Last update: 3 June 2013
We shall put a sheaf of rings on (where is any commutative ring) in such a way that the stalk of the sheaf at is the local ring (i.e. the local ring of with respect to the prime ideal For this we use the open sets and the rings of fractions (Chapter 3, Ex. 4 and Prop. (3.6)).
Suppose in are such that Then so that for some and some Define a ring homomorphism
as follows: Verify that is a well-defined ring homomorphism depending only on and (and not on the particular equation chosen), and that if then Then the assignment (and the homomorphisms forms a presheaf on the basis This presheaf on determines a presheaf on denoted by or by
|(i)||The stalk of at is isomorphic to|
|(ii)||is a sheaf on and hence for all In particular|
(i) is a straightforward verification: if and only if and maps to Check that this gives an isomorphism of onto
(ii) does require proof. We have to show that the condition of (4.3) is satisfied. First, by (3.6) is canonically homeomorphic to also it is easily checked that the presheaf on constructed as above is canonically isomorphic to Hence it is enough to show that i.e. that if is any covering of by basic open sets, and if are such that the images of in are the same for all such that then there exists a unique whose image in is for all
Uniqueness: if are solutions of this problem, then has zero image in each hence for each we have for some Since the cover and since the ideal generated by the is the whole of Consequently we have an equation of the form and hence Therefore
Existence: is quasi-compact by (3.3), hence there is a finite subset of such that Say where Since is finite we may suppose that all the are equal: say For each pair in the images of and in are the same, so that
for some integer Again, we may assume that all the are equal, say for all then, multiplying each by we reduce to the case i.e.
Now the cover hence the ideal generated by the is the whole of so that we have an equation of the form
so that the image of in is (for all On the face of it, depends on the finite subset but if is another finite subset of we construct satisfying the same conditions as and by the uniqueness of the solution must therefore be equal to
Thus, starting from an arbitrary commutative ring we have constructed a topological space and a sheaf of local rings (or on This is the basic construction on which all else is founded. The ringed space is called the affine scheme of the ring
Let be rings, and let be a ring homomorphism. We have seen in Chapter 3 that defines an associated continuous mapping In fact defines a morphism of ringed spaces as follows. Let then is a basic open set in and we have by (3.5). How induces a homomorphism namely is mapped to hence by (5.1) (ii) induces a homomorphism
Clearly the are compatible with the restriction homomorphisms, hence we have as required. induces a homomorphism of the stalks: if we have Now and the map is the obvious one: is mapped to
If are local rings, and their respective maximal ideals, a homomorphism is said to be local if the following equivalent conditions are satisfied:
|(i)||(i.e. the image of a non-unit is a non-unit);|
|(ii)||(i.e. the inverse image of a unit is a unit).|
If so, then induces a field monomorphism
Now in the case in point, the homomorphism is local, for the maximal ideal of is and that of is Hence the morphism has the property that the homomorphisms induced on the stalks are local homomorphisms.
Conversely, let be a morphism of ringed spaces (where and are the structure sheaves such that is a local homomorphism for each We have then, in particular, a ring homomorphism but and by (5.1), hence determines a ring homomorphism Since is local, it gives rise to an embedding
of the residue fields, such that for each we have Since is injective we have if and only if i.e. if and only if so that i.e. hence Moreover, the diagram
is commutative, hence is the homomorphism of into induced by but is uniquely determined by the homomorphisms and therefore We have therefore proved
Proposition (5.2). There is a one-to-one correspondence between the ring homomorphisms and the morphisms such that is a local homomorphism for each
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.
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".