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.

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

$T:=\left\{\left( \begin{array}{cc} u & 0 \\ 0 & u^{-1} \\ \end{array} \right) \;|\; u\in A^{\times}\right\}$.

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.

For the next homology $H_{1}(T, V)$, I have a non trivial action in this case, I tried by calculating the homology of the complex

$ A\otimes B_2\xrightarrow{1\otimes d} A\otimes B_1\xrightarrow{1\otimes d} A\otimes \mathbb{Z}A^{\ast}$

where the tensor is taken over $\mathbb{Z}A^{\ast}$. The module $B_1$ is the free $G$ module on the symbols $[a_1]$ for $a_1\in A^{\ast}$ and $a_1\not=1$. If we let $[]\in\mathbb{Z}G$ be the identity, then

$$d([a_1]) = a_1[] - []$$.

Similarly, $B_2$ is the free $G$ module on $[a_1|a_2]$ where neither are the identity, and

$$d([a_1|a_2]) = a_1[a_2] - [a_1a_2] + [a_1]$$.

Here $[a_1a_2] = 0$ if $a_1a_2 = 1$ in the group $A^{\ast}$.

We have that $a\otimes u\in Ker(1\otimes d)$ if and only if $0=u\cdot a-a=u^{2}a-a=(u^{2}-1)a$. Since $A$ is a domain we have that $a=0$ or $u=\pm 1$. From this I got that $Ker(1\otimes d)=\langle a\otimes u[-1]\rangle$. I was trying to check if $a\otimes [-1]$ is a boundary element and if the size of the residue field might influence that.

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

Thank you for your time!