alternative lecture notes - Rational points and algebraic cycles

Loosely speaking, given functors L: A → B and R: B → A, they are called adjoint
functors if there is a bijection Hom B (L(a), b) = Hom A (a, R(b)) varying functorially in a ∈ A
and b ∈ B.
Definition 3.1. A pair of functors L: A → B and R: B → A together with two natural
transformations u: id A → R ◦ L (called the unit) and c: L ◦ R → id B (called the co-unit) is
called an adjunction if for every a ∈ A the composition
is id L(a) , and for every b ∈ B the composition
is id R(b) .
and
Given an adjunction as above, we get
L(a) L◦u
−→ LRL(a) c◦L
−→ L(a)
R(b) u◦R
−→ RLR(b) R◦c
−→ R(b)
Hom(L(a), b) → Hom(RL(a), R(b) u → Hom(a, R(b))
Hom(L(a), b) c ← Hom(L(a), LR(b)) ← Hom(a, R(b)).
Example 3.2. Let U be the functor Sets ← Groups forgetting the group structure. (Do
not confuse U with the unit.) Let Fr: Sets → Groups be the functor sending a set to the
free group on the set. Then
and in fact we get an adjunction.
Hom Groups (Fr(S), G) = Hom Sets (S, U(G)),
Example 3.3. Let R be commutative ring. Let A be an R-module. Then ⊗A: R-modules →
R-modules and Hom(A, −) in the other direction are adjoint functors: in particular,
Hom(B ⊗ A, C) = Hom(B, Hom(A, C)).
Example 3.4. Let R → S be a ring homomorphism. Then ⊗ R S : R-modules → S-modules
is left adjoint to the functor U in the opposite direction that maps an S-module to the
R-module obtained by composing the action with R → S.
Let D be a small category. Let const: C → C D be the functor taking an object to the
constant diagram. A limit functor is a functor lim: C D → C that is right adjoint to const.
←−
Notation: Write L ⊣ R if (L, R) are a pair of adjoint functors. Thus const ⊣ lim. Similarly
←−
lim
−→ ⊣ const.
A limit functor does not always exist. In fact, individual limits need not exist. For
example: let D be the 3-object category with two morphisms to the same object. If C is the
same category, the identity functor D → C does not have a limit.
Call C complete if all limits of diagrams D → C exist, for every small category D. A
functor R: C → D is called continuous if it respects limits. There is always a morphism
R ( lim F ) → lim R◦F , from the universal property of the latter. To say that R is continuous
←− ←−
means that this morphism is always an isomorphism.
Theorem 3.5. If R has a left adjoint, then R is continuous.
6