On the relative K -group in the ETNC, Part II
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On the relative K -group in the ETNC, Part II Oliver Braunling1 Received: 1 July 2019 / Accepted: 10 September 2020 © The Author(s) 2020
Abstract In a previous paper we showed that, under some assumptions, the relative K -group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K -group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass– Swan, applied to categories of finitely generated projective modules; and one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism. Keywords Equivariant Tamagawa number conjecture · ETNC · Locally compact modules Mathematics Subject Classification Primary 11R23, 11G40; Secondary 11R65, 28C10
1 Introduction The equivariant Tamagawa number conjecture (ETNC) for possibly non-commutative coefficients postulates the equality of two elements in a certain K -theory group. We shall recall the wider background of the conjecture below in Sect. 2, or see Burns–Flach [1, Sect. 4.3, Conjecture 4]. Let A be a finite-dimensional semisimple Q-algebra and A ⊂ A an order. We write AR := A ⊗Q R. Let K n (−) denote n-th K -group. Only K 0 and K 1 are truly important
Communicated by Charles Weibel. The author was supported by DFG (Deutsche Forschungsgemeinschaft) GK1821 “Cohomological Methods in Geometry” and a Junior Fellowship at the Freiburg Institute for Advanced Studies (FRIAS).
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Oliver Braunling [email protected]; [email protected] Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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for this article. By general principles, there is a long exact sequence · · · −→ K 1 (A) −→ K 1 (AR ) −→ K 0 (A, R) −→ K 0 (A) −→ · · · ,
(1.1)
where K n (A, R) denotes certain groups which are designed to make this sequence exact. These groups are called ‘relative K -theory’, but there is not much behind it. They simply come from the fiber belonging to the map between the other two K -theory spectra involved in the sequence. In the previous paper [5] we have shown, assuming A to be regular, that there is a canonical isomorphism K n (A, R) ∼ = K n+1 (LCAA ),
(1.2)
where LCAA is the exact category of locally compact A-modules: Its objects are locally compact topological right A-modules, morphisms are continuous A-module morphisms. There is no finite generation assumption. The exact structure is such that admissible monics are the closed injections, and admissible epics are the open surjections. For example, this implies that cokernels in this category always carry precisely the quotient topology. The category is not abelian, but it has all kernels and cokernels. We are mostly interested in the case n = 0. In the formulation of the ETNC in [1], the equivariant Tamagawa numbers live in the relative K -group K 0 (A, R). This group has an explicit presentation due to Bass and Swan, based on generators [P, ϕ, Q], ∼
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