I haven't gotten a chance to search through the literature yet, but I was wondering if anyone knows anything about or has any thoughts about this idea:
So, if we have a module $M$, then we can easily talk about $Ext^n_R(M,N)$ and we know that if $M$ is projective that $Ext^n(M,N)=0$ for all $n \geq 1$. That is, the Ext groups measure how projective/not projective $M$ is.
Further, if consider $kG$ where $k$ is a field and $G$ is a finite group, then we can talk about the vertex $D \leq G$ (where $D$ is a $p$-group) of the module $M$. That is, $M$ is $H$-projective if and only if $H$ contains a conjugate of $D$. So, $M$ is projective exactly when $D$ is trivial. That is, the size of $D$ gives us a measure of how not projective $M$ is.
What I'm wondering is whether or not we can find a relationship between Ext and $D$. For example, if we were able to put a bound on the order of $D$ would we be able to say something about $Ext^n(M,N)$. Or, vice-versa, if we knew how many non-zero Ext groups we have does that tell us something about $D$.
There are a number of results that lead me to expect some kind of connection between this kind of homological characterization and this group theoretic characterization and I'd like to know what people think/know of.
If $\text{Ext}^i_{kG}(M,N)$ is nonzero for some $i>0$, then it is nonzero for infinitely many $i$, so there is no sensible statement about the number of nonzero Ext groups.
However, you can say something about the rate of growth of the dimension of $\text{Ext}^i_{kG}(M,N)$ as $i$ increases (I'll assume we're considering finite dimensional modules) in terms of the vertex of $M$.
The $p$-rank of a group is the largest integer $d$ such that there is a subgroup isomorphic to $C_p^d=C_p\times\dots\times C_p$. If the vertex $D$ of $M$ has $p$-rank $d$, then for any given module $N$, $\dim_k \text{Ext}^i_{kG}(M,N)$ is bounded by a polynomial function in $i$ of degree $d-1$.
However, there's no simple converse statement. The module $M$ could have large vertex (e.g., a Sylow $p$-subgroup of $G$), with $\dim_k\text{Ext}^i_{kG}(M,N)$ bounded for each $N$.