Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$; i.e. 


$V:=\left\{\left(
                \begin{array}{cc}
                  1 & a \\
                  0 & 1 \\
                \end{array}
              \right)
\;|\; a\in A\right\}$.


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. 

For the $H_{0}(T, V)=V_{T}$ , since $T\cong A^{\ast}$ and $V\cong A$, through this identifications, we get that $A^{\ast}$ acts on $A$ by multiplication of a square unit. 

Thus for $V_{T}$ I need to check  the elements $ua-a=(u^{2}-1) a$ with $u$ a unit in $A$ and $a\in A$.

If $u^{2}-1$ is unit for some $u\in A^{\ast}$ then $V_{T}=0$. 

From this I found that if the residue field of the local domain (it can be a local ring) $A$  has at least four elements, then $H_{0}(T, V)=V_{T}$ is equal to zero.

Now I am trying to find the homology of $H_{0}(T,H_{2}(V,\mathbb{Z}))=H_{0}(A^{\ast},H_{2}(A,\mathbb{Z}))=(A\bigwedge_{\mathbb{Z}}A)_{A^{\ast}}$ which are the $A^{\ast}$- coinvariants of $A\bigwedge_{\mathbb{Z}}A$ where the actions is given by

$u\cdot(a\wedge b)=u^{2}a\wedge u^{2}b$. 

I am trying to find out when $(A\bigwedge_{\mathbb{Z}}A)_{A^{\ast}}$ is zero, so I need to check if the ideal $I$ generated by the elements  

$u^{2}a\wedge u^{2}b  - a\wedge b$   $(1)$ .

It has a unit.

I was trying some cases for $A$, for instance the case of $p$- adic integers
and if $u$ is unit and also and integers the condition $(1)$ reduces to

$(u^{4}-1)(a\wedge b)$

Thus I need to check if there is a unit $u$ and also and integers such that $u^{4}-1$ is a unit. When $p\geq 7$ is true since $2^{4}-1$ is unit. For the cases $p=2,3,5$ , $u^{4}-1$ is not a unit.

For the local ring $\mathbb{Z}/4\mathbb{Z}$ the ideal $I$ is zero. So I have not found the importance of the residue field for these cases.


Thank you for your time!