Suppose we have a short exact sequence of modules, where one module is unknown. Then what can we know about the unknown module?

Question

Suppose $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is a short exact sequence of $R$-modules, where $R$ is a ring with unity.

If $A$ and $B$ are known, then $C$ is unique up to isomorphism, because by the definition of short exact sequences, $C \cong B/A,$ where $A$ is considered to be a submodule of $B$.

But what about the other cases?

Suppose $A$ and $C$ are known. What choices do we have for the module $B$? I know that $B$ is the direct sum of $A$ and $C$ if and only if the sequence at the top is a split exact sequence. But otherwise, is there anything we can say about the structure of $B$?

Or if $B$ and $C$ are known, what can we derive about the structure of $A$?

(I feel like there was a specific name for this type of problem which I once saw on Wikipedia, but have forgotten it, so it would be great if somebody could remind me of it!)

Answer 1

Certainly if $B$ and $C$ are known, then $A$ is just the kernel of $B\rightarrow C$.

If only $A$ and $C$ are known, then $B$ is an extension of $C$ by $A$, and cannot in general be recovered from $A$ and $C$ alone. For example, one can choose $A = C = \mathbb{Z}/2$, then $B$ could either be $\mathbb{Z}/2\times\mathbb{Z}/2$ or $\mathbb{Z}/4$. In this example you can take $R = \mathbb{Z}$.

The set of all extensions of $C$ by $A$ (up to isomorphism, see this wikipedia article on the Ext functor) forms an abelian group $Ext^1(A,C)$.

If we're working instead in the category of groups, and $A$ is abelian, then the set of possible choices of $B$ form the group $H^2(C,A)$, where $H^2$ denotes the second group cohomology. However, note that in $H^2(C,A)$, one must specify an action of $C$ on $A$. Given your exact sequence, this action is given by conjugating elements of $A$ by lifts of elements of $C$ to $B$. Since $A$ is abelian, you can easily prove that this action is well-defined (does not depend on the choice of lift).

In general, describing the groups $Ext^1(A,C)$ or $H^2(C,A)$ are difficult problems, though there are many cases where they are understood.

Note that you may have heard of (c)homology from a topological setting, and group cohomology, despite the (initially) strange definition, is very much related to cohomology in topology.

In any case, from what I can tell of your background, cohomology is probably going to seem a bit arcane to you (it certainly did to me). However, if you keep doing math, especially at the graduate level and beyond, it is going to be pretty difficult to avoid, so the earlier you start reading about it, the more familiar it will be when you finally need to learn it!