Chapter 5

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 June 2013

Affine schemes

The structure sheaf of Spec(A)

We shall put a sheaf of rings on X=Spec(A) (where A is any commutative ring) in such a way that the stalk of the sheaf at xX is the local ring Ax (i.e. the local ring of A with respect to the prime ideal jx). For this we use the open sets D(f) (fA) and the rings of fractions Af (Chapter 3, Ex. 4 and Prop. (3.6)).

Suppose f,g in A are such that D(f)D(g). Then r(f)r(g), so that gn=sf for some sA and some n>0. Define a ring homomorphism


as follows: ρg,g(a/fm)= asm/gmnAg. Verify that ρg,f is a well-defined ring homomorphism depending only on f and g (and not on the particular equation gn=sf chosen), and that if D(f)D(g)D(h) then ρh,gρg,f=ρh,f. Then the assignment D(f)Af (and the homomorphisms ρg,f) forms a presheaf on the basis =(D(f))fA. This presheaf on determines a presheaf on X, denoted by 𝒪X or by A.

Proposition (5.1).

(i) The stalk of 𝒪X at xX is isomorphic to Ax.
(ii) 𝒪X is a sheaf on X, and hence Γ(D(f),𝒪X)Af for all fA. In particular Γ(X,𝒪X)A.


(i) is a straightforward verification: xD(f) if and only if fjx, and a/fnAf maps to a/fnAjx=Ax. Check that this gives an isomorphism of limAf onto Ax.

(ii) does require proof. We have to show that the condition of (4.3) is satisfied. First, by (3.6) D(f) is canonically homeomorphic to Spec(Af); also it is easily checked that the presheaf Af on Spec(Af) constructed as above is canonically isomorphic to A|D(f). Hence it is enough to show that Γ(X,𝒪X)=A, i.e. that if (D(fi))iI is any covering of X by basic open sets, and if siAfi are such that the images of si,sj in Ag are the same for all gA such that D(g)D(fi)D(fj), then there exists a unique sA whose image in Afi is si, for all iI.

Uniqueness: if s,sA are solutions of this problem, then t=s-s has zero image in each Afi, hence for each iI we have tfini=0 for some ni>0. Since the D(fi) cover X and since D(fi)=D(fini), the ideal generated by the fini is the whole of A. Consequently we have an equation of the form 1=aifini (aiA), and hence t=aitfini=0. Therefore s=s.

Existence: X is quasi-compact by (3.3), hence there is a finite subset J of I such that X=iJD(fi). Say si=zi/fini (iJ), where ziA. Since J is finite we may suppose that all the mi are equal: say si=zi/fim (iJ). For each pair i,j in J the images of si and sj in Afifj are the same, so that

zifjm/ (fifj)m= zjfim/ (fifj)m inAfifj,


(zifjm-zjfim) (fifj)mij =0inA,

for some integer mij. Again, we may assume that all the mij are equal, say mij=n for all i,jJ; then, multiplying each zi by fin, we reduce to the case n=0, i.e.

zifjm=zj fim.

Now the D(fi)=D(fim) (iJ) cover X, hence the ideal generated by the fim is the whole of A, so that we have an equation of the form

1=iJ gims fim (giA).

Put sJ=iJgizi; then

sJfkm=igi zifkm=igi zjfim=zjinA,

so that the image of sJ in Afj is aj/fjm=sj (for all jJ). On the face of it, sJ depends on the finite subset J; but if JJ is another finite subset of I, we construct sJ, satisfying the same conditions as sJ, and by the uniqueness of the solution sJ, must therefore be equal to sJ.

Thus, starting from an arbitrary commutative ring A, we have constructed a topological space X=Spec(A) and a sheaf of local rings 𝒪X (or A) on X. This is the basic construction on which all else is founded. The ringed space (X,𝒪X) is called the affine scheme of the ring A.

Morphisms of affine schemes

Let A,B be rings, X=Spec(A), Y=Spec(B), and let φ:BA be a ring homomorphism. We have seen in Chapter 3 that φ defines an associated continuous mapping aφ:XY. In fact φ defines a morphism of ringed spaces (aφ,φ): (X,𝒪X) (Y,𝒪Y), as follows. Let gB, then D(g) is a basic open set in Y, and we have aφ-1(D(g))= D(φ(g)) by (3.5). How φ induces a homomorphism BgAφ(g), namely b/gn is mapped to φ(b)/φ(g)n; hence by (5.1) (ii) φ induces a homomorphism

φD(g): Γ(D(g),𝒪) γ ( aφ-1 (D(g)), 𝒪X ) .

Clearly the φD(g) are compatible with the restriction homomorphisms, hence we have φ:𝒪Y𝒪X as required. φ induces a homomorphism of the stalks: if y=aφ(x), we have φx#:𝒪Y,y𝒪X,x. Now 𝒪Y,yBy, 𝒪X,xAx, and the map ByAx is the obvious one: b/sBy is mapped to φ(b)/φ(s)Ax.

If P,Q are local rings, m and n their respective maximal ideals, a homomorphism f:PQ is said to be local if the following equivalent conditions are satisfied:

(i) f(m)n (i.e. the image of a non-unit is a non-unit);
(ii) f-1(n)=m (i.e. the inverse image of a unit is a unit).

If so, then f induces a field monomorphism P/mQ/n.

Now in the case in point, the homomorphism ByAx is local, for the maximal ideal of Ax is jxAx and that of By is jyBy=φ-1(jx)Ax. Hence the morphism (aφ,φ): (X,𝒪X) (Y,𝒪Y) has the property that the homomorphisms induced on the stalks are local homomorphisms.

Conversely, let (ψ,θ): (X,𝒪X) (Y,𝒪Y) be a morphism of ringed spaces (where X=Spec(A), Y=Spec(B) and 𝒪X,𝒪Y are the structure sheaves A,B) such that θx#:ByAx is a local homomorphism for each xX (y=ψ(x)). We have then, in particular, a ring homomorphism θ(Y): Γ(Y,𝒪Y) Γ(X,𝒪X); but Γ(Y,𝒪Y)B and Γ(X,𝒪X)A by (5.1), hence (ψ,θ) determines a ring homomorphism φ:BA. Since θx# is local, it gives rise to an embedding


of the residue fields, such that for each gB we have θx(g(y))=ϕ(g)(x). Since θx is injective we have g(y)=0 if and only if φ(g)(x)=0, i.e. gjy if and only if φ(g)jx, so that jy=φ-1(jx), i.e. y=aφ(x); hence ψ=aφ. Moreover, the diagram

B φ A By θx# Ax

is commutative, hence θx# is the homomorphism of By into Ax induced by φ; but θ is uniquely determined by the homomorphisms θx#, and therefore (ψ,θ)=(aφ,φ). We have therefore proved

Proposition (5.2). There is a one-to-one correspondence between the ring homomorphisms BA and the morphisms (ψ,θ): (X,𝒪X) (Y,𝒪Y) such that θx# is a local homomorphism for each xX.

So far, this is the basic local theory. The next step is to define the global objects. By analogy with Serre's definition of an algebraic variety (Chapter 1), it is clear what the general definition should be.

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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