Chapter 1 Sheaves and cohomology

SKETCH:LECTURE 2

1. Finish direct limit 2. Sheafification of a presheaf a. Stalks and germs of sections; b. Examples: Smooth functions and holomorphic functions c. The sheafification functor. d. Explicit description

1.1 Lecture 2

1.1.1 Sheafification of a presheaf

Given an abelian presheaf ℱ on a space X and a point x ∈ X, one can proceed in a similar fashion to Exercise ?? and U consider the directed system {ρV : ℱ(U) → ℱ(V) ∣ V ⊂ U ∈ Ngd(x)}.

Definition 1. The colimit ℱx := colimU∈Ngd(x) ℱ(U) is called the stalk of the presheaf ℱ at x ∈ X and we denote by U ρx : ℱ(U) → ℱx the structural map.

Exercise 1. Given a presheaf ℱ on X and x ∈ X define an equivalence relation on the set Lℱ,x := {(U,s) ∣ s ∈ U ℱ(U),U ∈ Ngd(x)} by (U,s) ∼ (V,t) iff there is W ∈ Ngd(x) and σ ∈ ℱ(W) such that W ⊂ U ∩V and σ = ρW (s) = V ρW (t). Show that Lℱ,x/ ∼ can be naturally identified with the stalk ℱx. Example 1. Let an denote the sheaf of holomorphic functions on . Given a point a ∈ , the stalk an is identified Oℂ ℂ ℂ Oℂ,a j with the ring of “convergent power series” ∑ j≥0 a j(z − a) , i.e. those power series around a that have a positive convergence radius. It is a simple exercise to verify that the assignment that sends a germ sa represented by a section n s ∈ an(U) to the Taylor series 1 d s (z − a)n, gives an explicit isomorphism. Oℂ ∑ j≥0 j! dzn ∣a

U Definition 2. Given x ∈ X, U ∈ Ngdx and s ∈ ℱ(U), the equivalence class of (U,s) in ℱx is precisely ρx (s) and is called the germ of the section s at x. Sometimes one denotes the germ of s at x by sx.

Exercise 2. Let F be an abelian sheaf on X, and let U ⊂ X be open. Show that a section s ∈ F (U) is 0 if and only U if the germs sx = ρx (s) ∈ Fx are 0 for all x ∈ X.

1 2 1 Sheaves and cohomology

Proposition 1. Given an abelian presheaf ℱ on X, there is a sheaf ℱ+ together with a morphism of presheaves θ : ℱ → ℱ+ satisfying the following universal property: Given any sheaf G on X and a morphism of presheaves f : ℱ → G there is a unique morphism f + : ℱ+ → G such that f = f + ∘θ. The pair (ℱ+,θ) is unique up to a unique isomorphism.

+ x Proof. Given U ⊂ X, let ℱ (U) ⊂ ∏x∈U ℱx be the subgroup consisting of of those elements (s )x∈U such that for V x each a ∈ U there is a neighborhood V of a and σ ∈ ℱ(V) such that V ⊂ U and for each x ∈ V one has ρx (σ) = s . If W ⊂ U is an open subset, then the natural projection ∏x∈U ℱx −→ ∏x∈W ℱx clearly restricts to give a homomor- U + + U U V U + phism ρW : ℱ (U) −→ ℱ (W), satisfying ρU = Id and ρW = ρW ∘ ρV , whenever W ⊂ V ⊂ U. This shows that ℱ + is a presheaf of abelian groups. It is easy to see that, for each open U, the assignment θU : ℱ(U) → ℱ (U) sending U + s ∈ ℱ(U) to (ρx (s))x∈U ∈ ℱ (U) gives a presheaf morphism:

θ ℱ : ℱ −→ ℱ+. (1.1)

Furthermore, if ϕ : P → ℱ is a morphism of presheaves, one has a commutative diagram, for x ∈ V ⊂ U:

ϕU P(U) / ℱ(U) DD U DD ˆU DDρx DDρx D ˆU D DD ρV DD D ∃ ! ϕ D U ! x ! ρV Px ______/ ℱx V = = ρx zz zz zz zz zz zz V  zz  zz ρˆx P(V) / ℱ(V) ϕV

showing the existence of induced stalk homomorphisms ϕx : Px → ℱx. The product (∏x∈U ϕx) : ∏x∈U Px → + + ∏x∈U ℱx of the stalk homomorphisms ϕx restrict to give a natural homomorphisms ϕb: P → ℱ , making the fol- lowing diagram of presheaf morphisms commute:

ϕ P / ℱ (1.2)

θ P θ ℱ  ϕb  P+ / ℱ+.

Exercise 3. Given an abelian presheaf ℱ: 1. Show that ℱ+ is an abelian sheaf. 2. Show that if ℱ is a sheaf then θ ℱ : ℱ → ℱ+ is a natural isomorphism. Given a morphism f : ℱ → G from a presheaf ℱ to a sheaf G , define

f + := (θ G )−1 ∘ fb : ℱ+ → G .

It follows from its definition that f + satisfies the necessary conditions, and this completes the proof of the proposi- tion. Definition 3. The sheaf ℱ+ associated to a presheaf ℱ is called the sheaffification of ℱ, and sometimes we denote ℱ+ = Sheaf{U 7→ ℱ(U)}, or simply ℱ+ = Sheaf{ℱ}. Example 2. i) The sheaffification of the constant presheaf A is the sheaf whose sections over U form the group AX (U) consisting of continuous homomorphisms s: U → A, where A has the discrete topology. n ii) The sheaffificatino of the presheaf Bℂn of bounded holomorphic functions on ℂ is the sheaf of holomorphic functions an . Oℂn