Tensor Triangular Chow Groups

We propose a definition of the Chow group of a rigid tensor triangulated category. The basic idea is to allow “generalized” cycles, with nonintegral coefficients. The precise choice of relations is open to some fine-tuning. Hypothesis 1. Let K be an essentially small tensor triangulated category. Let us assume that its triangular spectrum in the sense of [1], Spc(K) = { P ⊂ K ∣∣P is prime}, is a noetherian topological space, i.e. that every open of Spc(K) is quasi-compact. Let us also assume that K is rigid, as explained in [4] (or [2], where this property was called strongly closed). These hypotheses allow us to use the techniques of filtration of K by (generalized) dimension of the support. Definition 2. As in [2, Def. 3.1], let us consider dim : Spc(K)−→Z ∪ {±∞} a dimension function, meaning that P ⊆ Q =⇒ dim(P) ≤ dim(Q), with equality in the finite range only if P = Q (i.e. P ⊆ Q and dim(P) = dim(Q) ∈ Z forces P = Q). Examples are the Krull dimension of {P} in Spc(K), or the opposite of its Krull codimension. Assuming dim(−) is clear from the context, we shall use the notation Spc(K)(p) := { P ∈ Spc(K) ∣∣ dim(P) = p} . Remark 3. In my opinion, there is nothing conceptually remarkable about the free abelian group on Spc(K)(p). Therefore I propose another definition of p-dimensional cycles. This requires some preparation. Definition 4. Recall from [3, § 4] that a rigid tensor triangulated category L is called local if a⊗ b = 0 implies a = 0 or b = 0. Conceptually, this means that Spc(L) is a local space, i.e. that Spc(L) has a unique closed point ∗ := 0 ⊂ L, which is prime by assumption. Example 5. For every prime P ∈ Spc(K), the following tensor triangulated category is local in the above sense : KP := ( K/P )\ where K/P denotes the Verdier quotient and (−) the idempotent completion. We call KP the local category at P. There is an obvious (localization) functor q P : K K/P ↪→ KP composed of localization and idempotent completion. (The category KP can also be understood as the strict filtered colimit of the K(U) over those open subsets U ⊆ Spc(K) which contain P. See more in [4, § 2.2] if helpful.) We can identify Spc(KP) with the subspace { Q ∈ Spc(K) ∣∣P ∈ {Q}} of Spc(K), hence the space Spc(KP) remains noetherian. Date: October 8, 2012. 1991 Mathematics Subject Classification. 18E30. Research supported by NSF grant DMS-0969644.