– AARON PIXTON

JESSE FREEMAN

Contents 1. Lecture 2 – Morphisms, New Sheaves from Old, Stalks 2 2. Lecture 3 – Sheafification, Kernels and Cokernels, Abelian Categories 4

Date: September 18, 2016. 1 2 JESSE FREEMAN

1. Lecture 2 – Morphisms, New Sheaves from Old, Stalks Recall the definitions of a and a presheaf. We can have sheaves of abelian groups, rings, or sets.

Definition 1.1. If OX is a sheaf of rings on X, we call the pair (X, OX ) a .

Definition 1.2. The sheaf OX is called the structure sheaf and sections s ∈ OX (U) are called functions on U.

Definition 1.3. Let (X, OX ) be a ringed space. Then, an OX moduleis a sheaf of abelian groups F on X such that for each open set U, F(U) has the structure of an OX (U) module, and restriction maps respect this structure, i.e. the following diagram commutes:

OX (V ) × F(V ) F(V )

rVU rVU

OX (U) × F(U) F(U)

Example 1.1. Differential forms are a module over differentiable functions. The action is multiplication. Sections of a vector bundle are another example. 1.1. Morphisms of (pre)sheaves. Definition 1.4. Let F and G be presheaves on topological spaces X. Then, a morphism φ : F → G is given by maps φ(U): F(U) → G(U)

for each open U satisfying, for U ⊂ V , rVU ◦ φ(V ) = φ(U) ◦ rVU .

In other words, morphisms of presheaves are natural transformations between the F, G. We will use the following notation for categories of (pre)sheaves:

setsX : category of sheaves on X pre setsX : category of presheaves on X and so on. . .

1.2. New Sheaves from Old. • Restriction to an open subset.

U ⊂ X open, F a sheaf on X. Define a sheaf F|U (defined by open sets of X inside U)

F|U (V ) := F(V ).

We can also restrict morphisms of sheaves. We get that |U is a from setsX to setsU .

• Pushforward

π : X → Y continuous, F a sheaf on X. Define a sheaf π∗F on Y by −1 π∗(F)(V ) := F(π (V ))

and we can also push forward morphisms, giving a functor π∗ : setsX → setsY . Example 1.2. The skyscraper sheaf is a pushforward by a map {∗} −→ι X ALGEBRAIC GEOMETRY – AARON PIXTON 3

1.3. Stalks. Definition 1.5. Let F be a (pre)sheaf on X and let p ∈ X. Then the stalk of F at p is the set

Fp = {(s, u) | U 3 p open, s ∈ F(U)} / ∼ where ∼ is the equivalence relation given by (s, V ) ∼ (t, U) iff there eixsts W ⊂ U ∩ V such that s|W = t|W .

Elements of Fp are called germs of F at p. Remark 1.1. In category theory language, stalks are the colimits lim Fp = −−−→ F(U). p∈U The fact that this is an abelian group is due to the fact that this colimit exists. Definition 1.6. Let F be a sheaf of abelian groups on X. The support of a section s ∈ F(U) is

supp(s) = {p ∈ U | s 6= 0 ∈ Fp} Example 1.3. Let F be the skyscraper sheaf on R given by ( R 0 ∈ U F(U) = 0 else and the stalks are ( R p = 0 Fp = 0 else and supp(s) = {0} for any nonzero s ∈ F(R). Lemma 1.2 (Germs determine sections). Let F be a sheaf on X, U ⊂ X open. Then, the map Y F(U) → Fp p∈U is injective

Proof. Use the identity axiom.  Corollary 1.3. A nonzero section has nonempty support

Proof. Use the map above. At least one stalk must be nonzero.  We might ask what the image of the germ map looks like. Q Definition 1.7. An element (sp)p∈U ∈ p∈U Fp consists of ompatible germs if for all p ∈ U, there exists a neighborhood V of p, V ⊂ U, and a section f ∈ F(V ) such that fq = sq for all q ∈ V .

Lemma 1.4. Let F be a sheaf on X, U ⊂ X open, (sp)p∈U a collection of compatible germs. Then, (sp)p∈U lies in the image of the germ map

Proof. gluability axiom. 

Remark 1.5. A morphism of (pre)-sheaves F → G induces maps of stalks Fp → Gp for each p ∈ X (basically by definition of the equivalence relation on germs). Hence, the mantra “stalks determine morphisms”. If two morphisms φ1, φ2 : F → G induce the same maps of stalks over every point p ∈ X, then φ1 = φ2. 4 JESSE FREEMAN

2. Lecture 3 – Sheafification, Kernels and Cokernels, Abelian Categories

Recall Fp is the stalk of F at p. We have a germ map Y F(U) → Fp. p∈U If F is a sheaf. This map (a) is injective (b) has image given by the compatible germs condition.

Proposition 2.1. If φ1, φ2 : F → G are morphisms of sheaves inducing the same map on stalks, then φ1 = φ2.

Proof. Recall a morphism is a collecion {φ(U)}U⊂X commuting with restriction. We have the following commutative diagram: φ(U) F(U) G(U)

g g (2.1) Q φ Q p∈U Fp p∈U Gp where g is the germ map, which is injective. If φ1, φ2 agree on stalks, then

φ1 ◦ g = φ2 ◦ g. and hence

g ◦ φ1(U) = g ◦ φ2(U) for all U. But, g is injective. So, this implies that φ1(U) = φ2(U). 

In fact, many other properties of morphisms sheaves are determined on the level of stalks.

Example 2.2. Morphisms of sheaves are  ⇔ they induce isomorphisms on each stalks  ⇔ injections on each stalk  ⇔ surjections on each stalk

2.1. Sheafification.

2.1.1. . Let F be a presheaf. A (in fact “the”) sheafification of F is a sheaf F sh along with a morphism of presheaves sh : F → F sh satisfying whenever g : F → G is a morphism of presheaves and G is a sheaf, there exists a unique f such that g = f ◦ sh F sh F sh g f G

Remark 2.2. (1) F sh is unique up to unique . (2) Assuming sheafification exists, F → F sh is a functor from presheaves on X to sheaves on X. ALGEBRAIC GEOMETRY – AARON PIXTON 5

2.1.2. Construction. Let F be a presheaf on X. Define a sheaf F sh on X by   sh  Y  F (U) = (sp)p∈U ∈ Fp | (sp)p∈U satisfies the compatible germs condition  p∈U 

The restriction map simply forgets about some of the sp, those that are not stalks of points in the smaller set.

One can check: (1) This is a sheaf. The content here is that one must check after gluing the germs are still compatible. (2) There is a map F → F sh (it’s the germ map). (3) If F is a sheaf, F sh =∼ F via a unique isomorphism. Example 2.3. Let F be the constant presheaf on X with values in S, i.e. F(U) = S (for all U)

rV,U = idS The compatible germs condition says that each point has an open neighborhood where the function is con- stant. Then, F sh will be the sheaf of locally constant functions with values in S.

We call this the and denote it by S. Example 2.4. Let X be a manifold, F the presheaf of bounded continuous functions U → R. Sheafification is the sheaf of continuous functions, intuitively because if a function is continuous it is locally bounded. 2.2. Abelian Categories. Intuition: These are categories “like” the category of R-modules. Definition 2.1 (Partial Definition). An abelian category C is a category such that • Mor(A, B) is an abelian group whose addition commutes with restriction. •C has a zero object an object with exactly one morphism to or from any object. • Any morphism has a kernel and cokernel. • Three more conditions we won’t list. Here, the kernel of a morphism A → B is the limit of the diagram 0

A B and the cokernel is the colimit A B

0 pre Proposition 2.3. AbX is an abelian category. Kernels and cokernels can be constructed “open set by open set”, e.g. (ker : F → G)(U) = ker(F(U) → G(U)).

Lemma 2.4. Let φ : F → G be a morphism of sheaves. Then, the presheaf kernel, kerpreφ is actually a sheaf.

Proof. We check the sheaf axioms simultaneously. Suppose {Ui}i∈I is an open cover for U and let si ∈ (kerpreφ)(Ui) = ker(φ(Ui)) ⊂ F(Ui) be sections agreeing on pairwise intersections.

Since F is a sheaf, there is a unique s ∈ F(U) such that

rU,Ui (s) = si. 6 JESSE FREEMAN

We want to show s ∈ ker(φ(U)). But,

rU,Ui (φ(U)(s)) = φ(Ui)(rU,Ui (s)) = 0. So, the identity axiom on G implies s ∈ F(U).  The analogous lemma fails for cokernels – the resulting presheaf might not be a sheaf. We will give an example here. Vakil gives a different example. Example 2.5. Let X = R/Z = S1, and F be the sheaf of continuous functions U → Z and G the sheaf of continuous functions U → R.

pre There is a natural map F → G. The map F → G has cokernel in AbX continuous functions U → R/Z factoring through R → R/Z. This is not a sheaf. There is no map X R R/Z id so we cannot patch together local identity functions.

The cokernel in AbX will be continuous functions U → R/Z and factoring through R is automatic locally. Proposition 2.5. The sheafification of the sheaf cokernel of a morphism of sheaves is the cokernel. Proof. Let φ : F → G be a morphism of sheaves and H the sheafification of the cokernel presheaf of φ. Let E be any sheaf. We have the diagram φ F G

0 cokerpreφ H

∃ ! ∃ !

0 E

So, H satisfies the universal property of cokernel sheaves. 

Theorem 2.6. AbX is an abelian category.