Algebraic geometry without prime ideals August 6, 2009Posted by Joel Kamnitzer in Algebraic Geometry, Anton Geraschenko, things I don't understand.
The first definition in “Grothendieck-style” algebraic geometry is the affine scheme for any ring . This is a topological space whose set of points in the set of prime ideals in . Then one defines a scheme to be a locally ringed space locally isomorphic to an affine scheme.
The definition of goes against intuition since it involves prime ideals, not just maximal ideals. Maximal ideals are more natural, since if for some alg closed field , then the set of maximal ideals of is in bijection with the vanishing set in the affine space of the ideal . (Of course one can give a geometric meaning to the prime ideals in terms of subvarieties, but it is less natural.)
However, in Daniel Perrin’s text Algebraic geometry, an introduction, he states/implies that one can define affine schemes just using maximal ideals (at least for finitely-generated algebras) and still get a good theory of schemes and varieties. Is this true? If so why don’t we all learn it this way? (One answer to the this latter question could be that some people are interested in non-algebraically closed fields.)
Let me be more specific. Fix an algebraically closed field . Suppose that is a finitely generated algebra. Then we define a locally ringed space whose points are the maximal ideals of . and we give it a sheaf of rings in the usual manner. Let us call these affine m-schemes over .
Then we define a “m-scheme over ” to be a locally ringed space which is locally isomorphic to an affine m-scheme over .
I believe we have a functor from finite type schemes over to m-schemes over . which on the level of topological spaces is taking closed points.
Is the following true:
This gives an equivalence of categories between the category of m-schemes over and the category of finite type schemes over .
Following a reference in Perrin, I looked at EGA IV, section 10.9 which discusses these issues. In particular, what I call “m-schemes” above seems to be called “ultra-schemes” there (or rather ultra-preschema). Proposition 10.9.6 shows that the functor from Jacobson schemes to ultra-schemes is an equivalence. (This differs in two ways from what I ask above, first I have no idea what a Jacobson scheme is, second this is not dealing with finite type schemes.)
One last question. One way to justify the usual is to say that is the adjoint functor to the global sections functor which goes from locally ringed spaces to rings. Is there a similar way to justify ?
(By the way, I’m asking these questions since I’m teaching introductory algebraic geometry in the fall and I’m planning on using Perrin’s book.)