Enochs conjecture for cotorsion pairs over recollements of exact categories

Lemma 2.1

[20, Proposition 2.3] The following are true for any WIC exact category:

1. If gf is an admissible monomorphism, then f is an admissible monomorphism.

2. If gf is an admissible epimorphism, then g is an admissible epimorphism.

Definition 2.2

Let ℰ be a WIC exact category. A sequence A → f B → g C in ℰ is said to be right exact if there exist an admissible exact sequence K ↣ h 2 B ↠ g C and an admissible epimorphism h 1 : A ↠ K such that f = h 2 h 1 . Moreover, an additive covariant functor F : ℰ → ℰ ′ between exact categories is called a right exact functor if it takes those right exact sequences in ℰ to sequences of the same ilk in ℰ ′ . Dually, one can also define the left exact sequences and left exact functors.

A 4-term sequence A → f B → g C → h D in ℰ is called exact, denoted by A ↣ f B → g C ↠ h D , if there exists admissible exact sequences A ↣ f B ↠ g 1 K and K ↣ g 2 C ↠ h D such that g = g 2 g 1 .

Lemma 2.3

Let ( X , Y ) be a cotorsion pair in ℰ . If ℰ has enough projective objects or enough injective objects, then ( X , Y ) is hereditary if and only if Ext ℰ 2 ( X , Y ) = 0 for every X ∈ X and Y ∈ Y .

Definition 2.4

[17, Definition 3.1] Let A , ℬ , and C be three WIC exact categories. A recollement of C relative to A and ℬ , denoted by ( A , C , ℬ ) , is a diagram

given by two exact functors i * and j * , two right exact functors i * and j ! and two left exact functors i ! and j * , which satisfy the following conditions:

1. ( i * , i * ) , ( i * , i ! ) , ( j ! , j * ) , and ( j * , j * ) are adjoint pairs;

2. i * , j ! and j * are fully faithful;

3. im i * = ker j * ;

4. For any C ∈ C , there exists an exact sequence in C

   i * i ! ( C ) ↣ σ C C → η C j * j * ( C ) ↠ i * ( A ) ,

   with A ∈ A , where σ C and η C are given by the adjunction morphisms;

5. For any C ∈ C , there exists an exact sequence in C

   i * ( A ′ ) ↣ j ! j * ( C ) → ε C C ↠ δ C i * i * ( C ) ,

   with A ′ ∈ A , where ε C and δ C are given by the adjunction morphisms.

Theorem 2.5

[19, Theorem 1.1] Let ( A , C , ℬ ) be a recollement of exact categories with i ! an exact functor. Assume that ( U ′ , V ′ ) and ( U ″ , V ″ ) are complete hereditary cotorsion pairs in A and ℬ , respectively. If C has enough projective and injective objects and j ! is U ″ -exact, then ( U , V ) is a complete hereditary cotorsion pair in C .

Lemma 3.1

Let A be an object in A .

1. If f : U → i * ( A ) is a U -cover, then δ U : U → i * i * ( U ) is an isomorphism.

2. If g : i * ( A ) → V is a V -envelope, then σ V : i * i ! ( V ) → V is an isomorphism.

Proof

We only prove (1), and the proof of (2) is similar. By Definition 2.4 (5), it suffices to show that δ U : U → i * i * ( U ) is an admissible monomorphism. Let f : U → i * ( A ) be a U -cover. Thus, we have the following commutative diagram:

Note that δ i * ( A ) is an isomorphism. It follows that f = δ i * ( A ) − 1 i * i * ( f ) δ U . Since U ∈ U , we have i * i * ( U ) ∈ U . Thus, there exists a morphism α : i * i * ( U ) → U such that δ i * ( A ) − 1 i * i * ( f ) = f α , and hence, f = δ i * ( A ) − 1 i * i * ( f ) δ U = f α δ U . So α δ U is an isomorphism and δ U is an admissible monomorphism. This completes the proof.□

Proposition 3.2

If U is covering in C , then U ′ is covering in A .

Proof

Let A be an object in A . Then, i * ( A ) has a U -cover f : U → i * ( A ) . It follows from Lemma 3.1 (1) that i * i * ( f ) : i * i * ( U ) → i * i * i * ( A ) is a U -cover. Since A ≅ i * i * ( A ) , it suffices to show that i * ( f ) : i * ( U ) → i * i * ( A ) is a U ′ -cover. Let α : U ′ → i * i * ( A ) be a morphism in A with U ′ ∈ U ′ . Hence, i * ( U ′ ) ∈ U , and there exists a morphism β : i * ( U ′ ) → i * i * ( U ) such that i * ( α ) = i * i * ( f ) β . Since i * is fully faithful, there exists a morphism γ : U ′ → i * ( U ) such that β = i * ( γ ) , which shows α = i * ( f ) γ . So i * ( f ) : i * ( U ) → i * i * ( A ) is a U ′ -precover. Assume there exists a morphism h : i * ( U ) → i * ( U ) such that i * ( f ) = i * ( f ) h . Then, i * i * ( f ) = i * i * ( f ) i * ( h ) , and so i * ( h ) is an isomorphism. This implies that h is an isomorphism. So U ′ is covering in A .□

Lemma 3.3

Let ( X , Y ) be a complete hereditary cotorsion pair in an exact category ℰ . Then,

1. X is covering if and only if each object in Y has an X -cover;

2. Y is enveloping if and only if each object in X has an Y -envelope.

Proof

We only prove (1), and the proof of (2) is similar. It suffices to show the “if” part. Assume that each object in Y has an X -cover. Let M be an object in ℰ . Then, there is an admissible exact sequence M ↣ α Y ↠ β X in ℰ with Y ∈ Y and X ∈ X . Thus, we have an admissible Y ′ ↣ X ′ ↠ f Y in ℰ with Y ′ ∈ Y and X ′ ∈ X ∩ Y such that f is an X -cover in ℰ . Consider the following pullback diagram in ℰ :

Since X , X ′ ∈ X , so is P and g : P → M is a special X -precover. Next, we need to show that g is an X -cover. Assume there is a morphism h : P → P such that g = g h . Since Ext B 1 ( X , X ′ ) = 0 , there exists a morphism h ′ : X ′ → X ′ such that h ′ q = q h . Note that ( f h ′ − f ) q = ( f h ′ ) q − f q = f ( h ′ q ) − α g = f ( q h ) − α ( g h ) = ( f q ) h − ( α g ) h = 0 . Then, there is a morphism λ : X → Y such that f h ′ − f = λ π . Note that f : X ′ → Y is an X -cover and X ∈ X . Then, there exists a morphism γ : X → X ′ such that λ = f γ .

We set φ = h ′ − γ π . Then, f φ = f h ′ − f γ π = f h ′ − λ π = f and φ q = ( h ′ − γ π ) q = h ′ q − γ π q = h ′ q = q h . Thus, we the following commutative diagram in ℰ with exact rows:

Since f is an X -cover and f φ = f , it follows that φ is an isomorphism. Thus, π φ = ψ π = ψ β f = ψ β f φ = ψ π φ , and hence, ψ π = π . So ψ = id X and h is an isomorphism by the snake lemma (see [21, Corollary 8.13]). This completes the proof.□

Remark 3.4

We note that Lemma 3.3 has been proved by Bazzoni and Positselski in any abelian category (see [8, Lemma 10.3]), and the proof here is different from that of [8, Lemma 10.3].

Lemma 3.5

Let B be an object in ℬ and i ! j ! ( U ″ ∩ V ″ ) ⊆ V ′ .

1. If f : U → j * ( B ) is a U -cover in C with U ∈ U ∩ V , then ε U : j ! j * ( U ) → U is an isomorphism.

2. If g : j ! ( B ) → V is a V -envelope in C with V ∈ U ∩ V , then ε V : j ! j * ( V ) → V is an isomorphism.

Proof

(1) Since U ∈ U by hypothesis, there exists an admissible exact sequence j ! j * ( U ) ↣ ε U U ↠ i * i * ( U ) in C . Note that i ! j ! ( U ″ ∩ V ″ ) ⊆ V ′ and U ∈ U ∩ V . One can show j ! j * ( U ) ∈ V and i * i * ( U ) ∈ U . Thus, the admissible exact sequence j ! j * ( U ) ↣ ε U U ↠ i * i * ( U ) is split, whence there exists a morphism ε U − 1 : U → j ! j * ( U ) such that ε U − 1 ε U = 1 j ! j * ( U ) . This implies that i * i * ( U ) ↣ U ↠ ε U − 1 j ! j * ( U ) is admissible exact in C . Note that Hom C ( i * i * ( U ) , j * ( B ) ) ≅ Hom ℬ ( j * i * i * ( U ) , B ) = 0 . Thus, there exists a morphism φ : j ! j * ( U ) → j * ( B ) such that f = φ ε U − 1 and φ = f ε U , whence f = f ε U ε U − 1 . Since f is a U -cover, it follows that ε U ε U − 1 is an isomorphism. So ε U is an isomorphism.

(2) By a similar proof of (1), we obtain j ! j * ( V ) ∈ V and i * i * ( V ) ∈ U . Thus, the admissible exact sequence j ! j * ( V ) ↣ ε V V ↠ i * i * ( V ) in C is split, whence there exists a morphism ε V − 1 : V → j ! j * ( V ) such that ε V − 1 ε V = 1 j ! j * ( V ) . Since Hom C ( j ! ( B ) , i * i * ( V ) ) ≅ Hom ℬ ( B , j * i * i * ( V ) ) = 0 , there exists a morphism ψ : j ! ( B ) → j ! j * ( V ) such that g = ε V ψ and ψ = ε V − 1 g , and therefore, we have g = ε V ε V − 1 g . Since g is a V -envelope, it follows that ε V ε V − 1 is an isomorphism. So ε V is an isomorphism.□

Proposition 3.6

If U is covering in C and i ! j ! ( U ″ ∩ V ″ ) ⊆ V ′ , then U ″ is covering in ℬ .

Proof

Let B be an object in V ″ . Then, j * ( B ) ∈ V . Thus, we have an admissible exact sequence K ↣ U ↠ f j * ( B ) in C with U ∈ U ∩ V and K ∈ V such that f is a U -cover in C , whence there exists an admissible exact sequence j * ( K ) ↣ j * ( U ) ↠ j * ( f ) j * j * ( B ) in ℬ . Since j * ( K ) ∈ V ″ and j * ( U ) ∈ U ″ ∩ V ″ , it follows that j * ( f ) : j * ( U ) → j * j * ( B ) is a U ″ -precover of j * j * ( B ) ≅ B in ℬ . Assume that φ is a morphism such that j * ( f ) φ = j * ( f ) . Then, j ! j * ( f ) j ! ( φ ) = j ! j * ( f ) . Note that we have the following commutative diagram:

Since ε j * ( B ) j ! j * ( f ) = ε j * ( B ) j ! j * ( f ) j ! ( φ ) , we have f ε U = f ε U j ! ( φ ) . Note that ε U is an isomorphism by Lemma 3.5 (1). It follows that f = f ε U j ! ( φ ) ε U − 1 . Hence, ε U j ! ( φ ) ε U − 1 is an isomorphism because f is a U -cover. So j ! ( φ ) is an isomorphism. Since j ! is a fully faithful functor, we know that φ is an isomorphism, which implies that j * ( f ) : j * ( U ) → j * j * ( B ) is a U ″ -cover of j * j * ( B ) ≅ B in V ″ . So U ″ is covering in ℬ by Lemma 3.3 (1).□

Proof of Theorem 1.1

Note that ( U ′ , V ′ ) and ( U ″ , V ″ ) are complete cotorsion pairs by hypothesis. It follows from Theorem 2.5 that ( U , V ) is a complete hereditary cotorsion pair in C . Now, we assume that U is covering in C . Thanks to Propositions 3.2 and 3.6, U ′ is covering in A and U ″ is covering in ℬ . Hence, both U ′ and U ″ are closed under direct limits by hypothesis. So U is closed under direct limits by noting that i * , j * , and j ! commute with direct limits since all of them have right adjoint functors. This completes the proof.□

Example 3.7

[26, Example 2.2]

1. Let R and S be two rings, R M S an R - S -bimodule, and Λ = R M 0 S the triangular matrix ring. If we define T ≅ M ⊗ S − : Mod S → Mod R , then we obtain that Mod Λ is equivalent to the comma category ( T ↓ Mod R ) .

2. Let Λ = R M 0 S be a triangular matrix ring. If we define T ≅ M ⊗ S − : Ch ( S ) → Ch ( R ) , then Ch ( Λ ) is equivalent to the comma category ( T ↓ Ch ( R ) ) .

3. Let A  = Mod R and ℬ  = Ch ( R ) . If we define e : ℬ → A via C • ↦ C 0 for any C • ∈ ℬ , then e is an exact functor and we have a comma category ( e ↓ A ) .

Corollary 3.8

Let T : ℬ → A be a compatible right exact functor between abelian categories with enough projective and injective objects. Assume that the direct limit of a system of admissible monomorphisms in the comma category C ≔ ( T ↓ A ) , if it exists in C , is again an admissible monomorphism. Denote by P ( A ) (resp., P ( ℬ ) and P ( C ) ) the subcategory of A (resp., ℬ and C ) consisting of projective objects. If P ( A ) and P ( ℬ ) satisfy Enochs conjecture in A and ℬ , respectively, then so is P ( C ) in C .

Proof

We set U ′ = P ( A ) , V ′ = A , U ″ = P ( ℬ ) , and V ″ = ℬ . By [26, Propositions 3.4 and 2.5], we obtain that U = P ( C ) = { C ∈ C ∣ i * ( C ) ∈ P ( A ) , j * ( C ) ∈ P ( ℬ ) , ε C : j ! j * ( C ) → C is a monomorphism } and V = C = { C ∈ C ∣ i ! ( C ) ∈ A , j * ( C ) ∈ ℬ } in the recollement (3.1). Note that U ″ = P ( ℬ ) and V ′ = A . It follows that j ! is U ″ -exact and i ! j ! ( U ″ ∩ V ″ ) ⊆ V ′ . So the result follows from Theorem 1.1.□

Corollary 3.9

Let Λ = R M 0 S be a triangular matrix ring. If d w P ˜ R and d w P ˜ S satisfy Enochs conjecture in Ch ( R ) and Ch ( S ) , respectively, then d w P ˜ Λ satisfies Enochs conjecture in Ch ( Λ ) .

Proof

We set U ′ = d w P ˜ R , V ′ = W ctr , R , U ″ = d w P ˜ S , and V ″ = W ctr , S . Since each entry of a complex X • in U ″ is projective, it follows that j ! is U ″ -exact. Now, Theorem 2.5 yields that

By [28, Theorem 3.1], we obtain that a left Λ -module X = X Y φ is projective if and only if Y is projective in Mod S , coker φ is projective in Mod R , and φ : M ⊗ S Y → X is monic. It follows that U = d w P ˜ Λ .

Let P • be a complex in U ″ ∩ V ″ . Then, P • is an exact complex in Ch ( S ) with cycle projective. Thus, P • is a contractible complex, whence i ! j ! ( P • ) = M ⊗ S P • is also a contractible complex in Ch ( R ) . It follows from [29, Proposition 3.2] that M ⊗ S P • ∈ W ctr , R , and therefore, i ! j ! ( P • ) is in V ′ . Consequently, i ! j ! ( U ″ ∩ V ″ ) ⊆ V ′ . So the result follows from Theorem 1.1.□