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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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I guess there's not a single correct answer to this question. Shouldn't this be community wiki? –  Fernando Muro Aug 19 '13 at 14:36
Is there a way to tag it as a community wiki? I'm not sure how that works. –  Zach Flores Aug 28 '13 at 4:14

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