Brief aside on schemes

Let us briefly review some definitions.

Definition 10.1

A presheaf of sets (rings/modules) on a topological space is a set (ring/module) for every open set , together with the following data:

  1. For every inclusion of open sets in we have a restriction map which is a map of sets (rings/modules).
  2. For open sets in , .
  3. For all open sets , the restriction map is the identity.

A presheaf is a sheaf if we have an additional two axioms:

  1. Identity axiom: If is an open set, , and is an open covering of , such that for all ,

then .

  1. Gluability axiom: If is an open set in , and is an open covering of , and for every we have such that for all ,

then there exists an such that for all .

A morphism of sheaves is a morphism for every open set, such that for every inclusion of open sets in the following diagram commutes:

Tikz diagram 4.png
 

For , the stalk at is the set of equivalence classes under the equivalence relation if there exists an open such that . We denote the germ as .

Presheaves can be made into sheaves by sheafification. This can be expressed by a universal property, and hence is unique up to unique isomorphism.

Let be a continuous map of topological spaces, and a sheaf on . The direct image of is the presheaf on given by, for open, . This is in fact a sheaf. There is also a pullback of a sheaf, .

Definition 10.2

A ringed space is a topological space together with a sheaf of rings on . A morphism of ringed spaces is a continuous map of topological spaces , and a morphism of sheaves on , . Equivalently, we could define to be a morphism of sheaves on , . A locally ringed space is a ringed space such that ring of germs at each point is local. A morphism of locally ringed spaces is a morphism of ringed spaces, with the additional requirement that it takes the maximal ideal of the germ in to the maximal ideal of the germ in for every . Morphisms of locally ringed spaces induce maps of stalks. That is, if , there is induced morphism of rings , where as needed.

 

A commutative ring with unity can be made into a locally ringed space using the Spec functor. Let be a ring. As a topological space, let . Define maps

The Zariski topology on is defined by saying is closed for every . The following lemma lists some well-known facts about these maps.

Lemma 10.1

Let be a ring, an ideal in , and a closed set.

  1. , where .
 

The topology on has as an open basis for all . We think of elements in as functions on , where the value of at is the projection of in . However, because of nilpotents (which are precisely elements in ), functions may not be determined by their values at points. In the particular case of the ring , where is an algebraically closed field, functions (elements in ) are determined by their values on the spectrum, and moreover, they are determined by their value at the maximal ideals of , which are in one to one correspondence with elements in .

The ringed space is as a topological space, together with the sheaf on the base of distinguished open set (sets of the form ), where is the localization of at the set of all elements . This in fact defines a sheaf on a base. An affine scheme is a ringed space which is isomorphic to for some ring . A scheme is a ringed space which can be covered by open sets such that is an affine scheme. If is a morphism of commutative rings, then it induces a morphisms of affine sheaves . We want morphisms of schemes to locally look like the morphisms that arise in this way. One can define morphisms of schemes like this, but equivalently, morphisms of locally ringed spaces coincide with them, which gives an alternative definition.

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