TENSOR TRIANGULAR CHOW GROUPS

PAUL BALMER

Abstract. 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 \overline{\{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 :

𝒦_{𝒫} := (𝒦/𝒫)^{♮}

where 𝒦/𝒫 denotes the Verdier quotient and (−)^{♮} the idempotent completion. We call 𝒦_{𝒫} the local category at 𝒫. There is an obvious (localization) functor

q_{𝒫} : 𝒦 ↠ 𝒦/𝒫 ↪ 𝒦_{𝒫}

composed of localization and idempotent completion. (The category 𝒦_{𝒫} 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(𝒦_{𝒫} ) with the subspace \{ 𝒬 ∈ Spc(𝒦) | 𝒫 ∈ \overline{\{𝒬\}} \} of Spc(𝒦), hence the space Spc(𝒦_{𝒫} ) remains noetherian.

Date: October 8, 2012.

1991 Mathematics Subject Classification. 18E30.

Research supported by NSF grant DMS-0969644.

Definition 6. Assuming that ℒ is local and that Spc(ℒ) is noetherian, the open complement of the unique closed point {∗} in Spc(ℒ) is quasi-compact, i.e. {∗} is a “Thomason (closed) subset”. Under the classification of thick ⊗-ideals of ℒ, see [1], this one-point subset corresponds to the minimal non-zero thick ⊗-ideal

Min(ℒ) := ℒ_{{∗}} = { a ∈ ℒ | supp(a) ⊆ {∗} } .

These are the objects with minimal possible support (empty or a point).

Remark 7. Some comments are in order :

(1) This subcategory was called the subcategory of finite-length objects in [2] and denoted FL(ℒ). As far as I know, there is no reason for objects of Min(ℒ) to have finite-length (in the categorical sense that they admit a finite filtration with simple subquotients). The present notation, Min(ℒ), is less biased towards commutative algebra and therefore probably preferable. It is however an interesting question to find some structure theorems about Min(ℒ).

(2) As the previous comment suggests, if we take ℒ = K^{b}(R – proj) the category of perfect complexes for R noetherian and local, then ℒ is local and Min(ℒ) is the subcategory of perfect complexes with finite-length homology.

(3) One can of course consider Min(ℒ) even if ∗ is not Thomason but in that case it would just be the zero subcategory 0 = ℒ_{∅}.

Definition 8. Let p ∈ ℤ. We define the group of generalized p-cycles to be

Z_{p}(𝒦) := ⨁_{𝒫∈Spc(𝒦)_{(p)}} K_{0}( Min(𝒦_{𝒫} )) ,

where K_{0} is the Grothendieck K-group (the quotient of the monoid of isomorphism classes [a] of objects under ⊕, by the submonoid of those [a] + [Σb] + [c] for which there exists a distinguished triangle a → b → c → Σa).

Out of nostalgia for usual cycles, a generalized p-cycle can be written ∑_{𝒫} λ_{𝒫} · 𝒫 or ∑_{𝒫} λ_{𝒫} · \overline{\{𝒫\}}, for λ_{𝒫} ∈ K_{0}( Min(𝒦_{𝒫} )). This is a purely notational choice. The non-trivial point is that we allow coefficients λ_{𝒫} to live in other abelian groups than ℤ, namely the Grothendieck groups of the minimal categories at every 𝒫.

Example 9. Let X be a (topologically) noetherian scheme and 𝒦 = D^{perf}(X) the derived category of perfect complexes, whose spectrum Spc(𝒦) ≅ X recovers the underlying space of X. Let dim(−) be the (opposite of the) Krull (co)dimension. Then we recover the usual p-dimensional (resp. (−p)-codimensional) cycles. Indeed, we have by Thomason that 𝒦_{𝒫} ≅ K^{b}(O_{X,x} – proj) if 𝒫 ∈ Spc(𝒦) corresponds to x ∈ X. The reason why integral coefficients suffice over regular schemes is that the group homomorphism defined by alternate sum of length of homology groups

K_{0}( Min(K^{b}(O_{X,x} – proj))) ⟶ ℤ ,

is an isomorphism if X is regular (at x). However, in general, the left-hand group could be tricky, as discussed for instance in Roberts-Srinivas [6].

Now to the relations. There might be several definitions of relations. The most flexible and most obvious one is the following.

Definition 10. For a (specialization) closed subset Y ⊂ Spc(𝒦), we set dim(Y) = sup { dim(𝒫) | 𝒫 ∈ Y } and consider the filtration ⋯ ⊂ 𝒦_{(p)} ⊂ 𝒦_{(p+1)} ⊂ ⋯ ⊂ 𝒦 by dimension of support

𝒦_{(p)} := { a ∈ 𝒦 | dim(supp(a)) ≤ p } .

By [2, Thm. 3.24], localization induces an equivalence

(11)  (𝒦_{(p)}/𝒦_{(p−1)})^{♮} ⟶^{∼} ∐_{𝒫∈Spc(𝒦)_{(p)}} Min(𝒦_{𝒫} )

and consequently Z_{p}(𝒦) ≅ K_{0}((𝒦_{(p)}/𝒦_{(p−1)})^{♮}). Note that this definition of Z_{p}(𝒦) does not need Spc(𝒦) being noetherian. It also allows the definition of the pboundaries B_{p}(𝒦) as the image in Z_{p}(𝒦) of Ker (K_{0}(𝒦_{(p)}) → K_{0}(𝒦_{(p+1)})). In other words we have the diagram with exact rows

[figure: Ker(ι); K_{0}(𝒦_{(p)}); ι; K_{0}(𝒦_{(p+1)}); B_{p}(𝒦); Z_{p}(𝒦); CH_{p}(𝒦)]

in which we define CH_{p}(𝒦) := Z_{p}(𝒦)/ B_{p}(𝒦) to be the quotient of p-cycles by pboundaries. These groups could be called the (K-theoretic) Chow groups of p-cycles in 𝒦, with respect to the chosen dimension function dim.

Remark 12. The above Ker(ι) is an ad hoc replacement for the maybe more natural image of K_{1}(𝒦_{(p+1)}/𝒦_{(p)}) by a connecting homomorphism. The reason for the above definition is that triangulated categories do not behave well with higher Ktheory. However, with this definition, it is not too hard to check that CH_{p}(𝒦) = CH_{p}(X) when X is a regular scheme and 𝒦 = D^{perf}(X). See more in Klein [5].

It is however tempting to give another definition of p-boundaries, closer to the classical ideas of equivalence of p-cycles by means of divisors of rational functions on (p + 1)-dimensional varieties. We need a preparation.

Lemma 13. Let a ∈ 𝒦_{(p+1)} be an object with support of dimension at most p+1 and

let γ : a →^{∼} a be an automorphism in 𝒦_{(p+1)}/𝒦_{(p)}. Choose a fraction a⟶^{α}b ⟵^{β} a in 𝒦_{(p+1)} representing γ, so that cone(α) and cone(β) both belong to 𝒦_{(p)}. Then the difference [cone(α)] − [cone(β)] in K_{0}(𝒦_{(p)}) belongs to Ker (ι : K_{0}(𝒦_{(p)}) → K_{0}(𝒦_{p+1})) and is independent of the choice of α and β.

Proof. This is an immediate verification : In K_{0}(𝒦_{(p+1)}), we have [cone(α)] = [b] − [a] = [cone(β)], hence the first statement. Independence on the choice of the fraction up to amplification by a morphism s : b → b′ with cone in 𝒦_{(p)} follows by the octahedron axiom : [cone(sα)] = [cone(s)] + [cone(α)] and [cone(sβ)] = [cone(s)] + [cone(β)], so [cone(sα)] − [cone(sβ)] = [cone(α)] − [cone(β)]. □

Definition 14. Let a ∈ 𝒦_{(p+1)} and let γ : a →^{∼} a be an automorphism in 𝒦_{(p+1)}/𝒦_{(p)}.

Choose a fraction a⟶^{α}b ⟵^{β} a in 𝒦_{(p+1)} representing γ, and let

div(a →^{γ} a) = [q(cone(α))] − [q(cone(β))] ∈ B_{p}(𝒦)

where q : 𝒦_{(p)} → (𝒦_{(p)}/𝒦_{(p−1)})^{♮} is the canonical functor. We might call this element the divisor of γ : a →^{∼} a. This generalized p-cycle is a p-boundary by construction.

Remark 15. Of course, in view of the equivalence (11), we can also write

div(γ) = ∑_{𝒫∈Spc(𝒦)_{(p)}} [q_{𝒫} (cone(α))] − [q_{𝒫} (cone(β))]

where q_{𝒫} : 𝒦 ⟶ 𝒦_{𝒫} is the localization and where γ = (a⟶^{α}b ⟵^{β} a) as before. The above formula for the divisor might look more familiar to the reader.

Remark 16. A priori, there might be more p-boundaries than the ones coming from the above divisors div(γ). This means that one might have a different Chow group CH′_{p}(𝒦) defined as the quotient of Z_{p}(𝒦) by the subgroup generated by those div(γ). This group CH′_{p}(𝒦) would surject the group CH_{p}(𝒦) of Definition 10. However, in the case of 𝒦 = D^{perf}(X) for a (nice) regular scheme X, it might well be that CH′_{p} coincides with CH_{p} because all relations coming from K_{1} seem to be captured by divisors. This point requires further investigation and we refer the interested reader to the forthcoming [5].

Acknowledgements : I would like to thank Jean Fasel for (old) discussions and Gunther Cornelissen and Sebastian Klein for reviving my interest in the subject and for precious interaction.

References

[1] P. Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005.

[2] P. Balmer. Supports and filtrations in algebraic geometry and modular representation theory. Amer. J. Math., 129(5):1227–1250, 2007.

[3] P. Balmer. Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol., 10(3):1521–1563, 2010.

[4] P. Balmer. Tensor triangular geometry. In International Congress of Mathematicians, Hyderabad (2010), Vol. II, pages 85–112. Hindustan Book Agency, 2010.

[5] S. Klein. Chow groups of tensor triangulated categories. In preparation 2012.

[6] P. C. Roberts and V. Srinivas. Modules of finite length and finite projective dimension. Invent. Math., 151(1):1–27, 2003.

Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA

   E-mail address: balmer@math.ucla.edu
   URL: http://www.math.ucla.edu/∼balmer