# 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

## 全文

(1)

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 Mx. 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 My.

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 ρUV(s) = {s(x)}x∈V.

We set ρU : 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 {Ui} be an open covering of U. Let s ∈ fM (U ). Assume that ρUU

i(s) = 0 for all i. Since ρUU

i(s) = {s(x)}x∈Ui, by assumption, ρUU

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

Let U and {Ui} be as above. Assume that si ∈ fM (Ui) so that ρUUi

i∩Uj(si) = ρUUj

i∩Uj(sj).

By definition, we know ρUUi

i∩Uj(si) = {si(x)}x∈Ui∩Uj = {sj(x)}x∈Ui∩Uj = ρUUj

i∩Uj(sj).

This implies that si(x) = sj(x) for all x ∈ Ui∩ Uj. 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

ρUUi(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 OX. Since each Ax 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)

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

ρU = ρV ◦ ρUV. This observation leads to the following lemma.

Lemma 1.3. Let x ∈ X = Spec A. The stalk Ox = Ax 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, OSpec 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, OU) is isomorphic to an affine scheme. Here OU is the restriction of OX 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 OX(U )-structure such that for each open sets V ⊂ U of X,

ρUV(as) = σVU(a)ρUV(s),

where σVU : OX(U ) → OX(V ) and ρUV : 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 OX-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 OX(U )-module for each open set U of X. Moreover, for each open neighborhood U of x, there is a natural map ρU : fM (U ) → Mx by ρU(s) = s(x). It is easy to see that ρU = ρVρUV for every open sets V ⊂ U. As above, we can verify that Mx is the directed limit of the directed system ( fM (U ), ρUV). Then

Mfx∼= 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 Af-homomorphism.

Lemma 1.4. The Af-homomorphism ψ : Mf → fM (D(f )) is an isomorphism. When M = A, we have a natural ring isomorphism Af ∼= 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 Ui ∼= 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 Ai-modules Mi given above is finitely generated.

(3)

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.

Updating...

Updating...