1. Directed Limit A partially ordered set I is said to be a directed set if for each i; j in I; there exists k in I so that i ≤ k and j ≤ k: Let A be a ring, I a directed set, and fMi : i 2 Ig be a family of A-modules index by I: Assume that for each i ≤ j; there exists a A-homomorphism fij : Mi ! Mj such that (1) fii is the identity map on Mi for all i 2 I; (2) fik = fjk ◦ fik whenever i ≤ j ≤ k: Then the family fMig of A-modules together with A-homomorphisms fij is called a directed system of A-modules. Assume that (Mi; fij) is a directed system of A-modules. Let M be a A-module and fi : Mi ! M be a family of A-homomorphisms such that fi = fj ◦ fij: We say that (M; fi) is the directed limit (unique up to isomorphism) if for any (N; gi); where N is a A-module, gi : Mi ! N is a A-homomorphism so that gi = gj ◦ fij; there is a unique A-homomorphism : M ! N so that gi = ◦ fi: Or equivalently, let us consider the category whose objects are (N; gi) and morphisms 0 0 0 0 :(N; gi) ! (N ; gi) are A-homomorphisms : N ! N so that gi = ◦ gi: The direct limit of the directed system (Mi; fij) is the universal object in this category. ` On the disjoint union i2I Mi; we define a relation ∼ as follows. Suppose xi 2 Mi and xj 2 Mj: We say that xi ∼ xj if there exists k 2 I with i ≤ k and j ≤ k so that ` fik(xi) = fjk(xj): The quotient set Mi= ∼ is denoted by M and the composition ` ` i2I Mi ! i2I Mi ! i2I Mi= ∼ of the inclusion and the quotient map is denoted by fi: Let us define an A-module structure on M as follows. Let [xi]; [yj] 2 M: Since I is directed, choose k so that i ≤ k and j ≤ k: Define the sum of [xi] and [yj] by [xi] + [yj] = [fik(xi) + fjk(yj)]: If a 2 A; define a[xi] = [axi]: Lemma 1.1. M is an A-module and fi : Mi ! M is an A-homomorphism. Moreover, (M; fi) is the directed limit of the directed system (Mi; fij): Proof. You need to check that the addition and the scalar multiplication on M is well- defined. The A-module structure on M implies that fi is an A-homomorphism. The proof of (M; fi) being the directed system of (Mi; fij) is routine: If gi : Mi ! N is an A-homomorphism for each i so that gi = fij ◦ gj for i ≤ j; we set : M ! N; [xi] 7! gi(xi): Check that is a well-defined A-homomorphism. Then gi = ◦fi follows from the definition of : The directed limit of the directed system (M ; f ) is denoted by lim M : i ij −! i i2I Remark. A Z-module is simply an abelian group. We also obtain the notion of directed limit of directed system of abelian groups. 1 2 2. Stalk of a pre sheaf of abelian groups on a Space Let X be a topological space and F be a pre sheaf of abelian groups on X: For each pair of open sets V ⊂ U; we have a restriction map rU;V : F(U) !F(V ): Let x be a point and I be the family of open neighborhoods of x: If U; V 2 I; we say U ≤ V if U contains V: Then I forms a directed set and (F(U); rU;V ) forms a directed system of abelian groups. The directed limit of the direct system is denoted by F = lim F(U) x −! x2U called the stalk of F at x: An element of Fx; called a germ of F at x; is represented by an equivalent class (s; U); where s 2 F(U): It follows from definition that (s; U) is equivalent to (t; V ) if there exists W ⊂ U \ V so that rU;W s = rV;W t:.