math 216: foundations of algebraic geometry - Stanford University

40 Math 216: Foundations of Algebraic Geometry
The cokernel of a monomorphism is called the quotient. The quotient of a
monomorphism A → B is often denoted B/A (with the map from B implicit).
We will leave the foundations of abelian categories untouched. The key thing
to remember is that if you understand kernels, cokernels, images and so on in
the category of modules over a ring ModA, you can manipulate objects in any
abelian category. This is made precise by Freyd-Mitchell Embedding Theorem
(Remark 2.6.4).
However, the abelian categories we will come across will obviously be related
to modules, and our intuition will clearly carry over, so we needn’t invoke a theorem
whose proof we haven’t read. For example, we will show that sheaves of
abelian groups on a topological space X form an abelian category (§3.5), and the
interpretation in terms of “compatible germs” will connect notions of kernels, cokernels
etc. of sheaves of abelian groups to the corresponding notions of abelian
groups.
2.6.4. Small remark on chasing diagrams. It is useful to prove facts (and solve
exercises) about abelian categories by chasing elements. This can be justified by
the Freyd-Mitchell Embedding Theorem: If A is an abelian category such that
Hom(a, a ′ ) is a set for all a, a ′ ∈ A , then there is a ring A and an exact, fully
faithful functor from A into ModA, which embeds A as a full subcategory. A proof
is sketched in [W, §1.6], and references to a complete proof are given there. A proof
is also given in [KS, §9.7]. The upshot is that to prove something about a diagram
in some abelian category, we may assume that it is a diagram of modules over
some ring, and we may then “diagram-chase” elements. Moreover, any fact about
kernels, cokernels, and so on that holds in ModA holds in any abelian category.)
If invoking a theorem whose proof you haven’t read bothers you, a short alternative
is Mac Lane’s “elementary rules for chasing diagrams”, [Mac, Thm. 3,
p. 200]; [Mac, Lemma. 4, p. 201] gives a proof of the Five Lemma (Exercise 2.7.6)
as an example.
But in any case, do what you have to do to put your mind at ease, so you can
move forward. Do as little as your conscience will allow.
2.6.5. Complexes, exactness, and homology.
We say a sequence
(2.6.5.1) · · ·
A f
B g
C
· · ·
is a complex at B if g ◦ f = 0, and is exact at B if ker g = im f. (More specifically,
g has a kernel that is an image of f. Exactness at B implies being a complex at
B — do you see why?) A sequence is a complex (resp. exact) if it is a complex
(resp. exact) at each (internal) term. A short exact sequence is an exact sequence
with five terms, the first and last of which are zeroes — in other words, an exact
sequence of the form
For example, 0
A
0 → A → B → C → 0.
0
0 is exact if and only if A = 0;
f
A
B