The properties of give the geometry of lines meeting .
A trick
We consider a parameter :
If we send , we degenerate into a pair of lines, which might be a simpler object to study.
Now, we have a family of curves, which we call , fitting into the diagram
We write , and as , .
What is a line?
A dimension one subspace ? ❌
Consider the dual situation of quotients . ✅
Definition (Locally free of rank )
Let be a ring. An -module is locally free of rank if it is finitely-presented and for all . We note that if is a field, then is locally free of rank if and only if itself is a vector space of dimension .
Alternatively: such that .
is locally free of rank .
Definition (Projective space)
For , we define
where we say that two quotients and are isomorphic if there is an isomorphism such that the following diagram commutes:
Alternatively, we have that the kernels :
A module map is essentially a choice of elements of , with the condition that they generate . So when is a field, this means that the elements are not all . The equivalence condition above allows for the scaling of these quotients by some unit.
Pullbacks
For a ring homomorphism, we define a map via a base change
Note that is still locally free of rank : both conditions are stable under base change.
Consequences
The above defines a presheaf on affine schemes:
By YonedaYoneda, natural transformations are in bijection with isomorphism classes of quotients in . In particular,
as sets: characterises dimension quotients of and characterises rank quotients of .
Defining the charts
Recall that we had the map
This gave a compactification of the plane .
For a ring and , we define the following subsets of :
This is still stable under any base change. Therefore is a sub-functor/presheaf:
What does a chart look like?
Fix a choice of . Consider the following natural transformations:
Claim: We claim that these are natural isomorphisms. Proof. The maps for each object in are given by
The map , lands in because the composition
is an isomorphism. For the inverse map, note that an element of is given by a quotient , where the composition
is an isomorphism. So define (an isomorphism) and similarly for all (not necessarily isomorphisms). We define
□
How do the charts glue together?
Consider the sub-presheaf :
We want to work out how they overlap. For each object in , we have
is a unit in because is and isomorphism and is given by multiplication by . There is a bijection:
This is
What is ?
What are the functions on , e.g., what is ?
Now because the composition
equalises , there is a map of to the equaliser:
But the equaliser consists only of constants: for if
by a degree argument. It remains to work out if is injective:
We will show this later, but . We conclude that is not representable by an affine, since
by YonedaYoneda, but it is “covered” by things that are.
Definition (Separated presheaf, sheaf)
Let be a topological space, and another one. Let be an open set and be an open cover of . Then the following are true:
(Separated) For continuous, if for all then .
(Gluing) For a collection of continuous functions such that for all , then there exists a unique such that .
Alternatively, let . We can express the above conditions as the fact that