t-STRUCTURES, TORSION THEORIES AND DG ALGEBRAS

JUN-ICHI MIYACHI

In this note, for a ring A Mod A (resp., mod A) is the category of right A-modules (resp., finitely generated right A-modules), and Proj A (resp., proj A) the category of projective right A-modules (resp., finitely generated projective right A-modules).

1. t-structures

We recall the notion of t-structures which was introduced by Beilinson, Bernstein and Deligne. In this section, T is a triangulated category, C is a full subcategory of T satisfying

Hom_{T} (C, C[i]) = 0 (i < 0).

Proposition 1.1. For a morphism f : X → Y in C, suppose that there are N, C ∈ C such that

\xymatrix{N \ar[d] \ar[dr]^{α[−1]} &  &  & N[1] \ar[d] \\ S[−1] \ar[d] \ar[r] & X \ar[r]^{f} & Y \ar[dr]_{β} \ar[r] & S \ar[d] \\ C[−1] &  &  & C}

where all vertical and horizontal sequences are distinguished triangles. Then we have ker f = α[−1], Cok f = β in C.

Definition 1.2. A morphism f : X → Y in C is called C-admissible if there exist N, C ∈ C satisfying Proposition 1.1. A sequence X → Y → Z in C is called an admissible short exact sequence if X → Y → Z → X[1] is a distinguish triangle for some Z → X[1].

Proposition 1.3. Suppose that C is stable under finite coproducts. Then the following are equivalent.

1. C is abelian, and all short exact sequences are admissible.

2. All morphisms in C are C-admissible.

Definition 1.4. A full subcategory C of T is called an admissible abelian category if C satisfy the equivalent conditions in Proposition 1.3.

Definition 1.5. Let T be a triangulated category. For full subcategories T^{≤0} and T^{≥0}, (T^{≤0}, T^{≥0}) is called a t-structure on T provided that

(i) Hom_{T} (T^{≤0}, T^{≥1}) = 0; (ii) T^{≤0} ⊂ T^{≤1} and T^{≥0} ⊃ T^{≥1}; (iii) for any X ∈ T , there exists a distinguished triangle

This is a seminar note of which I gave a lecture at Osaka City University in March 2001.

X′ → X → X′′ →

with X′ ∈ T^{≤0} and X′′ ∈ T^{≥1}, where T^{≤n} = T^{≤0}[−n] and T^{≥n} = T^{≥0}[−n].

The core of this t-structure is C = T^{≤0} ∩ T^{≥0}.

Proposition 1.6. For n ∈ ℤ, the following hold.

1. The inclusion T^{≤n} → T has a right adjoint σ_{≤n} : T → T^{≤n}.

2. The inclusion T^{≥n} → T has a left adjoint σ_{≥n} : T → T^{≥n}.

3. For any X ∈ T , there exists a unique d ∈ Hom_{T} (σ_{≥1}X, σ_{≤0}X[1]) such that

σ_{≤0}X → X → σ_{≥1}X ⟶^{d} σ_{≤0}X[1]

is a distinguished triangle.

4. Let A → X → B → A[1] be a distinguished triangle with A ∈ T^{≤0}, B ∈ T^{≥1}. Then this triangle is isomorphic to σ_{≤0}X → X → σ_{≥1}X ⟶^{d} σ_{≤0}X[1].

Remark 1.7. For X ∈ T , the following hold.

1. σ_{≥n}X = O iff X ∈ T^{≤n−1}.

2. σ_{≤n}X = O iff X ∈ T^{≥n+1}.

Proposition 1.8. For a ≤ b, X ∈ T , there is an isomorphism σ_{≥a}σ_{≤b}X ⟶^{∼} σ_{≤b}σ_{≥a}X such that

\xymatrix{σ_{≥a}σ_{≤b}X \ar[rr]^{∼} &  & σ_{≤b}σ_{≥a}X \ar[d] \\ σ_{≤b}X \ar[u] \ar[r] & X \ar[r] & σ_{≥a}X}

is commutative.

Theorem 1.9. The core C = T^{≤0} ∩ T^{≥0} is an admissible abelian category which is stable under extensions, and H^{0} = σ_{≥0}σ_{≤0} : T → C is a cohomological functor.

Definition 1.10. A t-structure (T^{≤0}, T^{≥0}) on T is called non-degenerate provided that ⋂_{n∈ℤ}T^{≤n} = ⋂_{n∈ℤ}T^{≥n} = {0}.

Proposition 1.11. Let (T^{≤0}, T^{≥0}) be a non-degenerate t-structure. For X ∈ T , the following hold.

1. H^{i} X = O for any n iff X = O.

2. H^{i} X = O for any i > n (resp., i < n) iff X ∈ T^{≤n} (resp., X ∈ T^{≥n}). Here H^{i} X = H^{0}(X[i]).

2. t-structures Induced by Compact Objects

A triangulated category T is said to contain coproducts if coproducts of objects indexed by any set exist in T . An object C of T is called compact if Hom_{T} (C, −) commutes with coproducts. Furthermore, a collection S of compact objects of T is called a generating set provided that X = 0 whenever Hom_{T} (S, X) = 0, and that S is stable under suspensions. In this case, T is called compactly generated (see [Ne] for details). For an object C ∈ T and an integer n, we denote by T^{≥n}(C) (resp., T^{≤n}(C)) the full subcategory of T consisting of X ∈ T with Hom_{T} (C, X[i]) = 0 for i < n (resp., i > n), and set T^{0}(C) = T^{≤0}(C) ∩ T^{≥0}(C).

For an abelian category A, we denote by C(A) the category of complexes of A, and denote by D(A) (resp., D^{+}(A), D^{−}(A), D^{b}(A)) the derived category of complexes of A (resp., complexes of A with bounded below homologies, complexes of A with bounded above homologies, complexes of A with bounded homologies). For an additive category B, we denote by K(B) (resp., K^{−}(B), K^{b}(B)) the homotopy category of complexes of B (resp., bounded above complexes of B, bounded complexes of B) (see [RD] for details).

Proposition 2.1. Let T be a triangulated category which contains coproducts, C a compact object satisfying Hom_{T} (C, C[n]) = 0 for n > 0. Then for any r ∈ ℤ and any object X ∈ T , there exist an object X^{≥r} ∈ T^{≥r}(C) and a morphism α^{≥r} : X → X^{≥r} in T such that

(i) for any i ≥ r, Hom_{T} (C, α^{≥r}[i]) is an isomorphism, (ii) for every object Y ∈ T^{≥r}(C), Hom_{T} (α^{≥r}, Y) is an isomorphism.

Theorem 2.2. Let T be a triangulated category which contains coproducts, C a compact object satisfying Hom_{T} (C, C[n]) = 0 for n > 0, and B = End_{T} (C). If {C[i] | i ∈ ℤ} is a generating set, then the following hold.

(1) (T^{≤0}(C), T^{≥0}(C)) is a non-degenerate t-structure on T .

(2) T^{0}(C) is admissible abelian.

(3) The functor

Hom_{T} (C, −) : T^{0}(C) → Mod B

is an equivalence.

3. Torsion Theories for Abelian Categories

Throughout this section, we fix the following notation. Let A be an abelian category satisfying the condition Ab4 (i.e. direct sums of exact sequences are exact), and let d_{P}^{−1} : P^{−1} → P^{0} be a morphism in A with the P^{i} being small projective objects of A, and denote by P^{▪} the mapping cone of d_{P}^{−1}. We set C(P^{▪}) = D(A)^{0}(P^{▪}), B = End_{D(A)}(P^{▪}), and define a pair of full subcategories of A

X (P^{▪}) = {X ∈ A | Hom_{D(A)}(P^{▪}, X[1]) = 0},
Y(P^{▪}) = {X ∈ A | Hom_{D(A)}(P^{▪}, X) = 0}.

For any X ∈ A, we define a subobject of X

τ(X) = ∑_{f∈Hom_{A}(H^{0}(P^{▪}),X)} Im f

and an exact sequence in A

(e_{X}) : 0 → τ(X) ⟶^{j_{X}} X → π(X) → 0.

Remark 3.1. It is easy to see that P^{▪} is a compact object of D(A), and we have X (P^{▪}) = D(A)^{≤0}(P^{▪}) ∩ A and Y(P^{▪}) = D(A)^{≥1}(P^{▪}) ∩ A.

Lemma 3.2. For any X^{▪} ∈ D(A) and n ∈ ℤ, we have a functorial exact sequence

0 → Hom_{D(A)}(P^{▪}, H^{n−1}(X^{▪})[1]) → Hom_{D(A)}(P^{▪}, X^{▪}[n]) → Hom_{D(A)}(P^{▪}, H^{n}(X^{▪})) → 0.

Moreover, the above short exact sequence commutes with coproducts.

Definition 3.3. A pair (X , Y) of full subcategories X , Y in an abelian category A is called a torsion theory for A provided that the following conditions are satisfied (see e.g. [Di] for details):

(i) X ∩ Y = {0}; (ii) X is closed under factor objects; (iii) Y is closed under subobjects; (iv) for any object X of A, there exists an exact sequence 0 → X′ → X → X′′ → 0

in A with X′ ∈ X and X′′ ∈ Y.

Remark 3.4. Let A be an abelian category and (X , Y) a torsion theory for A. Then for any Z ∈ A, the following hold.

(1) Z ∈ X if and only if Hom_{A}(Z, Y) = 0.

(2) Z ∈ Y if and only if Hom_{A}(X , Z) = 0.

Theorem 3.5. The following are equivalent for a complex P^{▪} : P^{−1} ⟶^{d_{P}^{−1}} P^{0} with the P^{i} being small projective objects of A.

(1) {P^{▪}[i] | i ∈ ℤ} is a generating set for D(A) and Hom_{D(A)}(P^{▪}, P^{▪}[i]) = 0 for all i > 0.

(2) X (P^{▪}) ∩ Y(P^{▪}) = {0} and H^{0}(P^{▪}) ∈ X (P^{▪}).

(3) X (P^{▪}) ∩ Y(P^{▪}) = {0} and τ(X) ∈ X (P^{▪}), π(X) ∈ Y(P^{▪}) for all X ∈ A.

(4) (X (P^{▪}), Y(P^{▪})) is a torsion theory for A.

Lemma 3.6. Assume X (P^{▪}) ∩ Y(P^{▪}) = {0}. Then for any X^{▪} ∈ D(A), the following are equivalent.

(1) X^{▪} ∈ C(P^{▪}).

(2) H^{n}(X^{▪}) = 0 for n > 0 and n < −1, H^{0}(X^{▪}) ∈ X (P^{▪}) and H^{−1}(X^{▪}) ∈ Y(P^{▪}).

Remark 3.7. Let A be an abelian category and X , Y full subcategories of A. Then the pair (X , Y) is a torsion theory for A if and only if the following two conditions are satisfied:

(i) Hom_{A}(X , Y) = 0; (ii) for any object X in A, there exists an exact sequence 0 → X′ → X → X′′ → 0

in A with X′ ∈ X and X′′ ∈ Y.

Theorem 3.8. Let P^{▪} be a complex P^{−1} ⟶^{d_{P}^{−1}} P^{0} with the P^{i} being small projective objects of A. Assume X (P^{▪})∩Y(P^{▪}) = {0} and H^{0}(P^{▪}) ∈ X (P^{▪}). Then the following hold.

(1) C(P^{▪}) is admissible abelian.

(2) The functor

Hom_{D(A)}(P^{▪}, −) : C(P^{▪}) → Mod B

is an equivalence.

(3) (Y(P^{▪})[1], X (P^{▪})) is a torsion theory for C(P^{▪}).

Proposition 3.9. Assume P^{▪} satisfies the conditions (i) {P^{▪}[i] | i ∈ ℤ} is a generating set for D(A), (ii) Hom_{D(A)}(P^{▪}, P^{▪}[i]) = 0 for i ≠ 0. If A has either enough projectives or enough injectives, then we have an equivalence of triangulated categories

D^{b}(A) ≅ D^{b}(Mod B).

Example 3.10 (cf. [HK]). Let A be a finite dimensional algebra over a field k given by a quiver

1 ⟶^{α} 2
δ↑  ↓^{β}
4 ⟵_{γ} 3

with relations βα = γβ = δγ = αδ = 0. For each vertex i, we denote by S(i), P (i) the corresponding simple and indecomposable projective left A-modules, respectively. Define a complex P^{▪} as the mapping cone of the homomorphism

d_{P}^{−1} = [ \begin{matrix}f & 0 & 0 & 0 \\ 0 & 0 & g & 0 \end{matrix} ] : P (2)^{2} ⊕ P (4)^{2} → P (1) ⊕ P (3),

where f and g denote the right multiplications of α and γ, respectively. Then P^{▪} is not a tilting complex. However, P^{▪} satisfies the assumption of Theorem 3.8 and hence we have an equivalence of abelian categories

Hom_{D(Mod A)}(P^{▪}, −) : C(P^{▪}) → Mod B,

where B = End_{D(Mod A)}(P^{▪}) is a finite dimensional k-algebra given by a quiver

1 ← 2 3 ← 4.

There exist exact sequences in C(P^{▪}) of the form

0 → S(1) → S(2)[1] → P (1)[1] → 0, 0 → S(3) → S(4)[1] → P (3)[1] → 0,

and these objects and morphisms generate C(P^{▪}).

In the rest of this section, we deal with the case where R is a commutative artin ring, I is an injective envelope of an R-module R/ rad(R) and A is a finitely generated R-module. We denote by mod A the full abelian subcategory of Mod A

consisting of finitely generated modules. P^{▪} is also a complex P^{−1} ⟶^{d_{P}^{−1}} P^{0} with the P^{i} being finitely generated projective A-modules. Note that H^{n}(P^{▪}), H^{n}(ν(P^{▪})) ∈ mod A for all n ∈ ℤ. We set

X_{c}(P^{▪}) = X (P^{▪}) ∩ mod A and Y_{c}(P^{▪}) = Y(P^{▪}) ∩ mod A.

Proposition 3.11. For any tilting complexes P_{1}^{▪} : P_{1}^{−1} → P_{1}^{0}, P_{2}^{▪} : P_{2}^{−1} → P_{2}^{0} for A of term length two, the following are equivalent.

(1) (X_{c}(P_{1}^{▪}), Y_{c}(P_{1}^{▪})) = (X_{c}(P_{2}^{▪}), Y_{c}(P_{2}^{▪})).

(2) add(P_{1}^{▪}) = add(P_{2}^{▪}) in K^{b}(proj A).

Proposition 3.12. The following are equivalent for a complex P^{−1} → P^{0} ∈ K^{b}(proj A)

(1) P^{▪} is a tilting complex.

(2) X_{c}(P^{▪}) ∩ Y_{c}(P^{▪}) = {0}, H^{0}(P^{▪}) ∈ X_{c}(P^{▪}) and H^{−1}(P^{▪}) ∈ Y_{c}(P^{▪}).

(3) (X_{c}(P^{▪}), Y_{c}(P^{▪})) is a torsion theory for mod A and H^{−1}(P^{▪}) ∈ Y_{c}(P^{▪}).

(4) (X_{c}(P^{▪}), Y_{c}(P^{▪})) is a torsion theory for mod A and X_{c}(P^{▪}) is stable under DA⊗_{A}−.

(5) (X_{c}(P^{▪}), Y_{c}(P^{▪})) is a torsion theory for mod A and Y_{c}(P^{▪}) is stable under Hom_{A}(DA, −).

Definition 3.13. Let A be an abelian category and C a full subcategory of A closed under extensions. Then an object X ∈ C is called Ext-projective (resp., Extinjective) if Ext_{A}^{1}(X, C) = 0 (resp., Ext_{A}^{1}(C, X) = 0).

Proposition 3.14. Assume P^{▪} is a tilting complex. Then the following hold.

(1) H^{0}(P^{▪}) ∈ X_{c}(P^{▪}) is Ext-projective and generates X_{c}(P^{▪}).

(2) H^{−1}(ν(P^{▪})) ∈ Y_{c}(P^{▪}) is Ext-injective and cogenerates Y_{c}(P^{▪}).

Theorem 3.15. Let (X , Y) be a torsion theory for mod A such that X contains an Ext-projective module X which generates X, Y contains an Ext-injective module Y which cogenerates Y, and X is stable under DA⊗_{A}−. Let M_{X}^{▪} be a minimal projective presentation of X and N_{Y}^{▪} a minimal injective presentation of Y. Then

P^{▪} = M_{X}^{▪} ⊕ Hom_{A}^{▪}(DA, N_{Y}^{▪})[1]

is a tilting complex such that X = X_{c}(P^{▪}) and Y = Y_{c}(P^{▪}).

Remark 3.16. Let

𝔖 = {P^{▪} : P^{−1} → P^{0} ∈ K^{b}(proj A) | P^{▪} is a tilting complex for A}

on which we define the equivalence relation P_{1}^{▪} ∼ P_{2}^{▪} provided add P_{1}^{▪} = add P_{2}^{▪} in K^{b}(proj A), and let 𝔗 be the collection of torsion theories (X , Y) for mod A such that X contains an Ext-projective module X which generates X, Y contains an Ext-injective module Y which cogenerates Y, and X is stable under DA⊗_{A}−. Set

Φ(P^{▪}) = ((X_{c}(P^{▪}), Y_{c}(P^{▪})) for P^{▪} ∈ 𝔖,
Ψ((X , Y)) = M_{X}^{▪} ⊕ Hom_{A}^{▪}(DA, N_{Y}^{▪})[1] for (X , Y) ∈ 𝔗.

Then, according to Propositions 3.11, 3.12, 3.14 and Theorem 3.15, Φ and Ψ induce a one to one correspondence between 𝔖/ ∼ and 𝔗.

4. Perverse t-structures Induced by Torsion Theories

We recall the notion of perverse t-structures which was introduced by [BBD] and was translated into the language of torsion theories by [VB], and show a relation to the results of Section 3. In this section, A is an abelian category, D = D^{∗}(A), where ∗ = nothing, +, − or b, and

D^{≤0} := {X ∈ D| H^{i}(X) = O for i > 0}
D^{≥0} := {X ∈ D| H^{i}(X) = O for i < 0}

Definition 4.1. Let (X , Y) be a torsion theory for A. We set

\begin{aligned}^{p}D^{≤0} & := \{X ∈ D^{≤0}| H^{0}(X) ∈ X \} \\ ^{p}D^{≥0} & := \{X ∈ D^{≥−1}| H^{−1}(X) ∈ Y\} \end{aligned}

Lemma 4.2. For X^{▪} ∈ D^{≤0}, we have a distinguished triangle

X_{1}^{▪} → X^{▪} → X_{2}^{▪} → X_{1}^{▪}[1]

with X_{1}^{▪} ∈ ^{p}D^{≤0}, X_{2}^{▪} ∈ ^{p}D^{≥1} ∩ D^{0}.

Sketch. We have

[figure: …; X^{−1}; X^{′0}; P B; τ H^{0} X^{▪}; …; X^{−1}; X^{0}; H^{0} X^{▪}; O; π H^{0} X^{▪}; π H^{0} X^{▪}]

Proposition 4.3. Let (X , Y) be a torsion theory for A. Then (^{p}D^{≤0},^{p}D^{≥0}) is a non-degenerate t-structure in D.

Proof. For X^{▪} ∈ ^{p}D^{≤0}, Y^{▪} ∈ ^{p}D^{≥1}, we have

Hom_{D}(X^{▪}, Y^{▪}) ≅ Hom_{D}(σ_{≥0}X^{▪}, σ_{≤0}Y^{▪})

≅ Hom_{D}(H^{0} X^{▪}, H^{0} Y^{▪}) = 0

It is easy to see that ^{p}D^{≤0} ⊂ ^{p}D^{≤1} and ^{p}D^{≥1} ⊂ ^{p}D^{≥0}. Let Y^{▪} ∈ D. By Lemma 4.2, we have a commutative diagram

Y_{1}^{▪} ⟶ τ_{≤0}Y^{▪} ⟶ Y_{2}^{▪} ⟶ Y_{1}^{▪}[1]
∥  ↓  ↓  ∥
Y_{1}^{▪} ⟶ Y^{▪} ⟶ Z^{▪} ⟶ Y_{1}^{▪}[1]
↓  ↓
τ_{≥1}Y^{▪}  τ_{≥1}Y^{▪}
↓  ↓
τ_{≤0}Y^{▪}[1] ⟶ Y_{2}^{▪}[1]

where all vertical and horizontal sequences are distinguished triangles, and Y_{1}^{▪} ∈ pD^{≤0}, Y_{2}^{▪} ∈ ^{p}D^{≥1} ∩ D^{0}. Therefore Z^{▪} ∈ D^{≥0} and H^{0} Z^{▪} ≅ H^{0} Y_{2}^{▪} ∈ Y. Hence Z^{▪} ∈ ^{p}D^{≥1}. Since ^{p}D^{≤0} ⊂ D^{≤0} and ^{p}D^{≥0} ⊂ D^{≥−1}, it is non-degenerate.

Proposition 4.4. Let (X , Y) be a torsion theory for A, ^{p}C = ^{p}D^{≤0} ∩^{p}D^{≥0}. Then pC is admissible abelian and (Y[1], X) is a torsion theory for ^{p}C.

Proof. It is easy to see that Hom_{D}(Y[1], X) = {O}. X^{▪} ∈ ^{p}C iff X^{▪} ≅ Y^{▪} : Y^{−1} → Y^{0} with H^{0} Y^{▪} ∈ X and H^{−1} Y^{▪} ∈ Y. Then we have a distinguished triangle

H^{−1} Y^{▪}[1] → Y^{▪} → H^{0} Y^{▪} → H^{−1} Y^{▪}[2].

This means that we have an exact sequence in ^{p}C

O → F → Y^{▪} → T → O

with F ∈ Y[1], T ∈ X.

Proposition 4.5. Let P^{▪} be a complex P^{−1} → P^{0} with the P^{i} being small projective objects of A. Assume X (P^{▪}) ∩ Y(P^{▪}) = {0} and H^{0}(P^{▪}) ∈ X (P^{▪}). Then a perverse t-structure (^{p}D^{≤0},^{p}D^{≥0}) coincides with (D^{≤0}(P^{▪}), D^{≥0}(P^{▪})).

Proof. By Lemma 3.2.

5. DG-Algebras and Derived Equivalences

Definition 5.1. A differential graded algebra (a DG algebra) B over a commutative ring k is a ℤ-graded k-algebra B = ∐_{n∈ℤ} B^{n} endowed with a differential d : B^{n} → B^{n+1} (n ∈ ℤ) such that

d(ab) = d(a)b + (−1)^{p}ad(b)

for a ∈ B^{p}.

A DG (right) B-module M is a ℤ-graded B-module M = ∐_{n∈ℤ} M^{n} endowed with a differential d : M^{n} → M^{n+1} (n ∈ ℤ) such that

d(ma) = d(m)a + (−1)^{p}md(a)

for m ∈ M^{p}, a ∈ B.

For DG B-module M, N and n ∈ ℤ,

Hom_{Gr B}(M, N)^{n} = the set of graded B-homomorphisms of degree n
Hom_{Gr B}(M, N) = ∐_{n∈ℤ} Hom_{Gr B}(M, N)^{n}
Hom_{Dif B}(M, N) = Hom_{Gr B}(M, N) endowed with the differential
∂ : Hom_{Gr B}(M, N)^{n} → Hom_{Gr B}(M, N)^{n+1}
(∂((f^{p})_{p∈ℤ}) = (d_{N}^{p+n} ∘ f^{p} + (−1)^{n+1}f^{p+1} ∘ d_{M}^{p})_{p∈ℤ})
Hom_{CB}(M, N) = Z^{0} Hom_{Dif B}(M, N)
Hom_{HB}(M, N) = H^{0} Hom_{Dif B}(M, N)

Definition 5.2. The suspension functor S : CB → CB is defined by

\begin{aligned}(SM)^{n} & = M^{n+1} \\ m · a & = ma \\ d_{SM}^{n} & = −d_{M}^{n+1} \end{aligned}

for M ∈ CB.

For u : M → N in CB, the mapping cone M(u) is defined by

\begin{aligned}M^{n}(u) & = N^{n} ⊕ M^{n+1} \\ [ \begin{matrix}n \\ m \end{matrix}] · a & = [ \begin{matrix}na \\ ma \end{matrix}] \\ d_{M(u)}^{n} & = [ \begin{matrix}d_{N}^{n} & u^{n+1} \\ 0 & −d_{M}^{n+1} \end{matrix} ] \end{aligned}

Proposition 5.3. The following hold.

1. Let S_{B} be the collection of exact sequences O → X → Y → Z → O in CB such that O → X^{n} → Y^{n} → Z^{n} → O is split exact in Mod k. Then (CB, S_{B}) is a Frobenius category.

2. Let T_{B} be the collection of sextuples (X, Y, Z, i, v, w) which are isomorphic to standard triangles in HB. Then (HB, T_{B}) is a triangulated category.

Concerning the notion of Frobenius categories, see [Ha], [Mi] Section 5.

Definition 5.4. For a DG algebra B, H^{▪} B = ∐_{n∈ℤ} H^{n} B. For DG B-module M, H^{▪} M = ∐_{n∈ℤ} H^{n} M. Then we have the functor H^{▪} : HA → Gr H^{▪} B. A morphism f : M → N is called quasi-isomorphism if H^{▪} f is isomorphism.

Let Σ be the collection of quasi-isomorphisms in HB, then DB is Σ^{−1}HB. In this case, the canonical functor CB → HB → DB commutes with coproducts.

Lemma 5.5. Let (F_{i}, S_{i}) be Frobenius categories (i = 1, 2). If a functor F : F_{1} → F_{2} satisfies that F (S_{1}) ⊂ S_{2} and that F Q is S_{2}-projective for every S_{1}-projective object Q, then F induces ∂-functor \underline{F} : \underline{F}_{1} → \underline{F}_{2}.

Definition 5.6. Let A be an abelian category. For a complexes X^{▪}, Y^{▪} ∈ C(A), we define the complex Hom_{A}^{▪}(X^{▪}, Y^{▪}) by

Hom_{A}^{p}(X^{▪}, Y^{▪}) = ∏_{n∈ℤ} Hom_{A}(X^{n}, Y^{n+p})
Hom_{A}^{▪}(X^{▪}, Y^{▪}) = ∐_{p∈ℤ} Hom_{A}^{p}(X^{▪}, Y^{▪})
d_{Hom_{A}^{▪}(X^{▪},Y^{▪})}^{p}((f^{n})_{n∈ℤ}) = (d_{Y}^{n+p} ∘ f^{n} − (−1)^{p}f^{n+1} ∘ d_{X}^{n})_{n∈ℤ}.

Proposition 5.7. Let A be an AB4-category, A′ thick abelian subcategory which is closed under coproducts. Let C^{▪} ∈ C_{A′}(A), B = End_{C(A)}^{▪}(C^{▪}). Then the following hold.

1. We have the functor Hom_{A}^{▪}(C^{▪}, −) : C_{A′}(A) → CB.

2. Hom_{A}^{▪}(C^{▪}, −) induces the ∂-functor Hom_{A}^{▪}(C^{▪}, −) : K_{A′}(A) → HB.

3. If there is a triangulated full subcategory L of K_{A′}(A) such that (a) every X^{▪} ∈ K_{A′}(A) has a quasi-isomorphic to some complex in L,

(b) Hom_{K(A)}(K_{A′}^{ϕ}(A), L) = 0,

then the ∂-functor Hom_{A}^{▪}(C^{▪}, −) : K_{A′}(A) → HB induces the right derived functor R Hom_{A}^{▪}(C^{▪}, −) : D_{A′}(A) → DB.

Here K_{A′}^{ϕ}(A) is the full subcategory of K_{A′}(A) consisting of acyclic complexes. In this case, we say that K_{A′}(A) has a K_{A′}^{ϕ}(A)-Bousfield localization.

Lemma 5.8. Let A be an AB4-category. Let C^{▪} ∈ C(A) which is a bounded complex of small projective objects, and B = End_{C(A)}^{▪}(C^{▪}). Then the following hold.

1. R Hom_{A}^{▪}(C^{▪}, −) commutes with coproducts.

2. Hom_{D(A)}(C^{▪}, X^{▪}) ≅ Hom_{DB}(R Hom_{A}^{▪}(C^{▪}, C^{▪}), R Hom_{A}^{▪}(C^{▪}, X^{▪})).

Lemma 5.9. Let A be an AB4-category, A′ thick abelian subcategory which is closed under coproducts. Assume that K_{A′}(A) has a K_{A′}^{ϕ}(A)-Bousfield localization. Let C^{▪} ∈ C_{A′}(A) which is K_{A′}^{ϕ}(A)-local and is compact in D_{A′}(A), and B = End_{C(A)}^{▪}(C^{▪}). Then the following hold.

1. R Hom_{A}^{▪}(C^{▪}, −) commutes with coproducts.

2. Hom_{D(A)}(C^{▪}, X^{▪}) ≅ Hom_{DB}(R Hom_{A}^{▪}(C^{▪}, C^{▪}), R Hom_{A}^{▪}(C^{▪}, X^{▪})).

Proposition 5.10. Under the condition of Lemma 5.8 (resp., Lemma 5.9), if {C^{▪}[i]|i ∈ ℤ} is a generating set for D(A) (resp., D_{A′}(A)), then R Hom_{A}^{▪}(C^{▪}, −) : D(A) → DB (resp., R Hom_{A}^{▪}(C^{▪}, −) : D_{A′}(A) → DB) is an equivalence.

Proof. By Theorem 6.3 of Appendix.

Corollary 5.11. Let P^{▪} be a bounded complex of finitely generated projective modules over a ring A, B = End_{C(A)}^{▪}(C^{▪}). If {P^{▪}[i]|i ∈ ℤ} is a generating set for D(Mod A), then R Hom_{A}^{▪}(P^{▪}, −) : D(Mod A) → DB is an equivalence.

Corollary 5.12. Let X be a quasi-compact separated scheme over an algebraically closed field. If a perfect complex C^{▪} ∈ C_{qc}^{+}(Inj X) satisfies that {C^{▪}[i]|i ∈ ℤ} is a generating set for D_{qc}(X) (or D(QCoh X)), then

D(QCoh X) ≅ DB

with B = End_{C(A)}^{▪}(C^{▪}).

Proof. According to [BN], K_{qc}(X) has a K_{qc}^{ϕ}(X)-Bousfield localization, and D_{qc}(X) ≅ D(QCoh X). By 5.10 we complete the proof.

Corollary 5.13. Let X be a projective scheme which embeds to P_{k}^{n}. If a complex C^{▪} ∈ C_{qc}^{+}(Inj X) which is quasi-isomorphic to ⊕_{i=0}^{n}O_{X}(−i), then

D(QCoh X) ≅ DB.

Sketch. Let V be an (n + 1)-dimensional k-vector space. In Mod P_{k}^{n} we have an exact sequence

\begin{aligned}O → ∧^{n+1}V ⊗ O_{P}(−n − 1) & → ∧^{n}V ⊗ O_{P}(−n) → … \\ & → ∧^{1}V ⊗ O_{P}(−1) → O_{P} → O \end{aligned}

Since the above sequence is locally split exact, we have an exact sequence in Mod X

\begin{aligned}O → ∧^{n+1}V ⊗ O_{X}(−n − 1) & → ∧^{n}V ⊗ O_{X}(−n) → … \\ & → ∧^{1}V ⊗ O_{X}(−1) → O_{X} → O \end{aligned}

Therefore ⊕_{i=0}^{n}O_{X}(−i) generates D(Qcoh X).

Corollary 5.14 ([Be]). Let B′ = End_{P_{k}^{n}}(⊕_{i=0}^{n}O_{P^{n}}(−i)), then

D(QCoh P_{k}^{n}) ≅ D(Mod B′)
D^{b}(Coh P_{k}^{n}) ≅ D^{b}(mod B′).

Proof. By Corollary 5.13, we have D(QCoh P_{k}^{n}) ≅ DB. Let B′′ = σ_{≤0}B and B′ = H^{0} B, we have morphisms B′ ← B′′ → B which induce B′ ←^{∼} H^{▪} B′′ ⟶^{∼} H^{▪} B. By [Ke1] 6.1 Example, we have D(QCoh P_{k}^{n}) ≅ D(Mod B′). Since Coh P_{k}^{n} and mod B′ have finite global dimensions, the full subcategories of D(QCoh P_{k}^{n}) and D(Mod B′) consisting of compact objects are equivalent to D^{b}(Coh P_{k}^{n}) and D^{b}(mod B′), respectively. By Theorem 6.3, we complete the proof.

Remark 5.15. Let (Q⃗, ρ) be the following quiver with relations:

[figure: 0; α_{0}^{0}; ⋮; α_{n}^{0}; 1; α_{0}^{1}; ⋮; α_{n}^{1}; 2; ···; n − 1; α_{0}^{n−1}; ⋮; α_{n}^{n−1}; n] ,

and ρ is the set of relations over k

α_{i}^{l+1}α_{j}^{l} = α_{j}^{l+1}α_{i}^{l} for 0 ≤ i < j ≤ n, 0 ≤ l < n − 1.

Then B′ of Corollary 5.14 is isomorphic to k(Q⃗, ρ).

Remark 5.16. Recently, Bondal and Van den Bergh showed that the derived category D(QCoh X) of quasi-coherent sheaves of a Noetherian scheme X has a compact generator. By using [Ke2], they also showed that D(QCoh X) ≅ DB for some DG algebra B.

Example 5.17. In Example 3.10, let B′ = End_{A}^{▪}(P^{▪}). Then we have

D(Mod A) ≅ DB′.

Let B′′ = B^{−1} ⊕ B^{0} with

B^{−1} → B^{0} : Hom_{A}(P^{0}, P^{−1}) → Hom_{C(Mod A)}(P^{▪}, P^{▪})

(f ↦ (f ∘ d^{−1} − f^{−1} ∘ f))

According to [Ke1] 6.1 Example, the natural inclusion B′′ → B′ induces the derived equivalence DB′′ ≅ DB′. Hence we have

D(Mod A) ≅ DB′′.

Example 5.18. In Proposition 5.13, let B′ = End_{A}^{▪}(P^{▪}). Then we have

D(Mod A) ≅ DB′.

Let B′′ = B^{−1} ⊕ B^{0} with

B^{−1} → B^{0} : Hom_{A}(P^{0}, P^{−1}) → Hom_{C(Mod A)}(P^{▪}, P^{▪})

(f ↦ (f ∘ d^{−1} − f^{−1} ∘ f))

According to [Ke1] 6.1 Example, the natural inclusion B′′ → B′ induces the derived equivalence DB′′ ≅ DB′. Hence we have

D(Mod A) ≅ DB′′.

6. Appendix

Throughout this section all triangulated categories contains arbitrary coproducts.

Definition 6.1. A triangulated full subcategory L of T is called localizing provided that every coproduct of objects in L is in L.

Lemma 6.2. Let T be a triangulated category, S a generating set. Let L be a localizing subcategory of T which contains S. Then L = T . Furthermore, for every X ∈ T , there are distinguished triangles

Z_{n} → X_{n} → X_{n+1} → Z_{n}[1]

with X_{0}, Z_{n} ∈ Sum S (n ≥ 0), such that

X ≅ hlim_{⟶} X_{n}

Here Sum S is the full subcategory of T consisting of coproducts of objects X ∈ S.

Proof. See [Ke1] 5.2 Theorem and [Ne] Theorem 4.1.

Theorem 6.3. Let F : T_{1} → T_{2} be a ∂-functor commuting with coproducts. Assume that there is a generating set S for T_{1} such that F S is a generating set for T_{2}. If F |_{S} is fully faithful, then F : T_{1} → T_{2} is an triangle equivalence. In this case, F induces the triangle equivalence T_{1}^{c} → T_{2}^{c}, where T_{i}^{c} is the triangulated full subcategory of T_{i} consisting of compact objects.

Proof. Step 1. We have Hom_{T_{1}}(C, Y) ≅ Hom_{T_{2}}(F C, F Y) for C ∈ S and Y ∈ Sum S.

Step 2. We have Hom_{T_{1}}(C, Y) ≅ Hom_{T_{2}}(F C, F Y) for C ∈ S and Y ∈ T_{1}.

∵ ) Given Y ∈ T_{1}, by Lemma 6.2, there are distinguished triangles

Z_{n} → Y_{n} → Y_{n+1} → Z_{n}[1]

with Y_{0}, Z_{n} ∈ Sum S (n ≥ 0), such that

Y ≅ hlim_{⟶} Y_{n}

By induction on n, we have Hom_{T_{1}}(C, X_{n}) ≅ Hom_{T_{2}}(F C, F Y_{n}). Since F C is compact, we have Hom_{T_{1}}(C, hlim_{⟶} Y_{n}) ≅ Hom_{T_{2}}(F C, F hlim_{⟶} Y_{n}).

Step 3. We have Hom_{T_{1}}(X, Y) ≅ Hom_{T_{2}}(F X, F Y) for X, Y ∈ T_{1}.

∵) It is similar to Step 2.

Step 4. Given M ∈ T_{2}, by Lemma 6.2, there are distinguished triangles

N_{n} → M_{n} → M_{n+1} → N_{n}[1]

with M_{0}, N_{n} ∈ Sum F S (n ≥ 0), such that

M ≅ hlim_{⟶} M_{n}

Since F is fully faithful, by induction there are distinguished triangles

Z_{n} → X_{n} → X_{n+1} → Z_{n}[1]

with X_{0}, Z_{n} ∈ Sum S (n ≥ 0), such that

F Z_{n} ⟶ F X_{n} ⟶ F X_{n+1} ⟶ F Z_{n}[1]
↓^{≀}  ↓^{≀}  ↓^{≀}  ↓^{≀}
N_{n} ⟶ M_{n} ⟶ M_{n+1} ⟶ N_{n}[1]

Hence

M ≅ hlim_{⟶} M_{n}

≅ F hlim_{⟶} X_{n}

≅ F X

Since the compactness of an object is the categorical property, the last assertion is trivial.

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J. Miyachi: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184-8501, Japan

   E-mail address: miyachi@u-gakugei.ac.jp