# Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let $M_1,\ldots, M_n$ be isomorphism classes of the indecomposable MCM $R$-modules. We choose $M = M_1\oplus M_1\oplus\cdots \oplus M_n$ to be a representation generator for MCM $R$ (ie. MCM $R$ = $\text{add}_RM$).

Question: What techniques exist for computing $\text{Aut}_R(M)_\text{ab}$, the abelianization of the automorphism group of $M$? Or for computing the commutator subgroup of $\text{Aut}_R(M)$?

My question stems mainly from interest in the paper K-groups For Rings of Finite Cohen-Macaulay Type. This paper also presents a good number of techniques presented for computing $\text{Aut}_R(M)_\text{ab}$. In fact, this paper tackles the case in which $n = 1$ and $M_1 = \mathfrak m$ (which is useful for rings such as $k[[t^2, t^3]]$).

As for myself, I've managed to to compute $\text{Aut}_R(M)_\text{ab}$ in its entirety in the case $R = k[x]/(x^n)$ and have derived some useful structural results in general. While I've perused the literature quite a bit, it's difficult when looking for something so specific. Any references to specific cases, as well as general results, would be greatly appreciated!