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 $\mathrm{SL}_{2}$.

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 $V\cong A$ 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 $V_{T}=0$.

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!