(N0, gi0) are A-homomorphisms ψ : N → N0 so that gi0 = ψ ◦ gi

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→ M_j such that (1) f_ii is the identity map on M_i for all i ∈ I,

(2) f_ik = f_jk◦ f_ik whenever i ≤ j ≤ k.

Then the family {M_i} of A-modules together with A-homomorphisms f_ij 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 f_i : M_i→ M be a family of A-homomorphisms such that f_i= f_j◦ f_ij. We say that (M, f_i) is the directed limit (unique up to isomorphism) if for any (N, g_i), where N is a A-module, gi: Mi → N is a A-homomorphism so that g_i= gj◦ f_ij, there is a unique A-homomorphism ψ : M → N so that g_i= ψ ◦ f_i.

Or equivalently, let us consider the category whose objects are (N, g_i) and morphisms ψ : (N, gi) → (N⁰, g_i⁰) are A-homomorphisms ψ : N → N⁰ so that g_i⁰ = ψ ◦ gi. The direct limit of the directed system (M_i, f_ij) is the universal object in this category.

On the disjoint union `

i∈IMi, we define a relation ∼ as follows. Suppose xi ∈ M_i and x_j ∈ M_j. We say that x_i ∼ x_j if there exists k ∈ I with i ≤ k and j ≤ k so that fik(xi) = fjk(xj). The quotient set `

i∈IMi/ ∼ is denoted by M and the composition M_i →`

i∈IM_i →`

i∈IM_i/ ∼ of the inclusion and the quotient map is denoted by f_i. 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 [x_i] and [y_j] by

[xi] + [yj] = [fik(xi) + fjk(yj)].

If a ∈ A, define a[x_i] = [ax_i].

Lemma 1.1. M is an A-module and fi : Mi → M is an A-homomorphism. Moreover, (M, f_i) is the directed limit of the directed system (M_i, f_ij).

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 g_i= f_ij◦ g_j for i ≤ j, we set

ψ : M → N, [xi] 7→ gi(xi).

Check that ψ is a well-defined A-homomorphism. Then g_i = ψ◦f_ifollows from the definition

of ψ. 

The directed limit of the directed system (Mi, fij) is denoted by lim−→

i∈I

Mi.

Remark. A Z-module is simply an abelian group. We also obtain the notion of directed limit of directed system of abelian groups.

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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

r_U,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_x = lim−→

x∈U

F (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,Ws = rV,Wt.