{'fraction_non_alphanumeric': 0.07874441570656007, 'fraction_numerical': 0.018598636256759934, 'mean_word_length': 3.5044482101249734, 'pattern_counts': {'":': 0, '<': 47, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Recollements of derived categories III: finitistic dimensions

In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel's reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.


Introduction
Recollements of triangulated categories have been introduced by Beilinson, Bernstein and Deligne in order to decompose derived categories of sheaves into two parts, an open and a closed one (see [6]), and thus providing a natural habitat for Grothendieck's six functors. Similarly, recollements of derived module categories can be seen as short exact sequences, describing a derived module category in terms of a subcategory and of a quotient, both of which may be derived module categories themselves, related by six functors that in general are not known. It turns out that recollements provide a very useful framework for understanding connections among three algebraic or geometric objects in which one is interested.
In a series of papers on recollements of derived module categories, we are addressing basic questions about recollements and the rings involved. Our starting point has been infinite-dimensional tilting theory (see [7]). While Happel's theorem establishes a derived equivalence between a given ring and the endomorphism ring of a finitely generated tilting object (see [15,13]), Bazzoni has shown that for large tilting modules one gets instead a recollement relating three triangulated categories, with two of them being the derived categories of the given ring and the endomorphism ring of the large tilting module. In [7] we have addressed the question of determining the third category in this recollement as a derived category of a ring and we have explained this ring in terms of universal localisations in the sense of Cohn (see [12,20] for definition). Among the applications has been a counterexample to the Jordan-Hölder problem for derived module categories. In [8] we have dealt with the problem of constructing recollements in order to relate rings. Our main construction, of exact contexts, can be seen as a far-reaching generalisation of pullbacks of rings. In [9] we have used this Then the following hold true: (1) Suppose that j ! restricts to a functor D b (R 3 ) → D b (R 2 ) of bounded derived module categories. If fin.dim(R 2 ) < ∞, then fin.dim(R 3 ) < ∞.
Note that the assumption of Theorem 1.1 on unbounded derived module categories is weaker than the one on bounded derived module categories, because the existence of recollements of bounded derived module categories implies the one of unbounded derived module categories. This is shown by a recent investigation on recollements at different levels in [2,18]. So, Theorem 1.1 (see also Corollary 3.13) generalizes the main result in [16] since for a recollement of D b (R j -mod) with R j a finite-dimensional algebra over a field for 1 ≤ j ≤ 3, one can always deduce that i * (R 1 ) is compact in D(R 2 ). Moreover, Theorem 1.1 extends and amplifies a result in [25] because we deal with arbitrary rings instead of Artin algebras, and also yields a generalization of a result in [22] for left coherent rings to the one for arbitrary rings (see Corollary 3.9 below).
To prove this result, we introduce homological widths (or cowidth) for complexes that are quasi-isomorphic to bounded complexes of projective (or injective) modules (see Section 3.1 for details). Broadly speaking, the homological width (respectively, cowidth) defines a map from homotopy equivalence classes of bounded complexes of projective (respectively, injective) modules to the natural numbers. It measures, up to homotopy equivalence, how large the minimal interval of such a complex is in which its non-zero terms are distributed. Particularly, if a module has finite projective dimension, then its homological width is exactly the projective dimension. Using homological widths, we will present a substantial and technical result, Theorem 3.11, which is a strengthened version of Theorem 1.1 and describes explicitly upper bounds for finitistic dimensions, so that Theorem 1.1 will become an easy consequence of Theorem 3.11. Note that, in [16], one of the key arguments in proofs is that finite-dimensional algebras have finitely many non-isomorphic simple modules, while in our general context we do not have this fact and therefore must avoid this kind of arguments. So, the idea of proving Theorems 3.11 and 1.1 will be completely different from the ones in [16] and [25]. Moreover, our methods also lead to results on upper bounds for big finitistic and global dimensions. For details, we refer the reader to Theorems 3.17 and 3.18. Now, let us utilize Theorem 1.1 to recollements constructed in [8] and establish relationships among finitistic dimensions of noncommutative tensor products and related rings. First of all, we recall some notions from [8]: Let R, S and T be associative rings with identity, and let λ : R → S and µ : R → T be ring homomorphisms. Suppose that M is an S-T -bimodule together with an element m ∈ M. We say that the quadruple (λ, µ, M, m) is an exact context if the following sequence is an exact sequence of abelian groups, where ·m and m· denote the right and left multiplication by m maps, respectively. There is a list of examples in [8] that guarantees the ubiquity of exact contexts. Given an exact context (λ, µ, M, m), there is defined a ring with identity in [8], called the noncommutative tensor product of (λ, µ, M, m) and denoted by T ⊠ R S if the meaning of the exact context is clear. This notion not only generalizes the one of usual tensor products over commutative rings and captures coproducts of rings, but also plays a key role in describing the left parts of recollements induced from homological exact contexts (see [8,Theorem 1.1]).
For an R-module R X , we denote by flat.dim( R X ) and proj.dim( R X ) the flat and projective dimensions of X , respectively.
From the proof of Theorem 1.1 or Theorem 3.11, we have the following result.

Theorem 1.2. Let (λ, µ, M, m) be an exact context with the noncommutative tensor product T ⊠ R S. Then
If the left R-module R S has a finite projective resolution by finitely generated projective modules, then the following hold true: (a) fin.dim(T ⊠ R S) ≤ fin.dim(S) + fin.dim(T ) + 1.
Note that for the triangular matrix algebra B := S M 0 T , it is known that fin.dim(B) ≤ fin.dim(S) + fin.dim(T ) + 1. But, Theorem 1.2(2)(b) provides us with a new upper bound for the finitistic dimension of B.
That is, the finiteness of fin.dim(B) can be seen from the one of fin.dim(T ⊠ R S) and fin.dim(R), involving the starting ring R but without information on fin.dim(S) and fin.dim(T ). This is non-trivial and somewhat surprising. Moreover, in Theorem 1.2(2), if λ : R → S is a homological ring epimorphism, then we even obtain better estimations: fin.dim(S) ≤ fin.dim(R) and fin.dim(T ⊠ R S) ≤ fin.dim(T ). In this case, T ⊠ R S can be interpreted as the coproduct S ⊔ R T of the R-rings of S and T . Now, let us state several consequences of Theorem 1.2. First, we utilize Theorem 1.2 to finitistic dimensions of ring extensions. This is of particular interest because the finitistic dimension conjecture can be reformulated over perfect fields in terms of ring extensions (see [24]). Note that, in the following result, we do not impose any conditions on the radicals of rings, comparing with [23,24]. Corollary 1.3. Suppose that S ⊆ R is an extension of rings, that is, S is a subring of R with the same identity. Let R ′ be the endomorphism ring of the S-module R/S, and let R ′ ⊠ S R be the noncommutative tensor product of the exact context determined the extension. Then (2) Suppose that the left S-module R is projective and finitely generated. Then the following hold true: Next, we apply Theorem 1.2 to trivial extensions. Recall that, given a ring R and an R-R-bimodule M, the trivial extension of R by M is a ring, denoted by R ⋉ M, with abelian group R ⊕ M and multiplication: (r, m)(r ′ , m ′ ) = (rr ′ , rm ′ + mr ′ ) for r, r ′ ∈ R and m, m ′ ∈ M. For consideration of Fin.dim(R ⋉ M), we refer the reader to [14,Chapter 4].

Corollary 1.4. Let λ : R → S be a ring epimorphism and M an S-S-bimodule such that Tor
If R S has a finite projective resolution by finitely generated projective R-modules, then Now, we apply Theorem 1.2 to pullback squares of rings and surjective homomorphisms.
The strategy of proving Corollaries 1.4 and 1.5 is as follows: First, we show that under the given assumptions we can get exact pairs, a class of special exact contexts, and then employ Theorem 1.2 by verifying the Tor-vanishing condition. At last, we have to describe noncommutative tensor products more substantially for the cases considered.
Finally, we mention a corollary on finitistic dimensions of algebras arising from idempotent ideals and almost split sequences (see [3] for definition).
(2) Let 0 → Z → Y → X → 0 be an almost split sequence of R-modules with R an Artin algebra. If The paper is sketched as follows: In Section 2, we first recall some necessary definitions and then prove two results on coproducts of rings. In Section 3, we provide all proofs of our results. Especially, we introduce homological widths of complexes and prove an amplified version, Theorem 3.11, of Theorem 1.1 phrased in terms of homological widths and finitistic dimensions of involved rings, such that Theorem 1.1 is deduced readily from Theorem 3.11. Moreover, the methods developed in this section also give similar upper bounds for global and big finitistic dimensions (see Theorems 3.17 and 3.18).

Definitions and conventions
In this section, we fix notation and briefly recall some definitions. For unexplained ones, we refer the reader to [8,9].
Throughout the paper, all notation and terminology are standard. For example, by a ring we mean an associative ring with identity. For a ring R, we denote by R-Mod the category of all left R-modues, and by C (R), K (R) and D(R) the unbounded complex, homotopy and derived categories of R-Mod, respectively. As usual, by adding a superscript * ∈ {−, +, b}, we denote their corresponding * -bounded categories, for instance, D b (R) is the bounded derived category of R-Mod. The full subcategory of compact objects in D(R) is denoted by D c (R). This category is also called the perfect derived module category of R. It is known that the localization functor K (R) → D(R) induces a triangle equivalence from the homotopy category of bounded complexes of finitely generated projective R-modules to D c (R).
As usual, we write a complex in C (R) as X • = (X i , d i X • ) i∈Z , and call d i X • : X i → X i+1 the i-th differential of X • . Sometimes, for simplicity, we shall write (X i ) i∈Z for X • without mentioning the morphisms d i X • . Given a chain map f • : X • → Y • in C (R), its mapping cone is denoted by Con( f • ). For an integer n, the n-th cohomology of X • is denoted by H n (X • ). Let sup (X • ) and inf (X • ) be the supremum and minimum is equivalent to the full subcategory of D(R) consisting of all complexes with finitely many nonzero cohomologies.
As a convention, we write the composite of two homomorphisms f : X → Y and g : Y → Z in R-Mod as f g. Thus the image of an element x ∈ X under f will be written on the opposite of the scalars as (x) f instead of f (x). This convention makes Hom R (X ,Y ) naturally a left End R (X )and right End R (Y )-bimodule. But, for two functors F : X → Y and G : Y → Z of categories, we write GF : X → Z for their composition.
Let us now recall the notion of recollements of triangulated categories, which was defined by Beilinson, Bernstein and Deligne in [6] to study derived categories of perverse sheaves over singular spaces. It may be thought as a kind of categorifications of exact sequences in abelian categories.
i * , j * and j ! are fully faithful functors, (3) j ! i ! = 0 (and thus also i ! j * = 0 and i * j ! = 0), and (4) for each object X ∈ D, there are two triangles in D induced by counit and unit adjunctions: where the shift functor of triangulated categories is denoted by [1].
Recall that the coproduct of a family {R i | i ∈ I} of R 0 -rings with I an index set is defined to be an R 0 -ring R together with a family {ρ i : R i → R | i ∈ I} of R 0 -homomorphisms of rings such that, for any R 0 -ring S with a family of R 0 -homomorphisms {τ i : R i → S | i ∈ I}, there exists a unique R 0 -homomorphism δ : R → S such that τ i = ρ i δ for all i ∈ I. It is well known that coproducts of rings exist (see [11]). However, this existence result does not provide us with a handy form of coproducts; therefore we need a concrete description of coproducts for our situations considered.
In the following we describe coproducts of rings for two cases in terms of some known constructions. This will be used in later proofs. The first one is for trivial extensions Lemma 2.2. Suppose that λ : R → S is a ring epimorphism and M is an S-S-bimodule. Let λ : R ⋉ M → S ⋉ M be the ring homomorphism induced by λ. Then the coproduct S ⊔ R (R ⋉ M) is isomorphic to S ⋉ M, that is, the inclusion S → S ⋉ M and λ define the coproduct.
Proof. Let µ : R → R ⋉ M and ρ : S → S ⋉ M be the inclusions of rings. Note that S and R ⋉ M are R-rings via λ and µ, respectively, and that λρ = µ λ : R → S ⋉ M. To prove that S ⋉ M, together with ρ and λ, is the coproduct of S and R ⋉ M over R, we suppose that Λ is an arbitrary ring and that f : R ⋉ M → Λ and g : S → Λ are arbitrary ring homomorphisms such that λ g = µ f . Then we have to show that there is a unique ring homomorphism h : S ⋉ M → Λ such that λ h = f and ρ h = g. Clearly, if such an h exists, then h must be defined by (s, m) → (m) f + (s)g for s ∈ S and m ∈ M. This shows the uniqueness of h. So, it suffices to show that the above-defined map h is a ring homomorphism. Certainly, h is a homomorphism of abelian groups. Hence, we have to show that h preserves multiplication.
Let s i ∈ S and m i ∈ M for i = 1, 2. On the one hand, to prove that h preserves multiplication, we need only to verify these additional conditions, that is, Since λg = µ f , one can check that both ϕ and ψ are homomorphisms of R-modules such that λϕ = λψ. But we do not know if they are homomorphisms of rings. Nevertheless, we can still have φ = ψ because λ : R → S being a ring epimorphism by assumption implies that the map Hom R (R, λ) : Hom R (S, Λ) → Hom R (R, Λ) is an isomorphism, and therefore it is injective. Thus φ = ψ. Similarly, we can show that (ms) f = (m) f (s)g. Consequently, the map h preserves multiplication and is actually a ring homomorphism.
The other description of coproducts is for quotients of rings by ideals, which applies to Milnor squares (see [19]).

Lemma 2.3.
Let R 0 be a ring, and let R i be an R 0 -ring with ring homomorphism λ i : R 0 → R i for i = 1, 2.
(1) If λ 1 : R 0 → R 1 is a ring epimorphism, then so is the canonical homomorphism ρ 2 : (2) Let I be an ideal of R 0 , and let J be the ideal of R 2 generated by the image (I)λ 2 of I under the map λ 2 . If R 1 = R 0 /I and λ 1 : Proof. (1) It follows from the definition of coproducts of rings that We point out that ρ 2 is a ring epimorphism. In fact, if f , g : R 1 ⊔ R 0 R 2 → S are two ring homomorphisms such that ρ 2 f = ρ 2 g, then λ 2 ρ 2 f = λ 2 ρ 2 g. This means that λ 1 ρ 1 f = λ 1 ρ 1 g, and therefore ρ 1 f = ρ 1 g since λ 1 is a ring epimorphism. By the universal property of coproducts, we have g = f . Thus ρ 2 is a ring epimorphism.

Proofs
This section is devoted to proofs of all results mentioned in the introduction. We start with introducing the so-called homological widths for complexes, and then prove a strengthened version, Theorem 3.11 below, of Theorem 1.1. As consequences of Theorem 3.11, we get proofs of Theorems 1.1 and 1.2. Finally, we apply Theorem 1.2 to give proofs of all corollaries, and mention two results on global and big finitistic dimensions.

Homological widths and cowidths of complexes
As a generalization of finite projective or injective dimensions of modules, we define, in this subsection, homological widths and cowidths for bounded complexes of projective and injective modules, respectively.
Let R be a ring. For an R-module M, we denote by proj.dim(M), inj.dim(M) and flat.dim(M) the projective, injective and flat dimension of M, respectively. As usual, R-Proj is the category of all projective left R-modules, and R-proj is the full subcategory of R-Proj consisting of all finitely generated projective left R-modules. If there is a projective resolution 0 → P n → · · · → P 1 → P 0 → M → 0 of M with all P i in R-proj, then we say that M is of finite type. The category of all R-modules of finite type will be denoted by P <∞ (R). Let We define the homological width of P • in the following way: and each term being projective. Clearly, the sequence The following result says that homological widths of bounded complexes of projective modules are preserved under homotopy equivalences.
. Thanks to Lemma 3.1, the definition of homological widths for complexes can be extended slightly to derived categories in the following sense: Given a complex . This is well defined: If there exists another So, for such a complex X • , its homological width w(X • ) can be characterized as follows: Clearly, if X ∈ R-Mod has finite projective dimension, then w(X ) = proj.dim(X ). Dually, we can define homological cowidths for bounded complexes of injective R-modules.
Let R-Inj denote the category of injective R-modules. Given a complex I • := (I n , d n I • ) n∈Z ∈ C b (R-Inj), we define the homological cowidth of I • as follows: Homological widths and cowidths will be used to bound homological dimensions in the next section.

Proof of Theoem 1.1
In this subsection, we shall first prove an amplified version of Theorem 1.1, namely Theorem 3.11 below, so that Theorem 1.1 becomes a straightforward consequence of Theorem 3.11. Recall that the finitistic dimension of a ring R, denoted by fin.dim(R), is defined as follows: For each n ∈ Z, we define For the convenience of later discussions, we also Lemma 3.2. Let m, n ∈ N. Then the following statements are true: is the stable category of the Frobenius category C b (R-proj) with projective objects being acyclic complexes.
So we can find two acyclic complexes Recall that the localization functor Thus the given triangle yields a distinguished triangle in D c (R): To investigate relationships among finitistic dimensions of rings in recollements, it may be convenient to introduce the notion of finitistic dimensions of functors.
Let R 1 and R 2 be two arbitrary rings. Suppose that X 1 and X 2 are full subcategories of D(R 1 ) and D(R 2 ), respectively, and that R 1 -Mod ⊆ X 1 . For a given additive functor F : Note that inf(F) = +∞ if and only if F(X ) = 0 in D(R 2 ) for all X ∈ R 1 -Mod. In fact, if there exits some X ∈ R 1 -Mod such that H n (F(X )) = 0 for some integer n, then inf(F) ≤ n. Moreover, by definition, we always have inf(F) ≤ fin.dim(F) and fin.dim(F) ∈ Z ∪ {−∞, +∞}.
be a triangle functor. Then the following statements are true: Since L(R 2 ) ∈ D − (R 1 ), we have s := sup(L(R 2 )) < +∞. Recall that the localization functor (2) To calculate cohomologies of complexes, we consider the functor This is an exact functor with the property that a Z-module U is zero if and only if so is

This implies that H n (F(M)) = 0 if and only if Hom
In the following, we will show the necessity of (2). Suppose that F restricts to a functor Actually, we have the following isomorphisms of abelian groups: , we have H n (F(R 1 )) = 0 for almost all n. Thus H n (W • ) = 0 for almost all n. In other words, W • is isomorphic in D b (R 1 ) to a bounded complex. Consequently, there exists a lower-bounded complex I • of injective R 1 -modules such that I • ≃ W • in D(R 1 ). In particular, we have H n (I • ) ≃ H n (W • ) for all n. To complete the proof of the necessity of (2), it remains to show that I • can be chosen to be a bounded complex. Note that we have the following isomorphisms: .
where all terms I ֒→ I t+1 , is homotopic to the zero map. Therefore, the canonical surjection I t ։ Im(d t ) must split. With I t then also Im(d t ) is an injective module. Since H i (I • ) = 0 for any i > m, we see that I • is isomorphic in D(R 1 ) to the following bounded complex: with all of its terms being injective. Thus, up to isomorphism in D(R 1 ), we can choose I • to be a bounded complex of injective modules. This completes the proof of the necessity of (2).
To show the last statement of (2), we note that the R 1 -module Ker(d m ) has a finite injective resolution since H i (I • ) = 0 for all i > m. Hence, up to isomorphism in D(R 1 ), we can replace I • by the following bounded complex of injective R 1 -modules: where  We remark that, in Lemma 3.3(2), the R 2 -module I := Hom Z (R 2 , Q/Z) can be replaced by any injective cogenerator of R 2 -Mod. This is due to the fact that G always commutes with direct products. Recall that an R 2 -module M is called a cogenerator of R 2 -Mod if any R 2 -module can be embedded into a direct product of copies of M. Clearly, I is an injective cogenerator of R 2 -Mod. In case that R 2 is an Artin algebra, there is another injective cogenerator, the usual dual module D(R 2 ) of the right regular module R 2 , where D is the usual duality of an Artin algebra. (1) F(P <∞ (R 1 )) ⊆ D c ≥s−t (R 2 ). (2) Let m ∈ Z. Then, for any X ∈ P <∞ (R 1 ) and for any Y Proof. Note that s = +∞ if and only if F(X ) = 0 for any X ∈ P <∞ (R 1 ). In this case, both (1) and (2) are true. Now, we assume s < +∞. Thus s is an integer. ( Since fin.dim(F) = s < ∞, we have H i (F(X )) = 0 for all i < s. Thus H i (Q • ) = 0 for all i < s. It follows that Y := Cok(d s−1 ) ∈ P <∞ (R 2 ), and therefore Q • is isomorphic in D(R 2 ) to the following canonical truncated complex: So the R 2 -module Y has a finite projective resolution: to the following complex Clearly, P • ∈ C b (R 2 -proj) and P i = 0 for i < s − t. This implies F(X ) ∈ D c ≥s−t (R 2 ). Hence, we have F(P <∞ (R 1 )) ⊆ D c ≥s−t (R 2 ). (2) Let X ∈ P <∞ (R 1 ) and Y • ∈ D(R 2 ) with sup(Y • ) ≤ m < ∞. Then H j (Y • ) = 0 for j > m, and therefore there exists a complex Z • ∈ C − (R 2 ) with Z r = 0 for r > m, such that Z • ≃ Y • in D(R 2 ). Moreover, by the proof of (1), there exists another complex P • ∈ C b (R 2 -proj) with P i = 0 for all i < s −t, such that P • ≃ F(X ) in D(R 2 ). It follows that be a fully faithful triangle functor. If fin.dim(F) = s is an integer, then fin.dim(R 1 ) ≤ fin.dim(R 2 ) − s + sup(F(R 1 )). In particular, if fin.dim(R 2 ) < ∞, then fin.dim(R 1 ) < ∞.
Proof. If fin.dim(R 2 ) is infinity, then the right-hand side of the inequality is infinity and the corollary is true. So we assume that fin.dim(R 2 ) = t < ∞. Further, we may assume R 1 = 0. Since F is fully faithful, we have 0 = F(R 1 ) ∈ D c (R 2 ). This implies that sup(F(R 1 )) < ∞. Moreover, it is known that, for any X ∈ P <∞ (R 1 ), if there is a natural number n such that Ext i R 1 (X , R 1 ) = 0 for all i > n, then proj.dim( R 1 X ) ≤ n. So, to show that fin.dim(R 1 ) ≤ n := t − s + sup(F(R 1 )) < ∞, it is enough to prove that Ext i R 1 (X , R 1 ) = 0 for all X ∈ P <∞ (R 1 ) and all i > n. In fact, since F is fully faithful, we see that
. Assume that the following conditions hold:

as rings (via multiplication), and Hom
, the functor F restricts to a functor F ′ : D c (R 1 ) → D c (R 2 ). Note that the condition (1) implies that F ′ is fully faithful. Further, since F commutes with direct sums and D(R 1 ) is compactly generated by R 1 , we see that F itself is also fully faithful. Now, we claim that F restricts to a functor D b (R 1 ) → D b (R 2 ). In fact, by Lemma 3.3(2), this is equivalent to saying that the complex G Hom Z (R 2 , Q/Z) is isomorphic in D(R 1 ) to a bounded complex of injective R 1 -modules.
To check the latter, we use the functor (−) ∨ := Hom Z (−, Q/Z) and apply G to the injective R 2 -module R ∨ 2 . Then we have the following isomorphisms in D(R 1 ): Note that (−) ∨ : R op 1 -Mod → R 1 -Mod is an exact functor, which sends flat R op 1 -modules to injective R 1modules. Thus the condition (2) implies that (P • ) ∨ is isomorphic in D(R 1 ) to the following bounded complex of injective R-modules: where (F s ) ∨ and (F r ) ∨ are of degrees −s and −r, respectively. Consequently, we have cw(G(R 2 ∨ )) = cw((P • ) ∨ ) = cw((F • ) ∨ ) ≤ s − r. Now, it follows from Corollary 3.6(2) that This completes the proof.
Recall that a ring epimorphism λ : R → S is homological if Tor R i (S, S) = 0 for all i > 0. This is equivalent to saying that the restriction functor D(λ * ) : D(S) → D(R) is fully faithful. Note that D(λ * ) always has a left adjoint functor S ⊗ L R − : D(R) → D(S). For some new advances on homological ring epimorphisms phrased in terms of recollements of derived categories, we refer the reader to [8,9,10]. Applying Corollary 3.6(1) to homological ring epimorphisms, we have the following simple result. Corollary 3.8. Let λ : R → S be a homological ring epimorphism such that R S ∈ P <∞ (R). Then fin.dim(S) ≤ fin.dim(R). In this case, if fin.dim(R) < ∞, then fin.dim(S) < ∞.
Let us point out a straightforward proof of Corollary 3.8: Let S X ∈ P <∞ (S). Since proj.dim( R S) < ∞, the Change of Rings Theorem implies that proj.dim( R X ) ≤ proj.dim( S X ) + proj.dim( R S) < ∞. Thus proj.dim( R X ) ≤ fin.dim(R). As λ is homological, we see that for all Y ∈ S-Mod and i ≥ 0. This implies that proj.dim( S X ) ≤ proj.dim( R X ). As a result, we have proj.dim( S X ) ≤ proj.dim( R X ) ≤ fin.dim(R). This shows fin.dim(S) ≤ fin.dim(R).
The following result extends [22, Theorem 1.1] on finitistic dimensions for derived equivalences of left coherent rings to those of arbitrary rings. F) and (F, G) are adjoint pairs. Clearly, G is also a triangle equivalence. Since both F and G preserve compact objects, they restrict to triangle equivalences of perfect derived categories: . By Corollary 3.6(1), we have fin.dim(R 1 ) ≤ fin.dim(R 2 )+ w(G(R 2 )) and fin.dim(R 2 ) ≤ fin.dim(R 1 )+ w(F(R 1 )). Thus, to complete the proof, it is enough to show that w(G(R 2 )) = w(F(R 1 )).
In fact, up to isomorphism in derived categories, we may assume that F( Without loss of generality, we suppose that such that r = w(F(R 1 ) ≥ 0. This implies that H 0 (F(R 1 )) = 0 and d −r is not a split injection. Since (F, G) is an adjoint pair, we always have for all n ∈ Z. It follows that H i (G(R 2 )) = 0 for i < 0 or i > r. Further, we claim that H r (G(R 2 )) = 0, and therefore sup(G(R 2 )) = r. Actually, since d −r is not a split injection, we have Hom K (R 2 ) (F(R 1 ), P −r [r]) = 0. Thus 0 = Hom K (R 2 ) (F(R 1 ), R 2 [r]) ≃ H r (G(R 2 )). So, up to isomorphism in K (R 1 ), the complex G(R 2 ) has the following form ). In particular, this implies that ϕ s is not a split injection. So, to show w(F(R 1 ) = w(G(R 2 )), we only need to show s = 0. Indeed, since (G, F) is an adjoint pair, we have for all n ∈ Z. On the one hand, if s < 0, then Hom K (R 1 ) (G(R 2 ), R 1 [−s]) ≃ H −s (F(R 1 )) = 0, and therefore Hom K (R 1 ) (G(R 2 ), Q s [−s]) = 0. This means that ϕ s is a split injection, a contradiction. On the other hand, if s > 0, then 0 = Hom K (R 1 ) (G(R 2 ), R 1 ) ≃ H 0 (F(R 1 )). This is also a contradiction. Thus s = 0 and w(F(R 1 )) = w(G(R 2 )), as desired.
The above result describes a relationship for finitistic dimensions of derived equivalent rings. If we weaken derived equivalences into half recollements of perfect derived module categories, we will obtain the following general result which provides a bound for the finitistic dimension of the middle ring by those of the other two rings.
Proof. The proof will be done in several steps. We may suppose that fin.dim(R 1 ) < ∞ and fin.dim(R 3 ) < ∞. Clearly, if one of R 1 and R 3 is zero, then Proposition 3.10 follows from Corollary 3.9. From now on, we assume that R 1 = 0 = R 3 .
Step 1. We claim that is an adjoint pair, one can follow the proof of Lemma 3.3 (1) . Clearly, the complex P • Y is of the following form: and that L i ∈ R 2 -proj for all i ≥ u 2 (see Section 3.1). This implies This induces a distinguished triangle in D c (R 2 ): Note that j ! P , one can apply Lemma 3.2(2) to show that j ! (P • Y ) ∈ D c ≥(u 1 +u 2 ) (R 2 ) by induction on the number of non-zero terms of a complex. It follows from Step 2. We show that i * i * (P <∞ (R 2 )) ⊆ D c ≥v (R 2 ), where v := fin.dim(R 1 ) + w i * (R 1 ) + u + 1. First of all, we claim that there is an integer m such that m ≤ fin.dim(i * ) ≤ +∞. Indeed, the given half recollement yields the following canonical triangle [1] in D(R 2 ), where η Y and ε Y stand for the counit and unit adjunction morphisms, respectively. Since By assumption, i * (R 1 ) ∈ D c (R 2 ) and therefore is isomorphic in D c (R 2 ) to a complex for all n < m. This implies that m ≤ fin.dim(i * ) ≤ +∞, as claimed.
Let v 1 := m − fin.dim(R 1 ). It follows from Lemma 3.4(1) that i * (P <∞ (R 2 )) ⊆ D c ≥v 1 (R 1 ). Now, replacing the pair ( j ! , j ! ) in the proof of Step 1 with (i * , i * ), one can similarly show that Step 3. We show that fin.dim( it follows from the triangle ( †) and Lemma 3.2(2) that Y ∈ D c ≥−v (R 2 ). Now, let P • := (P n , d n ) n∈Z ∈ C b (R 2 -proj) such that P n = 0 for all n < −v and that Y ≃ P • in D c (R 2 ). Since Y is an R 2 -module, we see that H n (P • ) = 0 for n = 0 and H 0 (P • ) ≃ Y . Consequently, Ker(d 0 ) ∈ R 2 -proj and the following complex Now, with the above preparations, we prove the following strong version of Theorem 1.1 .
Theorem 3.11. Let R 1 , R 2 and R 3 be rings. Suppose that there exists a recollement among the derived module categories D(R 3 ), D(R 2 ) and D(R 1 ) of R 3 , R 2 and R 1 : Then the following statements hold true: (1) Suppose that j ! restricts to a functor D b (R 3 ) → D b (R 2 ) of bounded derived module categories. Then fin.dim(R 3 ) ≤ fin.dim(R 2 ) + cw j ! (Hom Z (R 2 , Q/Z)) .
Proof. Note that the triangle functors j ! and i * in a recollement always take compact objects to compact objects and that i * (R 1 ) is compact if and only if j ! (R 2 ) is compact (for a reference of this fact, one may see, for example, [9, Lemma 2.2]). Thus we have a sequence of functors: where the functor j ! is fully faithful. Applying Corollary 3.6(2) to the adjoint pair ( j ! , j ! ), we then obtain (1).
. Then j ! (R 2 ) ∈ D c (R 3 ) and the given recollement in Theorem 3.11 induces a half recollment of prefect derived module categories: Now, the statements (a) and (b) in (2) follow from Corollary 3.6(1) and Proposition 3.10, respectively. This completes the proof of Theorem 3.11.
As a consequence of Theorem 3.11, we obtain the following corollary which extends the main result [25,Theorem] on finitistic dimensions of Artin algebras to the one of arbitrary rings.

Corollary 3.12. Let R be a ring and e an idempotent element of R. Suppose that the canonical surjection
Proof. Let J := ReR. Since the canonical surjection R → R/J is homological, there exists a recollement of derived module categories: Since R J ∈ P <∞ (R), we see that D(π * )(R/J) = R/J ∈ D c (R) and that w(R/J) = proj.dim( R R/J). Moreover, Re ⊗ L eRe eRe = Re and w( R Re) = 0. Now, Corollary 3.12 follows from Theorem 3.11(2)(b) and Corollary 3.8.
Since a recollement at D b -level induces a recollement at D-level, the following result is a straightforward consequence of Theorem 3.11, which also generalizes [16,Theorem 2]. Corollary 3.13. Let R 1 , R 2 and R 3 be rings. Suppose that there exists a recollement among the derived The existence of a recollement at D b -level occurs in the following special case (see [21], [18]): Let R be a ring and J = ReR be an ideal generated by an idempotent element e in R such that R J is projective and finitely generated and that J R has finite projective dimension. Then there exists a recollement among D b (R/J), D b (R) and D b (eRe). Remark that, without proj.dim(J R ) < ∞, we may not get a recollement at D blevel because the left-derived functor Re ⊗ L eRe − : D(eRe) → D(R) may not restrict to a functor of bounded derived categories. One can construct a desired counterexample from triangular matrix rings.
Applying Corollary 3.12 to triangular matrix rings, we re-obtain the following well-known result (for example, see [14,Corollary 4.21]).
where End C ,Y (X ) is the quotient ring of the endomorphism ring End C (X ) of X modulo the ideal generated by all those endomorphisms of X which factorize through the object Y .
Consequently, we have the following result which restates Corollary 1.6. In particular, if R I is projective and finitely generated, then fin.dim(R/I) ≤ fin.dim(R) ≤ fin.dim End R ( R I) + fin.dim(R/I) + 2.
(2) Let 0 → Z → Y f −→ X → 0 be an almost split sequence in R-mod with R an Artin algebra such that Hom R (Y, Z) = 0 (see [3] for definition). Then Proof. (1) Since the inclusion I ֒→ R is a covariant homomorphism in R-Mod and End R,I (R) ≃ R/I, we know that the first statement in (1) follows from Corollary 3.15 immediately. The last statement is a consequence of the fact that R is Morita equivalent to End R (R ⊕ I).
(2) Under the assumption, we know that f is a covariant map in R-mod, the category of finitely generated R-modules. So, by Corollary 3.15, it is sufficient to show that fin.dim(End R,Y (X )) = 0. In fact, since End R (X ) is a local algebra and since the ideal of End R (X ) generated by all homomorphisms which factorize through Y belong to the radical of End R (X ), the algebra End R,Y (X ) is local. Note that a local Artin algebra has finitistic dimension 0. Therefore fin.dim(End R,Y (X )) = 0. Now, (2) follows from Corollary 3.15.
Note that an alternative proof of Corollary 1.6(2) can be given by [17,Theorem 1.1] together with Corollary 3.14 and [22].
In the following we point out that the methods developed in this paper for little finitistic dimensions also work for big finitistic and global dimensions. Recall that, for an arbitrary ring R, we denote by Fin.dim(R) and gl.dim(R) the big finitistic and global dimensions of R, respectively. By definition, gl.dim(R) (respectively, Fin.dim(R)) is the supremum of projective dimensions of all left R-modules (respectively, which have finite projective dimension). Clearly, fin.dim(R) ≤ Fin.dim(R) ≤ gl.dim(R); and if gl.dim(R) < ∞, then Fin.dim(R) = gl.dim(R). However, the equality fin.dim(R) = Fin.dim(R) does not have to hold in general (see [26]).
As in Theorem 3.11, we have the following result on big finitistic dimensions of rings, in which the condition (2) is weaker than the one in Theorem 3.11 (2). Theorem 3.17. Let R 1 , R 2 and R 3 be rings. Suppose that there exists a recollement among the derived module categories D(R 3 ), D(R 2 ) and D(R 1 ) of R 3 , R 2 and R 1 : Then the following statements hold true: (1) Suppose that j ! restricts to a functor (2) Suppose that i * (R 1 ) is isomorphic in D(R 2 ) to a bounded complex of (not necessarily finitely generated) projective R 2 -modules. Then we have the following: Sketch of the proof. Let us consider the full subcategory X (R) of D(R) consisting of all those complexes which are isomorphic in D(R) to bounded complexes of projective R-modules. It is known that X (R) contains D c (R) and that the localization functor K (R) → D(R) induces a triangle equivalence Similarly, one can define big finitistic dimensions of functors, and establish several parallel results for Fin.dim(R), such as Lemma 3.5, Corollary 3.6 and Proposition 3.10. In the present situation, we shall replace D c (R) with X (R), and consider big finitistic dimensions of triangle functors which commute with direct sums. Further, to show Theorem 3.17, we observe the following facts for a given recollement: (i) The functors j ! , j ! , i * and i * commute with direct sums.
(ii) The functors j ! and i * preserve compact objects and restrict to triangle functors , then i * and j ! restrict to triangle functors Now, one can use the methods in the proof of Theorem 3.11 to show Theorem 3.17. Here, we omit the details.
Concerning global dimensions, we can describe explicitly upper bounds for the global dimension of a ring in terms of the ones of the other two rings involved in a recollement. These upper bounds imply the finiteness of global dimensions mentioned in [2, Proposition 2.14].
(2) If gl.dim(R 1 ) < ∞ and gl.dim Sketch of the proof. From [2, Proposition 2.14] and its proof, we observe the following two facts:

Proofs of Theorem 1.2 and Corollary 1.3
Now let us turn to proofs of our results on exact contexts that arise from different situations.

Proof of Theorem 1.2.
Given an exact context (λ, µ, M, m), we have defined its noncommutative tensor product T ⊠ R S and the following two ring homomorphisms ρ : S → T ⊠ R S, s → 1 ⊗ s for s ∈ S, and φ : T → T ⊠ R S, t → t ⊗ 1 for t ∈ T.
Note that T ⊠ R S has T ⊗ R S as its abelian groups, while its multiplication is different from the usual tensor product (see [8] for details). Let B := S M 0 T , C := M 2 (T ⊠ R S) and where β : M → T ⊗ R S is the unique R-R-bimodule homomorphism such that φ = (m·)β and ρ = (·m)β. Let ϕ : Then ϕ is a homomorphism of B-R-bimodules. Denote by P • the mapping cone of ϕ.
where j ! := B P • ⊗ L R −, j ! := Hom • B (P • , −) and D(θ * ) is the restriction functor induced from the ring homomorphism θ : B → C. First of all, we have two easy observations: (i) Since C := M 2 (T ⊠ R S) is Morita equivalent to T ⊠ R S, we have fin.dim(C) = fin.dim(T ⊠ R S).
(ii) Since B is a triangular matrix ring with the rings S and T in the diagonal, it follows from Corollary 3.14 that fin.dim(B) ≤ fin.dim(S) + fin.dim(T ) + 1.
We first apply Corollary 3.7 to show Theorem 1.2(1). In fact, by [8,Lemma 5.4], we see that R ≃ End D(B) (P • ) as rings (via multiplication) and that Hom D(B) P • , P • [n] = 0 for any n = 0. It remains to show that P • R is isomorphic in D(R op ) to a bounded complex of flat R op -modules. Since the sequence 0 → R If flat.dim(T R ) = ∞, then Theorem 1.2(1) is trivially true. So we may suppose flat.dim(T R ) < ∞. Let t := max{1, flat.dim(T R )}. Then P • is isomorphic in D(R op ) to a bounded complex such that F i are flat R op -modules for −t ≤ i ≤ 0. It follows from Corollary 3.7 that fin.dim(R) ≤ fin.dim(B) + t ≤ fin.dim(S) + fin.dim(T ) + t + 1. This shows Theorem 1.2(1).
Let us point out the following fact related to Theorem 1.2(2): Suppose that (λ, µ, M, m) is an exact content with Tor R i (T, S) = 0 for all i ≥ 1. If λ : R → S is a homological ring epimorphism such that R S ∈ P <∞ (R), then fin.dim(S) ≤ fin.dim(R) and fin.dim(T ⊠ R S) ≤ fin.dim(T ).
In fact, in this case, the Tor-vanishing condition, that is, Tor R i (T, S) = 0 for all i > 0, is equivalent to that φ : T → T ⊠ R S is a homological ring epimorphism (see [8, Theorem 1.1(1)] for details). Moreover, we have T ⊠ R S ≃ T ⊗ R S as T -S-bimodules. It follows that if R S ∈ P <∞ (R), then T T ⊠ R S ∈ P <∞ (T ) by the Tor-vanishing condition. Therefore the above-mentioned fact is a consequence of Corollary 3.8.

Proof of Corollary 1.3.
Let τ : S ⊆ R be the inclusion of from S into R, and let π : R → R/S be the canonical surjection. We define σ : S −→ R ′ = End S (R/S), s → (r → rs) for s ∈ S and r ∈ R/S to be the right multiplication map. Then the quadruple τ, σ, Hom S (R, R/S), π determined by the extension is an exact context (see the examples in [8,Section 3]) and its noncommutative tensor product R ′ ⊠ S R is defined. If S R is flat, then Tor S i (R ′ , R) = 0 for all i ≥ 1. Particularly, under the assumption on S R in Corollary 1.3 (2), the quadruple fulfills the Tor-vanishing condition in Theorem 1.2 (2). Now, we apply Theorem 1.2 to the exact context (τ, σ, Hom S (R, R/S), π), and see that the statements However, this can be concluded from the following exact sequence of right R ′ -modules (also right S-modules): 0 −→ R ′ −→ Hom S (R, R/S) −→ Hom S (S, R/S) −→ 0.

Proofs of Corollaries 1.4 and 1.5
In the following, we shall show that under the assumptions in Corollaries 1.4 and 1.5, we can get exact pairs, a special class of exact contents, which satisfy the Tor-vanishing condition in Theorem 1.2, and then apply Theorem 1.2 to each case. Here, noncommutative tensor products will be replaced by coproducts, and the latter will be interpreted further as some usual constructions of rings. Let λ : R → S and µ : R → T be ring homomorphisms, and let M be an S-T -bimodule with m ∈ M. Recall that an exact context (λ, µ, M, m) is called an exact pair if M = S ⊗ R T and m = 1 ⊗ 1. In this case, we simply say that (λ, µ) is an exact pair. By [8,Corollary 4.3], if the map λ in the exact context is a ring epimorphism, then the pair (λ, µ) is exact. Moreover, by [8,Remark 5.2], for an exact pair (λ, µ), we have T ⊠ R S ≃ S ⊔ R T , the coproduct of the R-rings of S and T . Note that, if we apply Theorem 1.2(2)(b) to the exact pair (λ, µ), then we only get fin.dim(S) ≤ fin.dim(R)+ fin.dim(S ⋉ M) + max{1, proj.dim( R S)} + 3. So, to obtain the better upper bound given in Corollary 1.4(b), we need the following statement: ( * ) Let f : Λ → Γ and g : Γ → Λ be ring homomorphisms such that f g = Id Λ . If fin.dim( Γ Γ ⊗ L Λ −) = s < +∞, then fin.dim(Λ) ≤ fin.dim(Γ) − s.
Applying Theorem 1.2 to the exact pair (λ, µ) in the proof of Lemma 3.19, we obtain the following estimations on finitistic dimensions, which can be applied to a class of examples of Milnor squares. Corollary 3.20. Let λ : R → S be a homological ring epimorphism. Suppose that I is an ideal of R such that the image J ′ of I under λ is a left ideal in S and that the restriction of λ to I is injective. Let J be the ideal of S generated by J ′ . Suppose that one of the conditions (1)-(4) in Lemma 3.19 holds. Then (1) fin.dim(R) ≤ fin.dim(S) + fin.dim(R/I) + max{1, flat.dim((R/I) R )} + 1.