Algebraic Analysis notes Lecture 6 (18 Jan 2019)

Notes for Lecture 5

Last time, for a given topological space X and a ring k, we define a presheaf to be a functor $F \colon Op(X)^{op} \to k-mod$ .

Definition: If F is a presheaf on X, and x is a point in X, then the stalk of F at x, is the k-module $F_x = \varinjlim_{x \in U} F(U)$ .

Let’s unpack this: an element of $F_x$ is an equivalence class of pairs (U, s) where U is an open neighborhood of x, and $s \in F(U)$ , and $(U, s) \sim (U', s')$ if there is a set V contained in $U \cap U'$ such that $s|_V = s'|_V$ , a “common refinement”.

If $x \in U$ , there is a canonical map $F(U) \to F_x$ , and the image of a section $s \in F(U)$ under this map is called the germ of s at x, denoted $s_x$ .

Definition: Let $F \in PreSh(X)$ , the support of F is $supp = \overline{ \{ x \in X | F_x \neq 0 \} }$ . For $s \in F(U)$ , let $supp(s) = \{x \in U | s_x \neq 0 \}$ .

Exercise: supp(s) is always closed (that’s why the definition doesn’t have to include taking the closure).

Definition: A presheaf F on X is a sheaf if it satisfies the following:
(S1) Gluing: given an open cover $(U_\alpha)_{\alpha \in I}$ of U open in X, and $s_\alpha \in F(U_\alpha)$ such that $s_\alpha |_{U_\alpha \cap U_\beta} = s_\beta |_{U_\alpha \cap U_\beta}$ , then there exists $s \in F(U)$ such that $s|_{U_\alpha} = s_\alpha$ .
(S2) Local Identity: given an open cover $(U_\alpha)_{\alpha \in I}$ of U open in X, and $s,t \in F(U)$ such that $s|_{U_\alpha} = t|_{U_\alpha}$ , then s=t.

We’re thinking of F(U) as the set of functions from U into k. So the first condition (S1) is saying that if we have a cover of U by open sets, and we have a function on each of these opens, and they all agree on overlaps, then we can glue them all together into a function on all of U. The second condition (S2) says that if two sections agree for any restriction to a set in the open cover, then they must be equal. Taken together, if you have two candidates for what the gluing of a compatible family of sections $s_\alpha$ could be, they must be equal, i.e. they glue to a unique section s on U.

A morphism of sheaves is just a morphism of presheaves. So the category of sheaves on X is the full subcategory of PreSh(X).

Notice that if F is a sheaf, then the local identity condition gives $F(\emptyset) = 0$ . If F is a sheaf, and $(U_\alpha)_{\alpha \in I}$ is an oper cover of U, we get a sequence

where $r_0$ is given by $s \mapsto (s|_{U_\alpha})_{\alpha \in I}$ , and $r_1$ is given by $(s_\alpha)_{\alpha \in I} \mapsto (s_\alpha|_{U_\alpha \cap U_\beta} - s_\beta|_{U_\alpha \cap U_\beta})$ . F being a presheaf gives that $r_1 \circ r_0 = 0$ . The gluing condition gives that $ker(r_1) \subset Im(r_0)$ . The local identity condition gives that $r_0$ is injective. Thus if F is a sheaf, then this sequence is exact.

Lemma: Let $F \in PreSh(X)$ . The following are equivalent:

F satisfies local identity (S2)
for any U open in X, and $s \in F(U)$ , if $s_x = 0$ for any point x in U, then s = 0.
for any U open in X, and $s,t \in F(U)$ , if $s_x = t_x$ for each point x in U, then s = t.

proof: (1 implies 2) Suppose $s \in F(U)$ and $s_x = 0$ for each x in U. Then for each x in U, there is a neighborhood $U^{(x)}$ of x such that $s|_{U^{(x)}} = 0$ . So we have an open cover $(U^{(x)})_{x \in U}$ of U, and $s|_{U^{(x)}} = 0$ , so local identity gives s=0.

(2 implies 3) Suppose we have $s,t \in F(U)$ with $s_x = t_x$ for each point x in U. Then if we consider s-t, it reduces to the statement of 2.

(3 implies 1) Suppose (S2) fails. Then there is some U open in X, and some $s \neq t \in F(U)$ , and some open cover $(U_\alpha)_{\alpha \in I}$ such that $s|_{U_\alpha} = t|_{U_\alpha}$ for every $\alpha \in I$ . Then the germs of s and t agree, $s_x = t_x$ , and thus s=t by 3. Contradiction!

Conclusion: any sheaf can be recovered from the germs of its sections!

Example: First a non-example. The constant presheaf $\underline M_{pre,X}$ is usually not a sheaf because $\underline M_{pre,X} (\emptyset) = M$ which is required to be trivial.

Definition: The constant sheaf, denoted $\underline M_X$ or $\underline M$ when the space is clear, is given by $\underline M(U) = \{\text{locally const functions } U \to M\}$ , and $res_{U,V}$ is just given by restriction of functions.

More generally, for all $Z \subset X$ closed, we write $\underline M_{Z \subset X}$ or just $\underline M_Z$ for $\underline M_{Z}(U) = \{ \text{loc const fun.s } U \cap Z \to M\}$ .