Assume we have a complete regular local ring $R$ and an $R$-algebra $S$.


Is there a class of such algebras $S$ with the following property:


Given two $S$-modules $M,N$, then the maps induced by the forgetful functor $S-Mod \rightarrow R-Mod$ give injections $Ext^i\_S(M,N)\rightarrow Ext\_R^i(M,N)$?


If this question is too broad, here are the special cases i'm especially interested in:


We start with $R=\mathbb{C}[[x,y]]$, on the algebra side one may choose $S=M\_2(R)$ or the subalgebra of the matrices given by:
\begin{pmatrix}R &R \\ xR &R \end{pmatrix}
and one may choose $N=S$ and $M=S/T$ to be an finite length quotient of $S$, i.e there is a sequence $0\rightarrow T\rightarrow S \rightarrow M\rightarrow 0$. Further more the case $i=2$ would be enough. 


So what i'm really interested in is:
Is the map $ Ext\_S^2(S/T,S)\rightarrow Ext\_R^2(S/T,S)$ injective in these cases?


Inducing injective maps on the $Ext$-groups is, i think, more than being a faithful functor. Maybe this property has already been studied and has its own name? I don't see any sequences which relate these two groups, so that one could see this by doing some kind of diagram chasing. 


Any hints or ideas are welcome!



When does the forgetful functor S-Mod -> R-Mod induce injective maps on Ext-groups?
Given a map or algebras $R\to S$, a left $R$-module $M$ and a left $S$-module $N$, there is a natural first quadrant, cohomologically graded spectral sequence with $$E\_2^{p,q}=\mathrm{Ext}^p\_S(\mathrm{Tor}\_R^q(S,M),N)$$ converging to $\mathrm{Ext}^\bullet\_R(M,N)$.


**If $S$ is flat as a left $R$-module**, this collapses to natural isomorphisms 
$$
\mathrm{Ext}\_S^\bullet(S\otimes\_RM,N)\cong\mathrm{Ext}^\bullet\_R(M,N). \qquad\qquad(\star)
$$


**Suppose now moreover that $S$ is a separable $R$-algebra,** and go to the situation of the question in which both $M$ and $N$ are a left $S$-modules. If $S\otimes\_RS\to S$ is the map induced by multiplication in $S$, which is a map of $S$-bimodules, and $\Omega$ is its kernel, we have a short exact sequence of $S$-bimodules $$0\to\Omega\to S\otimes\_RS\to S\to 0$$ which splits—this is separability. Tensoring this with $M$ over $S$ gives then a *split* short exact sequence $$0\to\Omega\otimes\_SM\to S\otimes\_RS\otimes\_SM\to S\otimes\_SM\to 0$$ which can be identified with $$0\to\Omega\otimes\_SM\to S\otimes\_RM\to M\to 0.$$ The long exact sequence obtained from it by applying $\mathrm{Ext}\_S^\bullet(\mathord-,N)$ splits into little exact sequences $$0\to\mathrm{Ext}\_S^p(M,N)\to\mathrm{Ext}\_S^p(S\otimes\_RM,N)\to \mathrm{Ext}^p\_S(\Omega\_S\otimes\_SM,N)\to 0.$$
In particular, the map $\mathrm{Ext}\_S^p(M,N)\to\mathrm{Ext}\_S^p(S\otimes\_RM,N)$ is injective, so when we compose it with $(\star)$ we get an injection
$$\mathrm{Ext}\_S^p(M,N)\to\mathrm{Ext}\_R^p(M,N).$$ It is not hard to see that this composition is the map you wanted.


This deals with your matrix algebra.


