Chapter 1 Sheaves and cohomology

SKETCH:LECTURE 5

1. Exactness: a. Examples: i. “Extensions by zero” b. Exactness of the sections functor c. f∗ is left exact; d. f −1 is exact; F −1 2. Injectivity of η : F → f∗ f F when f : X → Y is onto; a. Example: ı: Xδ → X. 3. Types of sheaves: a. Flabby sheaves b. Soft sheaves c. Injective sheaves

1.1 Lecture 5

Have defined: 1. Exact sequences 2. Exponential sequence ϕ ψ Proposition 1. Let 0 → A −→ B −→ C → 0 be a short exact sequence of abelian sheaves on X, and let U ⊂ X be an open set. Then the sequence of abelian groups

ϕ ψ 0 → A (U) −→U B(U) −−→U C (U)

is exact. Definition 1. For a fixed open set U ⊂ X one can define a functor:

Γ (U,−): AbSh/X −→ Ab

sending F to Γ (U,F ) := F (U). When U = X, one calls Γ (X,−) the global sections functor.

1 2 1 Sheaves and cohomology

Remark 1. Another way of stating the Proposition above is to say that the functor Γ (U,−): AbSh/X −→ Ab is a left exact functor.

Proof. Given x ∈ U we have a commutative diagram

ϕU ψU A (U) / B(U) / C (U)

U U 0U ρx ρˆx ρx    0 / Ax / Bx / Cx / 0, ϕx ψx

with exact bottom row. U Given s ∈ kerϕU the diagram above and the injectivity of ϕx show that ρx (s) = 0 for all x ∈ U. It follows from (??) that s = 0 and hence ϕU is injective. Note that this applies to any open subset of X. U Similarly, suppose that σ ∈ B(U) is in the kernel of ψU . Chasing the diagram one concludes that ρˆx (σ) = ϕx(sx) x for some sx ∈ Ax, for all x ∈ U. This means that one can find neighborhoods U ⊃ Nx ∈ Ngdx and τ ∈ A (Nx) such Nx x that ρx (τ ) = sx. Nx x Nx x U Since ρˆx (ϕNx (τ )) = ϕx(ρx (τ ) = ϕx(sx) = ρˆx (σ) one can find a smaller neighborhood Wx ⊂ U ∩ Nx of x so that the restriction ˆU ( ) coincides with ( Nx ( x)). In particular, this shows that ρWx σ ϕWx ρWx τ

0  Nx x Nx0 x  ϕW ∩W 0 ρ (τ ) − ρ (τ ) = 0. x x Wx∩Wx0 Wx∩Wx0

{W } U Now, using the fact (already proven) that ϕWx∩Wx0 is injective and that x x∈U forms an open cover of , one concludes that the sections Nx ( x) ∈ (W ) assemble to give a global section ∈ (U), since is a sheaf. ρWx τ A x τ A A Furthermore, by construction, one sees that ϕU (τ) and σ have the same germs at each x ∈ U, and hence ϕU (τ) = σ, concluding the proof of the proposition.

Definition 2. Let ϕ : A → B be a morphism of abelian sheaves on X. For each open subset U ⊂ X, define (kerϕ)U := ker{ϕU : A (U) → B(U)}. It follows easily that the assignment U 7→ (kerϕ)U defines an abelian sheaf on X. When trying to define the cokernel of a morphism of sheaves, one sees that the assignment U 7→ coker(ϕU ) defines only an abelian presheaf on X, and one defines the abelian sheaf

coker(ϕ) := Sheaf{U 7→ coker(ϕU )}.

Exercise 1. Show that if ϕ : A → B is a morphism of abelian presheaves, then one has a natural homomorphism ι : ker(ϕ) → A and a short exact sequence

ι π 0 → ker(ϕ) −→ A −→ coker(ι) → 0.

Proposition 2. Let f : X → Y be a map. 1. If 0 → A → B → C → 0 is an exact sequence of sheaves on X then

0 → f∗A → f∗B → f∗C

is exact. In other words, f∗ : AbSh/X → AbSh/Y is a left exact functor. 2. If 0 → F → G → H → 0 (1.1) is an exact sequence of sheaves on Y then

0 → f −1F → f −1G → f −1H → 0 (1.2) 1.1 Lecture 5 3

−1 is exact. In other words, f : AbSh/Y → AbSh/X is an exact functor. Furthermore, if f is surjective, then (1.1) is exact if and only if (1.2) is exact.

Proof. Given an open set W ⊂ Y, the previous proposition gives an exact sequence

0 / A ( f −1(W)) / B( f −1(W)) / C ( f −1(W)) .

f∗A (W) f∗B(W) f∗C (W)

This proves the first assertion. Given any sheaf P on Y, it is easy to see( Verify the details.) that for each x ∈ X one has a natural identification −1 of stalks ( f P)x ≡ P f (x). Therefore, for each x ∈ X the sequence

0 / F f (x) / G f (x) / H f (x) / 0 . (1.3)

−1
−1
−1
f F x f G x f H x is exact, thus showing that (1.2) is exact. Furthermore, if f is surjective, given y ∈ Y one can find x ∈ X so that f (x) = y and the sequence (1.1) is exact on stalks for every y ∈ Y, thus showing the converse of the second statement whenever f is surjective.

Let f : X → Y be a map and let F be an abelian sheaf on Y. The natural unit of the adjunction

F −1 η : F → f∗ f F (1.4) will be shown to have many fundamental applications.

Example 1 (Extension by zero). Let j : Z ,→ X be the inclusion of a closed subset and let i: U := X − Z ,→ X denote ( Ax , if x ∈ Z the inclusion of its open complement. Note that if A is an abelian sheaf on Z then ( j∗A )x = 0 , otherwise. (Verify the details.) For this reason, j∗A is said to be the extension by zero to X of the sheaf A on Z. On the other hand, if B is an abelian sheaf on U, the sheaf i∗B does not have this property. However one can define another ( B(W) , if W ⊂ U sheaf i!B on X as the sheaffification of the presheaf W 7→ 0 , otherwise. Observe that ( Bx , if x ∈ U (i!B)x = 0 , otherwise.

The sheaf i!B is also called the extension by zero to X of the sheaf B on U. Now, given a sheaf F on X it is a simple observation (just check what happens at the stalk level) that one has a natural short exact sequence of sheaves on X

F −1 η −1 0 → i!i F −→ F −−−→ j∗ j F → 0. (1.5)

(Verify the details.)

Exercise 2. Given an open subset U ⊂ X and an abelian sheaf F on X show that one has a natural identification ∗ F (U) ≡ HomAbSh/X (ZX ,i!i F ). 4 1 Sheaves and cohomology

A −1 Example 2. Let f : X → Y be a surjective map. We claim that η : A → f∗ f A is a monomorphism of sheaves on Y. It suffices to show that for all y ∈ Y the sequence

A ηy −1
0 → Ay −−→ f∗ f A y

y A y is exact. Indeed, pick s ∈ Ay such that ηy (s ) = 0 and let (W,τ) with W ∈ Ngd(y) and τ ∈ A (W) represent the y A germ s . Hence, by shrinking W if necessary, one may assume that ηW (τ) = 0. However,

A −1 −1 −1 ηW : A (W) → f∗ f A (W) = ( f A )( f (W)) ⊂ ∏ A f (x) x∈ f −1(W)   sends τ ∈ A (W) to ρW (τ) . In particular, since f is surjective, one can find x ∈ f −1(W) such that f (x) x∈ f −1(W) A W f (x) = y, and the condition ηW (τ) = 0 implies ρy (τ) = 0. Since y ∈ Y was arbitrary and A is a sheaf one concludes that τ = 0.

1.1.1 Special types of sheaves

Definition 3. Let A , I be abelian sheaves on X. 1 X i) A is called a flabby sheaf if for every open set U ⊂ X the restriction ρU : A (X) → A (U) is surjective. In other words, any section over an open set U can be extended to a global section. X ii) A is called a soft sheaf if for every closed set Z ⊂ X the restriction ρZ : A (X) → A (Z) is surjective. Observe that we have not previously defined the group A (Z) of sections of a sheaf over a closed subset, but this is simply −1 −1 −1 −1 defined as ( j A )(Z), the group of global sections of j B. Since j∗ j B(X) = j B(Z) := B(Z), the unit A −1 X of the adjunction η : A → j∗ j A gives the “restriction map” ρZ : A (X) → A (Z). Therefore, an abelian sheaf A is soft if any section of A over a closed subset Z ⊂ X can be extended to a global section. iii) I is called an injective sheaf if it safisfies the following extension property: given a morphism of abelian sheaves ι ϕ : A → I and an exact sequence 0 → A −→ B of sheaves on X, one can find a morphism ϕˆ : B → I such that ϕˆ ◦ ι = ϕ. In other words, the following extension problem has a solution

ι 0 / A / B . } ϕ } } ∃ ϕˆ  ~} I

Example 3. Let A be an abelian sheaf on a space X. i) If X is discrete, then A is always flabby. Indeed, any subset U ⊂ X is both open and closed and hence, if A ηX −1 j : U ,→ X is the inclusion then A (X) −−→ j∗ j A (X) = A (U) is onto. See (1.5). ii) If X is a normal topological space, then Urysohn’s lemma shows that the sheaf CX (of real or complex-valued continuous function) is a soft sheaf of rings. ∞ iii) If X is a smooth manifold, then the sheaf CX (of real or complex-valued smooth functions) is also soft. This is a consequence of the existence of smooth partitions of unity on X. an iv) The sheaf OX of analytic functions on a complex manifold X (e.g. on C) is not soft.

1 Flasque in French.