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}_{U}^{i}

i∩U_{j}(s_{i}) = ρ^{U}_{U}^{j}

i∩U_{j}(s_{j}).

By definition, we know
ρ^{U}_{U}^{i}

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.

1

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,

ρ^{U}_{V}(as) = σ_{V}^{U}(a)ρ^{U}_{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/f^{n} 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/f^{n}. 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/f^{n}) = 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.

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.