\(K\)-groups for rings of finite Cohen-Macaulay type (Q2353629)

Let \((R, \mathfrak{m}, k)\) be a commutative noetherian local Cohen-Macaulay \(k\)-algebra with \(\mathrm{char}(k) \neq 2\). In this interesting article, the author describes \(G_1(R) = K_1(\pmod R)\) and the map from \(K_1(R) \cong R^*\) under the further assumptions that \(R\) is Henselian, admits a dualizing module, has finite CM-type and the Auslander-Reiten homomorphism \(\Upsilon\) (i.e. the matrix describing the relations between the indecomposable maximal Cohen-Macaulay \(R\)-modules) is injective. Let \(M\) be a representation generator of the category of MCM \(R\)-modules and let \(E\) be its endomorphism algebra. The main theorem describes \(G_1(R)\) as a quotient of \(E^*_{\mathrm{ab}}\), the abelianization of the automorphism group of \(M\) by a subgroup \(\Xi\) generated by special elements obtained from \(\Upsilon\). The article is broken into several sections of which (as the author themselves comment) the first few (up to Section 8) are general and put together various facts, specialized to the case at hand and required in the proof of the main theorem. The key ideas here which make the proof work include using the Gersten-Sherman transformation to identify various \(K_1\)-groups to be the same, Yoneda's lemma to enable use of functor categories and then the association of these functor categories with actual module categories over \(E^{\mathrm{op}}\), the remarkable fact that \(E\) and hence \(E^{\mathrm{op}}\) have finite global dimension and finally the comparison between functor categories over \(MCM(R)\) and \(\mathrm{proj}(R)\) by considering the kernel category \(\mathcal{Y}\) and observing that it is a finite length category and studying its structure. In section 8, the author invokes the long exact sequence for \(K\)-theory on the short exact sequence of functor categories mentioned above and their kernel. This yields via the above mentioned identifications a surjection from \(E^*_{\mathrm{ab}}\) to \(G_1(R)\) and the kernel can be identified with the image of the previous term which is \(K_1(\mathcal{Y})\). The previous identifications can then be used to identify the map with the Auslander-Reiten homomorphism \(\Upsilon\) and its image as the subgroup \(\Xi\) thus proving the main theorem. The relation with \(K_1(R)\) is also checked to be via the natural map from \(R^*\) to \(E^*_{\mathrm{ab}}\) thus proving the main theorem. In the remaining 2 sections, the author works out the special case when the only non-free indecomposable MCM \(R\)-module is \(\mathfrak{m}\) and applies it to specific examples such as \(k[X]/(X^2)\) and \(k[[T^2, T^3]]\) to recover earlier known computations. It should be pointed out (as also mentioned by the author) that in [\textit{V. Navkal}, J. \(K\)-Theory 12, No. 3, 405--432 (2013; Zbl 1292.19001)] the \(K'\)-theory in a similar set-up is computed.