alternative lecture notes - Rational points and algebraic cycles

Proof. We have
Conclude by Yoneda.
Hom(A, lim R ◦ F ) = lim Hom(A, RF (d))
←− ←−
d∈D
= lim Hom(L(A), F (d))
←−
d∈D
= Hom(L(A), lim ←−
F (d))
= Hom(A, R lim ←−
F (d)).
□
Similarly, a functor is cocontinuous if it commutes with colimits. If L has a right adjoint,
then L is cocontinuous.
Does the converse to Theorem 3.5 hold? Not quite, but essentially yes.
Theorem 3.6. Let C, D be complete, and let R: C → D be continuous. If some smallness
requirement holds on C and D, then R has a left adjoint.
(Categories arising in practice satisfy the small requirement. The problem can be solved
also by using Grothendieck’s universes.)
Proof. Suppose that we have R: C → D. Given d ∈ D, define the comma category d/R
whose objects are pairs (c ∈ C, d → R(c)) and whose morphisms from c ′ 1 → R(c 1 ) to
c ′ f
2 → R(c 2 ) are morphisms c 1 → c2 such that the associated R(f): R(c 1 ) → R(c 2 ) makes a
triangle with the two given morphisms. Define L(d) as a limit over (c, d → R(c)) ∈ d/R of c;
this makes sense if the category d/R is small. We now construct the unit and co-unit. The
unit id D → R◦L is defined by d → R(lim c) = lim R(c). The co-unit L◦R → id C
←−d→R(c) ←−d→R(c)
is defined by lim ˜c → c.
□
←−R(c)→R(˜c)
Assume that D is a full subcategory of C. Let i: D → C be the inclusion functor. Assume
that C and D are complete and that i is continuous. Then we have a left adjoint L ⊣ i.
Then L is called a localization functor.
Example 3.7. Let C be the category of all groups. Let D be the full subcategory of abelian
groups. The inclusion functor is continuous (limits of diagrams of abelian groups in the
category of groups are abelian groups). Then L is the functor sending each group G to its
abelianization.
Example 3.8. If C is the category of presheaves and D is the full subcategory of sheaves,
then L is the sheafification functor.
Example 3.9. Let C be the category of topological spaces, and let D be the full subcategory
of compact Hausdorff topological spaces. Then L is the Stone-Čech compactification functor.
Example 3.10. Let C be the category of abelian groups. Let D be the full subcategory of
Q-vector spaces. Then L is ⊗Q.
f
Call c 0 → c1 in C a D-equivalence if L(f) is an isomorphism in D. Then inverting all the
D-equivalences in C yields the category D.
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