Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $SL_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $SL_{2}$.

We have that $T$ acts on $V$ by conjugation so $V$ is a $T$-module.

This  action is non-trivial, so I am trying to calculate $H_{p}(T, V)$ with $p\in\{0, 1,2,3\} $

For the zero homology, it is well known that $H_{0}(T, V)=V_{T}$ i.e. the co-invariants. 

 In this case I have to find all the matrices in $V$ such that the conjugation by matrices in $T$ are trivial. 

From this I got $u^{2}a=a$ where $a\in A$ and $u$ is a unit of $A$  and since $A$ is an integral domain, it follows that $a=0$ and therefore $H_{0}(T, V)=V_{T}=0$. I am not sure about this since I think it might also depend on the size of the units of $A$.


For the others homologies I was trying to find some literature and I only found cases when the action is only trivial. Any guideline or idea might be helpful. 

Thank you for your time!