FOM: Expert gets to the point

[Colin McLarty]

Reply to Friedman Wed, 14 Apr 1999 20:27:52  

        I asked for the name of any text on cohomological number theory that
does not use universes. I like the answer:

Friedman:
>Hartshorne, Algebraic Geometry, for one - says my third expert in number
>theory.

        Now turn to Chapter 3 of Hartshorne. This is no "few remarks at the
beginning". It is 80 pages in the middle and is used for key proofs in the rest.

        It deals with "the derived functor approach of Grothendieck" (page
201) and puts the issue very well. I quote one paragraph:

        "We will take as our basic definition the derived functors of the
global section functor. This definition is the most general and also best
suited for theoretical purposes, such as the proof of Serre duality in
section 7. However it is practically impossible to calculate, so we
introduce Cech cohomology..." (page 201)

        The specific calculations could, in principle, be done and justified
without the general theorems. No one actually does that.

        The key theorem ineliminably quantifies over universe sized sets.
You could eliminate the theorem from number theory if you liked, but not
universes from the theorem. Hartshorne defines the "right derived functors"
of any left exact functor F:A-->B between Abelian categories (when A has
enough injectives). They are a series of functors T^n(F):A-->B, one for each
natural number n, and with a little additional structure they form a single
"Delta-functor" from T*(F) from A to B. Each Delta functor from A to B is a
universe sized set.

        The theorem: For every Delta functor H from A to B, if H^n(X) = 0
for every n and every injective object X of A, and H^0 is naturally
isomorphic to F; then H is isomorphic as a delta functor to T*(F).

        I.e. the right derived functor T*(F) is defined up to isomorphism as
the delta functor whose 0-part is F and which kills injectives.

        That last definition is used in proofs throughout Chapter 3, first
to prove that derived functor cohomology agrees with Cech cohomology. Most
notably to prove Serre duality which relates the topology of a scheme X to
the algebra of certain Abelian sheaves on X. Serre duality is used to prove
Riemann-Roch (for curves and surfaces over any algebraically closed field).
Riemann-Roch is used to prove any curve embeds into 3-dim projective space,
and is birationally equivalent to a plane curve whose singularities are all
nodes. Cohomology is used on every page and virtually every paragraph. Other
derived functors besides cohomology are also important in the book.

        Hartshorne quantifies over many other universe sized sets, of
course, already in chapter 2. But the case above is central, and basic to
all derived functor cohomology, which is the basis of cohomological number
theory today. 

        Is this "excessive generality"? In a proof theoretic sense, if you
only want the theorems in number theory, it sure is. In a practical sense,
no one would pursue a career in algebraic number theory without studying
Hartshorne.