Math 55a: A preview of “abstract nonsense”

Suppose we have a sequence of vector spaces and linear maps
... → Vn-1VnVn+1 → ...
The sequence is said to be “exact at Vn” if the kernel of the map VnVn+1 equals the image of the map Vn-1Vn. [As a consequence, the composite map Vn-1Vn+1 must be the zero map.] The sequence is said to be “exact” if it is exact at each vector space with an incoming and outgoing arrow.

The simplest cases:

An exact sequence 0 → UVV/U → 0 is called a “short exact sequence”; an exact sequence involving four or more vector spaces between the initial and final zero is called a “long exact sequence”. In any exact sequence of finite-dimensional vector spaces with an initial and final zero, the dimensions of the even- and odd-numbered vector spaces in the sequence have the same sum; in other words, the alternating sum of the dimensions (a.k.a. the “Euler characteristic” of the sequence) vanishes. [Check that this holds for the above cases of sequences of length at most 3.]

(Much the same definitions are made for sequences in some other “categories”, such as groups, or modules over a given ring. For instance,

0 → ZZZ/NZ → 0
is a short exact sequence of commutative groups if we use multiplication by N as the map ZZ.)

If the vector spaces Vn in our exact sequence are finite dimensional then the dual spaces Vn* form an exact sequence with the arrows going in the opposite direction:

... ← Vn-1VnVn+1 ← ...
[A mathematician enamored with abstract nonsense would express this fact by saying that “duality is an exact contravariant functor on the catogery of finite-dimensional vector spaces and linear maps”; the canonical identification of the second dual V** of every finite-dimensiona vector space V with V would likewise give rise to an “exact covariant functor” on the same category.]

For instance, if we dualize

0 → UVV/U → 0
we get
0 ← U*V* ← (V/U)* ← 0
Which is to say that (V/U)* is the kernel of a surjective linear map from V* to U* obtained from the injection of U into V. This map is none other than the restriction map from linear functionals on V to linear functionals on U; the kernel of this map consists exactly of those linear functionals that vanish on all of U. So we recover the fact that (V/U)* is the annihilator of U in V*, and that the quotient of V* by its subspace (V/U)* is canonically identified with U*.

How much of this still works if we drop the requirement that V be finite dimensional?