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3.2

Keller’s cyclic homology of exact categories

In this section we recall Keller’s construction of Hochschild and cyclic homology for exact k-linear categories [Kel99]. In fact, Keller constructs a functor

C : Exk→ Mix

from the category Exk of small exact k-linear categories to the category Mix of Kassel’s

mixed complexes. The functor C is also called the cone functor because its definition uses the mapping cone for morphisms of mixed complexes. Hochschild and cyclic homology are homological invariants of mixed complexes, as explained in Definition 3.1.3. Hence, Keller’s Hochschild (or cyclic) homology of a k-linear exact category A is the Hochschild (or cyclic) homology of the mixed complex C(A) associated to A by the functor C.

Among the various properties of Keller’s Hochschild homology, we have the following [Kel99]:

Agreement : the Hochschild homology of the exact category of finitely generated projective modules over a (unital) algebra agrees with the Hochschild homology of the algebra.

Invariance: Hochschild homology is preserved by exact functors inducing equivalences in the bounded derived categories.

Localization: suitable sequences of exact categories (roughly, inducing short exact sequences of bounded derived categories) are sent to triangles of the triangulated category D(Λ) in a sense to be made precise.

Additivity : Hochschild homology is additive.

Trace maps: there are trace maps linking the K-theory of an exact category to its Hochschild homology.

We review and explain these properties below in Theorem 3.2.5 and Theorem 3.2.7. The relation with K-theory is discussed in Proposition 4.4.1. The most important property for us is the localization property: it allows us to deduce Mayer-Vietoris type conclusions (see Theorem 3.3.8) in the same way as it has been already done for other localizing

invariants [BEKW17, BC17].

We start with the definition of the functor C : Ex_k→ Mix.

Let E be a small k-linear exact category (see Definition A.1.5). The category Chb(E ) of bounded chain complexes in E and the sub-category Acyb(E ) of bounded acyclic chain complexes in E have the structure of dg-categories, as described in Example A.2.6. We denote these dg-categories by Chb_dg(E ) and Acyb_dg(E ) respectively.

For a small dg-category A, let Mix(A) be the mixed complex associated to A as described in Definition 3.1.9.

Definition 3.2.1. [Kel99, Sec. 1.4] Let E be an exact k-linear category. The mixed complex C(E ) associated to E is the cone

C(E ) := cone(Mix(Acyb_dg(E )) → Mix(Chb_dg(E ))) (3.2.1) of the morphism of mixed complexes Mix(Acyb_dg(E )) → Mix(Chb_dg(E )) induced by the inclusion Acyb_dg(E ) → Chb_dg(E ) of dg-categories.

Remark 3.2.2. [Kel99, Sec. 1.4] The cone construction C of Definition 3.2.1 is functorial with respect to (k-linear) exact functors between (k-linear) exact categories.

The cone construction C : Ex_k→ Mix is a functor from the category Ex_k of small exact k-linear categories to the category Mix of mixed complexes. We can describe the functor as follows.

For a category C, we denote by C∆1 the arrow category Fun({0 → 1}, C). Then, the cone functor of Definition 3.2.1 can be written as the composition:

C : Exk dgcat∆ 1 k Mix∆ 1 Mix Mix cone (3.2.2)

where the first functor associates to a k-linear exact category E the morphism of dg- categories Acyb_dg(E ) → Chb_dg(E ) in dgcat∆1 (and on morphisms it is defined in the canonical way), the second functor uses Definition 3.1.9 and the third functor is the cone in the category Mix.

Definition 3.2.3. Let E be a small k-linear exact category. Keller’s Hochschild (cyclic) homology of E is defined as the Hochschild (cyclic) homology of the mixed complex C(E ) associated to E by the functor C of Definition 3.1.3.

Before describing the properties of the functor C : Exk→ Mix, we need some more

terminology:

Definition 3.2.4. [Kel99, Sec. 1.5] Let A, B be additive categories and let T , T0 and T00 be triangulated categories.

(i) A factor-dense subcategory A0 of A is a full subcategory such that each object of A is a direct factor of a finite direct sum of objects of A0.

(ii) An equivalence up to factors is an additive functor A → B that induces an equivalence onto a factor-dense subcategory of B.

(iii) A sequence

T0→ T → T00

of triangulated categories is exact up to factors if the composition is zero, the functor T0 → T is fully faithful and the induced functor T /T0 → T00 from the Verdier quotient to T00 is an equivalence up to factors.

3.2. KELLER’S CYCLIC HOMOLOGY OF EXACT CATEGORIES 55

Let E be a small exact k-linear category and let Db(E ) be the derived category of E (see Definition A.1.10). We recall that, by Example 3.1.5, one can associate a mixed complex C(A) to every k-algebra A. Recall also that D(Λ) denotes the derived category of the dg-algebra Λ.

Theorem 3.2.5. [Kel99, Theorem 1.5] Let k be a commutative ring. 1. If A is a k-algebra, there is a natural isomorphism

C(A) → C(projA)

in D(Λ), where projA is the exact category of finitely generated projective modules. 2. If F : A → B is an exact functor between exact categories that induces an equivalence

up to factors Db(A) → Db(B), then F induces an isomorphism C(A) → C(B)

in D(Λ).

3. If F : A0 → A and G : A → A00 _{are exact functors between exact categories such that}

the sequence

Db(A0) → Db(A) → Db(A00)

is exact up to factors, then there is a canonical morphism ∂(F, G) such that the sequence

C(A0) C(A) C(A00) ∂(F,G) C(A0)[1]

is a triangle in D(Λ).

Remark 3.2.6. After application of the functor C, the inclusion of an exact k-linear category in its idempotent completion induces an isomorphism in D(Λ).

If E is a small k-linear exact category, let Ex E denote the category of admissible short exact sequences of E (which is again exact, provided that short exact sequences are defined component-wise). Consider the following exact functors:

I : E → Ex E A 7→ (A−→ A → 0)1 R : Ex E → E (A → B → C) 7→ A P : Ex E → E (A → B → C) 7→ C

S : E → Ex E C 7→ (0 → C −→ C)1

The functors I and P are left adjoint to the functors R and S, respectively. Keller proves that Hochschild and cyclic homology are additive invariants:

Theorem 3.2.7 (Theorem 1.12 [Kel99]). Let E be a small k-linear exact category. Then, the functors P and R induce an isomorphism

C(Ex E )−→ C(E) ⊕ C(E)∼ in D(Λ).

We now interpret Keller’s results in the context of ∞-categories.

Let dgcat_k be the category of small dg-categories over k. The category dgcat_k is endowed with the Morita model structure of Theorem A.2.16; we denote by dgcat_k,∞ the underlying ∞-category:

dgcat_k,∞:= N(dgcat_k)[W_Morita−1 ] (3.2.3) where WMorita is the class of Morita equivalences between dg-categories.

To every dg-category we can associate a cyclic module, i.e., its additive cyclic nerve of Definition A.4.7, hence a mixed complex by Remark 3.1.4. Let

Mix : dgcat_k→ Mix

be the functor of Definition 3.1.9. By Theorem 3.2.5 (2), the functor Mix sends Morita equivalences of dg-categories to quasi-isomorphisms of mixed complexes. A morphism in an ∞-category is an equivalence if its image in the homotopy category is an isomorphism, hence, the functor Mix descends to a functor

dgcat_k Mix dgcat_k,∞ Mix∞ loc Mix loc Mix∞ (3.2.4)

on the localizations. Observe that the functor Mix preserves filtered colimits and that quasi- isomorphisms of mixed complexes commute with filtered colimits of mixed complexes. Hence, loc ◦Mix preserves filtered colimits because the localization preserves filtered colimits as well.

Let Exk be the category of small exact k-linear categories. Let Wex be the class

of exact functors between exact k-linear categories inducing equivalences up to factors between the associated bounded derived categories (Definition 3.2.4). Let N(Ex_k)[W_ex−1] be the ∞-category associated to Exk by inverting the morphisms of Wex [Lur14, Def.

1.3.4.1]. By Theorem 3.2.5 (2), the cone functor C : Ex_k→ Mix (3.2.2) sends morphisms in W_ex to quasi-isomorphisms of mixed complexes, hence it descends to a functor between