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Fundamental isomorphism conjecture via non-commutative motives

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Abstract

Given a group, we construct a “fundamental localizing invariant” on its orbit category. We prove that any isomorphism conjecture valid for this fundamental invariant implies the same isomorphism conjecture for all localizing invariants, like non-connective K-theory, Hochschild homology, cyclic homology, and so on. Then, we discuss how to reduce such a fundamental isomorphism conjecture to essentially K-theoretic ones. Finally, we develop the analogue additive results.

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