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Algebraic Geometry Lecture 31 – Sheafification Joe Grant^1
Let X be a topological space.
Definition. A presheaf F of rings on X is:
(1) for every open subset U ⊆ X, a ring F(U ); (2) for every inclusion V ⊆ U a ring homomorphism ρU,V : F(U ) → F(V )
such that
(1) F(∅) = 0 (2) ρU,U = idF(U ) (3) W ⊆ V ⊆ U all open, then ρU,W = ρV,W ρU,V.
We call ρU,V the restriction maps, and for s ∈ F(U ) we sometimes write s |V for ρU,V (s). We call the elements of F(U ) the sections of F over U , and we sometimes write Γ(U, F) for F(U ).
Example. Let X = {x 1 , x 2 } with the discrete topology, i.e. all subsets are open. We write Xi = {xi}. We define some presheaves on X.
- F 1 : F 1 (Xi) = Z (1 6 i 6 2) F 1 (X) = Z ρF U,V^1 = 0 for U 6 = V.
- F 2 : F 2 (Xi) = Z (1 6 i 6 2) F 2 (X) = Z ρF U,V^2 = idZ for V 6 = ∅.
It’s easy to see that F 1 and F 2 are presheaves.
A sheaf is “a presheaf whose sections are determined by local data.”
Definition. A presheaf F on X is a sheaf if, for any open U ⊆ X, and any open covering {Vi} of U ,
(4) if, for s ∈ F(U ), s |Vi = 0, then s = 0; (5) if si ∈ F(Vi) such that for all i, j, si |Vi∩Vj = sj |Vi∩Vj , then there exists s ∈ F(U ) such that s |Vi = si for every i.
Note that (4) ⇒ the s in (5) is unique.
Example. • F 1 is not a sheaf because (4) fails: 2 ∈ F 1 (X), 2 |Xi = 0 for all i, but 2 6 = 0.
- F 2 is not a sheaf as (5) fails: 2 ∈ F 2 (X 1 ), 3 ∈ F 2 (X 2 ), but there is no s ∈ F 2 (X) = Z such that s |X 1 = s = 2 and s |X 2 = s = 3.
Example. Let
F 3 : F 3 (Xi) = Z (1 6 i 6 2) F 3 (X) = Z ⊕ Z ρX,X 1 = π 1 : Z ⊕ Z → Z : (m, n) 7 → m ρX,X 2 = π 2 : Z ⊕ Z → Z : (m, n) 7 → n.
Then F 3 is a sheaf. (We’ll see later that it’s a sheafification of F 2 .)
(^1) Typed by Lee Butler based on a talk by Joey G. 1
2
Definition. If F is a presheaf on X, and P ∈ X, then we define the stalk FP of F at P to be the direct limit of the rings F(U ) for all open U 3 P , via the restriction maps.
Unraveling this definition we see that an element of FP is represented by a pair 〈U, s〉 where U 3 P is open, s ∈ F(U ), and 〈U, s〉 ∼ 〈V, t〉
if and only if there exists some open set W ⊆ U ∩ V , P ∈ W , such that s |W = t |W. We call elements of FP germs of sections of F at the point P.
Example. As we have given X the discrete topology,
Fxi = F(Xi) (1 6 i 6 2)
and germs are just sections for F = F 1 , F 2 , F 3.
So the stalks here look uninteresting, but we’ll see later that we can still learn something from looking at them. Thinking of a sheaf as a (contravariant) functor F : Top(X)op^ → Rings
suggests the following definition.
Definition. If F, G are presheaves on X, a morphism ϕ : F → G consists of a ring homomorphism
ϕ(U ) : F(U ) → G(U )
for each open U such that whenever V ⊆ U is open, the diagram
F(U )
ϕ(U- ) G(U )
F(V )
ρF U,V
? (^) ϕ(V )
ρG U,V
?
commutes. I.e. ρG U,V ϕ(U )(s) = ϕ(V )ρF U,V (s),
or “ρϕ = ϕρ”.
A morphism of sheaves is just a morphism of the underlying presheaves. An isomorphism is just a morphism with a two-sided inverse.
Lemma. For any P ∈ X, a morphism ϕ : F → G of presheaves on X induces morphisms ϕP : FP → GP on the stalks, sending
〈U, s〉 → 〈U, ϕ(U )(s)〉.
Proof. Suppose 〈U, s〉 ∼ 〈V, t〉. We want 〈U, ϕ(U )(s)〉 ∼ 〈V, ϕ(V )(t)〉. By definition of ∼ there exists W ⊆ U ∩ V , P ∈ W such that s |W = t |W. Then
ρG U,W (ϕ(U )(s)) = ϕ(W )ρF U,W (s) = ϕ(W )(s |W ) = ϕ(W )(t |W ) = ϕ(W )ρF V,W (t) = ρG V,W (ϕ(V )(t)).
