be the trivial group and for each nonempty open set U of X, let fM (U ) be the set of all {s(x)}x∈U such that (1) for each x ∈ U, s(x

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AFFINE SCHEMES AND THEIR SHEAVES OF MODULES

All the rings are commutative with identity and all the modules are unitary.

1. Sheaves on Spec A

Let A be a ring and M be an A-module. Denote Spec A by X. For each x ∈ X, the localization of M at x is denoted by M_x. Let fM (∅) be the trivial group and for each nonempty open set U of X, let fM (U ) be the set of all {s(x)}_x∈U such that

(1) for each x ∈ U, s(x) ∈ Mx,

(2) for each x ∈ U, there is an open neighborhood V of x and element m ∈ M, f ∈ A such that for each y ∈ V, f 6∈ y and s(y) = m/f in M_y.

Since each Mx is a group, given s, t ∈ fM (U ), we can define

{s(x)}_x∈U + {t(x)}_x∈U = {s(x) + t(x)}_x∈U.

Then fM (U ) forms an abelian group, where the zero of this group is the zero function 0 = {0}x∈U.

Let V be a nonempty open subset of an open set U in X. We define ρ^U_V(s) = {s(x)}x∈V.

We set ρ^U_∅ : fM (U ) → fM (∅) be the zero homomorphism.

Lemma 1.1. The assignment U 7→ fM (U ) defines an abelian sheaf on X.

Proof. It follows from definition that the assignment is a pre sheaf of abelian groups.

Let U be an open set of X and {U_i} be an open covering of U. Let s ∈ fM (U ). Assume that ρ^U_U

i(s) = 0 for all i. Since ρ^U_U

i(s) = {s(x)}_x∈U_i, by assumption, ρ^U_U

i(s) = 0 implies that s(x) = 0 for all x ∈ U_i. Since {U_i} forms an open cover, s(x) = 0 for all x ∈ U. Hence s = 0.

Let U and {U_i} be as above. Assume that s_i ∈ fM (U_i) so that ρ^U_Uⁱ

i∩U_j(s_i) = ρ^U_U^j

i∩U_j(s_j).

By definition, we know ρ^U_Uⁱ

i∩Uj(s_i) = {s_i(x)}_x∈U_i∩U_j = {s_j(x)}_x∈U_i∩U_j = ρ^U_U^j

i∩Uj(s_j).

This implies that si(x) = sj(x) for all x ∈ Ui∩ U_j. Define s(x) = si(x) if x ∈ Ui. Then s(x) is well-defined for each x ∈ U. We set s = {s(x)}_x∈U. Then we have

ρ^U_U_i(s) = {s(x)}x∈Ui = {si(x)}x∈Ui.

We complete the proof of our assertion.

Definition 1.1. The abelian sheaf fM is called the sheaf associated with the A-module M.

Let us take M = A. The associated abelian sheaf is denoted by O_X. Since each A_x is a ring, for each s, t ∈ OX(U ), we can define

{s(x)}_x∈U · {t(x)}_x∈U = {s(x)t(x)}_x∈U. This gives a ring structure on OX(U ).

Lemma 1.2. The abelian sheaf OX on X is a sheaf of rings on X.

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2 AFFINE SCHEMES AND THEIR SHEAVES OF MODULES

This gives us a ringed space (X, OX).

For each open neighborhood U of x ∈ Spec A, we define a ring homomorphism ρU : OX(U ) → Ax

by ρ_U(s) = s(x). Then for V ⊂ U, we know

ρ_U = ρ_V ◦ ρ^U_V. This observation leads to the following lemma.

Lemma 1.3. Let x ∈ X = Spec A. The stalk O_x = A_x and hence is a local ring.

The ringed space (X, OX), where X = Spec A, is a locally ringed space. A locally ringed space is called an affine scheme if it is isomorphic to (Spec A, O_{Spec A}) in the category of locally ringed spaces. A scheme is a locally ringed space (X, OX) with the property that each point x ∈ X has an open neighborhood U such that the locally ringed space (U, O_U) is isomorphic to an affine scheme. Here O_U is the restriction of O_X to U. An open subset U of a scheme X is called affine if U ∼= Spec A.

Let (X, OX) be a ringed space. An abelian sheaf F on X is called a sheaf of OX-module if for each open set U of X, the group F (U ) as a O_X(U )-structure such that for each open sets V ⊂ U of X,

ρÛ_V(as) = σ_VÛ(a)ρÛ_V(s),

where σ_V^U : OX(U ) → OX(V ) and ρ^U_V : F (U ) → F (V ) are the corresponding restriction maps.

Corollary 1.1. Let M be an A-module. The abelian sheaf fM on X = Spec A is a sheaf of O_X-module. Moreover, the stalk fMx at x is isomorphic to Mx.

Proof. Since each Mx is a Ax-module, we find fM (U ) is a O_X(U )-module for each open set U of X. Moreover, for each open neighborhood U of x, there is a natural map ρ_U : fM (U ) → Mx by ρU(s) = s(x). It is easy to see that ρU = ρVρ^U_V for every open sets V ⊂ U. As above, we can verify that M_x is the directed limit of the directed system ( fM (U ), ρ^U_V). Then

Mf_x∼= Mx.

Let m/fⁿ be an element of Mf for f ∈ A. For each x ∈ D(f ), we know f 6∈ x We define s(x) ∈ Mxby s(x) = m/fⁿ. This gives an element s = {s(x)}_{x∈D(f )}in fM (D(f )). We obtain a map

ψ : Mf → fM (D(f ))

by ψ(m/fⁿ) = s. This map ψ is obvious a A_f-homomorphism.

Lemma 1.4. The A_f-homomorphism ψ : M_f → fM (D(f )) is an isomorphism. When M = A, we have a natural ring isomorphism A_f ∼= OX(D(f )).

Let X be a scheme. A sheaf F of OX-modules on X is a quasi-coherent sheaf if there is an affine open covering {Ui} of X with U_i ∼= Spec Ai and an Ai-module Mi so that F |Ui ∼= fMi. A coherent sheaf F on X is a quasi-coherent sheaf so that each A_i-modules M_i given above is finitely generated.

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AFFINE SCHEMES AND THEIR SHEAVES OF MODULES 3

Theorem 1.1. Let X = Spec A for a ring A. The functor M 7→ fM

gives an equivalence of categories between the category of A-modules and the category of quasi-coherent sheaves on X. If A is noetherian, the same functor gives an equivalence between the category of finite A-modules and the category of coherent sheaves on X.