Algebraic theory of complex multiplication
The theory of complex multiplication. . . is not only the most beautiful
part of mathematics but also of all science.
David Hilbert
1.1. Introduction
1.1.1. Lifting questions. A natural question early in the theory of abelian vari-
eties is whether every abelian variety in positive characteristic admits a lift to char-
acteristic 0. That is, given an abelian variety A0 over a field κ with char(κ) 0,
does there exist a local domain R of characteristic zero with residue field κ and
an abelian scheme A over R whose special fiber is isomorphic to A0? We may
also wish to demand that a specified polarization of A0 or subring of the endomor-
phism algebra of A0 (or both) also lifts to A. (The functor A from abelian
R-schemes to abelian varieties over κ is faithful, by consideration of finite ´etale
torsion levels; see the beginning of 1.4.4.)
Suppose there is an affirmative solution A to such a lifting problem over some
local domain R as above. Let’s see that we can arrange for a solution to be found
over a local noetherian domain (that is even complete). This rests on a direct limit
technique (that is very useful throughout algebraic geometry), as follows. Observe
that for the directed system of noetherian local subrings Ri with local inclusions
Ri R, we have R = lim

Ri. In [34, IV3, §8–§12; IV4, §17] there is an exhaustive
development of the technique of descent through direct limits. The principle is
that if {Di} is a directed system of rings with limit D, and if we are given a
“finitely presented” algebro-geometric situation over D (a diagram of finitely many
D-schemes of finite presentation, equipped with with finitely many D-morphisms
among them and perhaps some finitely presented quasi-coherent sheaves on them,
some of which may be D-flat, etc.) then the entire structure descends to Di for
sufficiently large i. Moreover, if we increase i enough then we can also descend
“reasonable” properties (such as flatness for morphisms or sheaves, and properness,
surjectivity, smoothness, and having geometrically connected fibers for morphisms),
any two descents become isomorphic after increasing i some more, and so on.
The results of this direct limit formalism are intuitively plausible, but their
proofs can be rather non-obvious to the uninitiated (e.g., descending the properties
of flatness and surjectivity). We will often use this limit formalism without much
explanation, and we hope that the plausibility of such results is sufficient for a non-
expert reader to follow the ideas. Everything we need is completed proved in the
cited sections of [34]. As a basic example, since the condition of being an abelian
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