Recollements of derived categories III: finitistic dimensions
aa r X i v : . [ m a t h . R A ] M a y Recollements of derived categories III: finitistic dimensions
Hong Xing Chen and
Chang Chang Xi ∗ Abstract
In this paper, we study homological dimensions of algebras linked by recollements of derived modulecategories, and establish a series of new upper bounds and relationships among their finitistic or globaldimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, inrepresentation theory and homological algebra. Further, we apply our results to a series of situations ofparticular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebrasinduced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel’s re-duction techniques for finitistic dimenson conjecture to more general contexts, but also generalise somerecent results in the literature.
Contents
Recollements of triangulated categories have been introduced by Beilinson, Bernstein and Deligne in order todecompose derived categories of sheaves into two parts, an open and a closed one (see [6]), and thus providinga natural habitat for Grothendieck’s six functors. Similarly, recollements of derived module categories canbe seen as short exact sequences, describing a derived module category in terms of a subcategory and of aquotient, both of which may be derived module categories themselves, related by six functors that in generalare not known. It turns out that recollements provide a very useful framework for understanding connectionsamong 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 questionsabout 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 endomorphismring of a finitely generated tilting object (see [15, 13]), Bazzoni has shown that for large tilting modules onegets instead a recollement relating three triangulated categories, with two of them being the derived categoriesof the given ring and the endomorphism ring of the large tilting module. In [7] we have addressed the questionof determining the third category in this recollement as a derived category of a ring and we have explainedthis ring in terms of universal localisations in the sense of Cohn (see [12, 20] for definition). Among theapplications has been a counterexample to the Jordan-H ¨older 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 ∗ Corresponding author. Email: [email protected]; Fax: +86 10 68903637.2010 Mathematics Subject Classification: Primary 18E30, 16G10, 13B30; Secondary 16S10, 13E05.Keywords: Derived category; Exact context; Finitistic dimension; Homological width; Recollement. K -theory of different rings. It turned out that under mild assumptions, the K -theory of an algebra can be fully decomposed under a sequence of recollements.For cohomology and for homological invariants of algebras, such a complete decomposition is not pos-sible. Nevertheless, results by Happel in [16] for the case of bounded derived categories (when fewer rec-ollements exist than in the unbounded case) show that finiteness of finitistic dimension of an algebra canbe reduced along a recollement; if such an invariant is finite for the two outer terms, then it is finite for themiddle term, too. Note that the particular values of these invariants depend on the ring and are not invariantsof the derived category. The present paper aims at extending Happel’s reduction techniques for homologicalconjectures. As in Happel’s paper [16] we will focus on finitistic dimensions, which include finite globaldimensions as a special case.Recall that the finitistic dimension of a ring R , denoted by fin.dim ( R ) , is by definition the supremumof projective dimensions of those left R -modules having a finite projective resolution by finitely generatedprojective modules. The well-known finitistic dimension conjecture states that any Artin algebra should havefinite finitistic dimension (see, for instance, [3, conjecture (11), p.410]). This conjecture is a longstandingquestion ([4, Bass, 1960]) and has still not been settled. It is closely related to at least seven other mainconjectures in homological representation theory of algebras (see [3, p. 409-410]). In the literature, thereis another definition of big finitistic dimension of a ring R , denoted by Fin.dim ( R ) , which is the supre-mum of projective dimensions of all those left R -modules which have finite projective dimension. Clearly,fin.dim ( R ) ≤ Fin.dim ( R ) . Usually, they are quite different (see [26]).There are two main directions in this article. First, we provide reduction techniques for homologicalinvariants of unbounded derived module categories, that is, for the most general possible setup (which alsohas been covered in the preceding articles in this series). In the first main result, Theorem 1.1, we givecriteria for the finiteness of finitistic dimension for each of the three rings in a recollement of derived modulecategories, in terms of the other two. The criteria aim to be applicable by putting conditions on particularobjects, not on the whole category. The second main result, Theorem 1.2, applies the first main result to thegeneral contexts of the so-called exact contexts introduced in [7], and in addition provides upper and lowerbounds for the finitistic dimensions of the three rings involved. A series of corollaries then applies the generalresults to classes of examples of particular interest, such as ring extensions, trivial extensions, quotient ringsand endomorphism rings of modules related by an almost split sequence. Theorem 1.1.
Let R , R and R be rings. Suppose that there exists a recollement among the derived modulecategories D ( R ) , D ( R ) and D ( R ) of R , R and R : D ( R ) i ∗ / / D ( R ) j ! / / i ! f f i ∗ x x D ( R ) j ∗ f f j ! x x Then the following hold true: ( ) Suppose that j ! restricts to a functor D b ( R ) → D b ( R ) of bounded derived module categories. If fin . dim ( R ) < ¥ , then fin . dim ( R ) < ¥ . ( ) Suppose that i ∗ ( R ) is a compact object in D ( R ) . Then we have the following: ( a ) If fin . dim ( R ) < ¥ , then fin . dim ( R ) < ¥ . ( b ) If fin . dim ( R ) < ¥ and fin . dim ( R ) < ¥ , then fin . dim ( R ) < ¥ . Note that the assumption of Theorem 1.1 on unbounded derived module categories is weaker than theone on bounded derived module categories, because the existence of recollements of bounded derived modulecategories implies the one of unbounded derived module categories. This is shown by a recent investigationon recollements at different levels in [2, 18]. So, Theorem 1.1 (see also Corollary 3.13) generalizes themain result in [16] since for a recollement of D b ( R j -mod ) with R j a finite-dimensional algebra over a fieldfor 1 ≤ j ≤
3, one can always deduce that i ∗ ( R ) is compact in D ( R ) . Moreover, Theorem 1.1 extends2nd amplifies a result in [25] because we deal with arbitrary rings instead of Artin algebras, and also yieldsa generalization of a result in [22] for left coherent rings to the one for arbitrary rings (see Corollary 3.9below).To prove this result, we introduce homological widths (or cowidth) for complexes that are quasi-isomorphicto 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 boundedcomplexes of projective (respectively, injective) modules to the natural numbers. It measures, up to ho-motopy equivalence, how large the minimal interval of such a complex is in which its non-zero terms aredistributed. Particularly, if a module has finite projective dimension, then its homological width is exactly theprojective dimension. Using homological widths, we will present a substantial and technical result, Theorem3.11, which is a strengthened version of Theorem 1.1 and describes explicitly upper bounds for finitisticdimensions, so that Theorem 1.1 will become an easy consequence of Theorem 3.11. Note that, in [16], oneof the key arguments in proofs is that finite-dimensional algebras have finitely many non-isomorphic simplemodules, while in our general context we do not have this fact and therefore must avoid this kind of argu-ments. 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 amongfinitistic dimensions of noncommutative tensor products and related rings. First of all, we recall some notionsfrom [8]:Let R , S and T be associative rings with identity, and let l : 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 ( l , µ , M , m ) is an exact context if the following sequence0 −→ R ( l , µ ) −→ S ⊕ T ( · m − m · ) −→ M −→ · 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 ( l , µ , M , m ) , there is defined a ring with identity in [8], called the noncommutativetensor product of ( l , µ , M , m ) and denoted by T ⊠ R S if the meaning of the exact context is clear. This notionnot only generalizes the one of usual tensor products over commutative rings and captures coproducts ofrings, but also plays a key role in describing the left parts of recollements induced from homological exactcontexts (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 ( l , µ , M , m ) be an exact context with the noncommutative tensor product T ⊠ R S. Then ( ) fin . dim ( R ) ≤ fin . dim ( S ) + fin . dim ( T ) + max { , flat . dim ( T R ) } + . ( ) Suppose that
Tor Ri ( T , S ) = for all i ≥ . If the left R-module R S has a finite projective resolution byfinitely generated projective modules, then the following hold true: ( a ) fin . dim ( T ⊠ R S ) ≤ fin . dim ( S ) + fin . dim ( T ) + . ( b ) fin . dim ( S ) ≤ fin . dim (cid:18) S M T (cid:19) ≤ fin . dim ( R ) + fin . dim ( T ⊠ R S ) + max { , proj . dim ( R S ) } + . Note that for the triangular matrix algebra B : = (cid:18) S M T (cid:19) , 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 ) , involvingthe starting ring R but without information on fin . dim ( S ) and fin . dim ( T ) . This is non-trivial and somewhatsurprising. Moreover, in Theorem 1.2(2), if l : R → S is a homological ring epimorphism, then we even3btain 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 di-mensions of ring extensions. This is of particular interest because the finitistic dimension conjecture can bereformulated over perfect fields in terms of ring extensions (see [24]). Note that, in the following result, wedo 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 productof the exact context determined the extension. Then ( ) fin . dim ( S ) ≤ fin . dim ( R ) + fin . dim ( R ′ ) + max (cid:8) , flat . dim (( R / S ) S ) , flat . dim (cid:0) Hom S ( R , R / S ) S (cid:1)(cid:9) + . ( ) Suppose that the left S-module R is projective and finitely generated. Then the following hold true: ( a ) fin . dim ( R ′ ⊠ S R ) ≤ fin . dim ( R ) + fin . dim ( R ′ ) + . ( b ) fin . dim ( R ) ≤ fin . dim ( S ) + fin . dim ( R ′ ⊠ S R ) + . 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 thereader to [14, Chapter 4]. Corollary 1.4.
Let l : R → S be a ring epimorphism and M an S-S-bimodule such that
Tor Ri ( M , S ) = forall i ≥ . If R S has a finite projective resolution by finitely generated projective R-modules, then ( a ) fin . dim ( S ⋉ M ) ≤ fin . dim ( S ) + fin . dim ( R ⋉ M ) + . ( b ) fin . dim ( S ) ≤ fin . dim ( R ) + fin . dim ( S ⋉ M ) . Now, we apply Theorem 1.2 to pullback squares of rings and surjective homomorphisms.
Corollary 1.5.
Let R be a ring, and let I and I be ideals of R such that I ∩ I = . Then ( ) fin . dim ( R ) ≤ fin . dim ( R / I ) + fin . dim ( R / I ) + max { , flat . dim (( R / I ) R ) } + . ( ) Suppose that
Tor Ri ( I , I ) = for all i ≥ . If the left R-module R / I has a finite projective resolutionby finitely generated projective modules, then ( a ) fin . dim ( R / ( I + I )) ≤ fin . dim ( R / I ) + fin . dim ( R / I ) + . ( b ) fin . dim ( R / I ) ≤ fin . dim ( R ) + fin . dim ( R / ( I + I )) + max { , proj . dim ( R ( R / I )) } + . The strategy of proving Corollaries 1.4 and 1.5 is as follows: First, we show that under the given assump-tions we can get exact pairs, a class of special exact contexts, and then employ Theorem 1.2 by verifying theTor-vanishing condition. At last, we have to describe noncommutative tensor products more substantially forthe cases considered.Finally, we mention a corollary on finitistic dimensions of algebras arising from idempotent ideals andalmost split sequences (see [3] for definition).
Corollary 1.6. ( ) If I is an idempotent ideal in a ring R, then fin . dim ( R / I ) ≤ fin . dim (cid:0) End R ( R ⊕ I ) (cid:1) ≤ fin . dim (cid:0) End R ( R I ) (cid:1) + fin . dim ( R / I ) + . ( ) Let → Z → Y → X → be an almost split sequence of R-modules with R an Artin algebra. If Hom R ( Y , Z ) = , then fin . dim ( End R ( Y ⊕ X )) ≤ fin . dim ( End R ( Y )) + . The paper is sketched as follows: In Section 2, we first recall some necessary definitions and then provetwo results on coproducts of rings. In Section 3, we provide all proofs of our results. Especially, we introducehomological widths of complexes and prove an amplified version, Theorem 3.11, of Theorem 1.1 phrased interms of homological widths and finitistic dimensions of involved rings, such that Theorem 1.1 is deducedreadily from Theorem 3.11. Moreover, the methods developed in this section also give similar upper boundsfor global and big finitistic dimensions (see Theorems 3.17 and 3.18).4
Definitions and conventions
In this section, we fix notation and briefly recall some definitions. For unexplained ones, we refer the readerto [8, 9].Throughout the paper, all notation and terminology are standard. For example, by a ring we mean anassociative 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, forinstance, 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 knownthat the localization functor K ( R ) → D ( R ) induces a triangle equivalence from the homotopy category ofbounded 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 iX • ) i ∈ Z , and call d iX • : X i → X i + the i -th differentialof X • . Sometimes, for simplicity, we shall write ( X i ) i ∈ Z for X • without mentioning the morphisms d iX • .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 minimumof indices i ∈ Z such that H i ( X • ) =
0, respectively. If X • is acyclic, that is, H i ( X • ) = i ∈ Z , thenwe understand that sup ( X • ) = − ¥ and inf ( X • ) = + ¥ . If X • is not acyclic, then inf ( X • ) ≤ sup ( X • ) , and H n ( X • ) = ( X • ) is an integer and n > sup ( X • ) or if inf ( X • ) is an integer and n < inf ( X • ) . For acomplex X • in C ( R ) , if H n ( X • ) = n , then X • is isomorphic in D ( R ) to a bounded complex.So, D b ( R ) is equivalent to the full subcategory of D ( R ) consisting of all complexes with finitely manynonzero 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 insteadof f ( x ) . This convention makes Hom R ( X , Y ) naturally a left End R ( X ) - and right End R ( Y ) -bimodule. But, fortwo 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 bethought as a kind of categorifications of exact sequences in abelian categories. Definition 2.1.
Let D , D ′ and D ′′ be triangulated categories. We say that D is a recollement of D ′ and D ′′ if there are six triangle functors among the three categories: D ′′ i ∗ = i ! / / D j ! = j ∗ / / i ! ^ ^ i ∗ (cid:3) (cid:3) D ′ j ∗ ^ ^ j ! (cid:4) (cid:4) such that ( ) ( i ∗ , i ∗ ) , ( i ! , i ! ) , ( j ! , j ! ) and ( j ∗ , j ∗ ) are adjoint pairs, ( ) i ∗ , j ∗ and j ! are fully faithful functors, ( ) j ! i ! = i ! j ∗ = i ∗ j ! = ( ) for each object X ∈ D , there are two triangles in D induced by counit and unit adjunctions: i ! i ! ( X ) −→ X −→ j ∗ j ∗ ( X ) −→ i ! i ! ( X )[ ] , j ! j ! ( X ) −→ X −→ i ∗ i ∗ ( X ) −→ j ! j ! ( X )[ ] , where the shift functor of triangulated categories is denoted by [1].Recall that the coproduct of a family { R i | i ∈ I } of R -rings with I an index set is defined to be an R -ring R together with a family { r i : R i → R | i ∈ I } of R -homomorphisms of rings such that, for any R -ring S with5 family of R -homomorphisms { t i : R i → S | i ∈ I } , there exists a unique R -homomorphism d : R → S suchthat t i = r i d for all i ∈ I . It is well known that coproducts of rings exist (see [11]). However, this existenceresult does not provide us with a handy form of coproducts; therefore we need a concrete description ofcoproducts 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 l : R → S is a ring epimorphism and M is an S-S-bimodule. Let e l : R ⋉ M → S ⋉ Mbe the ring homomorphism induced by l . Then the coproduct S ⊔ R ( R ⋉ M ) is isomorphic to S ⋉ M, that is,the inclusion S → S ⋉ M and e l define the coproduct.Proof. Let µ : R → R ⋉ M and r : S → S ⋉ M be the inclusions of rings. Note that S and R ⋉ M are R -ringsvia l and µ , respectively, and that lr = µ e l : R → S ⋉ M . To prove that S ⋉ M , together with r and e l , isthe coproduct of S and R ⋉ M over R , we suppose that L is an arbitrary ring and that f : R ⋉ M → L and g : S → L are arbitrary ring homomorphisms such that l g = µ f . Then we have to show that there is a uniquering homomorphism h : S ⋉ M → L such that e l h = f and r h = g . Clearly, if such an h exists, then h must bedefined by ( s , m ) ( m ) f + ( s ) g for s ∈ S and m ∈ M . This shows the uniqueness of h . So, it suffices to showthat 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 = ,
2. On the one hand, (cid:0) ( s , m ) ( s , m ) (cid:1) h = ( s s , s m + m s ) h = ( s m + m s ) f + ( s s ) g = ( s m ) f + ( m s ) f + ( s ) g ( s ) g . On the other hand, (cid:0) ( s , m ) (cid:1) h (cid:0) ( s , m ) (cid:1) h = (cid:0) ( m ) f +( s ) g (cid:1)(cid:0) ( m ) f + ( s ) g (cid:1) = ( m ) f ( m ) f + ( m ) f ( s ) g + ( s ) g ( m ) f + ( s ) g ( s ) g = ( m m ) f + ( m ) f ( s ) g +( s ) g ( m ) f + ( s ) g ( s ) g = ( m ) f ( s ) g + ( s ) g ( m ) f + ( s ) g ( s ) g since m m =
0. This implies that if ( s m ) f = ( s ) g ( m ) f and ( m s ) f = ( m ) f ( s ) g , then (cid:0) ( s , m ) ( s , m ) (cid:1) h = (( s , m )) h (( s , m )) h . So,to prove that h preserves multiplication, we need only to verify these additional conditions, that is, ( sm ) f = ( s ) g ( m ) f and ( ms ) f = ( m ) f ( s ) g for s ∈ S and m ∈ M . To show the former, we fix an m ∈ M and define two maps: j : S → L , s ( sm ) f and y : S → L , s ( s ) g ( m ) f . Since l g = µ f , one can check that both j and y are homomorphisms of R -modules such that lj = ly . Butwe do not know if they are homomorphisms of rings. Nevertheless, we can still have f = y because l : R → S being a ring epimorphism by assumption implies that the map Hom R ( R , l ) : Hom R ( S , L ) → Hom R ( R , L ) isan isomorphism, and therefore it is injective. Thus f = y . Similarly, we can show that ( ms ) f = ( m ) f ( s ) g .Consequently, the map h preserves multiplication and is actually a ring homomorphism. (cid:3) The other description of coproducts is for quotients of rings by ideals, which applies to Milnor squares(see [19]).
Lemma 2.3.
Let R be a ring, and let R i be an R -ring with ring homomorphism l i : R → R i for i = , . ( ) If l : R → R is a ring epimorphism, then so is the canonical homomorphism r : R → R ⊔ R R . ( ) Let I be an ideal of R , and let J be the ideal of R generated by the image ( I ) l of I under the map l . If R = R / I and l : R → R is the canonical surjective map, then R ⊔ R R = R / J.Proof. ( ) It follows from the definition of coproducts of rings that l r = l r : R → R ⊔ R R . Wepoint out that r is a ring epimorphism. In fact, if f , g : R ⊔ R R → S are two ring homomorphisms suchthat r f = r g , then l r f = l r g . This means that l r f = l r g , and therefore r f = r g since l is aring epimorphism. By the universal property of coproducts, we have g = f . Thus r is a ring epimorphism. ( ) Let r : R → R / J be the canonical surjection, and let r : R → R / J be the ring homomorphisminduced by l since J = R ( I ) l R ⊇ ( I ) l . Now, we claim that R / J together with r and r is thecoproduct of R and R over R . Clearly, we have l r = l r : R → R / J . Further, assume that t : R → S t : R → S are two ring homomorphisms such that l t = l t . Then ( I ) l t = ( I ) l t =
0, andtherefore ( J ) t =
0. This means that there is a unique ring homomorphism d : R / J → S such that t = r d .It follows that l t = l t = l r d = l r d . Since l is surjective, we have t = r d . This shows that R ⊔ R R = R / J . (cid:3) This section is devoted to proofs of all results mentioned in the introduction. We start with introducing theso-called homological widths for complexes, and then prove a strengthened version, Theorem 3.11 below, ofTheorem 1.1. As consequences of Theorem 3.11, we get proofs of Theorems 1.1 and 1.2. Finally, we applyTheorem 1.2 to give proofs of all corollaries, and mention two results on global and big finitistic dimensions.
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 pro-jective, 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 → P → M → 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 P • : = ( P n , d nP • ) n ∈ Z ∈ C b ( R -Proj ) . We define the homological width of P • in the following way: w ( P • ) : = ( P • is acyclic , sup ( P • ) − inf ( P • ) + proj . dim (cid:0) Cok ( d inf ( P • ) − P • ) (cid:1) otherwise.Clearly, 0 ≤ w ( P • ) < ¥ . Moreover, P • is isomorphic in K b ( R -Proj ) to a complex Q • : 0 −→ Q t − p d t − p −→ Q t − p + d t − p + −→ · · · −→ Q t − d t − −→ Q t −→ P t + d t + P • −→ · · · −→ P s − d s − P • −→ Ker ( d sP • ) −→ s : = sup ( P • ) , t : = inf ( P • ) , p : = proj . dim (cid:0) Cok ( d t − P • ) (cid:1) and each term being projective. Clearly, thesequence 0 −→ Q t − p d t − p −→ Q t − p + d t − p + −→ · · · −→ Q t − d t − −→ Q t −→ Cok ( d t − P • ) −→ R -module Cok ( d t − P • ) . Note that if P • ∈ C b ( R -proj ) , we can choose Q • ∈ C b ( R -proj ) .The following result says that homological widths of bounded complexes of projective modules are pre-served under homotopy equivalences. Lemma 3.1.
Let M • , N • ∈ C b ( R -Proj ) . If M • ≃ N • in K b ( R -Proj ) , then w ( M • ) = w ( N • ) .Proof. Recall that K b ( R -Proj ) is the stable category of the Frobenius category C b ( R -Proj ) with projec-tive objects being acyclic complexes. Assume that M • ≃ N • in K b ( R -Proj ) . Then there exist two acycliccomplexes P • and Q • in C b ( R -Proj ) such that M • ⊕ P • ≃ N • ⊕ Q • in C b ( R -Proj ) . This implies that H i ( M • ) ≃ H i ( N • ) and Cok ( d iM • ) ⊕ Cok ( d iP • ) ≃ Cok ( d iN • ) ⊕ Cok ( d iQ • ) for all i ∈ Z . Thus sup ( M • ) = sup ( N • ) andinf ( M • ) = inf ( N • ) . Moreover, since Cok ( d iP • ) and Cok ( d iQ • ) belong to R -Proj, we have proj . dim (cid:0) Cok ( d iM • ) (cid:1) = proj . dim (cid:0) Cok ( d iN • ) (cid:1) . It follows that w ( M • ) = w ( N • ) . (cid:3) Thanks to Lemma 3.1, the definition of homological widths for complexes can be extended slightly toderived categories in the following sense: Given a complex X • ∈ D ( R ) , if there is a complex P • ∈ C b ( R -Proj ) such that X • ≃ P • in D ( R ) , then we define w ( X • ) : = w ( P • ) . This is well defined: If there exists another7omplex Q • ∈ C b ( R -Proj ) such that X • ≃ Q • in D ( R ) , then P • ≃ Q • in K b ( R -Proj ) and w ( P • ) = w ( Q • ) byLemma 3.1. So, for such a complex X • , its homological width w ( X • ) can be characterized as follows: w ( X • ) = min (cid:26) a P • − b P • ≥ (cid:12)(cid:12)(cid:12)(cid:12) P • ≃ X • in D ( R ) for P • ∈ C b ( R -Proj ) with P i = i < b P • or i > a P • (cid:27) . 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 nI • ) n ∈ Z ∈ C b ( R -Inj ) ,we define the homological cowidth of I • as follows: cw ( I • ) : = ( I • is acyclic , sup ( I • ) − inf ( I • ) + inj . dim (cid:0) Ker ( d sup ( I • ) I • ) (cid:1) otherwise.Similarly, if a complex Y • is isomorphic in D ( R ) to a bounded complex I • ∈ C b ( R -Inj ) , then we define cw ( Y • ) : = cw ( I • ) . In particular, if Y ∈ R -Mod has finite injective dimension, then cw ( Y ) = inj . dim ( Y ) . Also,we have the following characterization of cw ( Y • ) : cw ( Y • ) = min (cid:26) a I • − b I • ≥ (cid:12)(cid:12)(cid:12)(cid:12) I • ≃ Y • in D ( R ) for I • ∈ C b ( R -Inj ) with I i = i < b I • or i > a I • (cid:27) . Homological widths and cowidths will be used to bound homological dimensions in the next section.
In this subsection, we shall first prove an amplified version of Theorem 1.1, namely Theorem 3.11 below, sothat 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:fin . dim ( R ) : = sup { proj . dim ( R X ) | X ∈ P < ¥ ( R ) } . For each n ∈ Z , we define D c ≥ n ( R ) : = { X • ∈ D c ( R ) | X • ≃ P • in D c ( R ) with P • ∈ C b ( R -proj ) such that P i = i < n } . From this definition, we have D c ≥ n ( R ) ⊆ D c ≥ n ′ ( R ) whenever n ≥ n ′ . Since the localization functor K ( R ) → D ( R ) induces a triangle equivalence K b ( R -proj ) ≃ −→ D c ( R ) , we have D c ( R ) = [ n ∈ Z D c ≥ n ( R ) . Clearly, if fin . dim ( R ) = m < ¥ , then P < ¥ ( R ) ⊆ D c ≥− m ( R ) . For the convenience of later discussions, we alsoformally set D c ≥− ¥ ( R ) : = D c ( R ) and D c ≥ + ¥ ( R ) : = { } . Lemma 3.2.
Let m , n ∈ N . Then the following statements are true: ( ) The full subcategory D c ≥ n ( R ) of D c ( R ) is closed under direct summands in D c ( R ) . ( ) Let X • ∈ D c ≥ n ( R ) , Z • ∈ D c ≥ m ( R ) and s = min { n , m } . Then, for any distinguished triangle X • → Y • → Z • → X • [ ] in D c ( R ) , we have Y • ∈ D c ≥ s ( R ) .Proof. ( ) Let M • ∈ K b ( R -proj ) , and let N • : = ( N i ) i ∈ Z ∈ K b ( R -proj ) such that N i = i < n . Suppose that M • is a direct summand of N • in K b ( R -proj ) , or equivalently, there is a complex L • ∈ C b ( R -proj ) such that M • ⊕ L • ≃ N • in K b ( R -proj ) . Hence H i ( M • ) = i < n . Note that K b ( R -proj ) is the stable category of the Frobenius category C b ( R -proj ) with projective objects being acyclic complexes.8o we can find two acyclic complexes U • and V • in C b ( R -proj ) such that M • ⊕ L • ⊕ U • ≃ N • ⊕ V • in C b ( R -proj ) . This implies thatCok ( d n − M • ) ⊕ Cok ( d n − L • ) ⊕ Cok ( d n − U • ) ≃ Cok ( d n − N • ) ⊕ Cok ( d n − V • ) = N n ⊕ Cok ( d n − V • ) . Since N n ⊕ Cok ( d n − V • ) ∈ R -proj, we have Cok ( d n − M • ) ∈ R -proj. It follows that M • is isomorphic in K b ( R -proj ) to the following truncated complex0 −→ Cok ( d n − M • ) −→ M n + −→ M n + −→ · · · −→ . Recall that the localization functor K ( R ) → D ( R ) induces a triangle equivalence K b ( R -proj ) ≃ −→ D c ( R ) .Thus ( ) follows. ( ) Since X • ∈ D c ≥ n ( R ) , there exists a complex P • ∈ C b ( R -proj ) with P i = i < n such that X • ≃ P • in D c ( R ) . Similarly, there exists another complex Q • ∈ C b ( R -proj ) with Q i = i < m such that Z • ≃ Q • in D c ( R ) . It follows from the triangle equivalence K b ( R -proj ) ≃ −→ D c ( R ) thatHom K b ( R -proj ) ( Q • [ − ] , P • ) ≃ Hom D c ( R ) ( Q • [ − ] , P • ) ≃ Hom D c ( R ) ( Z • [ − ] , X • ) . Thus the given triangle yields a distinguished triangle in D c ( R ) : Q • [ − ] f • −→ P • −→ Y • −→ Q • with f • a chain map in C ( R ) . Then Y • ≃ Con ( f • ) in D c ( R ) . Since Con ( f • ) i = Q i ⊕ P i for any i ∈ Z , wehave Con ( f • ) ∈ C b ( R -proj ) and Con ( f • ) i = i < s . This implies Y • ∈ D c ≥ s ( R ) . (cid:3) To investigate relationships among finitistic dimensions of rings in recollements, it may be convenient tointroduce the notion of finitistic dimensions of functors .Let R and R be two arbitrary rings. Suppose that X and X are full subcategories of D ( R ) and D ( R ) , respectively, and that R -Mod ⊆ X . For a given additive functor F : X → X , we defineinf ( F ) : = inf { n ∈ Z | H n ( F ( X )) = X ∈ R -Mod } , fin . dim ( F ) : = inf { n ∈ Z | H n ( F ( X )) = X ∈ P < ¥ ( R ) } . Note that inf ( F ) = + ¥ if and only if F ( X ) = D ( R ) for all X ∈ R -Mod. In fact, if there exits some X ∈ R -Mod such that H n ( F ( X )) = n , then inf ( F ) ≤ n . Moreover, by definition, we alwayshave inf ( F ) ≤ fin . dim ( F ) and fin . dim ( F ) ∈ Z ∪ {− ¥ , + ¥ } . Lemma 3.3.
Let F : D ( R ) → D ( R ) be a triangle functor. Then the following statements are true: ( ) If F has a left adjoint L : D ( R ) → D ( R ) with L ( R ) ∈ D − ( R ) , then inf ( F ) ≥ − sup ( L ( R )) . ( ) If F has a right adjoint G : D ( R ) → D ( R ) , then F restricts to a functor D b ( R ) → D b ( R ) if andonly if G (cid:0) Hom Z ( R , Q / Z ) (cid:1) is isomorphic in D ( R ) to a bounded complex I • of injective R -modules. In thiscase, inf ( F ) ≥ − (cid:0) m + inj . dim ( Ker ( d mI • )) (cid:1) , where m : = sup ( I • ) and d mI • : I m → I m + is the m-th differential ofI • . Proof. ( ) For each n ∈ Z and M ∈ R -Mod, we have H n ( F ( M )) ≃ Hom D ( R ) ( R , F ( M )[ n ]) ≃ Hom D ( R ) ( L ( R ) , M [ n ]) . Since L ( R ) ∈ D − ( R ) , we have s : = sup ( L ( R )) < + ¥ . Recall that the localization functor K ( R ) → D ( R ) induces a triangle equivalence K − ( R -Proj ) ≃ −→ D − ( R ) . So there is a complex P • : = ( P j ) j ∈ Z ∈ C − ( R -Proj ) with P j = j > s such that P • ≃ L ( R ) in D ( R ) . It follows that H n ( F ( M )) ≃ Hom D ( R ) ( L ( R ) , M [ n ]) ≃ Hom D ( R ) ( P • , M [ n ]) ≃ Hom K ( R ) ( P • , M [ n ]) = n < − s . Thus inf ( F ) ≥ − s . ( ) To calculate cohomologies of complexes, we consider the functor ( − ) ∨ : = Hom Z ( − , Q / Z ) : Z -Mod −→ Z -Mod . This is an exact functor with the property that a Z -module U is zero if and only if so is U ∨ , because Q / Z isan injective cogenerator for Z -Mod.Let X • ∈ D ( R ) . Then H ( X • ) ∨ = Hom Z ( H ( X • ) , Q / Z ) ≃ Hom K ( Z ) ( X • , Q / Z ) ≃ Hom K ( Z ) ( R ⊗ R X • , Q / Z ) ≃ Hom K ( R ) ( X • , R ∨ ) . Since R ∨ is an injective R -module, we have Hom K ( R ) ( X • , R ∨ ) ≃ Hom D ( R ) ( X • , R ∨ ) . Thus H ( X • ) ∨ ≃ Hom D ( R ) ( X • , R ∨ ) . Now, let M ∈ R -Mod and n ∈ Z . Then H n ( F ( M )) ∨ ≃ Hom D ( R ) (cid:0) F ( M )[ n ] , R ∨ ) . Since ( F , G ) is an adjointpair, we have Hom D ( R ) (cid:0) F ( M )[ n ] , R ∨ ) ≃ Hom D ( R ) ( M [ n ] , G ( R ∨ ) (cid:1) . This implies that H n ( F ( M )) = D ( R ) ( M [ n ] , G ( R ∨ ) (cid:1) = W = G ( R ∨ ) . To check the sufficiency of ( ) , it is enough to show that Hom D ( R ) ( M [ n ] , W • (cid:1) = n . In fact, if W • is isomorphic in D ( R ) to a bounded complex I • of injective R -modules, thenHom D ( R ) ( M [ n ] , W • (cid:1) ≃ Hom D ( R ) ( M [ n ] , I • (cid:1) ≃ Hom K ( R ) ( M [ n ] , I • (cid:1) = n .In the following, we will show the necessity of ( ) . Suppose that F restricts to a functor D b ( R ) → D b ( R ) . We first claim that H n ( W • ) = n , that is W • ∈ D b ( R ) .Actually, we have the following isomorphisms of abelian groups: H n ( W • ) ≃ Hom D ( R ) ( R , G ( R ∨ )[ n ]) ≃ Hom D ( R ) ( F ( R ) , R ∨ [ n ]) ≃ H − n ( F ( R )) . Since F ( R ) ∈ D b ( R ) , we have H n ( F ( R )) = n . Thus H n ( W • ) = n . Inother words, W • is isomorphic in D b ( R ) to a bounded complex. Consequently, there exists a lower-boundedcomplex I • of injective R -modules such that I • ≃ W • in D ( R ) . In particular, we have H n ( I • ) ≃ H n ( W • ) forall n . To complete the proof of the necessity of ( ) , it remains to show that I • can be chosen to be a boundedcomplex.Note that we have the following isomorphisms:Hom D ( R ) ( F ( M ) , R ∨ [ n ]) ≃ Hom D ( R ) ( M , W • [ n ]) ≃ Hom D ( R ) ( M , I • [ n ]) ≃ Hom K ( R ) ( M , I • [ n ]) . As F : D ( R ) → D ( R ) restricts to a functor D b ( R ) → D b ( R ) by assumption, we get F ( M ) ∈ D b ( R ) . Upto isomorphism in D ( R ) , we may assume that F ( M ) ∈ C b ( R ) . Since R ∨ is an injective R -module, we seethat Hom D ( R ) ( F ( M ) , R ∨ [ n ]) ≃ Hom K ( R ) ( F ( M ) , R ∨ [ n ]) = n . Thus Hom K ( R ) ( M , I • [ n ]) = n . Particularly, there is a natural number d M (depending on M ) such that Hom K ( R ) ( M , I • [ n ]) = n > d M . We may suppose that the complex I • is of the following form:0 −→ I s d s −→ I s + d s + −→ · · · −→ I m d m −→ I m + d m + −→ · · · −→ I i d i −→ I i + −→ · · · where all terms I i are injective and where s ≤ m : = sup ( I • ) and H i ( I • ) = i > m . Let V : = L i ≥ m Im ( d i ) . Then Hom K ( R ) ( V , I • [ n ]) = n > d V . Now we define t : = max { m , d V } . Then Hom K ( R ) ( Im ( d t ) , I • [ t + ]) =
0. This implies that the chain mapIm ( d t ) → I • [ t + ] , induced from the inclusion Im ( d t ) ֒ → I t + , is homotopic to the zero map. Therefore, the10anonical surjection I t ։ Im ( d t ) must split. With I t then also Im ( d t ) is an injective module. Since H i ( I • ) = i > m , we see that I • is isomorphic in D ( R ) to the following bounded complex:0 −→ I s d s −→ I s + d s + −→ · · · −→ I m d m −→ I m + d m + −→ · · · −→ I t d t −→ Im ( d t ) −→ D ( R ) , we can choose I • to be a boundedcomplex of injective modules. This completes the proof of the necessity of ( ) .To show the last statement of ( ) , we note that the R -module Ker ( d m ) has a finite injective resolutionsince H i ( I • ) = i > m . Hence, up to isomorphism in D ( R ) , we can replace I • by the followingbounded complex of injective R -modules:0 −→ I s d s −→ I s + d s + −→ · · · −→ I m − −→ e I m e d m −→ e I m + e d m + −→ · · · −→ e I m + p − e d m + p − −→ e I m + p −→ ( e d m ) = Ker ( d m ) and p : = inj . dim ( Ker ( d m )) ≤ t . This implies that Hom K ( R ) ( M [ n ] , I • ) = n < − ( m + p ) . Since H n ( F ( M )) ∨ ≃ Hom D ( R ) ( M [ n ] , W • ) ≃ Hom D ( R ) ( M [ n ] , I • ) ≃ Hom K ( R ) ( M [ n ] , I • ) , we have H n ( F ( M )) = n < − ( m + p ) . Thus inf ( F ) ≥ − ( m + p ) . (cid:3) We remark that, in Lemma 3.3(2), the R -module I : = Hom Z ( R , Q / Z ) can be replaced by any injectivecogenerator of R -Mod. This is due to the fact that G always commutes with direct products. Recall that an R -module M is called a cogenerator of R -Mod if any R -module can be embedded into a direct product ofcopies of M . Clearly, I is an injective cogenerator of R -Mod. In case that R is an Artin algebra, there isanother injective cogenerator, the usual dual module D ( R ) of the right regular module R , where D is theusual duality of an Artin algebra. Lemma 3.4.
Let F : D c ( R ) → D c ( R ) be a triangle functor. Suppose that fin . dim ( F ) = s > − ¥ and fin . dim ( R ) = t < ¥ . Then we have the following: ( ) F ( P < ¥ ( R )) ⊆ D c ≥ s − t ( R ) . ( ) Let m ∈ Z . Then, for any X ∈ P < ¥ ( R ) and for any Y • ∈ D ( R ) with sup ( Y • ) ≤ m, we have Hom D ( R ) ( F ( X ) , Y • [ i ]) = for all i > t − s + m.Proof. Note that s = + ¥ if and only if F ( X ) = X ∈ P < ¥ ( R ) . In this case, both ( ) and ( ) are true. Now, we assume s < + ¥ . Thus s is an integer.(1) Since F ( X ) ∈ D c ( R ) , there exists a complex Q • = ( Q j , d j ) j ∈ Z ∈ C b ( R -proj ) such that F ( X ) ≃ Q • in D c ( R ) . In particular, H i ( F ( X )) ≃ H i ( Q • ) for all i ∈ Z . Since fin . dim ( F ) = s < ¥ , we have H i ( F ( X )) = i < s . Thus H i ( Q • ) = i < s . It follows that Y : = Cok ( d s − ) ∈ P < ¥ ( R ) , and therefore Q • isisomorphic in D ( R ) to the following canonical truncated complex:0 −→ Y −→ Q s + d s + −→ Q s + −→ · · · −→ . Since fin . dim ( R ) = t < ¥ , we have proj . dim ( R Y ) ≤ t . So the R -module Y has a finite projective resolution:0 −→ P s − t −→ · · · −→ P s − −→ P s −→ Y −→ P j ∈ R -proj for s − t ≤ j ≤ s . Consequently, F ( X ) is isomorphic in D ( R ) to the following complex P • : 0 −→ P s − t −→ · · · −→ P s − −→ P s −→ Q s + d s + −→ Q s + −→ · · · −→ . Clearly, P • ∈ C b ( R -proj ) and P i = i < s − t . This implies F ( X ) ∈ D c ≥ s − t ( R ) . Hence, we have F ( P < ¥ ( R )) ⊆ D c ≥ s − t ( R ) . ( ) Let X ∈ P < ¥ ( R ) and Y • ∈ D ( R ) with sup ( Y • ) ≤ m < ¥ . Then H j ( Y • ) = j > m , and thereforethere exists a complex Z • ∈ C − ( R ) with Z r = r > m , such that Z • ≃ Y • in D ( R ) . Moreover, by the11roof of ( ) , there exists another complex P • ∈ C b ( R -proj ) with P i = i < s − t , such that P • ≃ F ( X ) in D ( R ) . It follows thatHom D ( R ) ( F ( X ) , Y • [ i ]) ≃ Hom D ( R ) ( P • , Z • [ i ]) ≃ Hom K ( R ) ( P • , Z • [ i ]) = i > t − s + m . This shows ( ) . (cid:3) . Lemma 3.5.
Let F : D c ( R ) → D c ( R ) be a fully faithful triangle functor. If fin . dim ( F ) = s is an integer,then fin . dim ( R ) ≤ fin . dim ( R ) − s + sup ( F ( R )) . In particular, if fin . dim ( R ) < ¥ , then fin . dim ( R ) < ¥ .Proof. If fin . dim ( R ) is infinity, then the right-hand side of the inequality is infinity and the corollary istrue. So we assume that fin . dim ( R ) = t < ¥ . Further, we may assume R =
0. Since F is fully faithful,we have 0 = F ( R ) ∈ D c ( R ) . This implies that sup ( F ( R )) < ¥ . Moreover, it is known that, for any X ∈ P < ¥ ( R ) , if there is a natural number n such that Ext iR ( X , R ) = i > n , then proj . dim ( R X ) ≤ n .So, to show that fin . dim ( R ) ≤ n : = t − s + sup ( F ( R )) < ¥ , it is enough to prove that Ext iR ( X , R ) = X ∈ P < ¥ ( R ) and all i > n . In fact, since F is fully faithful, we see thatExt iR ( X , R ) ≃ Hom D ( R ) ( X , R [ i ]) ≃ Hom D ( R ) ( F ( X ) , F ( R )[ i ]) . Due to Lemma 3.4 (2), we have Hom D ( R ) ( F ( X ) , F ( R )[ i ]) = i > n . Thus Ext iR ( X , R ) = X ∈ P < ¥ ( R ) and all i > n . (cid:3) Summarizing Lemmas 3.3 and 3.5 together, we obtain the following useful result, in which w and cw denote the homological width and cowidth of a complex, respectively. Corollary 3.6.
Let F : D ( R ) → D ( R ) be a fully faithful triangle functor such that F ( R ) ∈ D c ( R ) . Thenthe following statements hold true: ( ) If F has a left adjoint L : D ( R ) → D ( R ) with L ( R ) ∈ D − ( R ) , then fin . dim ( R ) ≤ fin . dim ( R ) + sup ( L ( R )) + sup ( F ( R )) . If moreover L ( R ) ∈ D c ( R ) , then fin . dim ( R ) ≤ fin . dim ( R ) + w ( L ( R )) . ( ) If F has a right adjoint G : D ( R ) → D ( R ) and restricts to a functor D b ( R ) → D b ( R ) , then fin . dim ( R ) ≤ fin . dim ( R ) + cw (cid:0) G ( Hom Z ( R , Q / Z )) (cid:1) .Proof. Clearly, if fin . dim ( R ) is infinity, then the two statements (1) and (2) are trivially true. So, weassume that fin . dim ( R ) = t < ¥ . We further assume that R i = i = , . By assumption, we have F ( R ) ∈ D c ( R ) , and therefore F restricts to a functor D c ( R ) → D c ( R ) . Since F is fully faithful and R =
0, we have F ( R ) =
0. This leads to fin . dim ( F ) = + ¥ . Thus fin . dim ( F ) ∈ Z ∪ {− ¥ } . ( ) Since ( L , F ) is an adjoint pair, we have H n ( F ( R )) ≃ Hom D ( R ) ( R , F ( R )[ n ]) ≃ Hom D ( R ) ( L ( R ) , R [ n ]) .It follows from 0 = F ( R ) ∈ D ( R ) that L ( R ) = D ( R ) . Since L ( R ) ∈ D − ( R ) , we know thatsup ( L ( R )) is an integer. By Lemma 3.3(1), we see that inf ( F ) ≥ − sup ( L ( R )) > − ¥ , and thereforefin . dim ( F ) ≥ inf ( F ) > − ¥ . Combining this with Lemma 3.5, we havefin . dim ( R ) ≤ t − fin . dim ( F ) + sup ( F ( R )) ≤ t + sup ( L ( R )) + sup ( F ( R )) . This shows the first part of ( ) . For the second part of ( ) , we only need to check that w (cid:0) L ( R ) (cid:1) = sup ( L ( R )) + sup ( F ( R )) .In fact, it follows from L ( R ) ∈ D c ( R ) that the homological width of L ( R ) is well defined and thereexists a complex P • : 0 −→ P r d r −→ P r − −→ · · · −→ P s − −→ P s −→ C b ( R -proj ) with s = sup ( L ( R )) and s − r = w (cid:0) L ( R ) (cid:1) such that L ( R ) ≃ P • in D ( R ) (see Section 3.1).In this case, d r is not a split injection. Since ( L , F ) is an adjoint pair, we haveHom D ( R ) ( P • , R [ n ]) ≃ Hom D ( R ) ( L ( R ) , R [ n ]) ≃ Hom D ( R ) ( R , F ( R )[ n ]) ≃ H n ( F ( R )) n ∈ Z . This implies that H n ( F ( R )) = n > − r . Moreover, since the map d r is not a splitinjection, we have Hom D ( R ) ( P • , P r [ − r ]) =
0. Thus H − r ( F ( R )) ≃ Hom D ( R ) ( P • , R [ − r ]) =
0. This showssup ( F ( R )) = − r . It follows that w (cid:0) L ( R ) (cid:1) = s − r = sup ( L ( R )) + sup ( F ( R )) . ( ) Under the assumption of ( ) , we see from Lemma 3.3(2) that inf ( F ) ≥ − (cid:0) m + inj . dim ( Ker ( d mI • )) (cid:1) ,where I • ∈ C b ( R -Inj ) is defined in Lemma 3.3(2) and m : = sup ( I • ) . Thus fin . dim ( F ) ≥ inf ( F ) > − ¥ andfin . dim ( R ) ≤ t + m + inj . dim ( Ker ( d mI • )) + sup ( F ( R )) < ¥ by Lemma 3.5. Define W • : = G ( Hom Z ( R , Q / Z )) . By the proof of Lemma 3.3(2), we see that W • ≃ I • in D ( R ) and that H n ( W • ) ≃ H − n ( F ( R ) for all n ∈ Z . This implies that sup ( F ( R )) = − inf ( W • ) = − inf ( I • ) .Thus cw ( W • ) = sup ( I • ) − inf ( I • ) + inj . dim ( Ker ( d mI • )) = m + sup ( F ( R )) + inj . dim ( Ker ( d mI • )) . So fin . dim ( R ) ≤ t + cw ( W • ) . (cid:3) As a consequence of Corollary 3.6, we have the following applicable fact.
Corollary 3.7.
Let P • ∈ C ( R ⊗ Z R op1 ) such that R P • ∈ D c ( R ) . Assume that the following conditions hold: ( ) R ≃ End D ( R ) ( P • ) as rings (via multiplication), and Hom D ( R ) ( P • , P • [ n ]) = for all n = . ( ) P • R is isomorphic in D ( R op1 ) to a bounded complexF • : 0 −→ F r −→ F r − −→ · · · −→ F s − −→ F s −→ of flat R op1 -modules, where r , s ∈ Z and r ≤ s.Then fin . dim ( R ) ≤ fin . dim ( R ) + s − r. In this case, if fin . dim ( R ) < ¥ , then fin . dim ( R ) < ¥ .Proof. Let F : = P • ⊗ L R − : D ( R ) → D ( R ) . Then F ( R ) ≃ R P • ∈ D c ( R ) and F has a right adjoint G : = R Hom R ( P • , − ) : D ( R ) → D ( R ) . Since R P • ∈ D c ( R ) , the functor F restricts to a functor F ′ : D c ( R ) → D c ( R ) . Note that the condition ( ) implies that F ′ is fully faithful. Further, since F commuteswith direct sums and D ( R ) is compactly generated by R , we see that F itself is also fully faithful.Now, we claim that F restricts to a functor D b ( R ) → D b ( R ) . In fact, by Lemma 3.3(2), this is equivalentto saying that the complex G (cid:0) Hom Z ( R , Q / Z ) (cid:1) is isomorphic in D ( R ) to a bounded complex of injective R -modules.To check the latter, we use the functor ( − ) ∨ : = Hom Z ( − , Q / Z ) and apply G to the injective R -module R ∨ . Then we have the following isomorphisms in D ( R ) : G ( R ∨ ) = R Hom R ( P • , R ∨ ) = Hom • R ( P • , R ∨ ) ≃ Hom • Z ( R ⊗ R P • , Q / Z ) ≃ ( P • ) ∨ . Note that ( − ) ∨ : R op1 -Mod → R -Mod is an exact functor, which sends flat R op1 -modules to injective R -modules. Thus the condition ( ) implies that ( P • ) ∨ is isomorphic in D ( R ) to the following bounded complexof injective R -modules: ( F • ) ∨ : = −→ ( F s ) ∨ −→ ( F s − ) ∨ −→ · · · −→ ( F r − ) ∨ −→ ( F r ) ∨ −→ ( F s ) ∨ and ( F r ) ∨ are of degrees − s and − r , respectively. Consequently, we have cw ( G ( R ∨ )) = cw (( P • ) ∨ ) = cw (( F • ) ∨ ) ≤ s − r . Now, it follows from Corollary 3.6(2) thatfin . dim ( R ) ≤ fin . dim ( R ) + cw ( G ( R ∨ )) ≤ fin . dim ( R ) + s − r . This completes the proof. (cid:3)
Recall that a ring epimorphism l : R → S is homological if Tor Ri ( S , S ) = i >
0. This is equivalentto saying that the restriction functor D ( l ∗ ) : D ( S ) → D ( R ) is fully faithful. Note that D ( l ∗ ) always has a leftadjoint functor S ⊗ L R − : D ( R ) → D ( S ) . For some new advances on homological ring epimorphisms phrasedin terms of recollements of derived categories, we refer the reader to [ , , ] . Applying Corollary 3.6(1) tohomological ring epimorphisms, we have the following simple result.13 orollary 3.8. Let l : 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 ) < ¥ .Proof. If we take F : = D ( l ∗ ) and L : = S ⊗ L R − in Corollary 3.6(1), then fin . dim ( S ) ≤ fin . dim ( R ) + w ( L ( S )) . Since w ( L ( S )) = proj . dim ( S S ) =
0, we have fin . dim ( S ) ≤ fin . dim ( R ) . (cid:3) 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 l is homological, we see thatExt iS ( X , Y ) ≃ Ext iR ( X , Y ) for all Y ∈ S -Mod and i ≥
0. This implies that proj . dim ( S X ) ≤ proj . dim ( R X ) . As aresult, we have proj . dim ( S X ) ≤ proj . dim ( R X ) ≤ fin . dim ( R ) . This shows fin . dim ( S ) ≤ fin . dim ( R ) . (cid:3) The following result extends [22, Theorem 1.1] on finitistic dimensions for derived equivalences of leftcoherent rings to those of arbitrary rings.
Corollary 3.9.
Suppose that F : D ( R ) → D ( R ) is a triangle equivalence. Then | fin . dim ( R ) − fin . dim ( R ) |≤ w ( F ( R )) . Proof.
Suppose that G : D ( R ) → D ( R ) is a quasi-inverse of F . Then ( G , F ) and ( F , G ) are adjointpairs. Clearly, G is also a triangle equivalence. Since both F and G preserve compact objects, they restrict totriangle equivalences of perfect derived categories: F : D c ( R ) ≃ −→ D c ( R ) and G : D c ( R ) ≃ −→ D c ( R ) . ByCorollary 3.6(1), we have fin . dim ( R ) ≤ fin . dim ( R ) + w ( G ( R )) and fin . dim ( R ) ≤ fin . dim ( R ) + w ( F ( R )) .Thus, to complete the proof, it is enough to show that w ( G ( R )) = w ( F ( R )) .In fact, up to isomorphism in derived categories, we may assume that F ( R ) ∈ C b ( R -proj ) and G ( R ) ∈ C b ( R -proj ) .Without loss of generality, we suppose that F ( R ) is a complex in C b ( R -proj ) of the form0 −→ P − r d − r −→ P − r + −→ · · · −→ P − −→ P −→ r = w ( F ( R ) ≥
0. This implies that H ( F ( R )) = d − r is not a split injection. Since ( F , G ) isan adjoint pair, we always have Hom K ( R ) ( F ( R ) , R [ n ]) ≃ Hom D ( R ) ( F ( R ) , R [ n ]) ≃ Hom D ( R ) ( R , G ( R )[ n ]) ≃ H n ( G ( R )) for all n ∈ Z . It follows that H i ( G ( R )) = i < i > r . Further, we claim that H r ( G ( R )) =
0, andtherefore sup ( G ( R )) = r . Actually, since d − r is not a split injection, we have Hom K ( R ) ( F ( R ) , P − r [ r ]) = = Hom K ( R ) ( F ( R ) , R [ r ]) ≃ H r ( G ( R )) . So, up to isomorphism in K ( R ) , the complex G ( R ) hasthe following form0 −→ Q s j s −→ Q s + −→ Q s + −→ · · · −→ Q r − −→ Q r −→ ∈ C b ( R -proj ) such that 0 ≤ r − s = w ( G ( R )) . In particular, this implies that j s is not a split injection. So, to show w ( F ( R ) = w ( G ( R )) , we only need to show s = ( G , F ) is an adjoint pair, we haveHom K ( R ) ( G ( R ) , R [ n ]) ≃ Hom D ( R ) ( G ( R ) , R [ n ]) ≃ Hom D ( R ) ( R , F ( R )[ n ]) ≃ H n ( F ( R )) for all n ∈ Z . On the one hand, if s <
0, then Hom K ( R ) ( G ( R ) , R [ − s ]) ≃ H − s ( F ( R )) =
0, and thereforeHom K ( R ) ( G ( R ) , Q s [ − s ]) =
0. This means that j s is a split injection, a contradiction. On the other hand, if s >
0, then 0 = Hom K ( R ) ( G ( R ) , R ) ≃ H ( F ( R )) . This is also a contradiction. Thus s = w ( F ( R )) = w ( G ( R )) , as desired. (cid:3) The above result describes a relationship for finitistic dimensions of derived equivalent rings. If weweaken derived equivalences into half recollements of perfect derived module categories, we will obtain thefollowing general result which provides a bound for the finitistic dimension of the middle ring by those ofthe other two rings. 14 roposition 3.10.
Suppose that there is a half recollement of perfect derived module categories of the ringsR , R and R D c ( R ) i ∗ / / D c ( R ) j ! / / i ∗ w w D c ( R ) j ! w w . Then fin . dim ( R ) ≤ fin . dim ( R ) + fin . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + w (cid:0) j ! ( R ) (cid:1) + . In particular, if fin . dim ( R ) < ¥ and fin . dim ( R ) < ¥ , then fin . dim ( R ) < ¥ . Proof.
The proof will be done in several steps. We may suppose that fin . dim ( R ) < ¥ and fin . dim ( R ) < ¥ . Clearly, if one of R and R is zero, then Proposition 3.10 follows from Corollary 3.9. From now on, weassume that R = = R .Step 1. We claim that j ! j ! ( P < ¥ ( R )) ⊆ D c ≥− u ( R ) where u : = fin . dim ( R ) + w (cid:0) j ! ( R ) (cid:1) ≥ j ! : D c ( R ) → D c ( R ) is fully faithful, we have 0 = j ! ( R ) ∈ D c ( R ) . This impliessup ( j ! ( R )) < ¥ . As ( j ! , j ! ) is an adjoint pair, one can follow the proof of Lemma 3.3(1) to show that − sup ( j ! ( R )) ≤ inf ( j ! ) . Note that inf ( j ! ) ≤ fin . dim ( j ! ) . Thus − ¥ < − sup ( j ! ( R )) ≤ fin . dim ( j ! ) ≤ + ¥ .Define u : = − sup ( j ! ( R )) − fin . dim ( R ) . Then u ≤ fin . dim ( j ! ) − fin . dim ( R ) . It follows from Lemma3.4(1) that j ! ( P < ¥ ( R )) ⊆ D c ≥ u ( R ) . In other words, for any Y ∈ P < ¥ ( R ) , there exists a complex P • Y : = ( P nY ) n ∈ Z ∈ C b ( R -proj ) with P nY = n < u such that j ! ( Y ) ≃ P • Y in D c ( R ) . Clearly, the complex P • Y is of the following form:0 −→ P u Y −→ P u + Y −→ P u + Y −→ · · · −→ P s ( Y ) Y −→ , where s ( Y ) depends on Y and u ≤ s ( Y ) . Since j ! ( R ) ∈ D c ( R ) by the half recollement, we see that j ! ( R ) is isomorphic in D c ( R ) to a bounded complex L • of the form0 −→ L u −→ L u + −→ L u + −→ · · · −→ u = sup ( j ! ( R )) − w (cid:0) j ! ( R ) (cid:1) and that L i ∈ R -proj for all i ≥ u (see Section 3.1). This implies j ! ( R ) ∈ D c ≥ u ( R ) . Since D c ≥ u ( R )) is closed under direct summands in D c ( R ) by Lemma 3.2(1), we have j ! ( R -proj ) ⊆ D c ≥ u ( R ) .Note that u = fin . dim ( R ) + w (cid:0) j ! ( R ) (cid:1) = fin . dim ( R ) + sup ( j ! ( R )) − u = − ( u + u ) . Now, we claimthat j ! j ! ( P < ¥ ( R )) ⊆ D c ≥− u ( R ) = D c ≥ ( u + u ) ( R ) .Actually, for the complex P • Y ∈ C b ( R -proj ) , there is a canonical distinguished triangle in D c ( R ) : P s ( Y ) Y [ − s ( Y )] −→ P • Y −→ P • Y ≤ s ( Y ) − −→ P s ( Y ) Y [ − s ( Y )] where P • Y ≤ s ( Y ) − is truncated from P • Y by replacing P s ( Y ) Y with 0, that is, P • Y ≤ u − : 0 −→ P u Y −→ P u + Y −→ · · · −→ P s ( Y ) − Y −→ −→ . This induces a distinguished triangle in D c ( R ) : j ! (cid:0) P s ( Y ) Y (cid:1) [ − s ( Y )] −→ j ! (cid:0) P • Y (cid:1) −→ j ! (cid:0) P • Y ≤ s ( Y ) − (cid:1) −→ j ! (cid:0) P s ( Y ) Y (cid:1) [ − s ( Y )] . Note that j ! (cid:0) P s ( Y ) Y (cid:1) [ − s ( Y )] ∈ D c ≥ s ( Y )+ u ( R ) ⊆ D c ≥ ( u + u ) ( R ) due to u ≤ s ( Y ) . Since P iY ∈ R -proj for all u ≤ i ≤ s ( Y ) , one can apply Lemma 3.2(2) to show that j ! ( P • Y ) ∈ D c ≥ ( u + u ) ( R ) by induction on the numberof non-zero terms of a complex. It follows from j ! ( Y ) ≃ P • Y that j ! j ! ( Y ) ≃ j ! ( P • Y ) ∈ D c ≥ ( u + u ) ( R ) . Thisimplies that j ! j ! ( P < ¥ ( R )) ⊆ D c ≥ ( u + u ) ( R ) . 15tep 2. We show that i ∗ i ∗ ( P < ¥ ( R )) ⊆ D c ≥ v ( R ) , where v : = fin . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + u + m such that m ≤ fin . dim ( i ∗ ) ≤ + ¥ . Indeed, the given halfrecollement yields the following canonical triangle ( † ) j ! j ! ( Y ) h Y −→ Y e Y −→ i ∗ i ∗ ( Y ) −→ j ! j ! ( Y )[ ] in D ( R ) , where h Y and e Y stand for the counit and unit adjunction morphisms, respectively. Since j ! j ! ( Y ) ∈ D c ≥− u ( R ) ⊆ D c ( R ) , we can find a complex U • : = ( U n ) n ∈ Z ∈ C b ( R -proj ) with U n = n < − u ≤ j ! j ! ( Y ) ≃ U • in D ( R ) . It follows thatHom D ( R ) ( j ! j ! ( Y ) , Y ) ≃ Hom D ( R ) ( U • , Y ) ≃ Hom K ( R ) ( U • , Y ) . So there exists a chain map f • : U • → Y such that its mapping cone V • is isomorphic to i ∗ i ∗ ( Y ) in D ( R ) .Clearly, V = U ⊕ Y and V j = U j + for any j =
0. In particular, V j = j < − u −
1. Since i ∗ : D c ( R ) → D c ( R ) is fully faithful, we have H n ( i ∗ ( Y )) ≃ Hom D ( R ) ( R , i ∗ ( Y )[ n ]) ≃ Hom D ( R ) ( i ∗ ( R ) , i ∗ i ∗ ( Y )[ n ]) ≃ Hom D ( R ) ( i ∗ ( R ) , V • [ n ]) . By assumption, i ∗ ( R ) ∈ D c ( R ) and therefore is isomorphic in D c ( R ) to a complex Q • : 0 −→ Q v −→ Q v + −→ · · · −→ Q b −→ C b ( R -proj ) , where b : = sup ( i ∗ ( R )) and b − v = w (cid:0) i ∗ ( R ) (cid:1) (see Section 3.1). Let m : = − u − − b . Then H n ( i ∗ ( Y )) ≃ Hom D ( R ) ( i ∗ ( R ) , V • [ n ]) ≃ Hom D ( R ) ( Q • , V • [ n ]) ≃ Hom K ( R ) ( Q • , V • [ n ]) = n < m . This implies that m ≤ fin . dim ( i ∗ ) ≤ + ¥ , as claimed.Let v : = m − fin . dim ( R ) . It follows from Lemma 3.4(1) that i ∗ ( P < ¥ ( R )) ⊆ D c ≥ v ( R ) . Now, replacingthe pair ( j ! , j ! ) in the proof of Step 1 with ( i ∗ , i ∗ ) , one can similarly show that i ∗ i ∗ ( P < ¥ ( R )) ⊆ D c ≥ v + v ( R ) . Note that − ( v + v ) = fin . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + u + = v ≥ u + ≥ . dim ( R ) ≤ v = fin . dim ( R ) + fin . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + w (cid:0) j ! ( R ) (cid:1) + j ! j ! ( Y ) ⊆ D c ≥− u ( R ) and i ∗ i ∗ ( Y ) ∈ D c ≥− v ( R ) for Y ∈ P < ¥ ( R ) with u < v , it follows from thetriangle ( † ) and Lemma 3.2(2) that Y ∈ D c ≥− v ( R ) . Now, let P • : = ( P n , d n ) n ∈ Z ∈ C b ( R -proj ) such that P n = n < − v and that Y ≃ P • in D c ( R ) . Since Y is an R -module, we see that H n ( P • ) = n = H ( P • ) ≃ Y . Consequently, Ker ( d ) ∈ R -proj and the following complex0 −→ P − v d − v −→ P − v + d − v + −→ · · · −→ P − d − −→ Ker ( d ) −→ Y −→ . dim ( R Y ) ≤ v and therefore fin . dim ( R ) ≤ v < ¥ . (cid:3) Now, with the above preparations, we prove the following strong version of Theorem 1.1 .
Theorem 3.11.
Let R , R and R be rings. Suppose that there exists a recollement among the derivedmodule categories D ( R ) , D ( R ) and D ( R ) of R , R and R : D ( R ) i ∗ / / D ( R ) j ! / / i ! f f i ∗ x x D ( R ) j ∗ f f j ! x x Then the following statements hold true: ( ) Suppose that j ! restricts to a functor D b ( R ) → D b ( R ) of bounded derived module categories. Then fin . dim ( R ) ≤ fin . dim ( R ) + cw (cid:0) j ! ( Hom Z ( R , Q / Z )) (cid:1) . ( ) Suppose that i ∗ ( R ) is a compact object in D ( R ) . Then ( a ) fin . dim ( R ) ≤ fin . dim ( R ) + w ( i ∗ ( R )) . ( b ) fin . dim ( R ) ≤ fin . dim ( R ) + fin . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + w (cid:0) j ! ( R ) (cid:1) + . roof. Note that the triangle functors j ! and i ∗ in a recollement always take compact objects to compactobjects and that i ∗ ( R ) is compact if and only if j ! ( R ) is compact (for a reference of this fact, one may see,for example, [9, Lemma 2.2]). Thus we have a sequence of functors: D c ( R ) D c ( R ) i ∗ o o D c ( R ) , j ! o o where the functor j ! is fully faithful.Applying Corollary 3.6(2) to the adjoint pair ( j ! , j ! ) , we then obtain ( ) .Suppose i ∗ ( R ) ∈ D c ( R ) . Then j ! ( R ) ∈ D c ( R ) and the given recollement in Theorem 3.11 induces ahalf recollment of prefect derived module categories: D c ( R ) i ∗ / / D c ( R ) j ! / / i ∗ w w D c ( R ) j ! w w . Now, the statements ( a ) and ( b ) in ( ) follow from Corollary 3.6(1) and Proposition 3.10, respectively. Thiscompletes the proof of Theorem 3.11. (cid:3) 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 surjectionR → R / ReR is homological with R ReR ∈ P < ¥ ( R ) . Then fin . dim ( R / ReR ) ≤ fin . dim ( R ) ≤ fin . dim ( eRe ) + fin . dim ( R / ReR ) + proj . dim ( R R / ReR ) + .Proof. Let J : = ReR . Since the canonical surjection R → R / J is homological, there exists a recollementof derived module categories: D ( R / J ) D ( p ∗ ) / / D ( R ) eR ⊗ L R − / / e e R / J ⊗ L R − ~ ~ D ( eRe ) Re ⊗ L eRe − ~ ~ e e Since R J ∈ P < ¥ ( R ) , we see that D ( p ∗ )( 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. (cid:3)
Since a recollement at D b -level induces a recollement at D -level, the following result is a straightforwardconsequence of Theorem 3.11, which also generalizes [16, Theorem 2]. Corollary 3.13.
Let R , R and R be rings. Suppose that there exists a recollement among the derivedmodule categories D b ( R ) , D b ( R ) and D b ( R ) : D b ( R ) i ∗ / / D b ( R ) j ! / / i ! g g i ∗ v v D b ( R ) j ∗ g g j ! v v such that i ∗ ( R ) ∈ D c ( R ) . Then fin . dim ( R ) < ¥ if and only if max { fin . dim ( R ) , fin . dim ( R ) } < ¥ . 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 projectiveand finitely generated and that J R has finite projective dimension. Then there exists a recollement among17 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 b -level because the left-derived functor Re ⊗ L eRe − : D ( eRe ) → D ( R ) may not restrict to a functor of boundedderived 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 (forexample, see [14, Corollary 4.21]). Corollary 3.14.
Let R and S be rings, and let M be an S-T -bimodule. Set B : = (cid:18) S M T (cid:19) . Then fin . dim ( S ) ≤ fin . dim ( B ) ≤ fin . dim ( S ) + fin . dim ( T ) + .Proof. Let e = (cid:18) (cid:19) . Then BeB = Be , eBe ≃ T , B / BeB ≃ S = B ( − e ) and B B = B S ⊕ Be . Thus B BeB ∈ B -proj and the canonical surjection B → S is homological. Now, Corollary 3.14 follows from Corol-lary 3.12. (cid:3) Recall from [10] that a morphism l : Y → X of objects in an additive category C is said to be co-variant if the induced map Hom C ( X , l ) : Hom C ( X , Y ) → Hom C ( X , X ) is injective, and the induced mapHom C ( Y , l ) : Hom C ( Y , Y ) → Hom C ( Y , X ) is a split epimorphism of End C ( Y ) -modules. Covariant morphismscapture traces of modules, which guarantee the ubiquity of covariant morphisms (see [10]).For covariant morphisms, we have the following result which follows from Corollary 3.12 and [10,Lemma 3.2]. Corollary 3.15.
Let f : Y → X be a covariant morphism in an additive category C . Then fin . dim (cid:0) End C , Y ( X ) (cid:1) ≤ fin . dim (cid:0) End C ( Y ⊕ X ) (cid:1) ≤ fin . dim (cid:0) End C ( Y ) (cid:1) + fin . dim (cid:0) End C , Y ( X ) (cid:1) + , where End C , Y ( X ) is the quotient ring of the endomorphism ring End C ( X ) of X modulo the ideal generatedby all those endomorphisms of X which factorize through the object Y . Consequently, we have the following result which restates Corollary 1.6.
Corollary 3.16. ( ) Let I be an idempotent ideal in a ring R. Then fin . dim ( R / I ) ≤ fin . dim (cid:0) End R ( R ⊕ I ) (cid:1) ≤ fin . dim (cid:0) End R ( R I ) (cid:1) + fin . dim ( R / I ) + . In particular, if R I is projective and finitely generated, then fin . dim ( R / I ) ≤ fin . dim ( R ) ≤ fin . dim (cid:0) End R ( R I ) (cid:1) + fin . dim ( R / I ) + . ( ) Let → Z → Y f −→ X → be an almost split sequence in R -mod with R an Artin algebra such that Hom R ( Y , Z ) = (see [3] for definition). Then fin . dim (cid:0) End R ( Y ⊕ X ) (cid:1) ≤ fin . dim (cid:0) End R ( Y ) (cid:1) + . 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 aconsequence 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 finitisticdimension 0. Therefore fin . dim ( End R , Y ( X )) =
0. Now, (2) follows from Corollary 3.15. (cid:3)
Note that an alternative proof of Corollary 1.6(2) can be given by [17, Theorem 1.1] together with Corol-lary 3.14 and [22]. 18n the following we point out that the methods developed in this paper for little finitistic dimensions alsowork 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 ) (respec-tively, Fin . dim ( R ) ) is the supremum of projective dimensions of all left R -modules (respectively, which havefinite projective dimension). Clearly, fin . dim ( R ) ≤ Fin . dim ( R ) ≤ gl . dim ( R ) ; and if gl . dim ( R ) < ¥ , thenFin . 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 thecondition (2) is weaker than the one in Theorem 3.11(2). Theorem 3.17.
Let R , R and R be rings. Suppose that there exists a recollement among the derivedmodule categories D ( R ) , D ( R ) and D ( R ) of R , R and R : D ( R ) i ∗ / / D ( R ) j ! / / i ! f f i ∗ x x D ( R ) j ∗ f f j ! x x Then the following statements hold true: ( ) Suppose that j ! restricts to a functor D b ( R ) → D b ( R ) of bounded derived module categories. Then Fin . dim ( R ) ≤ Fin . dim ( R ) + cw (cid:0) j ! ( Hom Z ( R , Q / Z )) (cid:1) . ( ) Suppose that i ∗ ( R ) is isomorphic in D ( R ) to a bounded complex of (not necessarily finitely gener-ated) projective R -modules. Then we have the following: ( a ) Fin . dim ( R ) ≤ Fin . dim ( R ) + w ( i ∗ ( R )) . ( b ) Fin . dim ( R ) ≤ Fin . dim ( R ) + Fin . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + w (cid:0) j ! ( R ) (cid:1) + .Sketch of the proof. Let us consider the full subcategory X ( R ) of D ( R ) consisting of all those com-plexes 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 K b ( R -Proj ) ≃ −→ X ( R ) .Similarly, one can define big finitistic dimensions of functors, and establish several parallel results forFin . dim ( R ) , such as Lemma 3.5, Corollary 3.6 and Proposition 3.10. In the present situation, we shallreplace D c ( R ) with X ( R ) , and consider big finitistic dimensions of triangle functors which commute withdirect 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 X ( R ) j ! −→ X ( R ) and X ( R ) i ∗ −→ X ( R ) . ( iii ) If i ∗ ( R ) ∈ X ( R ) , then i ∗ and j ! restrict to triangle functors X ( R ) i ∗ −→ X ( R ) and X ( R ) j ! −→ X ( R ) . Now, one can use the methods in the proof of Theorem 3.11 to show Theorem 3.17. Here, we omit the details. (cid:3)
Concerning global dimensions, we can describe explicitly upper bounds for the global dimension of aring in terms of the ones of the other two rings involved in a recollement. These upper bounds imply thefiniteness of global dimensions mentioned in [2, Proposition 2.14].
Theorem 3.18.
Let R , R and R be rings. Suppose that there exists a recollement among the derivedmodule categories D ( R ) , D ( R ) and D ( R ) of R , R and R :19 ( R ) i ∗ / / D ( R ) j ! / / i ! f f i ∗ x x D ( R ) j ∗ f f j ! x x Then we have the following: ( ) If gl . dim ( R ) < ¥ , then gl . dim ( R ) ≤ gl . dim ( R ) + cw (cid:0) j ! ( Hom Z ( R , Q / Z )) (cid:1) and gl . dim ( R ) ≤ gl . dim ( R ) + w ( i ∗ ( R )) . ( ) If gl . dim ( R ) < ¥ and gl . dim ( R ) < ¥ , then gl . dim ( R ) ≤ gl . dim ( R ) + gl . dim ( R ) + w (cid:0) i ∗ ( R ) (cid:1) + w (cid:0) j ! ( R ) (cid:1) + .Sketch of the proof. From [2, Proposition 2.14] and its proof, we observe the following two facts: ( i ) If gl . dim ( R ) < ¥ or gl . dim ( R ) < ¥ , then i ∗ ( R ) is isomorphic in D ( R ) to a bounded complex ofprojective R -modules. ( ii ) gl . dim ( R ) < ¥ if and only if both gl . dim ( R ) < ¥ and gl . dim ( R ) < ¥ . In this case, the recollementamong unbounded derived categories can restrict to a recollement of bounded derived categories.Moreover, for a ring R , if gl . dim ( R ) < ¥ , then gl . dim ( R ) = Fin . dim ( R ) . Now, Theorem 3.18 becomes aconsequence of Theorem 3.17. (cid: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 ( l , µ , M , m ) , we have defined its noncommutative tensor product T ⊠ R S and thefollowing two ring homomorphisms r : S → T ⊠ R S , s ⊗ s for s ∈ S , and f : T → T ⊠ R S , t t ⊗ t ∈ T . Note that T ⊠ R S has T ⊗ R S as its abelian groups, while its multiplication is different from the usual tensorproduct (see [8] for details). Let B : = (cid:18) S M T (cid:19) , C : = M ( T ⊠ R S ) and q : = (cid:18) r b f (cid:19) : B −→ C , where b : M → T ⊗ R S is the unique R - R -bimodule homomorphism such that f = ( m · ) b and r = ( · m ) b .Let j : (cid:18) S (cid:19) −→ (cid:18) MT (cid:19) , (cid:18) s (cid:19) (cid:18) sm (cid:19) for s ∈ S . Then j is a homomorphism of B - R -bimodules. Denote by P • the mapping cone of j . Then P • ∈ C b ( B ⊗ Z R op ) and B P • ∈ C b ( B -proj ) . In particular, P • ∈ D c ( B ) .By [8, Theorem 1.1], if Tor Ri ( T , S ) = i ≥
1, then there is a recollement of derived categories: D ( C ) D ( q ∗ ) / / D ( B ) j ! / / f f C ⊗ L B − x x D ( R ) f f j ! x x where j ! : = B P • ⊗ L R − , j ! : = Hom • B ( P • , − ) and D ( q ∗ ) is the restriction functor induced from the ring homo-morphism q : B → C . First of all, we have two easy observations:(i) Since C : = M ( T ⊠ R S ) is Morita equivalent to T ⊠ R S , we have fin . dim ( C ) = fin . dim ( T ⊠ R S ) .20ii) Since B is a triangular matrix ring with the rings S and T in the diagonal, it follows from Corollary3.14 that fin . dim ( B ) ≤ fin . dim ( S ) + fin . dim ( T ) + R ≃ End D ( B ) ( P • ) as rings (via multiplication) and that Hom D ( B ) (cid:0) P • , P • [ n ] (cid:1) = n =
0. It remains to showthat P • R is isomorphic in D ( R op ) to a bounded complex of flat R op -modules.Since the sequence 0 → R ( l , µ ) −→ S ⊕ T ( · m − m · ) −→ M → ( · m ) ≃ Con ( µ ) in D ( R op ) . Thisimplies that P • R ≃ T ⊕ Con ( · m ) ≃ T ⊕ Con ( µ ) in D ( R op ) , where Con ( µ ) is the complex 0 → R µ −→ T → T in degree 0. If flat . dim ( T R ) = ¥ , then Theorem 1.2(1) is trivially true. So we may suppose flat . dim ( T R ) < ¥ .Let t : = max { , flat . dim ( T R ) } . Then P • is isomorphic in D ( R op ) to a bounded complex F • : = −→ F − t −→ F − t + −→ · · · −→ F − −→ F −→ 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 + . This shows Theorem 1.2(1).Next, we shall apply Theorem 3.11 to the above recollement and give a proof of Theorem 1.2(2).By the proof of [9, Theorem 1.3(2)], we see that D ( q ∗ )( C ) = B C ∈ P < ¥ ( B ) if and only if R S ∈ P < ¥ ( R ) .Suppose R S ∈ P < ¥ ( R ) . It follows from [8, Corollary 5.8(1)] thatproj . dim ( B C ) ≤ max { , proj . dim ( R S ) + } . Since C ⊗ L B B ≃ C in D ( C ) , we see from Theorem 3.11(2)(a) that fin . dim ( C ) ≤ fin . dim ( B ) . Note thatfin . dim ( T ⊠ R S ) = fin . dim ( C ) and fin . dim ( B ) ≤ fin . dim ( S ) + fin . dim ( T ) +
1. Thus ( a ) holds.Since D ( q ∗ )( C ) = B C and j ! ( R ) ≃ B P • in D ( B ) , we know that w ( P • ) = w (cid:0) D ( q ∗ )( C ) (cid:1) = w ( B C ) = proj . dim ( B C ) ≤ max { , proj . dim ( R S ) + } . Now, it follows from Theorem 3.11(2)(b) thatfin . dim ( B ) ≤ fin . dim ( R ) + fin . dim ( T ⊠ R S ) + max { , proj . dim ( R S ) + } + + . Clearly, fin . dim ( S ) ≤ fin . dim ( B ) . Thus ( b ) holds. (cid:3) Let us point out the following fact related to Theorem 1.2(2): Suppose that ( l , µ , M , m ) is an exact contentwith Tor Ri ( T , S ) = i ≥
1. If l : 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 Ri ( T , S ) = i >
0, is equivalent tothat f : T → T ⊠ R S is a homological ring epimorphism (see [8, Theorem 1.1(1)] for details). Moreover, wehave 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 theTor-vanishing condition. Therefore the above-mentioned fact is a consequence of Corollary 3.8. Proof of Corollary 1.3.
Let t : S ⊆ R be the inclusion of from S into R , and let p : R → R / S be the canonical surjection. We define s : 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 (cid:0) t , s , Hom S ( R , R / S ) , p (cid:1) determined by the extensionis an exact context (see the examples in [8, Section 3]) and its noncommutative tensor product R ′ ⊠ S R isdefined. If S R is flat, then Tor Si ( R ′ , R ) = i ≥
1. Particularly, under the assumption on S R in Corollary1.3(2), the quadruple fulfills the Tor-vanishing condition in Theorem 1.2(2).Now, we apply Theorem 1.2 to the exact context ( t , s , Hom S ( R , R / S ) , p ) , and see that the statements ( a ) and ( b ) in Corollary 1.3 follow from the statements ( a ) and ( b ) in Theorem 1.2, respectively. To showCorollary 1.3(1), we shall apply Theorem 1.2(1). For this aim, we shall proveflat . dim ( R ′ S ) ≤ max { flat . dim (cid:0) Hom S ( R , R / S ) S (cid:1) , flat . dim (cid:0) ( R / S ) S (cid:1) } . R ′ -modules (also right S -modules):0 −→ R ′ −→ Hom S ( R , R / S ) −→ Hom S ( S , R / S ) −→ . which is obtained by applying Hom S ( − , R / S ) to the exact sequence 0 → S → R → R / S →
0. Now, thestatement ( ) follows from Theorem 1.2(1). (cid:3) 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 applyTheorem 1.2 to each case. Here, noncommutative tensor products will be replaced by coproducts, and thelatter will be interpreted further as some usual constructions of rings.Let l : R → S and µ : R → T be ring homomorphisms, and let M be an S - T -bimodule with m ∈ M . Recallthat an exact context ( l , µ , M , m ) is called an exact pair if M = S ⊗ R T and m = ⊗
1. In this case, we simplysay that ( l , µ ) is an exact pair. By [8, Corollary 4.3], if the map l in the exact context is a ring epimorphism,then the pair ( l , µ ) is exact. Moreover, by [8, Remark 5.2], for an exact pair ( l , µ ) , we have T ⊠ R S ≃ S ⊔ R T ,the coproduct of the R -rings of S and T . Proof of Corollary 1.4 .We define T : = R ⋉ M , µ : R → T to be the inclusion from R into T , and e l : R ⋉ M → S ⋉ M to be thecanonical map induced from l . By Lemma 2.2, the ring S ⋉ M , together with the inclusion r : S → S ⋉ M and e l : T → S ⋉ M , is the coproduct of S and T over R .Now, we show that ( l , µ ) is an exact pair. Actually, the split exact sequence 0 → R µ −→ T → M → R - R -bimodules implies that R T R ≃ R ⊕ M as R - R -bimodules. Since l is a ring epimorphism and M is an S - S -bimodule, the map S ⊗ R T −→ S ⋉ M , s ⊗ ( r , m ) ( sr , sm ) for s ∈ S and m ∈ M , is an isomorphism of S - T -bimodules. Under this isomorphism, we can identify themap µ ′ = id S ⊗ µ : S → S ⊗ T with the inclusion r : S → S ⋉ M , and the map l ′ = l ⊗ id T : T → S ⊗ R T with e l . Note that 0 → S r −→ S ⋉ M → M → S - S -bimodules. It follows thatCok ( µ ) ≃ Cok ( r ) ≃ M as R - R -bimodules, and therefore the sequence of R - R -bimodules:0 −→ R ( l , µ ) −→ S ⊕ T (cid:16) r − e l (cid:17) −→ S ⋉ M −→ ( l , µ ) is exact.Consequently, we know that T ⊠ R S ≃ S ⊔ R T ≃ S ⋉ M as rings. Note that Tor Ri ( T , S ) ≃ Tor Ri ( R ⊕ M , S ) ≃ Tor Ri ( M , S ) = i ≥
1. Thus Corollary 1.4(a) follows immediately from Theorem 1.2(2)(a).Now we turn to the proof of Corollary 1.4(b).Note that, if we apply Theorem 1.2(2)(b) to the exact pair ( l , µ ) , then we only get fin . dim ( S ) ≤ fin . dim ( R )+ fin . dim ( S ⋉ M ) + max { , 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 : L → G and g : G → L be ring homomorphisms such that f g = Id L . If fin . dim ( G G ⊗ L L − ) = s < + ¥ , then fin . dim ( L ) ≤ fin . dim ( G ) − s .To show ( ∗ ) , we set F : = G G ⊗ L L − : D ( L ) → D ( G ) . If fin . dim ( F ) = − ¥ , then ( ∗ ) is automatically true.So, we suppose that fin . dim ( F ) = s is an integer and L =
0. Since F ( L ) ≃ G =
0, we have s ≤
0. Let X ∈ P < ¥ ( L ) . Then there exists a finite projective resolution 0 → P n → · · · → P → P → X → L X with all P i in L -proj. Now we define Y : = W − s L ( X ) , the ( − s ) -th syzygy module of L X . Thus Y ∈ P < ¥ ( L ) .Since fin . dim ( F ) = s , we see that Tor L j ( G , Y ) = Tor L j − s ( G , X ) ≃ H s − j ( F ( X )) = j >
0. It followsthat G ⊗ L Y ∈ P < ¥ ( G ) and G ⊗ L W i L ( Y ) = W i G ( G ⊗ L Y ) ⊕ Q i for all i ≥
0, where all Q i are finitely generatedprojective G -modules. Further, we may suppose that fin . dim ( G ) = t < ¥ . Then proj . dim ( G G ⊗ L Y ) ≤ t , and22herefore G ⊗ L W t L ( Y ) = W t G ( G ⊗ L Y ) ⊕ Q t ∈ G -proj. Since f g = Id L , we have W t L ( Y ) ≃ L ⊗ G ( G ⊗ L W t L ( Y )) ∈ L -proj. Consequently,proj . dim ( L X ) ≤ proj . dim ( L Y ) − s ≤ proj . dim ( L W t L ( Y )) + t − s ≤ t − s . Thus fin . dim ( L ) ≤ fin . dim ( G ) − s . This finishes the proof of ( ∗ ) .Now, we take L : = S and G : = S ⋉ M . Let f : S → S ⋉ M and g : S ⋉ M → S be the canonical injectionand surjection, respectively. Clearly, we have f g = Id S . We assume S =
0. Then G G ⊗ L L L = G =
0, andfin . dim ( G G ⊗ L L − ) ≤
0. Suppose fin . dim ( R ) = m < ¥ . Due to ( ∗ ) , in order to show Corollary 1.4(b), we onlyneed to prove that fin . dim ( G G ⊗ L S − ) ≥ − m . This is equivalent to saying that Tor Sn ( G , X ) ≃ Tor Sn ( M , X ) = X ∈ P < ¥ ( S ) and for all n > m .To check the latter, we first prove that Tor Sj ( M , N ) ≃ Tor Rj ( M , N ) for any S -module N and for all j ≥ P • be a deleted projective resolution of the R op -module M . Since Tor Ri ( M , S ) = i ≥ P • ⊗ R S is a deleted projective resolution of the S op -module M ⊗ R S . Note that M ⊗ R S ≃ M as S op -modules since l : R → S is a ring epimorphism and M is an S op -module. It follows that P • ⊗ R S is adeleted projective resolution of the S op -module M . Since ( P • ⊗ R S ) ⊗ S N ≃ P • ⊗ R N as complexes, we haveTor Sj ( M , N ) ≃ Tor Rj ( M , N ) for all j ≥ 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 ) < ¥ . Hence proj . dim ( R X ) ≤ m = fin . dim ( R ) and Tor Sn ( M , X ) ≃ Tor Rn ( M , X ) = n > m . This implies that fin . dim ( G G ⊗ L S − ) ≥ − m . Now, by the result ( ∗ ) , we obtain fin . dim ( S ) ≤ fin . dim ( G ) + m = fin . dim ( S ⋉ M ) + fin . dim ( R ) . This completes the proof of Corollary 1.4(b). (cid:3)
We remark that the statement ( ∗ ) also implies that for any trivial extension of R by an R - R -bimodule M ,we always have fin . dim ( R ) ≤ fin . dim ( R ⋉ M ) + flat . dim ( M R ) . Proof of Corollary 1.5 :Let l : R → S : = R / I and µ : R → T : = R / I be the canonical surjective ring homomorphisms. Since I ∩ I =
0, we see that ( l , µ , R / ( I + I ) , ) is an exact context, where 1 is the identity of the ring R / ( I + I ) .Even more, since R is a pullback of the surjective maps R → R / I i over R / ( I + I ) , the pair ( l , µ ) is exact (forexample, see [8, Section 3]). So T ⊠ R S ≃ S ⊔ R T as rings. Note that S ⊔ R T = ( R / I ) ⊔ R ( R / I ) = R / ( I + I ) by Lemma 2.3(2). Thus T ⊠ R S ≃ R / ( I + I ) as rings.Now, we apply Theorem 1.2 to show Corollary 1.5. Clearly, it remains to check that if Tor Ri ( I , I ) = i ≥
0, then Tor Ri ( R / I , R / I ) = i >
0. In fact, for i >
2, we have Tor Ri ( R / I , R / I ) ≃ Tor Ri − ( I , I ) = R ( R / I , R / I ) ≃ ( I ∩ I ) / ( I I ) = R ( R / I , R / I ) ≃ Tor R ( I , R / I ) = Ker ( f ) where f : I ⊗ R I → I I is the multiplication map. Since I ⊗ R I =
0, we have Tor R ( R / I , R / I ) ≃ Ker ( f ) =
0. Thus Tor Ri ( R / I , R / I ) = i > (cid:3) Finally, we apply our results to exact contexts related to homological ring epimorphisms. First of all, weestablish a method to construct new homological ring epimorphisms from given ones.
Lemma 3.19.
Let l : R → S be a homological ring epimorphism. Suppose that I is an ideal of R such thatthe image J ′ of I under l is a left ideal in S and that the restriction of l to I is injective. Let J be the ideal ofS generated by J ′ . Then the following statements are equivalent: ( ) The homomorphism e l : R / I → S / J induced from l is homological. ( ) Tor R / Ii (cid:0) J / J ′ , S / J (cid:1) = for all i ≥ . ( ) The multiplication map I ⊗ R S → J is an isomorphism and
Tor Rj ( I , S ) = for all j ≥ . ( ) Tor Rj ( R / I , S ) = for all j ≥ .Let B : = (cid:18) S S / J ′ R / I (cid:19) . If one of the above statements holds true, then there is a recollement of derivedmodule categories: D ( S / J ) / / D ( B ) / / g g w w D ( R ) f f x x . roof. We take T : = R / I and choose µ : R → T to be the canonical surjective homomorphism of rings.Since J ′ is a left ideal of S , we have S ⊗ R T = S ⊗ R ( R / I ) ≃ S / ( S · I ) = S / J ′ . On the one hand, the pair ( l , µ ) is exact if and only if l | I : I → J ′ is an isomorphism. On the other hand, by Lemma 2.3(2), we see that S ⊔ R T = S ⊔ ( R / I ) = S / J with J = J ′ S , and that the ring homomorphism f : T → S ⊔ R T in [8, Theorem 1.1]can be chosen as the canonical map e l : R / I → S / J induced from l . Thus ( ) and ( ) are equivalent by [8,Theorem 1.1(1)]. Moreover, the recollement follows from [8, Theorem 1.1(2)].In the following, we shall show that ( ) and ( ) are equivalent.Applying the tensor functor − ⊗ R S to the exact sequence 0 → I → R → R / I →
0, we obtainTor R ( R / I , S ) ≃ Ker ( d ) and Tor Rj + ( R / I , S ) ≃ Tor Rj ( I , S ) for all j ≥ , where d : I ⊗ R S → J is the multiplication map defined by x ⊗ s ( x ) l s for x ∈ I and s ∈ S . Clearly, thisimplies that ( ) is equivalent to ( ) .Now we show that ( ) and ( ) are equivalent.According to Lemma 2.3(1) and the fact that l is a ring epimorphism, it follows that e l is a ring epimor-phism. By assumption, J ′ is a left ideal of S , and therefore S ⊗ R ( R / I ) ≃ S / ( S · I ) = S / J ′ . Thanks to thegeneral result proved in the last part of the proof of [8, Lemma 5.6], we seeTor R / Ii ( S / J ′ , W ) ≃ Tor R / Ii ( S ⊗ R ( R / I ) , W ) = i ≥ S / J -modules W . It then follows that Tor R / Ii ( S / J ′ , S / J ) = i ≥
1. Consider theshort exact sequence of right R / I -modules:0 −→ J / J ′ −→ S / J ′ −→ S / J −→ . If we apply the functor − ⊗ R / I ( S / J ) to this sequence, then Tor R / Ii ( J / J ′ , S / J ) ≃ Tor R / Ii + ( S / J , S / J ) for all i ≥ R / I ( S / J , S / J ) → ( J / J ′ ) ⊗ R / I ( S / J ) is injective.Clearly, if Tor R / I ( S / J , S / J ) =
0, then Tor R / Ij ( S / J , S / J ) = j ≥ R / Ii ( J / J ′ , S / J ) = i ≥
1. This will imply that ( ) and ( ) are equivalent. So it is enough to demonstrate thatTor R / I ( S / J , S / J ) = ( J / J ′ ) ⊗ R / I ( S / J ) = C → D is a ring epimorphism, then D ⊗ C X ≃ X as D -modules for any D -module X , and Y ⊗ C D ≃ Y as right D -modules for any right D -module Y . This fact, together with properties of ring epimorphisms,implies the following isomorphisms: ( J / J ′ ) ⊗ R / I ( S / J ) ≃ ( J / J ′ ) ⊗ R ( S / J ) ≃ ( J / J ′ ) ⊗ R (cid:0) S ⊗ R ( S / J ) (cid:1) ≃ (cid:0) ( J / J ′ ) ⊗ R S (cid:1) ⊗ R ( S / J ) . Since SJ ′ = J ′ and JJ ′ ⊆ J ′ , we deduce (cid:0) ( J / J ′ ) ⊗ R S (cid:1) J ′ =
0. This means that ( J / J ′ ) ⊗ R S is a right S / J -module.Clearly, the composite of the two ring epimorphisms R → S and S → S / J is again a ring epimorphism. Itfollows that (cid:0) ( J / J ′ ) ⊗ R S (cid:1) ⊗ R ( S / J ) ≃ ( J / J ′ ) ⊗ R S as right S / J -modules.In the following, we shall show ( J / J ′ ) ⊗ R S =
0. Actually, applying the functor − ⊗ R S to the exactsequence 0 −→ J ′ a −→ J −→ J / J ′ −→ R -modules, we get an exact sequence J ′ ⊗ R S a ⊗ R S −→ J ⊗ R S −→ ( J / J ′ ) ⊗ R S −→ S -modules. Since J is a right S -module and l : R → S is a ring epimorphism, the multiplicationmap y : J ⊗ R S → J , defined by x ⊗ s xs for x ∈ J and s ∈ S , is an isomorphism. Note that the map ( a ⊗ R S ) y : J ′ ⊗ R S → J is surjective. This yields that a ⊗ R S is surjective and ( J / J ′ ) ⊗ R S =
0. HenceTor R / I ( S / J , S / J ) =
0. This finishes the proof. (cid:3)
24 special case of Lemma 3.19 appears in trivial extensions. Let l : R → S be a homomorphism of ringsand M be an S - S -bimodule. Then l is homological if and only if e l : R ⋉ M → S ⋉ M is homological. Thenecessity of this condition follows from [8, Theorem 1.1(1)] and the proof of Corollary 1.4. The sufficiencycan be seen from Lemma 3.19.Applying Theorem 1.2 to the exact pair ( l , µ ) in the proof of Lemma 3.19, we obtain the followingestimations on finitistic dimensions, which can be applied to a class of examples of Milnor squares. Corollary 3.20.
Let l : R → S be a homological ring epimorphism. Suppose that I is an ideal of R such thatthe image J ′ of I under l is a left ideal in S and that the restriction of l to I is injective. Let J be the ideal ofS generated by J ′ . Suppose that one of the conditions (1)-(4) in Lemma 3.19 holds. Then ( ) fin . dim ( R ) ≤ fin . dim ( S ) + fin . dim ( R / I ) + max { , flat . dim (( R / I ) R ) } + . ( ) If R S ∈ P < ¥ ( R ) , then ( a ) fin . dim ( S ) ≤ fin . dim ( R ) and fin . dim ( S / J ) ≤ fin . dim ( R / I ) . ( b ) fin . dim ( B ) ≤ fin . dim ( R ) + fin . dim ( S / J ) + max { , proj . dim ( R S ) } + , where B : = (cid:18) S S / J ′ R / I (cid:19) . Acknowledgement.
The both authors would like to express heartfelt gratitude to Steffen Koenig for hiskind help with the presentation of the results in this paper. The research work of the corresponding authorCCX is partially supported by NSF (KZ201410028033).
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Invent. Math. (1992) 369-383.Hongxing Chen,School of Mathematical Sciences, BCMIIS, Capital Normal University, 100048 Beijing, People’s Republic of China
Email: [email protected]
Changchang Xi,School of Mathematical Sciences, BCMIIS, Capital Normal University, 100048 Beijing, People’s Republic of China