$, and since $s_i|_{U_i \cap U_{i_0}} = s_{i_0}|_{U_i \cap U_{i_0}}$ with these restrictions being the identity, we find $s_i = s_{i_0} = s = s|_{U_i}$, as required. This was a lot of words to say a simple thing: we have a sheaf.

The “skyscraper” in the name is explained by the following fact:

The stalk of $i_{p,*}S$ at a point $q \in X$ is $S$ if $q$ is in the closure of $p$, and is the singleton set $1$ otherwise.

That’s not very hard to show. Suppose that $q$ is not in the closure of $p$. Then there exists some open set around $q$ that does not contain $p$. Hence any germ in the stalk at $q$ can be represented by an element in the singleton set $1$, which means all germs are equal and the stalk may be identified with $1$. On the other hand, suppose that $q$ is in the closure of $p$. This means all open sets which contain $q$ also contain $p$. The stalk is a colimit, and now we’re saying it’s a colimit over a constant diagram (every object in the diagram is $S$). Therefore, the colimit is $S$.

Note that we can argue more abstractly for the first case, when $q$ is not in the closure, in a way that shows the stalk is the terminal object $1$ in other categories (abelian groups, rings, etc). The stalk is a direct limit which is computed over the directed set of all opens containing $p$, ordered by reverse inclusion. Recall that a directed set is a poset $(A, \leq)$ in which every pair of elements has an upper bound. A subset $(B,\leq)$ of a poset is said to be cofinal in $A$ if, for every $a \in A$, it is possible to find some $b \in B$ such that $a \leq b$. For instance, when $q$ is not contained in the closure of $p$, the set of open neighborhoods of $q$ that do not contain $p$ is cofinal in the directed set of all open neighborhoods of $q$, ordered by reverse inclusion. Note that any cofinal set in a directed set is also directed. One can show that the direct limit computed over a directed set is equal (or more precisely, isomorphic up to a unique canonical isomorphism) to the direct limit computed over the “smaller” cofinal set. In our example, this means the stalk at $q$ is $1$ in any category with such a terminal object, because the direct limit can be computed over the cofinal set of neighborhoods not containing $p$, and that’s a constant diagram with all objects equal to $1$.

From the previous discussion, skyscraper sheaves look like a skyscraper towering above a point, and this mental picture is accurate when the point $p$ is closed. When a point is not closed (such a situation happens frequently in algebraic geometry), there are some points “nearby” over which the stalk is also $S$, so it looks like a city’s downtown more than a single skyscraper.

What About the Weird Notation?

The notation $i_{p,*}S$ is weird, but it makes sense in light of the following construction. Let $f : X \to Y$ be a (continuous) map of topological spaces, and let $\mathscr{F}$ be a sheaf on $X$. We define the pushforward of $\mathscr{F}$ along $f$ to be the sheaf defined by the equation $$(f_*\mathscr{F})(U) = \mathscr{F}(f^*U),$$ where $f^*$ denotes the inverse image (or preimage) of $f$ (it’s more often written as $f^{-1}$ but I prefer the notation with a star). Because $f$ is continuous, the inverse image of an open set is an open set, so the previous equation makes sense. Given $V \subseteq U$ an inclusion of open sets in $Y$, the restriction from $U$ to $V$ is defined by the equation $$\rho_{U,V}^{f_*\mathscr{F}} = \rho_{f^*U, f^*V}^{\mathscr{F}}.$$ This defines a presheaf, simply because $\mathscr{F}$ itself is a presheaf: clearly the restriction from an open set to itself is the identity, and $$\begin{align*} \rho_{V,W}^{f_*\mathscr{F}} \circ \rho_{U,V}^{f_*\mathscr{F}} &= \rho_{f^*V,f^*W}^{\mathscr{F}} \circ \rho_{f^*U,f^*V}^{\mathscr{F}} \\ &= \rho_{f^*U,f^*W}^{\mathscr{F}} \\ &= \rho_{U,W}^{f_*\mathscr{F}}. \end{align*}$$

The fact $\mathscr{F}$ is a sheaf is also sufficient to make its pushforward a sheaf as well. Suppose $U$ is an open set in $Y$, and $\{U_i\}_{i \in I}$ is an open cover of $U$. For each $i \in I$, let $s_i \in f_*\mathscr{F}(U_i)$ and suppose further that, for any $i, j \in I$, we have $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$. We want to show the existence of a unique section $s \in f_*\mathscr{F}(U)$ such that $s|_{U_i} = s_i$ for each $i \in I$. Each section $s_i$ is an element of $\mathscr{F}(f^*U_i)$, and the fact these sections all agree on overlaps $U_i \cap U_j$ together with the fact $f^*(U_i \cap U_j) = f^*U_i \cap f^*U_j$ means there exists a unique $s \in \mathscr{F}(f^*U)$ with the desired property. Notice that a key part of why the pushforward is a sheaf, is the fact the inverse image preserves both arbitrary unions and intersections (union is because we need the collection $\{f^*U_i\}_{i \in I}$ to be an open cover of $f^*U$).

To make sense of the notation for skyscraper sheaves, we also need to talk about the constant sheaf. Let $S$ be any set. The constant sheaf associated to $S$, denoted $\underline{S}$, is defined by labeling each open set $U$ with the set of functions $U \to S$ that are locally constant (i.e. around each point of $U$ there exists some open set contained in $U$ on which the function is constant – this is the same as requiring the function to be constant on connected components of $U$). Restriction is the usual restriction of maps, which obviously respects the presheaf condition. The sheaf axiom is not hard to check either.

Back to skyscrapers. Let $i_p : 1 \to X$ be the “inclusion map” which points to $p \in X$. We consider $\underline{S}$ as a sheaf over the topological space $1$. Let $U$ be an open set in $X$. If $p \in U$, then $i_p^*U$ is the unique point of $1$, while on the other hand if $p \notin U$ then $i_p^*U$ is the empty set. Hence we see the pushforward $i_{p,*}\underline{S}$ of the constant sheaf $\underline{S}$ is isomorphic in some obvious sense to the skyscraper sheaf $i_{p,*}S$ as defined earlier.