math 216: foundations of algebraic geometry - Stanford University

32 Math 216: Foundations of Algebraic Geometry
If I is a set (i.e. the only morphisms are the identity maps), then the limit is
called the product of the Ai, and is denoted
i Ai. The special case where I has
two elements is the example of the previous paragraph.
If I has an initial object e, then Ae is the limit, and in particular the limit
always exists.
2.4.3. Unimportant Example: the p-adic integers. For a prime number p, the p-adic
integers (or more informally, p-adics), Zp, are often described informally (and
somewhat unnaturally) as being of the form Zp = a0 + a1p + a2p 2 + · · · (where
0 ≤ ai < p). They are an example of a limit in the category of rings:
Zp
· · ·
3 Z/p
2 Z/p
Z/p.
(Warning: Zp is sometimes is used to denote the integers modulo p, but Z/(p) or
Z/pZ is better to use for this, to avoid confusion. Worse: by §2.3.3, Zp also denotes
those rationals whose denominators are a power of p. Hopefully the meaning of
Zp will be clear from the context.)
Limits do not always exist for any index category I . However, you can often
easily check that limits exist if the objects of your category can be interpreted as
sets with additional structure, and arbitrary products exist (respecting the set-like
structure).
2.4.A. IMPORTANT EXERCISE. Show that in the category Sets,
(ai)i∈I ∈
Ai : F(m)(aj) = ak for all m ∈ MorI (j, k) ∈ Mor(I ) ,
i
along with the obvious projection maps to each Ai, is the limit lim
←−I Ai.
This clearly also works in the category ModA of A-modules (in particular Veck
and Ab), as well as Rings.
From this point of view, 2 + 3p + 2p 2 + · · · ∈ Zp can be understood as the
sequence (2, 2 + 3p, 2 + 3p + 2p 2 , . . . ).
2.4.4. Colimits. More immediately relevant for us will be the dual (arrowreversed
version) of the notion of limit (or inverse limit). We just flip the arrows
fi in (2.4.1.1), and get the notion of a colimit, which is denoted lim
−→ I Ai. (You
should draw the corresponding diagram.) Again, if it exists, it is unique up to
unique isomorphism. (In some cases, the colimit is sometimes called the direct
limit, inductive limit, or injective limit. We won’t use this language. I prefer using
limit/colimit in analogy with kernel/cokernel and product/coproduct. This
is more than analogy, as kernels and products may be interpreted as limits, and
similarly with cokernels and coproducts. Also, I remember that kernels “map to”,
and cokernels are “mapped to”, which reminds me that a limit maps to all the objects
in the big commutative diagram indexed by I ; and a colimit has a map from
all the objects.)