 respectively. So the dual version of Lemma 2.7 holds true.

Derived functors of the module categories of rings
Let R be a (unitary associative) ring. The full subcategories of projective and injective R-modules are denoted by R-Proj and R-Inj, respectively. If M is an R-module and I is a nonempty set, then M (I) and M I denote the direct sum and product of I copies of M, respectively. The projective dimension and the endomorphism ring of M are denoted by proj.dim( R M) and End R (M), respectively.
A full subcategory T of R-Mod is called a thick subcategory if it is closed under direct summands in R-Mod and has the two out of three property: for any short exact sequence 0 → X → Y → Z → 0 in R-Mod with two terms in T, the third term belongs to T as well.
In the sequel, we simply write C (R), K (R) and D(R) for C (R-Mod), K (R-Mod) and D(R-Mod), respectively, and regard R-Mod as the subcategory of D(R) consisting of all stalk complexes concentrated in degree zero.
Let K (R) P (respectively, K (R) I ) be the smallest full triangulated subcategory of K (R), which contains all the bounded-above (respectively, bounded-below) complexes of projective (respectively, injective) Rmodules, and is closed under arbitrary direct sums (respectively, direct products). Since K (R-Proj) satisfies theses two properties, we have. For example, if X is an R-module, then p X can be chosen as a deleted projective resolution of R X. Dually, (K ac (R), K (R) I ) is a semi-orthogonal decomposition of K (R). In particular, there exists a quasi-isomorphism β X • : Now, let S be another ring. For a triangle functor F :

Symmetric subcategories of abelian categories
Now we introduce n-symmetric subcategories of an ableian category for n ≥ 0.
Recall that an abelian category is complete (respectively, cocomplete) if it has products (respectively, coproducts) indexed over sets; and bicomplete if it is complete and cocomplete. (2) Given an exact sequence Remark 3.2. Let B be an additive full subcategory of a bicomplete abelian category A.
(1) If B is an n-symmetric subcategory of A, then B is an exact, thick subcategory of A. It is also (n + 1)-symmetric.
Proof. It is easy to see that E is an additive subcategory and closed under extensions, coproducts and In particular, H i (F(Y )) = 0 by assumption. Thus H i (F(Y )) = 0 for all i ∈ Z and Y ∈ Ker(F). This also shows X ∈ Ker(F). Thus E is an n-symmetric subcategory of A.
As an application of Proposition 3.3, we consider the module categories of rings.
Example 3.4. Let R and S be rings, and let R M S be an R-S-bimodule.
(1) If M S has a finite projective resolution of length n by finitely generated projective right S-modules, that is, there is an extact sequence 0 → Q n → · · · → Q 1 → Q 0 → M S → 0 with all Q j being finitely generated projective right S-modules, then Ker(M ⊗ L S −) ∩ S-Mod is an n-symmetric subcategory of S-Mod. (2) If R M has a finite projective resolution of length n by finitely generated projective modules, then A further discussion on n-symmetric subcategories will be done in a forthcoming article.

Tilting modules and derived recollements
This entire section is devoted to a proof of Theorem 1.2. We first make a couple of preparations

Derived functors induced by good tilting modules
Throughout this section, A denotes a unitary ring, T a good n-tilting A-module with (T 1), (T 2), and (T 3) for a natural number n, and B := End A (T ). Further, we define to be the total left-and right-derived functors of A T B, respectively. We write by the unit adjunctions associated with the adjoint pairs (T ⊗ B −, Hom A (T, −)) and (G, H), respectively. Recall that the restriction of the localization functor K (B) → D(B) to K (B) p is a triangle equivalence. A quasiinverse of this equivalence is denoted by Without loss of generality, we assume that p M is a deleted projective resolution of M for each B-module M. The derived categories D(A) and D(B) are related by the following recollement (see [4,Theorem 2.2]). This is used in [11,Lemma 5.2] to understand when a tilting module is homological.
There exists a recollement of triangulated categories: Observe that the modules A T and T B have finite projective dimension. In fact, T B has a finite projective resolution by finitely generated projective right B-modules since the tilting module A T is good. The next result is deduced from [30, Theorem 10.5.9, Corollary 10.5.11].
Combining Theorem 4.1 with Lemma 4.2, we have the lemma.
Proof. (1) Since the canonical map Hom A (T, T ) (α) → Hom A (T, T (α) ) is injective for any nonempty set α, the map η ′ P is injective for any free B-module P, and therefore η ′ P is injective for any projective B-module P. By Lemma 4.2, H • G(P) ≃ Hom A (T, T ⊗ B P). It follows from Theorem 4.1 that there exists a triangle        .
with Q Y and Q E the localization functors. By Lemma 4.3(2), there is a commutative diagram of natural transformations among triangle endofunctors of D(B):

Moreover, Ψ, D( j) is an adjoint pair and D( j) is a fully faithful functor.
(  Lemma 4.3(2), and therefore there exists a diagram

2) Let κ : Ker(G) → D(B) and ν : Im(H) → D(B) be the inclusion functors. Then D(i) = κ • D(i) and D( j) = ν • D( j), namely the commutative diagrams exist:
in which the left square is commutative in D(B) and the right square is commutative in K (B). Note that the isomorphism ψ ′ H T (X) follows from Lemma 2.6(2) and the fact that K(H T )(X ) lies in Im(K(H T )). Since Im(H) = Im(D(H T )) by (1), we have Q • ≃ K(H T )(X ) ∈ Im(H). Recall that H is fully faithful. Thus η Q • is an isomorphism in D(B), and further, is represented by a chain map, it is a quasi-isomorphism. This shows that φ is a natural isomorphism.
(2) This follows from (1). The reason reads as follows: π can be embedded into a triangle is an isomorphism, then D(i)(W ) = 0. It follows from Ker(D(i)) = 0 that W = 0 and π is an isomorphism in D(E ).
Since D(i) commutes with direct products by Lemma 4.6(1), it is enough to show that the canonical mor- commutes with direct products. But this follows from the diagram (♯) and the fact that H and G commute with direct products. Thus Φ and therefore Φ commute with direct products. As a consequence of Theorem 4.1 and Lemma 4.7(1), we re-obtain [3,Theorem 4.5] which was proved by employing the theory of derivators in [29].  Moreover, by definition, Φ and Ψ can also be restricted to functors between bounded-above derived categories: Thus Lemma 4.7 hold true for bounded-above derived categories.

Fully faithful triangle functors between derived categories
In this section we develop properties of triangle functors needed in our proofs.
By the diagram (♯), there exists a triangle in D(B) for each M ∈ B-Mod: Taking homologies of the triangle, we get a 5-term exact sequence of B-modules: In the following, we consider the full subcategories of B-Mod: Next, we study properties of ε M and E. Let M ∈ E. Then X M = 0, which means that the homomorphism d −1 is surjective. Since E is closed under kernels of surjective homomorphisms in B-Mod by Lemma 4.6(1), we have Ker( (2) Since X is closed under cokernels in B-Mod and E ⊆ X, we have Coker (1) in D(B). Taking homology H 0 of this triangle yields an exact sequence (2) ε −1 M and ε 0 M in (1) give rise to isomorphisms of abelian groups for X ∈ X and Y ∈ Y : (3) The inclusion X → E