For simplicity let $\Bbbk$ be a field of characteristic $0$ and let $A$ be a finitely generated unital associative $\Bbbk$-algebra.
Is it true that for any two simple $A$-modules $S_1, S_2$, we have that $\operatorname{Ext}^1_A (S_1, S_2)$ is finite-dimensional?
If not, what would be a simple counterexample, and what sort of conditions do we need to ensure this?
(As far as I understand, the Weyl algebra $\Bbbk \langle x, y \rangle / (xy – yx – 1)$ has only infinite-dimensional simple modules, but their first extension groups are still finite-dimensional.)
Best Answer
Actually, your claim about the Weyl algebra is only true for simple holonomic modules. It turns out that there’s a simple non-holonomic module $M$ over the 2nd Weyl algebra $A=A_2(\Bbb C)$ for which $\operatorname{Ext}_A^1(M,M)$ is infinite dimensional. This is Corollary 1.3 of “Non-holonomic modules over Weyl algebras and enveloping algebras” by Stafford.