English version C. R. Acad. Sci. Paris 307 (1988), 543-546.

Grothendieck-Roos-Duality and Tilting

Bernhard Keller Dieter Vossieck

Abstract^{(1)} - We investigate the relations between the hearts of two t-structures on one triangulated category. Under suitable compatibility conditions, we obtain a common generalization of the duality theory developed by Grothendieck and Roos [10] for regular commutative rings and the tilting theory [3] [6] [4] used in the investigation of finite-dimensional algebras [5].

Dualité de Grothendieck-Roos et basculement

Résumé^{(1)} - Nous étudions les relations entre les coeurs de deux t-struc-tures d’une même catégorie triangulée. Des conditions de compatibilité appropriées nous permettent de généraliser à la fois la théorie de dualité développée par Grothendieck et Roos [10] et la théorie du basculement [3] [6] [4] utilisée dans l’étude des algèbres de dimension finie [5].

1. Let T be a triangulated category [2] [7] with suspension functor S and let (V^{<1}, V^{≥0}) be a t-structure on T in the sense of Beilinson-Bernstein-Deligne [2] [8]. The inclusion of V^{≥n} := S^{−n} V^{≥0} (resp. of V^{<n} := S^{−n+1} V^{<1}) into T admits a left adjoint τ^{≥n} : T → V^{≥n} (resp. a right adjoint τ^{<n} : T → V^{<n}) and gives rise to triangles of the form

τ^{<n}X → X → τ^{≥n}X → Sτ^{<n}X, X ∈ T .

We propose ourselves to compare (V^{<1}, V^{≥0}) to a second t-structure (U_{≥0} , U_{<1}) on T . We denote by τ_{≥n} : T → U_{≥n} and τ_{<n} : T → U_{<n} the left and right adjoints of the inclusions U_{≥n} := S^{n} U_{≥0} et U_{<n} := S^{n−1} U_{<1} into T . These give rise to triangles

τ_{≥n}X → X → τ_{<n}X → Sτ_{≥n}X, X ∈ T .

The hearts [2] of the two t-structures are the abelian categories A = U_{≥0} ∩ U_{<1} and B = V^{≥0} ∩ V^{<1}. They are related to T by the functors “homology”

H_{n} = τ_{<1}τ_{≥0}S^{−n} : T → A

and “cohomology”

H^{n} = τ^{<1}τ^{≥0}S^{n} : T → B

which transform triangles X → Y → Z → SX of T into long exact sequences [2]

… → H_{n+1}Z → H_{n}X → H_{n}Y → H_{n}Z → H_{n−1}X → …

and

… → H^{n−1}Z → H^{n}X → H^{n}Y → H^{n}Z → H^{n+1}X → …

2. In order to investigate the relations between the hearts A et B, we set

A^{≥n} := A ∩ V^{≥n} and B_{≥n} := B ∩ U_{≥n} ,

thus obtaining filtrations of A and B by subcategories which are full and stable under extensions

… ⊂ A^{≥n+1} ⊂ A^{≥n} ⊂ … ⊂ A and B ⊃ … ⊃ B_{≥n} ⊃ B_{≥n+1} ⊃ …

We say that the aisle [8] U_{≥0} is compatible with the co-aisle V^{≥0} if τ^{<n} U_{≥0} ⊂ U_{≥0} for all n ∈ Z. This implies τ^{≥n} U_{≥0} ⊂ U_{≥0}, H^{n} U_{≥0} ⊂ B_{≥n} and H_{m} H^{n}| A = 0 for all m, n such that m < n.

Proposition - If U_{≥0} is compatible with V^{≥0}, the filtration (B_{≥n}) of B has the following property (*) : For each morphism g : N → N′ of B such that N ∈ B_{≥n} and N′ ∈ B_{≥n+1}, we have Ker g ∈ B_{≥n} and Coker g ∈ B_{≥n+1}.

Indeed, consider the following octahedron where K = Ker g, I = Im g, C = Coker g [2].

[figure: I; N′; C; Y; SK; N]

The triangle N → N′ → Y → SN shows that Y ∈ U_{≥n+1}. The triangle SK → Y → C → S^{2}K yields the exact sequences 0 = H^{−2}C → K → H^{−1}Y → H^{−1}C = 0 and 0 = H^{1}K → H^{0}Y → C → H^{2}K = 0. The compatibility condition implies that K →^{∼} H^{−1}Y ∈ B_{≥n} and that C →^{∼} H^{0}Y ∈ B_{≥n+1}.

Dually, we shall say that the co-aisle V^{≥0} is compatible with the aisle U_{≥0} if τ_{<n}V^{≥0} ⊂ V^{≥0} for all n ∈ Z. This new definition entails the dual of the foregoing proposition.

3. We say that B generates T , if T coïncides with the smallest triangulated subcategory of T which is strictly full and contains B. Then each object X ∈ T is obtained by successive extensions from a finite number of shifted homology groups S^{−n}H^{n} X. In particular, the t-structure (V^{<1}, V^{≥0}) is non-degenerated [2]. Moreover, in this case U_{≥0} is compatible with V^{≥0} iff U_{≥0} = {X ∈ T : H^{n} X ∈ B_{≥n} for all n ∈ Z}.

Proposition - Suppose that T is generated by A, as well as by B. In order for U_{≥0} to be compatible with V^{≥0}, it is then necessary and sufficient that H_{m}H^{n}| A = 0 if m < n, and that the filtration (B_{≥n}) satisfies the condition (*) above.

Indeed, it remains to be shown that these conditions are sufficient. Since (U_{≥0} , U_{<1}) is not degenerated the first condition means that H^{n}A ⊂ B_{≥n}, ∀n. We conclude that H^{n}X ∈ B_{≥n} for each X ∈ U_{≥0}: if X ≠ 0, we proceed by recursion on the greatest r such that H_{r}X ≠ 0 (A generates T !). The triangle S^{r}H_{r}X → X → τ_{<r}X → S^{r+1}H_{r}X induces an exact sequence

H^{n−1}τ_{<r} X →^{f} H^{n+r}H_{r} X → H^{n}X → H^{n}τ_{<r}X →^{g} H^{n+r+1}H_{r}X ,

where H^{n+r}H_{r} X ∈ B_{≥n+r} by what we already showed and H^{n−1}τ_{<r} X ∈ B_{≥n−1} by the recursion hypthesis. The condition (*) implies Coker f ∈ B_{≥n}. Similarly, Ker g ∈ B_{≥n}, hence H^{n} X ∈ B_{≥n}, since B_{≥n} is stable under extensions.

Since T is also generated by B, we finally obtain τ^{<n} U_{≥0} ⊂ U_{≥0} for each n.

4. Example. (cf. [10]) Let Λ be a regular ring, i.e. Λ is commutative, noetherian and of finite global dimension. We recall that [9]:

a) For each finitely generated Λ-module M , the “codimension”

c (M) = inf {dim Λ_{℘} : ℘ ∈ Spec (Λ) , M_{℘} ≠ 0}

coïncides with the “grade”

g (M) = inf {i : Ext_{Λ}^{i} (M, N) ≠ 0}.

b) c (Ext_{Λ}^{n} (M, N)) ≥ n for all finitely generated Λ-modules M, N and each n.

The derived functor D = R Hom_{Λ}(?, Λ) induces a duality on the “bounded” derived category T := D^{b}(mod Λ) associated with the category mod Λ of finitely generated Λ-modules. We consider the natural co-aisle V^{≥0} = {X ∈ T : H^{n} X = 0, ∀n < 0} defined by means of the usual cohomology functor, and the aisle U_{≥0} = {Y ∈ T : ∃X ∈ V^{≥0}, Y →^{∼} DX}. We “identify” mod Λ with B = V^{≥0} ∩ V^{<1}, (mod Λ)op with A = U_{≥0} ∩ U_{<1}, and the functors H_{n} : B → A and H^{n} : A → B with Ext_{Λ}^{n} (?, Λ).

In this case, B_{≥n} is the Serre subcategory of B which is formed by the Λ-modules of codimension ≥ n and therefore satisfies (*). It is immediate from b) that H_{m} H^{n}| A = 0 for m < n, hence that U_{≥0} is compatible with V^{≥0}, and V^{≥0} with U_{≥0}.

5. Example. (cf. [3][6][4][5]) Let k be a commutative field, Λ a finitedimensional k-algebra, B_{≥n} = B = mod Λ for n < 0, B_{≥n} = 0 for n > 0 and B_{≥0} a torsion subcategory (i. e. full and closed under extensions and quotients) of B. Then U_{≥0} = {X ∈ D^{b}(B) : H^{n} X ∈ B_{≥n}, ∀n ∈ Z} is an aisle in D^{b}(B), which is compatible with the natural co-aisle V^{≥0} (§4). Also U_{<1} is compatible with V^{<1}, V^{≥0} with U_{≥0} and V^{<1} with U_{<1}.

Suppose that moreover B_{≥0} is generated by a tilting module T_{Λ} over Λ. If Γ = End (T_{Λ}), the derived functors R Hom_{Λ}(T,?) and L (? ⊗_{Γ} T) : D^{b}(mod Γ) → D^{b}(mod Λ) are quasi-inverse [1] [5] S-equivalences [7]. They allow us to identify U_{≥0} with the natural aisle of D^{b}(mod Γ), A with mod Γ, H_{n}| A with Tor_{n}^{Γ} (?, T) and H^{n}| B with Ext_{Λ}^{n} (T,?).

6. Example. (cf. [2] [8]) Let k be a commutative field, Q a finite quiver without oriented cycle, I an admissible ideal in the path category kQ, Λ the quotient kQ/I and mod Λ the category of Λ-“modules” M : Λop → mod k. We consider the natural co-aisle V^{≥0} of T = D^{b}(mod Λ).

In order to construct an (artificial) aisle in T , we start from a function p : { points of Λ} → Z such that p (x) ≥ p (y) if Hom (x, y) ≠ 0. We denote by Λ_{≥n} (resp. Λ_{<n}, resp. Λ_{n}) the full subcategory of Λ formed by the x ∈ Λ such that p (x) ≥ n (resp. p (x) < n, resp. p (x) = n) and we set

U′ = {X ∈ T : supp H^{n} X ⊂ Λ_{≥n}, ∀n}
U′′ = {X ∈ T : supp H^{n} X ⊂ Λ_{<n}, ∀n}

It is clear that U′ is stable under S and U′′ under S^{−1}. Moreover, Hom (X, Y) = 0 if X ∈ U′ and Y ∈ U′′: indeed, X ∈ U′ is equivalent to the existence of a quasi-isomorphism P → X, where the P^{n} are projective and such that supp P^{n} ⊂ Λ_{≥n}, ∀n. And Y ∈ U′′ means that there is a quasi-isomorphism Y → J, where the J^{n} are injective and such that supp J^{n} ⊂ Λ_{<n}, ∀n.

For each X ∈ T we denote by X′ the subcomplex such that X^{′n} = (X^{n}| Λ_{≥n+1})_{0} +(Z^{n} X| Λ_{≥n})_{0} , where the index 0 stands for the extension by zero. We obtain a triangle X′ → X → X′′ = X/X′ → SX′ of T where X′ ∈ U′ and X′′ ∈ U′′, as can be seen from the restrictions X′| Λ_{n} et X′′| Λ_{n}. We finally establish that:

• U_{≥0} = U′ is an aisle in T , and we recover the situation described in §1 with U_{<0} = U′′.

• τ_{≥0} and τ_{<0} can be chosen such that τ_{≥0}X = X′ et τ_{<0}X = X′′ with the above notations.

• The functor X ↦ (H^{n} X)_{n∈Z} induces an equivalence between the

heart

A = {X ∈ T : supp H^{n} X ⊂ Λ_{n}, ∀n}

and the direct sum of the categories mod Λ_{n} ; in particular, A generates T .

• If X ∈ T , the complex H_{0} X of mod Λ has homology groups

H^{n} H_{0} X →^{∼} (H^{n} X| Λ_{n})_{0}.

By construction, U_{≥0} is compatible with V^{≥0} and U_{<1} and V^{<1}. The description of τ_{≥0} and τ_{<1} shows that V^{≥0} is also compatible with U_{≥0} and V^{<1} with U_{<1}. In particular, for each N ∈ mod Λ, we have H^{n} H_{n} N →^{∼} (N| Λ_{n})_{0} for each n and H^{m} H_{n} N = 0 if m ≠ n.

7. We return to the situation of §1. In the following paragraphs, we always suppose that U_{≥0} is compatible with V^{≥0} and V^{≥0} with U_{≥0}.

Each N ∈ B gives rise to a triangle τ_{≥n}N → N → τ_{<n}N → Sτ_{≥n}N of T . Since τ_{≥n}N ∈ V^{≥0} and τ_{<n}N ∈ V^{≥0}, the associated cohomology sequence reduces to

0 → H^{0}τ_{≥n}N → N → H^{0}τ_{<n}N → H^{1}τ_{≥n}N → 0

and to the isomorphisms H^{i}τ_{<n}N →^{∼} H^{i+1}τ_{≥n}N (i > 0). Since τ_{≥n}N ∈ U_{≥n} , it follows that H^{0}τ_{≥n}N ∈ B_{≥n} and H^{1}τ_{≥n}N ∈ B_{≥n+1}.

8. Proposition -

a) The subcategory B_{≥n} of B contains with N all the quotients of N.

b) For each N ∈ B, N_{≥n} := H^{0}τ_{≥n}N is the largest subobject of N belonging to B_{≥n}.

Indeed, each morphism N′ → N of B such that N′ ∈ B_{≥n} factors through N_{≥n} → N, which proves b) and a).

9. We say that a morphism t : N → N′ of B is a n-quasi-isomorphism if Ker t ∈ B_{≥n} and Coker t ∈ B_{≥n+1} . An objet B ∈ B is called nclosed if for each n-quasi-isomorphism t : N → N′ the map Hom (t, B) : Hom (N′, B) → Hom (N, B) is bijective.

Proposition - The inclusion of the full subcategory of B formed by the n-closed objects has the functor N ↦ H^{0}τ_{<n}N (= n-closure of N) as a left adjoint.

Indeed, the canonical morphism N → H^{0}τ_{<n}N is an n-quasi-isomorphism (§7). Thus it suffices to show that H^{0}τ_{<n}N is n-closed. Now let t : Q → Q′ be an n-quasi-isomorphism. Since τ_{<n}N ∈ V^{≥0}, H^{0}τ_{<n}N coïncides with τ^{<1}τ_{<n}N, and Hom (t, H^{0}τ_{<n}N) identifies with Hom (t, τ_{<n}N). The lower triangle of the diagram

[figure: Q; Q′; C; SK; Y; Δ; Δ; =; =; t]

where K = Ker t, C = Coker t and Y ∈ U_{≥n+1} (cf. §2), yields an exact sequence

0 = Hom (Y, τ_{<n}N) → Hom (Q′, τ_{<n}N) →
Hom (Q, τ_{<n}N) → Hom (S^{−1}Y, τ_{<n}N) = 0 ,

which proves our assertion.

Dually, one defines “n-coquasi-isomorphisms” and n-co-closed objects in A.

10. Consider the following strictly full subcategories of A and B:

\underline{A}^{n} = \{M ∈ A^{≥n} : M is (n + 1)-co-closed \}

\overline{B}_{n} = \{N ∈ B_{≥n} : N is (n + 1)-closed \}

Proposition - The functors H_{n} et H^{n} induce a pair of adjoint functors

H_{n} : B_{≥n} → A^{≥n} , H^{n} : A^{≥n} → B_{≥n}

and of quasi-inverse equivalences

H_{n} : \overline{B}_{n} →^{∼} \underline{A}^{n} , H^{n} : \underline{A}^{n} →^{∼} \overline{B}_{n}.

Indeed, we know that H_{n} B_{≥n} ⊂ A^{≥n} et H^{n} A^{≥n} ⊂ B_{≥n}. For each N ∈ B_{≥n}, we have H_{n} N = τ_{<1}S^{−n}N; also, H^{n} M = τ^{<1}S^{n}M for each M ∈ A^{≥n}. Thus we obtain a functorial isomorphism

Hom (H_{n} N, M) →^{∼} Hom (S^{−n}N, M) →^{∼} Hom (N, S^{n}M)

→^{∼} Hom (N, H^{n} M),

which the first assertion.

If n ∈ B_{≥n}, we also have S^{n}H_{n} N = τ_{<n+1}N, and the adjunction morphism N → H^{n} H_{n} N coïncides with the canonical morphism N → H^{0}τ_{<n+1}N to the (n + 1)-closure of N. It follows that N → H^{n} H_{n} N is invertible iff N if (n+1)-closed . By duality, one sees that H_{n} H^{n} M → M is invertible iff M is (n + 1)-co-closed.

(^{1}) Wir danken P. Gabriel für Vorlesungen zu diesem Thema.

References

[1] A. A. Beilinson, Coherent sheaves on P^{n} and problems of linear algebra, Funct. Anal. and Appl., vol. 12, 1979, p. 214-216.

[2] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100, 1982.

[3] S. Brenner, M. C. R. Butler, Generalization of the Bernstein-Gelfand-Ponomarev reflection functors, Representation theory II, Proc. Ottawa 1979, Lecture Notes in Math., 832, Springer, 1980, p. 399-443.

[4] K. Bongartz, Tilted algebras, Representations of Algebras, Proceedings Puebla 1980, Lecture Notes in Math., 903, Springer, 1982, p. 26-38.

[5] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62, 1987 , p. 339-389.

[6] D. Happel, C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274, 1982, p. 399-443.

[7] B. Keller, D. Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris 305, série I, 1987, p. 225-228.

[8] B. Keller, D. Vossieck, Aisles in derived categories, Bulletin de la Soc. Math. de Belgique, to appear

[9] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.

[10] J.-E. Roos, Bidualité et structure des foncteurs dérivées de lim_{←} dans la catégorie des modules sur un anneau régulier, C. R. Acad. Sci. Paris, 280, série I, 1962, p. 1556-1558.

Bernhard Keller : Mathematik, G 28.2, E.T.H.-Zentrum,

8092 Zürich, Switzerland;

Dieter Vossieck : Mathematisches Institut, Universität Zürich, Rämistrasse 74, 8001 Zürich, Switzerland.