CHAPTER 1
Schemes
It is the purpose of this first chapter to give the necessary introduction to
schemes following the functorial approach of [DG]. This approach appears to be
most suitable when dealing with group schemes later on. After trying to motivate
the definitions in 1.1, we discuss affine schemes in 1.2-1.6. What is done there is
fundamental for the understanding of everything to follow.
As far as arbitrary schemes are concerned, it is most of the time enough to know
that they form a certain class of functors that includes the affine schemes and so that
over an algebraically closed field any variety gives rise to a scheme in a canonical
way. Sometimes, e.g., when dealing with quotients, it is useful to know more. So we
give the appropriate definitions in 1.7-1.9 and mention the comparison with other
approaches to schemes and with varieties in 1.11. The elementary discussion of a
base change in 1.10 is again necessary for many parts later on.
There is also a discussion of closed subfunctors and of closures (1.12-1.14).
Finally, we describe the functor of morphisms between two functors (1.15) and
prove some of its properties. These results are used only in few places.
A ring or an associative algebra will always be assumed to have a 1, and
homomorphisms are assumed to respect this 1. Let H e a fixed commutative ring.
Notations of linear algebra (like Horn, ®) without special reference to a ground
ring always refer to structures as ^-modules. A fc-algebra is always assumed to
be commutative and associative. (For noncommutative algebras we shall use the
terminology: algebras over k.)
1.1. Before giving the definitions, I want to point out how functors arise naturally
in algebraic geometry. Assume for the moment that k is an algebraically closed field.
Consider a Zariski closed subset X of some
kn
and denote by / the ideal of
all polynomials / G /c[Ti,T2,... ,Tn] (over k in n variables Ti,T
2
,... ,Tn) with
f(X) 0. Instead of looking at the zeroes of I only over k, we can also look at the
zeroes over any /c-algebra A, i.e., at X(A) = {x G An \ f(x) = 0 for all / G / } . The
map A i— » X(A) from { A;-algebras } to {sets} is a functor: Any homomorphism
p : A A! of ^-algebras induces a map
ipn : An - (A')n, (ai, a
2
,..., an) •- (p(ai), p(a2), •, p{an))
with f(pn(x)) = p(f(x)) for all x G An and / G fc[Ti,T2,... ,T
n
]. Therefore (pn
maps X(A) to X(A'). Denote its restriction by X(ip) : X(A) - X(A'). For another
homomorphism
(pf
: A' » A" of /c-algebras, one has obviously
X((pf)
o X(tp) =
X(ip' o £), proving that X is indeed a functor.
A regular map / from A T to a Zariski closed subset Y of some
km
is given by
m polynomials /1? /
2
, . . . , fm G k[Tu T
2
,..., Tn] as
/ : X - y , x»(h(x)J2{x\...Jm(x)).
3
http://dx.doi.org/10.1090/surv/107/01
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