On the relative \(K\)-group in the ETNC. II. (English) Zbl 1466.19005

Let \(\mathfrak A\) be an order in a finite-dimensional, semisimple \(\mathbb Q\)-algebra. In [O. Braunling, New York J. Math. 25, 1112–1177 (2019; Zbl 1451.19012)], the author identified the homotopy fibre \(K(\mathfrak A, \mathbb R)\) of the canonical map \(K(\mathfrak A) \to K(A \otimes_{\mathbb{Q}} \mathbb R)\) as a shift of the algebraic \(K\)-theory of the exact category \(\mathsf{LCA}_{\mathfrak A}\) of locally compact \(\mathfrak A\)-modules. In a section complementing the introduction, the author explains how this relates to the equivariant Tamagawa number conjecture.
The article then proceeds to establish an explicit isomorphism \[ K_0(\mathfrak A, \mathbb R) \cong K_1(\mathsf{LCA}_{\mathfrak A}), \] where the left hand side is modelled following [R. G. Swan, Algebraic \(K\)-theory. Berlin-Heidelberg-New York: Springer-Verlag (1968; Zbl 0193.34601)], while the right hand side is described using A. Nenashev’s presentation of \(K_1\) of an exact category [J. Pure Appl. Algebra 131, No. 2, 195–212 (1998; Zbl 0923.19001)].


19F99 \(K\)-theory in number theory
11R23 Iwasawa theory
11R65 Class groups and Picard groups of orders
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
22B05 General properties and structure of LCA groups
18E10 Abelian categories, Grothendieck categories
19A31 \(K_0\) of group rings and orders
Full Text: DOI


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