$\DeclareMathOperator{\Hom}{\operatorname{Hom}}\DeclareMathOperator{\Ext}{\operatorname{Ext}}$I am wondering what sort of basic basic intuitive meaning $\Ext^1(M,N)$ has.
As a base case: if $M$ and $N$ are say, (finite-dimensional) vector spaces (with a compatible group/algebra action), and $M$ and $N$ are indecomposable inequivalent (so $\Hom(M,N)={0}$), can I somehow conclude that $\Ext^1(M,N)$ is zero?
All I can get from Weibel/Wikipedia is that $\Ext^1(M,N)$ is a group under the Baer sum operation, and is in bijection with the set of solutions $X$ to the short exact sequence $0\to N\to X\to M\to 0$. I don't know how to use this second meaning, but it seems the most hands-on.
Full disclosure: If this sounds like a homework exercise, it (almost) was — past tense. Although, the problem/text/instructor had no desire for use of $\Ext$ or $\Hom$, I just want to know how to use these functors (better). I could give character references (even from some past Berkeley grads) to allay fears….
I would be happy to have a good reference to look this up myself. I hear Rotman's first book was good, but I've only read negative responses to the new edition (and the old one isn't for sale anywhere I've seen), and Weibel is apparently too abstract for me, in some way. I'll post a separate question for that, in fact.
Best Answer
It seems like you already can see this, but $Ext^1(M,N)$ is measuring all the ways to form distinct short exact sequences $0\to N\to ?\to M\to 0$
If you are looking for intuition, what you should do is think about picking a set of generators $G$ for $M$. This is equivalent to choosing a surjection from the free module $R^G \to M$, whose kernel is the 1st syzygy module S (which you should think of as relations between the generators). Then $Ext^1(M,N)$ is going to be all the maps from $S$ to $N$, modulo those which come from a map $R^G \to N$. Therefore, $Ext^1$ is something like all the ways of assigning an element in $N$ to every relation between generators in $M$, modulo dumb ways of doing this assignment (those coming from sending generators to $N$). This perspective can make it easy to guess what $Ext^1$ should be.