Categorifying Rationalization

We solve a problem proposed by Khovanov by constructing, for any set of primes S, a triangulated category (in fact a stable∞-category) whose Grothendieck group is S−1Z. More generally, for any exact∞-category E, we construct an exact∞-category S−1E of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this∞-category is precisely the result of categorifying division by the primes in S. In particular, Kn(SE) ≅ SKn(E). It is a peculiar fact that rationalized algebraic K-groups have largely remained out of reach of algebraic techniques. For example, the rationalized K-groups of a number field F were computed by Borel [4]: for n ≥ 2, dimKn(F) ⊗Q = {{ {{ { 0 if n ≡ 0 mod 2; 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. But Borel’s proof depends upon a delicate analysis of invariant differential forms on the Borel–Serre compactification of a symmetric space. As far as we know, no algebraic approach to this computation has appeared in the literature. For function fields, the situation is at least as dire. For example, we have the following. Conjecture (Parshin). IfX is a smooth projective variety over a finite field, thenKn(X)⊗Q = 0 for any n ≥ 1. But only when the dimension ofX is 0 or 1 is this assertion known. The task of this paper is to categorify rationalization, in order to get a more explicit grasp on rational K-theory classes. That is, we introduce explicit categories of divisible objects whose K-theory gives the rational K-theory directly. More precisely, if S is a set of prime numbers, then for any exact∞-category E (in particular, for any exact ordinary category or any stable∞-category [2]), we construct here an exact∞-category S−1E such that K(S−1E) ≃ S−1K(E) as spectra, and, consequently, K∗(SE) ≅ SK∗(E) as graded abelian groups. When E is an idempotent-complete stable∞-category, we can offer an explicit – though perhaps unwieldy – characterization of S−1E: it is an∞-category of what we call S-divisible objects. These are sequences {Xi} of objectsXi of IndE, indexed over the various products i of the primes in S, along with suitably compatible identifications, whenm divides n, between the object Xm and the n/m-fold direct sum Xn ⊕ Xn ⊕ ⋯ ⊕ Xn, all subject to a finiteness condition. Our main theorem goes a step still further, and identifies S−1E as an ∞-category of sheaves of objects of IndE on the Cantor space Ω that are equivariant with respect to a free