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

Home , Directed set

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 {Mi : i ∈ I} 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 ∈ I, (2) fik = fjk ◦ fik whenever i ≤ j ≤ k.

Then the family {Mi} 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 ψ : M → N so that gi = ψ ◦ fi.

Or equivalently, let us consider the category whose objects are (N, gi) and morphisms 0 0 0 0 ψ :(N, gi) → (N , gi) are A-homomorphisms ψ : N → N so that gi = ψ ◦ gi. The direct limit of the directed system (Mi, fij) is the universal object in this category. ` On the disjoint union i∈I Mi, we define a relation ∼ as follows. Suppose xi ∈ Mi and xj ∈ Mj. We say that xi ∼ xj if there exists k ∈ I with i ≤ k and j ≤ k so that ` fik(xi) = fjk(xj). The quotient set Mi/ ∼ is denoted by M and the composition ` ` i∈I Mi → i∈I Mi → i∈I 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] ∈ 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 ∈ 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

ψ : M → N, [xi] 7→ gi(xi).

Check that ψ is a well-deﬁned A-homomorphism. Then gi = ψ◦fi follows from the deﬁnition of ψ. The directed limit of the directed system (M , f ) is denoted by lim M . i ij −→ i i∈I

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 ∈ 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 −→ x∈U 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 ∈ 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.