Good source of exercises in computing $\operatorname{Tor}$ and $\operatorname{Ext}$ groups

Question

I am planning to attend special lectures on geometry and mathematical physics, and the professor advised me in advance to make sure I am familiar with various computations of $\operatorname{Tor}$ and $\operatorname{Ext}$ arising from algebraic topology.

Since I am already familiar to the (graduate level) conceptual/theoretical constructions of homological algebra, I would like to further my understanding by solving many examples.

Any recommendation on a source of exercises containing explicit computation of $\operatorname{Tor}$ and $\operatorname{Ext}$ groups will be appreciated. It would be even better if the problems contain concrete geometric/topological context.

Answer 1

Here are some nice exercises for working with Ext groups. They will take place in the ring of graded $R=k[x_1,x_2,..,x_n]$ modules. You can set $n=1$ or $n=2$ if the general case is too imposing too.

1. Compute all the Ext groups $Ext^i(R,R\langle n\rangle)$, where $\langle n \rangle$ denotes shifting the grading by $n$.

2. Compute all the self extensions of the trivial $R$ module $k$ by itself, $Ext^i(k,k)$. For bonus points, compute the algebra structure on this graded ring $S=Ext^*(k,k)$.

3. Once you work out what $S$ is as a graded ring (or extrapolate from the $n=1$ case) compute the extension groups for the trivial graded $S$ module $k$: $$R'=Ext^*_S(k,k).$$ It turns out this ring $R'$ is isomorphic to the polynomial ring $R$ you started with! This phenomenon is known as Koszul duality, and is completely computable in this setup, just set $n=1$ if you don't want to deal with indices.

Now you do say that you want familiarity with Ext and Tor from algebraic topology, so you should definitely compute Ext and Tor groups between all cyclic groups. One more type of concrete computation to do is to compute $Tor^i(M,N)$ for two modules, first by resolving $M$ then tensoring with $N$, then taking homology, or switching the roles of $M$ and $N$. The result you get is the same, even though the resolutions can look very different. Proving this is a nice homological algebra exercise, but is less hands on.