TITLE: Deformations of Ext rings
QUESTION [9 upvotes]: Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$.  Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$.  Then for each element $q \in k$, we obtain specialisations
$$A_{q} := A\otimes_{k[x]} k, \; M_{q} := M\otimes_{k[x]} k,\; \mbox{ and } N_{q} := N\otimes_{k[x]} k$$ 
where $k$ becomes a $k[x]$-module by sending $x$ to $q$.  Then $A_q$ is a $k$-algebra, and $M_{q}$ and $N_{q}$ are $A_q$-modules.
I'm interested in how the groups $Ext_{A_q}^*(M_q, N_q)$ vary as $q$ ranges over $k$.  
My somewhat vague question is: if I know that a property holds for these Ext groups for a Zariski dense subset of elements $q \in U \subseteq \mathbb{A}^1(k)$ (which I suppose is simply fancy language for "all but finitely many elements of $k$"), can I conclude anything about these groups for $q \notin U$?  Properties that I'd be interested in include: finite cohomological dimension, finite generation, etc.
Perhaps less vague: are there any known techniques for getting at these groups for $q \notin U$?

REPLY [2 votes]: Let $A = k[x, y, z]/(yz - x^2)$ with augmentation $x \mapsto x$, $y \mapsto x$, and $z \mapsto x$ and with $M = N = k[x]$. Then $M_q = N_q = k$ and we are taking the ext groups over a nonsingular algebra when $q \not = 0$, but over a singular one if $q = 0$. Namely, for $q = 0$ we get $k[y, z]/(yz)$ and infinitely many of the Ext groups are nonzero, whereas for $q \not = 0$ you only have $2$ nonzero Ext groups.
This is not an answer but just an example showing that things can jump.