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1 The Basics For the purposes of this discussion X will always denote a topological space. I will only deal with presheaves and sheaves of abelian groups, but it is worth noting that we can discuss sheaves of objects with more or less structure. For example we can talk of sheaves of sets, rings, or modules just as easily. 1.1 Presheaf A presheaf A on X is a function associating an abelian group (set, ring, module, ...) A(U ) to each open U ⊂ X such that for each pair U ⊂ V of open sets in X, there is a homomorphism (called restriction) ρU V : A(V ) → A(U ) such that ρU U = 1. 1.1.1 Examples 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of a presheaf on X 3. pth cohomology. Let U ⊂ X be an open subset, and let H p (U ) be the pth cohomology group. Each cohomology group will give the structure of a presheaf on X. An equivalent definition can be given using category theory. It is more concise (certainly) it may be more straightforward to check, later we will will want categorical formulations of most of these statements. A presheaf A of abelian groups (rings, modules, etc...) on X is a contravarient functor from the category of open subsets of X (with inclusions) to the category of abelian groups (rings, modules, etc...) If we fix a topological space X we can consider the category of presheaves of X. If F and G are presheaves, a morphism ϕ : F → G is a family of group homomorphisms ϕ(U ) making the following diagram commute. F (U ) ϕ(U ) ρU V ρU V F (V ) / G(U ) ϕ(V ) 1 / G(V ) 1.2 Sheaves As a general principle groups are good, but topologies are better. So we can require that a presheaf have a topology. Again, let X be a topological space. Let (A, π) be a pair where • A is a topological space • π : A → X is a local homeomorphism • each Ax = π −1 (x) for x ∈ X, is an abelian group • the group operations are continuous We call the pair (A, π) a sheaf (of Abelian Groups). π −1 (x) is called the “stalk” at x. If A is Hausdorff, then the sheaf is considered to be Hausdorff. 1.2.1 Examples 1. Non-Hausdorff sheaves. Let X = R and let ( 0 if x 6= 0 −1 π (x) = Z/2 if x = 0 2. Constant Sheaf. Let A be any abelian group, given the discrete topology. X × A then forms a sheaf over X, with projection map p(x, a) = x 3. Let X be a manifold. For each open U in X, let C ∞ (U ) be the set of smooth functions from U to R. C ∞ (X) is a Sheaf abelian groups on X. If we restrict to germs we have a sheaf of modules on X As immediate consequences of the definition of sheaves we have the following: a) π is an open map b) Any section of A is an open map. c) Any element of A is in the range of some section over some open set. d) The set of all images of sections over open sets is a base for the topology of A. e) For any two sections s ∈ A(U ) and t ∈ A(V ), U and V open, the set W of points x ∈ U ∩ V such that s(x) = t(x) is open. Just as we used a categorical definition of a presheaf we can now give as similar definition of a sheaf. A sheaf is a presheaf under which the following sequence is exact. 0 / F (U ) / Q F (Ui ) / dif fQ i<j F (Ui ∩ Uj ) We can consider the category of sheaves of X, and morphisms will be defined the same as morphisms in the category of presheaves of X. 2 1.3 Constructions, Remarks, and Examples It is clear that sheaves and presheaves are closely related, and any sheaf is necessarily a presheaf. We can make this stronger my making a sheaf from any presheaf. The additional topological information needed to move from a presheaf to a presheaf will come from the local structure of X in terms of a germ. Let M be the set of all elements s ∈ A(U ) for all open neighborhoods U of x in X. We say s ∈ A(U ) and t ∈ A(V ) are equivalent if there exists a neighborhood of W of x in X such that W ⊂ U ∩ V and ρW V (s) = ρW V (t). The germ of A at x is given by Ax = M/ ∼ It is worth noting (but non-trivial to see) that Ax is given as Ax = lim A(U ) −→ where U ranges over all neighborhoods of x ∈ X ( with inclusion as morphisms). Fix a topological space X, and let A be a presheaf on X. Each open U is a topological space (with the subspace topology) and each A(U ) can be considered a topological space with the quotient topology. Construct the coproduct a (U × A(U )) U ⊂X and say that a = (x, s) ∈ U × A(U ) and b = (y, t) ∈ V × A(V ) are equivalent if and only if there exists W ⊂ U ∩ V such that s|W = t|W . Now let A = E/ ∼ and note that the fallowing diagram commutes [email protected] @@ p @@ @@ X /A ~ ~ ~ ~~π ~ ~~ where π : A → X is the map induced from p(x, y) = x. q is open so π is continuous. p is locally injective so π is a local homeomorphism. It is simple to see that A is a sheaf of abelian groups on X. We refer to A as the sheaf generated by the presheaf A. In review we have defined sheaves and presheaves of topological spaces. We have seen several examples, and discussed categories of sheaves and presheaves. We will now consider the kernel’s and cokernels of sheaves. g : A → B is a presheaf homomorphism h : A → B is a sheaf homomorphism 3 Presheaves Sheaves ker {a ∈ A : such that g(a) = 0} {α ∈ A : such that h(α) = 0} coker B/g(A) B/h(A) While this table looks symmetric, we have no reason to believe the quotient in presheaves(X) will equal the quotient in sheaves(X). This will be illustrated by the following example and be the basis for sheaf cohomology. For any topological space X let O be the sheaf such that O(U ) is the group of continuous maps from U into C. Likewise let O be the sheaf such that O(U ) is the group of continuous maps from U into C× . There is then a short exact sequence in Sheaves(X). 0 /Z 2πi /O exp / O× /0 When X is the space C× , this sequence is not exact in Presheaves(X) because exp : C → is not onto; the cokernel is Z = H 1 (X, Z), generated by the global unit 1/z. 4