Abstract. We generalize the linear algebra setting of Tate's central extension to arbitrary dimension. In general, one obtains a Lie (n + 1)-cocycle. We compute it to some extent. The construction is based on a Lie algebra variant of Beilinson's adelic multidimensional residue symbol, generalizing Tate's approach to the local residue symbol for 1-forms on curves.

We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different: Instead of a homotopy coherent cone construction in ∞-categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact which might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory. 2000 Mathematics Subject Classification. Primary 22B05; Secondary 19D10. The category LCA O is of glaring beauty: LCA O is a quasi-abelian exact category with duality. The Minkowski embedding gives rise to an exact sequence O ֒→ σ∈S R σ ։ T O (S the set of infinite places, R σ the codomain of σ, and T O the torus quotient). In LCA O this sequence is simultaneously, (1) an injective resolution of O, (2) a projective resolution of T O , and (3) the Pontryagin dual of the sequence is isomorphic to the sequence itself. So the Minkowski embedding is hardcoded in the homological algebra of LCA O .Similarly for the Dirichlet embedding: For algebraic K-theory, the sequence in Theorem 1.1 in degree one yieldswhich essentially identifies K 1 (LCA O ) as an extension of the class group with R × >0 × (torus), naturally containing the torus arising from Dirichlet's Unit Theorem. The free real factor R × >0 corresponds to choosing a normalization for the Haar measure the underlying LCA group.These facts resemble regulator constructions. Indeed, making use of properties of the Borel regulator, we get:Theorem 1.2. Let F be a number field, O its ring of integers, and K (non-connective) algebraic K-theory. With rational coefficients, the long exact sequence induced from the theorem above splits into short exact sequencesSee Theorem 5.1. Our methods are quite different from Clausen's in [Cla17]. While Clausen sets up a homotopy coherent cone construction in the context of stable ∞-categories, this paper exclusively deals with exact categories. Our method is based on techniques of Schlichting (namely left and right s-filtering subcategories) in order to show that certain quotient constructions do not only make sense on a derived level, but admit concrete exact category models. The main technical tool is the following: Theorem 1.3. Let F be a number field, O its ring of integers. Then (1) The compactly generated O-modules LCA O,cg are left s-filtering in LCA O . (2) The O-modules LCA O,nss without small subgroups are right s-filtering in LCA O .These results essentially amount to setting up a calculus of left or right fractions with certain additional properties. They might be of independent interest. Thus, even in the case O = Z, our proof is different from the one in [Cla17].Needless to say, this article is heavily inspired by Clausen's work. Conventions:We shall work a lot with exact categories. We follow the nomenclature of Bühler's survey [Büh...

In a previous paper we showed, under some assumptions, that 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. This viewpoint, as well as the usual one, come with generator-relator presentations (due to Bass-Swan and Nenashev) and in this paper we provide an explicit map.

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Clausen has constructed a homotopical enrichment of the Artin reciprocity symbol in class field theory. On the Galois side, Selmer K-homology replaces the abelianized Galois group, while on the automorphic side the K-theory of locally compact vector spaces replaces classical idelic objects. We supply proofs for some predictions of Clausen regarding the automorphic side.