The Lumenary · First-year graduate

Sheaves

Name: ______________________

Date: ______________________

You will learn to

Define presheaves as functors $\mathcal{O}(X)^{\mathrm{op}} \to \cat{Set}$ and state the sheaf condition both as locality-plus-gluing and as an equaliser condition
Prove that continuous functions form a sheaf, that constant presheaves usually do not, and that a section of a sheaf is determined by its germs
Compute stalks, prove that a morphism of sheaves is an isomorphism if and only if it is one on every stalk, and sheafify simple presheaves such as constant presheaves
Explain how sheaves of continuous, smooth and holomorphic functions encode local-to-global structure, with the logarithm and the exponential sequence as examples
Compute a first Čech cohomology group and explain how it measures the failure of local solutions to glue

1.
Which of these presheaves of real-valued functions on $\R$ are sheaves? (Select all that apply.)
- A. $U \mapsto$ continuous functions on $U$
- B. $U \mapsto$ smooth functions on $U$
- C. $U \mapsto$ bounded continuous functions on $U$
- D. $U \mapsto$ constant functions on $U$
- E. $U \mapsto$ locally constant functions on $U$
- F. $U \mapsto$ restrictions of polynomials to $U$
2.
How many sections does the sheaf $\underline{\Z/2}$ of locally constant $\Z/2$ -valued functions have over $U = (0, 1) \cup (2, 3) \cup (4, 5) \subset \R$ ?
3.
In the sheaf of continuous functions on $\R$ , which pair of functions has the same germ at $0$ ?
- A. $f(x) = \max(0, x - 1)$ and $g(x) = 0$
- B. $f(x) = x$ and $g(x) = \sin x$
- C. $f(x) = 0$ and $g(x) = x^2$
- D. $f(x) = |x|$ and $g(x) = x$
4.
On $X = \C \setminus \{0\}$ , the morphism $\exp : \mathcal{O} \to \mathcal{O}^\times$ is surjective on every stalk. Is $\exp : \mathcal{O}(X) \to \mathcal{O}^\times(X)$ surjective?
- A.No: $z \in \mathcal{O}^\times(X)$ has no global logarithm; the obstruction is its image in $H^1(X, \underline{\Z}) \cong \Z$ , the winding number.
- B.Yes, by Theorem 4: surjective on stalks implies surjective on sections.
- C.Yes, because every non-vanishing holomorphic function has a logarithm.
- D.No, because $\exp$ is not surjective on stalks at $0$ .
5.
In the proof that a morphism of sheaves $\varphi : F \to G$ which is bijective on stalks is surjective on sections, local preimages $s^{(x)} \in F(V_x)$ of $t$ are glued. Why do $s^{(x)}$ and $s^{(y)}$ agree on $V_x \cap V_y$ ?
- A.Both map to $t|_{V_x \cap V_y}$ , and $\varphi$ is injective on sections (proved first, from injectivity on stalks and Theorem 3).
- B.Because $G$ is a sheaf.
- C.Because they have the same value at the points of $V_x \cap V_y$ .
- D.Because $\varphi_x$ is surjective.
6.
Cover $S^1$ by three open arcs $U_1, U_2, U_3$ such that each pairwise intersection is a single (connected) arc and the triple intersection is empty. What is $\dim_{\Z/2} \check H^1(\mathcal{U}, \underline{\Z/2})$ ?
7.
Given a cover U : ι → Opens X of V with hU : ∀ i, U i ≤ V and a compatible family s, which conclusion in the definition of IsSheaf captures both locality and gluing?
- A.∃! t : F.sec V, ∀ i, F.res (hU i) t = s i
- B.∃ t : F.sec V, ∀ i, F.res (hU i) t = s i
- C.∀ t t' : F.sec V, (∀ i, F.res (hU i) t = F.res (hU i) t') → t = t'
- D.∀ i j, F.res inf_le_left (s i) = F.res inf_le_right (s j)
8.
Sections of the sheaf of all functions on a finite set are dictionaries. Write glue(sections) that takes a list of dictionaries (functions on subsets) and returns the single dictionary on the union that restricts to each of them, or None if two of them disagree on a common point.
Computational — Practice

Answer key — Sheaves

1.
Continuous, smooth and locally constant functions form sheaves: their conditions are local. Bounded, constant and polynomial functions do not: each condition is global, and gluing fails (for an unbounded union, resp. a disconnected one).
2.
$\underline{\Z/2}(U) = (\Z/2)^{\pi_0(U)}$ and $U$ has three components, so there are $2^3 = 8$ sections.
3.
Only $\max(0, x - 1)$ and $0$ agree on an open neighbourhood of $0$ (namely $(-\infty, 1)$ ). The other pairs agree at $0$ , or to first order, or on a half-line, but germs are equal only if the functions agree on a whole neighbourhood.
4.
No. If $e^{\ell} = z$ on $X$ , then $\ell' = 1/z$ , but $\oint_{|z| = 1} dz/z = 2\pi i \neq 0$ , so $1/z$ has no primitive on $X$ . Sheaf-theoretically, $z$ maps to a generator of $H^1(X, \underline{\Z})$ in the long exact sequence of the exponential sequence.
5.
On $V_x \cap V_y$ , $\varphi(s^{(x)}) = t = \varphi(s^{(y)})$ . Since $\varphi$ is injective on sections (because it is injective on stalks and sections are determined by germs), $s^{(x)} = s^{(y)}$ there, so they glue.
6.
Each $U_i$ and each $U_i \cap U_j$ is connected, so $\check C^0 \cong \check C^1 \cong (\Z/2)^3$ and $\check C^2 = 0$ . The kernel of $\delta^0$ is the constants, of dimension $1$ , so $\delta^0$ has rank $2$ and $\check H^1 = (\Z/2)^3 / \operatorname{im}\delta^0$ has dimension $1$ — as it must, since this is a good cover and $H^1(S^1; \Z/2) \cong \Z/2$ .
7.
Option (a), used in Lumenary.Sheaves.IsSheaf. Existence of $t$ is the gluing axiom and its uniqueness is the locality axiom; (b) and (c) are each one half, and (d) is the hypothesis.
8.
```
def glue(sections):
    result = {}
    for s in sections:
        for x, v in s.items():
            if x in result and result[x] != v:
                return None
            result[x] = v
    return result
```
Compatibility on overlaps is exactly the condition that no point gets two values; the glued function is unique because a function is determined by its values. The empty family glues to the empty function, the unique section over $\varnothing$ (Proposition 1).