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The recent paper K-Groups for Rings of Finite Cohen-Macaulay Type by H. Holm allows us to compute the Quillen $K$-group $K_1(\text{mod}\hspace{.1 cm}R)$ as a quotient of the abelianization of the automorphism group of a representation generator of the category of maximal Cohen-Macaulay $R$-modules (that is, a module $M$ such $\text{add}_RM = \text{MCM}\hspace{.1 cm}R)$) under some restrictions (that can be found in the paper).

While I'm a bit of a newcomer to algebraic K-theory, I understand that the computation of (Quillen) K-groups is a notoriously difficult task, which is why I find the main theorem of this paper so striking. The computation of $K_1$ in this context relies on being able to compute Auslander-Reiten sequences, which from my understanding, can be done readily. In fact, this paper seems to have almost an algorithmic feel (though, that may be stretching it!) to the computation of $K_1(\text{mod}\hspace{.1 cm}R)$. Which brings me to ask, are there any other such theorems out there? Specifically, what theorems are out there that say we can compute $K_1(\text{mod}\hspace{.1 cm}R)$ in a similarly concrete way?

Just to point out, in the case $R$ is Artinian, Quillen's Dévissage Theorem gives us that $K_1(\text{mod}\hspace{.1 cm}R)$ is isomorphic to $k^*$.

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    $\begingroup$ I guess there's not a single correct answer to this question. Shouldn't this be community wiki? $\endgroup$
    – Fernando Muro
    Aug 19, 2013 at 14:36
  • $\begingroup$ I would argue that the group $K_1(R)$ is concrete enough: it is the abelianization of $\mathrm{GL}_\infty(R)$ (and we even know the commutator subgroup: it's generated by the elementary matrices) $\endgroup$
    – Denis Nardin
    Aug 30, 2021 at 14:36
  • $\begingroup$ @denisnardin that K 1 is something different $\endgroup$
    – Fernando Muro
    Aug 30, 2021 at 15:34
  • $\begingroup$ Holm's theorem is about what is often called $G_1(R)$ (i.e., it's the K-group of all f.g. modules, not just projective f.g. modules), but if $R$ is a regular ring, then by a famous theorem of Quillen, $G_*(R)$ coincides with $K_*(R)$. If $R$ is commutative, then the determinant map from $K_1(R)$ to the group of units of $R$ splits, with kernel often called $SK_1(R)$. So if $R$ is commutative and regular and $SK_1(R)$ vanishes, then $G_1(R)$ is simply the group of units of $R$. Bass-Milnor-Serre proved $SK_1(R)$ vanishes for integrally closed subrings of number fields, for example. $\endgroup$
    – user164898
    Aug 30, 2021 at 17:04
  • $\begingroup$ Another example: $SK_1(R)$ is known to vanish for commutative semilocal $R$, so if $R$ is a regular local commutative ring, then $G_1(R)$ again agrees with the group of units of $R$. Anyway, I haven't looked at Holm's paper, but I presume (from the mention of MCM modules) that Holm is working with Gorenstein rings. Gorenstein rings will often be singular, so that $K_*(R)$ doesn't necessarily agree with $G_*(R)$, so all these old results about $SK_1$ aren't nearly so relevant, and you have to work harder. $\endgroup$
    – user164898
    Aug 30, 2021 at 17:09

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