Abstract

We construct, for any set of primes$S$, a triangulated category (in fact a stable$\infty$-category) whose Grothendieck group is$S^{-1}\mathbf{Z}$. More generally, for any exact$\infty$-category$E$, we construct an exact$\infty$-category$S^{-1}E$of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this$\infty$-category is precisely the result of categorifying division by the primes in$S$. In particular,$K_{n}(S^{-1}E)\cong S^{-1}K_{n}(E)$.

Highlights

  • ; dim Kn(F) ⊗ Q = r1 + r2 if n ≡ 1 mod 4; r2 if n ≡ 3 mod 4, where r1 is the number of real places and r2 is the number of complex places of F

  • We introduce explicit categories of divisible objects whose K -theory gives the rational K -theory directly

  • Our main theorem goes a step still further, and identifies S−1 E as an ∞category of sheaves of objects of Ind E on the Cantor space Ω that are equivariant with respect to a free action (Construction 4.2) of the S-adic circle

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Summary

Introduction

If C is an ∞-category that admits direct sums and filtered colimits, we write S−1 : C C for the functor E colim E[S]. If E is S-local, it follows from Proposition 1.4 (and the fact that the category ΦS is filtered and weakly contractible) that the natural map E S−1 E is an equivalence.

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