ALGEBRAIC GEOMETRY { 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 sheaf 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 ringed space. Definition 1.2. The sheaf OX is called the structure sheaf and sections s 2 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 functors 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 FjU (defined by open sets of X inside U) FjU (V ) := F(V ): We can also restrict morphisms of sheaves. We get that jU is a functor 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 2 X. Then the stalk of F at p is the set Fp = f(s; u) j U 3 p open; s 2 F(U)g = ∼ where ∼ is the equivalence relation given by (s; V ) ∼ (t; U) iff there eixsts W ⊂ U \ V such that sjW = tjW . 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): p2U 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 2 F(U) is supp(s) = fp 2 U j s 6= 0 2 Fpg Example 1.3. Let F be the skyscraper sheaf on R given by ( R 0 2 U F(U) = 0 else and the stalks are ( R p = 0 Fp = 0 else and supp(s) = f0g for any nonzero s 2 F(R). Lemma 1.2 (Germs determine sections). Let F be a sheaf on X, U ⊂ X open. Then, the germ map Y F(U) ! Fp p2U 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)p2U 2 p2U Fp consists of ompatible germs if for all p 2 U, there exists a neighborhood V of p, V ⊂ U, and a section f 2 F(V ) such that fq = sq for all q 2 V . Lemma 1.4. Let F be a sheaf on X, U ⊂ X open, (sp)p2U a collection of compatible germs. Then, (sp)p2U 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 2 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 2 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: p2U 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 fφ(U)gU⊂X commuting with restriction. We have the following commutative diagram: φ(U) F(U) G(U) g g (2.1) Q φ Q p2U Fp p2U 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 8 isomorphisms , they induce isomorphisms on each stalks <> monomorphisms , injections on each stalk :>epimorphisms , surjections on each stalk 2.1. Sheafification. 2.1.1. Universal Property. 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 isomorphism. (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 8 9 sh < Y = F (U) = (sp)p2U 2 Fp j (sp)p2U satisfies the compatible germs condition : p2U ; 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 constant sheaf 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 fUigi2I is an open cover for U and let si 2 (kerpreφ)(Ui) = ker(φ(Ui)) ⊂ F(Ui) be sections agreeing on pairwise intersections. Since F is a sheaf, there is a unique s 2 F(U) such that rU;Ui (s) = si: 6 JESSE FREEMAN We want to show s 2 ker(φ(U)). But, rU;Ui (φ(U)(s)) = φ(Ui)(rU;Ui (s)) = 0: So, the identity axiom on G implies s 2 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.