Filtered derived categories and their applications D. Kaledin∗ Contents 1 f-categories. 1 1.1 Definitions and examples. 1 1.2 Reflecting the filtration. 3 1.3 The comparison functor. 5 1.4 Further adjunctions. 7 1.5 Morphisms. 9 2 End(N)-action. 12 2.1 Prototype: End(N)-action on functor categories. 12 2.2 N-categories. 15 2.3 Uniqueness. 16 2.4 Full embeddings. 19 2.5 Bifibrations. 22 3 Λ∞-categories. 28 3.1 The category Λ∞. ........................ 28 3.2 Unbounded maps. 30 3.3 The bifibration ΛD: preliminaries. 32 3.4 The bifibration ΛD: the construction. 35 3.5 Functoriality. 37 Introduction. 1 f-categories. 1.1 Definitions and examples. Out starting point is a version of a definition given by Beilinson in [B]. ∗Partially supported by grant NSh-1987.2008.1 1 Preliminary version – please do not distribute, use at your own risk 2 Definition 1.1. An f-category is a triple hD, F, si of a triangulated cate- gory D, a fully faithful left-admissible triangulated embedding F : D → D, called the twist functor, and a morphism s : F → Id from F to the identity functor Id : D → D, such that 2 (i) for any M ∈ D, we have sF (M) = F (sM ): F (M) → F (M), and ⊥ (ii) for any M ∈ F (D), the cone of the map sM : F (M) → M lies in F (D)⊥. The quotient category D/F (D) is called the core of the f-category hD, F, si and denoted by C(D,F ), or simply C(D) when there is no danger of confu- sion. We note that the existence of the quotient C(D) is not a problem: it is canonically identified with ⊥F (D) ⊂ D, hence Hom-sets are small. Given an f-category hD, F, si, denote by L : D → D the left-adjoint functor to F . It immediately follows from the definition that for any integer n ≥ 2, the composition F n : D → D is also a left-admissible full triangulated embedding, with the left-adjoint functor Ln : D → D. We will denote the n n image of F by D≥n = F (D) ⊂ D, n ≥ 1, and we will set D≥0 = D. For n n any n ≥ 0, we will denote the composition F ◦ L : D → D≥n ⊂ D by σ≥n. By definition, for any M ∈ D, we have the adjunction map M → σ≥nM; the ⊥ cone of this map lies in the orthogonal D≥n, which we denote by D≤n−1. For any n ≥ 0, D≤n ⊂ D is a right-admissible full triangulated subcategory; we denote by σ≤n : D → D the composition of the embedding D≤n ⊂ D and its right-adjoint functor D → D≤n. For any M ∈ D, we have a functorial distiguished triangle (1.1) σ≤nM −−−−→ M −−−−→ σ≥n+1M −−−−→ . Definition 1.2. An f-category D is called non-degenerate if [ D = D≤n. n≥0 An f-structure on a triangulated category D is given by a non-degenerate f-category hDF, F, si and a triangulated equivalence D =∼ C(DF) between D and the core C(DF) of the f-category DF. To give a basic motivating example of an f-structure, we introduce some notation. Consider the ordered set N of natural numbers as a small category Preliminary version – please do not distribute, use at your own risk 3 in the usual way – objects are natural numbers, that is, non-negative inte- gers, the set N(n, m) is empty if m < n, and consists of exactly one element ιn,m otherwise. For any category A, Fun(N, A) is the category of diagrams E(0) → E(1) → · · · → E(n) → ... in A. Let Funf (N, A) ⊂ Fun(N, A) be the full subcategory spanned by all such diagrams that stabilize at some n – that is, there exists a number n ∈ N such that the map E(i) → E(i + 1) is an isomorphism for i ≥ n. Example 1.3. Let A be an abelian category, and consider its derived cat- egory D(A). Then the derived category DF(A) = D(Funf (N, A)) gives an f-structure on D(A). The twist functor F is induced by the functor Funf (N, A) → Funf (N, A) given by ( E(n − 1), n ≥ 1, F (E)(n) = 0, n = 0, and the morphism s : F (E) → E is induced by the natural maps E(n) → F (E)(n + 1) = E(n) → E(n + 1), n ≥ 0. It is well-known (and easy to check) that in the definition of the derived category D(Funf (N, A)), it suffices to consider such complexes E of functors from N to A that all the maps E (n) → E (n + 1), n ≥ 0, areq injective. Such a complex is nothing but a filteredq complexq in A, and DF(A) is usually called the filtered derived category of the abelian category A. In these terms, the twist functor F is given by the shift in the filtration. 1.2 Reflecting the filtration. Assume given a non-degenerate f-cate- gory hD, F, si. Lemma 1.4. For any n ≥ 0, the left-admissible embedding D≥n ⊂ D is also ⊥ ⊥ right-admissible, and the projection D≥n+1 → D → D≥n+1 = D≤n is an equivalence of categories. Proof. It is a standard fact that for any left-admissible embedding D ⊂ D0 of triangulated categories, the composition functor D⊥ → D0 → ⊥D is fully faithful, and it is an equivalence if and only if D ⊂ D0 is right- admissible. Thus the second claim is equivalent to the first. Moreover, since the composition of right-admissible embeddings is right-admissible, it suffices to consider the case n = 1. Thus all we have to prove is that the ⊥ projection D≥n+1 → D → D≤n is essentially surjective for n = 0. This is Preliminary version – please do not distribute, use at your own risk 4 0 Definition 1.1 (ii): for any M ∈ D≤0, any cone M of the map s : F (M) → M ⊥ 0 ∼ lies in D≥1, and we obviously have σ≤0(M ) = M. In particular, denote by π : D → D/D≥1 = C(D) the projection functor, ∼ ⊥ and let λ : C(D) = D≤0 = D≥1 ⊂ D be its left-adjoint; then we also have ∼ ⊥ the right-adjoint ρ : C(D) = D≥1 ⊂ D, and the map s : F → Id has a functorial cone: for every M ∈ D, we have a functorial distiguished triangle F (M) −−−−→s M −−−−→t ρ(π(M)) −−−−→ , where t : M → ρ(π(M)) is the adjunction map. Applying this to M = Λ(M 0), M 0 ∈ C(D), we obtain a functorial distiguished triangle (1.2) F (λ(M 0)) −−−−→s λ(M 0) −−−−→t ρ(M 0) −−−−→ . Denote by R : D → D the right-adjoint functor to F : D → D; for any n ≥ 2, Rn is then right-adjoint to F n. Denote by s0 : Id → R the map ≥n adjoint to s : F → Id. Denote by σ : D → D≥n ⊂ D the composition n n ≤n ⊥ F ◦ R , and denote D = D≥n+1 ⊂ D. This is a category equivalent to D≤n, and both are equivalent to the quotient category D/D≥n+1; the ∼ ∼ ≤n embeddings D/D≥n+1 = D≤n ⊂ D and D/D≥n+1 = D ⊂ D are left and right-adjoint to the projection D → D/D≥n+1. We will denote the ∼ ≥n ≤n composition D → D/D≥n+1 = D ⊂ D by σ . For any M ∈ D and any n ≥ 0, we have a functorial distinguished triangle (1.3) σ≥n+1M −−−−→ M −−−−→ σ≤nM −−−−→ , similar to (1.1). Loosely speaking, the collections σ≥nM gives a (gener- alized) descreasing filtration on any object M ∈ D, a “reflection” of the (generalized) increasing filtration σ≤nM. It is also possible to do a second reflection. 0 Lemma 1.5. For any object M ∈ D≤0 ⊂ D, the map s : M → R(M) is an isomorphism. Proof. By definition, s0 is the composition of the adjunction map M → R(F (M)) and the map R(s): F (M) → M. The adjunction map is an isomorphism. Since R is a triangulated functor, the cone M 0 of the map s0 : M =∼ R(F (M)) → R(M) is then isomorphic to R(M 00), where M 00 is the cone of the map s : F (M) → M. By Definition 1.1 M 00 lies in ≤0 0 00 D = Ker R ⊂ D, so that M = R(M ) = 0. Preliminary version – please do not distribute, use at your own risk 5 Corollary 1.6. The left-admissible subcategory D≤0 ⊂ D is right-admis- sible. Proof. As follows from the proof of Lemma 1.4, every object in D≤0 is a cone of the map s : F (M) → M for some M ∈ D≤0. In particular, we have ≤0 D ⊂ D≤1. Since D≤n ⊂ D is right-admissible for any n ≥ 0, it suffices to ≤0 ⊥ ⊥ prove that D is right-admissible in D≤1. Let D , D ⊂ D≤1 be its right and left orthogonal. As in the proof of Lemma 1.4, we have to prove that the projection p : D⊥ → ⊥D is essentially surjective. ⊥ We have D = D≥1 ∩ D≤1 = F (D≤0). Moreover, for any M ∈ D≤0, the ≤0 ∼ cone of the map sM : F (M) → M lies in D , so that p(M) = p(F (M)) = ⊥ F (M). Thus it suffices to prove that every object M ∈ D≤0 lies in D .