math 216: foundations of algebraic geometry - Stanford University

46 Math 216: Foundations of Algebraic Geometry
2.6.I. EXERCISE (KERNELS COMMUTE WITH LIMITS). Suppose C is an abelian
category, and a : I → C and b : I → C are two diagrams in C indexed by I .
For convenience, let Ai = a(i) and Bi = b(i) be the objects in those two diagrams.
Let hi : Ai → Bi be maps commuting with the maps in the diagram. (Translation:
h is a natural transformation of functors a → b, see §2.2.21.) Then the ker hi
form another diagram in C indexed by I . Describe a canonical isomorphism
lim
←− ker hi ∼ = ker(lim Ai → lim Bi).
←− ←−
2.6.J. EXERCISE. Make sense of the statement that “limits commute with limits” in
a general category, and prove it. (Hint: recall that kernels are limits. The previous
exercise should be a corollary of this one.)
2.6.13. Proposition (right-adjoints commute with limits). — Suppose (F : C →
D, G : D → C ) is a pair of adjoint functors. If A = lim Ai is a limit in D of a diagram
←−
indexed by I, then GA = lim GAi (with the corresponding maps GA → GAi) is a limit
←−
in C .
Proof. We must show that GA → GAi satisfies the universal property of limits.
Suppose we have maps W → GAi commuting with the maps of I . We wish to
show that there exists a unique W → GA extending the W → GAi. By adjointness
of F and G, we can restate this as: Suppose we have maps FW → Ai commuting
with the maps of I . We wish to show that there exists a unique FW → A extending
the FW → Ai. But this is precisely the universal property of the limit.
Of course, the dual statements to Exercise 2.6.J and Proposition 2.6.13 hold by
the dual arguments.
If F and G are additive functors between abelian categories, and (F, G) is an
adjoint pair, then (as kernels are limits and cokernels are colimits) G is left-exact
and F is right-exact.
2.6.K. EXERCISE. Show that in ModA, colimits over filtered index categories are
exact. (Your argument will apply without change to any abelian category whose
objects can be interpreted as “sets with additional structure”.) Right-exactness
follows from the above discussion, so the issue is left-exactness. (Possible hint:
After you show that localization is exact, Exercise 2.6.F(a), or sheafification is exact,
Exercise 3.5.D, in a hands-on way, you will be easily able to prove this. Conversely,
if you do this exercise, those two will be easy.)
2.6.L. EXERCISE. Show that filtered colimits commute with homology in ModA.
Hint: use the FHHF Theorem (Exercise 2.6.H), and the previous Exercise.
In light of Exercise 2.6.L, you may want to think about how limits (and colimits)
commute with homology in general, and which way maps go. The statement
of the FHHF Theorem should suggest the answer. (Are limits analogous to leftexact
functors, or right-exact functors?) We won’t directly use this insight.
2.6.14. ⋆ Dreaming of derived functors. When you see a left-exact functor, you
should always dream that you are seeing the end of a long exact sequence. If
0 → M ′ → M → M ′′ → 0