Standardly stratified lower triangular \mathbb{K}-algebras with enough idempotents
aa r X i v : . [ m a t h . R A ] J a n STANDARDLY STRATIFIED LOWER TRIANGULAR K -ALGEBRAS WITH ENOUGH IDEMPOTENTS E. MARCOS, O. MENDOZA, C. S ´AENZ AND V. SANTIAGO
Abstract.
In this paper we study the lower triangular matrix K -algebraΛ := (cid:2) T M U (cid:3) , where U and T are basic K -algebras with enough idempo-tents and M is an U - T -bimodule where K acts centrally. Moreover, wecharacterise in terms of U, T and M when, on one hand, the lower trian-gular matrix K -algebra Λ is standardly stratified in the sense of [13]; andon another hand, when Λ is locally bounded in the sense of [3]. Finally,it is also studied several properties relating the projective dimensionsin the categories of finitely generated modules mod( U ), mod( T ) andmod(Λ) . introduction The definitions of quasi-hereditary algebra and of standardly stratified al-gebra have been introduced in the context of Artin algebras or, more broadly,for semi-primary rings. Note that in these cases, all the rings involved haveunity and a finite number (up to isomorphisms) of simple modules. Quasi-hereditary algebras were introduce in [7] to deal with certain categories in therepresentation theory of Lie algebras and algebraic groups. They appear alsoin knot theory [17]. To see more details about standardly stratified algebras,we recommend the reader to see in [1], [8], [9], [15] and [18].In [13], the authors introduce the notion of standardly stratified ringoidwhich is a generalization of the classical notion of standardly stratified al-gebra for semi-primary rings with unity. Moreover, they study the modulecategory of standardly stratified ringoids (in particular, algebras with enoughidempotents) and show that, even though they have no unity and an infinitenumber (up to isomorphisms) of simple modules, many classic results aboutthe ∆- filtered representations of standardly stratified algebras can be gener-alized to representations of standardly stratified ringoids. It is also remarked
Key words and phrases. standardly stratified algebras, rings with enough idempotents,matrix algebras. . Primary 16G10, 16D90; Secondary 13E10.The authors thanks project PAPIIT-Universidad Nacional Aut´onoma de M´exico IN100520.The first mentioned author was supported by the thematic project of FAPESP 2014/09310-5, and research grant from CNPq 302003/2018-5 . that certain equivalent characterizations of standardly stratified algebras andquasi-hereditary algebras are not necessarily equivalent any more in the realmof ringoids. The theory developed in [13] is then specified to algebras (overcommutative rings) which have enough idempotents but need not have anunity. As a result, the notions of standardly stratified algebra and quasi-hereditary algebra are expanded in [13] to algebras which may not have anunity.In this paper we study the lower triangular matrix K -algebra Λ := [ T M U ] , where U and T are basic K -algebras with enough idempotents and M is an U - T -bimodule where K acts centrally. The motivation, for doing so, is togeneralize to the setting of [13] the results obtained by E. Marcos, H. Merklenand C. S´aenz in [11], where they studied the lower triangular matrix algebraΛ for U and T finite dimensional left standardly stratified algebras with unity.It is worth mentioning that some of the results in [11] where also obtained,independently, by B. Zhu for the matrix algebra Λ where U and T are quasi-hereditary algebras with unity [19].Throughout the paper, the term ring (algebra) means associative ring (al-gebra) which does not necessarily has a unity. We also fix a commutative ring K with unity and denote by f.ℓ. ( K ) the category of all the K -modules havingfinite length. Following [13], a K -algebra with enough idempotents (w.e.i K -algebra) is a pair ( R ; { e i } i ∈ I ) , where R is a K-algebra and { e i } i ∈ I is a familyof orthogonal idempotents of R such that R = ⊕ i ∈ I e i R = ⊕ i ∈ I Re i . For suchan algebra R, we denote by Mod ( R ) the category of unitary left R -modulesand by mod ( R ) the full subcategory of finitely generated left R -modules.It is said that a w.e.i. K -algebra ( R ; { e i } i ∈ I ) is basic if it is Hom-finite,each e i is primitive and the elements of the set { Re i } i ∈ I are pairwise non-isomorphic (for more details, see in Section 2). For such basic algebra R, eachpartition ˜ A of the set I induces (see Section 2) the set ˜ A ∆ of ˜ A -standardmodules. Thus, by Proposition 2.10, we have that ( R, ˜ A ) is a left standardlystratified K -algebra (in the sense of [13]) if, and only if, each Re i has a finite ˜ A ∆-filtration. We denote by F f ( ˜ A ∆) the class of all the left unitary R -modules which have a finite ˜ A ∆-filtration.Let us fix ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) basic w.e.i. K -algebras, an U - T -bimodule M (acting centrally on K ) such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) , ˜ A = { ˜ A i } i<α a partition of I, ˜ B = { ˜ B j } i<β a partition of J and ˜ C := ˜ A ∨ ˜ B apartition of the disjoint union I ∨ J (see Lemma 3.1). By Lemma 3.3, we havethat (Λ , { g r } I ∨ J ) is a basic w.e.i. K -algebra, where Λ is the lower triangularmatrix K -algebra [ T M U ] . The first main result of this paper is the following one, see the details inTheorem 3.9.
TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 3
Theorem A
For Λ = [ T M U ] , the following statements are equivalent.(a) (Λ , ˜ C ) is a left standardly stratified K -algebra.(b) { M f j } j ∈ J ⊆ F f ( ˜ A ∆) and the pairs ( T, ˜ B ) and ( U, ˜ A ) are left standardlystratified K -algebras.In [3], L. Angeleri H¨ugel and J. A. de la Pe˜na introduced locally bounded K -algebras and studied their category of finitely generated modules. We usethe developed theory in [3] to study mod (Λ) in terms of mod ( T ) and mod ( U ) . In Section 3, we give enough conditions to have that Λ , T and U are locallybounded, for more details see Proposition 4.9.In order to state the next two main results of this paper, we introducethe following notation. Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra. We denoteby P < ∞ R the class of all the N ∈ mod ( R ) with pd( N ) < ∞ , where pd( N )stands for the projective dimension of N. For a class
C ⊆
Mod ( R ) , we setpd ( C ) := sup { pd( C ) : C ∈ C} . The small global dimension of the ring R isgl . dim . ( R ) := pd (mod ( R )) and the small finitistic dimension of the ring R isfin . dim . ( R ) := pd ( P < ∞ R ) . The second main result of this paper is the following one, see the details inTheorem 4.14.
Theorem B
For Λ = [ T M U ] support finite, the following statements holdtrue.(a) Let L = ( Y, ϕ, X ) ∈ Mod (Λ) and pd(
M f j ) < ∞ ∀ j ∈ J. Then L ∈ P < ∞ Λ ⇔ Y ∈ P < ∞ T and X ∈ P < ∞ U . (b) Let m := max { pd( M f j ) } j ∈ J < ∞ , a := fin . dim . ( T ) , α := gl . dim . ( T ) ,b := fin . dim . ( U ) and β := gl . dim . ( U ) . Thenmax { β − m, α } ≤ gl . dim . (Λ) ≤ max { β, α + 1 + m } , max { b − m, a } ≤ fin . dim . (Λ) ≤ max { b, a + 1 + m } . The third main result of this paper is the following one, see the details inTheorem 4.15.
Theorem C
For Λ = [ T M U ] support finite and pd(
M f j ) < ∞ ∀ j ∈ J, thefollowing statements are equivalent.(a) F f ( ˜ C ∆) = P < ∞ Λ . (b) F f ( ˜ A ∆) = P < ∞ U and F f ( ˜ B ∆) = P < ∞ T . Moreover, if one of the above equivalent conditions holds true, then Λ , U and T are locally bounded and left standardly stratified K -algebras, fin . dim . (Λ) =pd( ˜ C ∆) , fin . dim . ( T ) = pd( ˜ B ∆) and fin . dim . ( U ) = pd( ˜ A ∆) . Preliminaries
In what follows, we recall some basic notations and results appearing in[13].
Functor categories and ringoids.
Let C be a category. It is said that C is a K -category if Hom C ( X, Y ) is a K -module for any ( X, Y ) ∈ C and the E. MARCOS, O. MENDOZA, C. S ´AENZ AND V. SANTIAGO composition of morphisms in C is K -bilinear. Following B. Mitchell in [12], a K -ringoid is just a skeletally small K -category. In case K = Z , we name it ringoid instead of Z -ringoid.Let R be a ringoid and Ab be the category of abelian groups. We denoteby Mod ( R ) the category of left R -modules whose objects are all the additive(covariant) functors M : R → Ab and morphisms are the natural transforma-tions of functors. Note that Mod ( R ) is abelian and bicomplete since Ab is so.Denote by R op the opposite category of R . The category of right R -modulesis by definition Mod ( R op ) . We denote by mod ( R ) the full subcategory ofMod ( R ) whose objects are all the finitely generated left R -modules; and byproj( R ) the class of all the finitely generated projective left R -modules.Let R be a K -ringoid. Following [13], We recall that M ∈ Mod ( R ) is finitely presented if there is an exact sequence P → P → M → R -modules such that P , P ∈ proj( R ) . We denote by fin . p . ( R ) the full sub-category of Mod ( R ) whose objects are all the finitely presented R -modules.The K -ringoid R is thick if it is an additive category whose idempotentssplit. It is said that R is Krull-Schmidt ( KS-ringoid , for short) if R is aKrull-Schmidt category (that is an additive category in which every non-zeroobject decomposes into a finite coproduct of objects having local endomor-phism ring). We say that R is Hom-finite if the K -module Hom R ( a, b ) is offinite length, for all ( a, b ) ∈ R . Finally, a K -ringoid which is Hom-finite andKrull-Schmidt is called locally finite K -ringoid. Algebras with enough idempotents.
Let R be a K -algebra such that R = R. We denote by MOD( R ) the category of all the left R -modules and letMod( R ) be the full subcategory of all the unitary left R -modules M, whereunitary means that RM = M. The full subcategory of Mod( R ) whose objectsare the finitely generated left R -modules will be denoted by mod( R ) . Theclass of all the finitely generated projective objects in Mod( R ) is denoted byproj( R ) . Following [13], we recall that a K -algebra with enough idempotents ( w.e.i K -algebra ) is a pair ( R, { e i } i ∈ I ) , where R is a K -algebra and { e i } i ∈ I is afamily of orthogonal idempotents of R such that R = ⊕ i ∈ I e i R = ⊕ i ∈ I Re i . In this case, it can be shown that R = R and R = ⊕ ( i,j ) ∈ I e i R e j . A w.e.i K -algebra ( R, { e i } i ∈ I ) is Hom-finite if { e j Re i } ( i,j ) ∈ I ⊆ f.ℓ. ( K ) . Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra and let R op be the opposite K -algebraof R. Note that ( R op , { e i } i ∈ I ) is also a w.e.i. K -algebra and it is known asthe opposite w.e.i K -algebra of ( R, { e i } i ∈ I ) . Definition 2.1.
Let ( R, { e i } i ∈ I ) and ( T, { f j } j ∈ j ) be w.e.i. K -algebras. Wesay that φ : ( R, { e i } i ∈ I ) → ( T, { f j } j ∈ j ) is a morphism of w.e.i. K -algebras if φ : R → T is a morphism of K -algebras such that φ ( { e i } i ∈ I ) ⊆ { f j } j ∈ j . Let φ : ( R, { e i } i ∈ I ) → ( T, { f j } j ∈ j ) be a morphism of w.e.i. K -algebras.Consider M ∈ Mod ( T ) . Note that M ∈ MOD( R ) , via the action of R on M TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 5 r · m := φ ( r ) m ∀ r ∈ R, ∀ m ∈ M. Unfortunately, M does not necessarily belong to Mod( R ) . In order to con-struct the so called change of rings functor φ ∗ : Mod( T ) → Mod( R ) , weneed to do some adjustment in the above construction. Indeed, for M ∈ Mod ( T ) , we set φ ∗ ( M ) := P i ∈ I φ ( e i ) M ; and for a morphism f : M → N inMod( T ) , we define φ ∗ ( f ) as the restriction of f on φ ∗ ( M ) . Proposition 2.2.
Let φ : ( R, { e i } i ∈ I ) → ( T, { f j } j ∈ j ) be a morphism of w.e.i. K -algebras. Then, the following statements hold true. (a) The above correspondence φ ∗ : Mod( T ) → Mod( R ) is a well defined K -linear functor. (b) For any M ∈ Mod ( T ) , the correspondence ε M : a i ∈ I Hom T ( T φ ( e i ) , M ) → φ ∗ ( M ) , ( θ i ) i ∈ I X i ∈ I θ i ( φ ( e i )) is an isomorphism of R -modules which is functorial on M. (c) φ ∗ : Mod( T ) → Mod( R ) is an exact functor.Proof. (a) Let M ∈ Mod( T ) . We start by showing that the abelian group φ ∗ ( M ) is an R -submodule of M. Indeed, consider r = P i ∈ I e i r i ∈ R = P i ∈ I e i R and x = P j ∈ I φ ( e j ) x j ∈ φ ∗ ( M ) = P j ∈ I φ ( e j ) M. Then, r · x = φ ( r ) x = φ ( X i ∈ I e i r i ) X j ∈ I φ ( e j ) x j = X i ∈ I φ ( e i )( X j ∈ I φ ( r i e j ) x j ) ∈ φ ∗ ( M )since P j ∈ I φ ( r i e j ) x j ∈ M ∀ i. We check now that Rφ ∗ ( M ) = φ ∗ ( M ) . Let x = P i ∈ I φ ( e i ) x i ∈ φ ∗ ( M ) . Then, by using that φ ( e i ) x i ∈ φ ∗ ( M ) ∀ i, we get that x = X i ∈ I φ ( e i ) x i = X i ∈ I φ ( e i )( φ ( e i ) x i ) = X i ∈ I e i · ( φ ( e i ) x i ) ∈ Rφ ∗ ( M ) . Therefore, we proved that φ ∗ ( M ) ∈ Mod( R ) . Let f : M → N be a morphism in Mod( T ) . We need to show that f ( φ ∗ ( M )) ⊆ φ ∗ ( N ) . Let x = P i ∈ I φ ( e i ) x i ∈ φ ∗ ( M ) . Then, f ( x ) = X i ∈ I f ( e i · x i ) = X i ∈ I e i · f ( x i ) = X i ∈ I φ ( e i ) f ( x i ) ∈ φ ∗ ( N ) . Thus the correspondence φ ∗ : Mod( T ) → Mod( R ) is well defined and we letthe reader to check that it is a K -linear functor.(b) Let M ∈ Mod( T ) and i ∈ I. Then e i · φ ∗ ( M ) = X i ′ ∈ I φ ( e i ) φ ( e i ′ ) M = φ ( e i ) M. Thus, for each i ∈ I, ε iM : Hom T ( T φ ( e i ) , M ) → e i φ ∗ ( M ) , f f ( φ ( e i )) , isan isomorphism of K -modules, which is functorial on M, and Hom T ( T φ ( e i ) , M ) E. MARCOS, O. MENDOZA, C. S ´AENZ AND V. SANTIAGO is an R -module since M ∈ MOD( R ) . Since ε M = ` i ∈ I ε iM and ε M is a mor-phism of R -modules, we get (b).(c) It follows from (b) since T φ ( e i ) is projective in Mod( T ) , for all i ∈ I. (cid:3) Definition 2.3. [13, Remark 6.8]
It is said that a w.e.i. K -algebra ( R, { e i } i ∈ I ) is basic if it is Hom-finite, each e i is primitive and Re i Re j for e i = e j . Remark 2.4.
Let ( R, { e i } i ∈ I ) be a Hom-finite w.e.i. K -algebra. Note that,from [13, Lemma 6.3] , we have that ( R, { e i } i ∈ I ) is basic if, and only if, each e i is primitive and e i R e j R for e i = e j . The following result will be used frequently. For the convenience of thereader, we give a proof.
Lemma 2.5.
Let ( R, { e i } i ∈ I ) be a w.e.i K -algebra and M ∈ Mod ( R ) . Then M = L i ∈ I e i M. Proof.
Since RM = M and R = ⊕ i ∈ I e i R, we have that M = P i ∈ I e i M. Let us show that e i M ∩ (cid:0) P j = i e j M (cid:1) = 0 for i = j. Indeed, let x ∈ e i M ∩ (cid:0) P j = i e j M (cid:1) . Then x = e i m = P j ∈ I −{ i } e j m j with m, m j ∈ M and hence x = e i m = e i m = P j ∈ I −{ i } e i e j m j = 0; proving the result. (cid:3) For a given w.e.i K -algebra ( R, { e i } i ∈ I ) , we have its associated K -ringoid R ( R ) whose objects are { e i } i ∈ I and the morphisms from e i to e j are given bythe set Hom R ( R ) ( e i , e j ) := e j Re i . The composition of morphisms in R ( R ) isgiven by the multiplication of the ring R. Note that R ( R ) op = R ( R op ) . We recall that Y : R (Λ) → Mod( R (Λ)) is the Yoneda’s contravariantfunctor, where Y ( e ) := Hom R (Λ) ( e, − ) for any e ∈ R . The following result is well-known in the mathematical folklore, see [13,Proposition 6.1] for a proof.
Proposition 2.6.
Let (Λ , { e i } i ∈ I ) be a w.e.i. K -algebra. Then, the functor δ : Mod( R (Λ)) → Mod(Λ) , M
7→ ⊕ i ∈ I M ( e i ) , is an isomorphism of categories and δ ( Y ( e i )) = Λ e i , for any i ∈ I. Let (Λ , { e i } i ∈ I ) be a w.e.i K -algebra and Θ be a class of R -modules inMod ( R ) . We denote by F f (Θ) the full subcategory of Mod ( R ) whose objectshave a finite Θ-filtration. That is, M ∈ F f (Θ) if there exists a finite chain0 = M ⊆ M ⊆ · · · ⊆ M t − ⊆ M t = M of submodules of M such that each factor M i /M i − is isomorphic to someobject in Θ . The modules in F f (Θ) are called Θ- good modules and the class F f (Θ) is known as the category of Θ-good modules.Given a set X ⊆
Mod ( R ) and M ∈ Mod ( R ) , we recall that the trace of X in M is the submodule Tr X ( M ) := X { f ∈ Hom(
X,M ) | X ∈X } Im( f ) . TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 7
In what follows, we define the notions of standard module and standardlystratified K -algebra. Since we are interested only in basic w.e.i. K -algebras,by taking into account the general definition of [13], these notions are givenbelow only for this class of algebras .Let ( R, { e i } i ∈ I ) be a basic w.e.i K -algebra. Then, by [13, Corollary 6.6(c)],the set of finitely generated indecomposable projective R -modules (up to iso-morphisms) ind proj( R ) is given by the set { Re i } i ∈ I . For an ordinal number α, choose a partition (of size α ) ˜ A = { ˜ A i } i<α of the set I. Define, for any t ∈ ˜ A i , the R -module P t ( i ) = ˜ A P t ( i ) := Re t which is known as the t -thindecomposable projective laying in the i -th level of the partition ˜ A . Let P = ˜ A P := { P ( i ) } i<α , where P ( i ) = ˜ A P ( i ) := { P t ( i ) } t ∈ ˜ A i . We say that ˜ A P is the ˜ A -arrangement, associated with the partition ˜ A , of the set ind proj( R ) . We define the family of ˜ A -standard left R -modules ∆ = ˜ A ∆ := { ∆( i ) } i<α , where ∆( i ) = ˜ A ∆( i ) := { ∆ t ( i ) } t ∈ ˜ A i is defined as follows∆ t ( i ) = ˜ A ∆ t ( i ) := P t ( i )Tr ⊕ j
With the notation above, we have that theclass F f ( ˜ A ∆) is closed under extensions in Mod ( R ) . Definition 2.8. [13, Definition 6.11]
Let ( R, { e i } i ∈ I ) be a basic w.e.i K -algebra. We say that the pair ( R, ˜ A ) is a left standardly stratified K -algebra if ˜ A = { ˜ A i } i<α is a partition of the set I such that Tr ⊕ j
For a basic w.e.i K -algebra ( R, { e i } i ∈ I ) and a partition ˜ A = { ˜ A i } i<α of the set I such that ( R, ˜ A ) is a left standardly stratified K -algebra, the following statements hold true. (a) End R ( ˜ A ∆ t ( i ))) is a local ring, for each i < α and t ∈ ˜ A i . (b) For any M ∈ F f ( ˜ A ∆) , the filtration multiplicity [ M : ˜ A ∆ t ( i ))] doesnot depend on a given ˜ A ∆ -filtration of M. (c) F f ( ˜ A ∆) ⊆ fin . p . ( R ) and it is a locally finite K -ringoid. (d) F f ( ˜ A ∆) is closed under kernels of epimorphisms in Mod ( R ) . Proof.
For the proof of (a), (b) and (c), see [13, Corollary 6.12]. Finally, forthe proof of (d) use [13, Remark 6.11, Proposition 4.12]. (cid:3)
The following is a very useful criterion in order to see that a basic w.e.i K -algebra is standardly stratified. Proposition 2.10.
For a basic w.e.i K -algebra ( R, { e i } i ∈ I ) and a partition ˜ A = { ˜ A i } i<α of the set I, the following statements are equivalent. E. MARCOS, O. MENDOZA, C. S ´AENZ AND V. SANTIAGO (a) ( R, ˜ A ) is a left standardly stratified K -algebra. (b) ˜ A P t ( i ) ∈ F f ( ˜ A ∆) for all i < α and t ∈ ˜ A i . Proof. (a) ⇒ (b): It follows from Remark 2.7 and the definition of the ˜ A -standard left R -modules.(b) ⇒ (a): Let i < α, t ∈ ˜ A i and M := ˜ A P t ( i ) ∈ F ( ˜ A ∆) . By following [13,Definition 4.2], for each l < α, we have the additive pre-radical functors τ l ( − ) := Tr ⊕ j ≤ l ˜ A P ( j ) ( − ) and τ l ( − ) := Tr ⊕ j
Let ( R, { e i } i ∈ I ) be a Hom-finite w.e.i. K -algebra, ˜ A = { ˜ A i } i<α be a partition of the set ind { e i } i ∈ I (as in the general setting of [13, Defini-tion 6.10] ) and let ˜ A ∆ be the ˜ A -standard left R -modules. Then, the proof ofProposition 2.10 allow us to conclude that the following statements are true. (a) ( R, ˜ A ) is a left standardly stratified K -algebra if, and only if, ˜ A P t ( i ) ∈F f ( ˜ A ∆) for all i < α and t ∈ ˜ A i . TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 9 (b) Tr ⊕ j Let I and J be non-empty sets. Thedisjoint union of I and J is the set I ∨ J := ( I × { } ) ∪ ( J × { } ) . Then, wehave two bijective functions p : I × { } → I, ( i, i, and p : J × { } → J, ( j, j. For two ordinal numbers α and β, we consider the intervals ofordinal numbers [0 , α ) and [0 , β ) . In the set [0 , α ) ∨ [0 , β ) , we consider anorder ≤ in which every element of [0 , α ) × { } is smaller than any element in[0 , β ) × { } . Thus, the ordinal type of the well-ordered set ([0 , α ) ∨ [0 , β ) , ≤ )is α + β. We denote by ℘ ( I ) the set of all the partitions ˜ A = { ˜ A i } i<α of the set I, where α is an ordinal number. The joint partition function ∨ : ℘ ( I ) × ℘ ( J ) → ℘ ( I ∨ J ) , ( ˜ A , ˜ B ) ˜ A ∨ ˜ B is constructed as follows. For ˜ A = { ˜ A i } i<α ∈ ℘ ( I ) and ˜ B = { ˜ B j } j<β ∈ ℘ ( J ) , we set ˜ A ∨ ˜ B = { ( ˜ A ∨ ˜ B ) t } t ∈ [0 ,α ) ∨ [0 ,β ) , where( ˜ A ∨ ˜ B ) t := ( ˜ A i × { } if t = ( i, , ˜ B i × { } if t = ( j, . We also have the splitting partition function spl : ℘ ( I ∨ J ) → ℘ ( I ) × ℘ ( J ) , ˜ C 7→ (spl ( ˜ C ) , spl ( ˜ C )) , which is constructed, by using the given above bijections p : I × { } → I and p : J × { } → J, as follows:spl ( ˜ C ) := p ( ˜ C ∩ ( I × { } ) and spl ( ˜ C ) := p ( ˜ C ∩ ( J × { } ) . Lemma 3.1. For any non-empty sets I and J, the joint partition function ∨ : ℘ ( I ) × ℘ ( J ) → ℘ ( I ∨ J ) is a bijection whose inverse is the splitting partitionfunction spl : ℘ ( I ∨ J ) → ℘ ( I ) × ℘ ( J ) . Proof. It is straightforward from the definitions involved. (cid:3) Notation used for lower triangular matrices. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be w.e.i. K -algebras and M be a U - T -bimodule acting centrally on K . We fix the following notation:(1) the lower triangular matrix K -algebraΛ := [ T M U ] := { [ t m u ] : t ∈ T, u ∈ U, m ∈ M } with sum and product defined as follows (a) [ t m u ] + (cid:2) t ′ m ′ u ′ (cid:3) = h t + t ′ m + m ′ u + u ′ i , (b) [ t m u ] (cid:2) t ′ m ′ u ′ (cid:3) = h tt ′ mt ′ + um ′ uu ′ i ;(2) e i := (cid:2) e i (cid:3) ∈ Λ , for each i ∈ I ;(3) f j = (cid:2) f j 00 0 (cid:3) ∈ Λ , for each j ∈ J ;(4) the family { g r } r ∈ I ∨ J in Λ , where g r = ( e i if r = ( i, ,f j if r = ( j, . By using the notation above, we can show the following lemma. Lemma 3.2. For the lower triangular matrix K -algebra Λ , the following equal-ities hold true. (a) e i Λ = (cid:2) e i M e i U (cid:3) and f j Λ = (cid:2) f j T 00 0 (cid:3) ∀ i ∈ I and ∀ j ∈ J. (b) Λ e i = (cid:2) Ue i (cid:3) and Λ f j = h T f j Mf j i ∀ i ∈ I and ∀ j ∈ J. (c) For i, i ′ ∈ I and j, j ′ ∈ J, we have that f j Λ e i = 0 and e i Λ e i ′ = (cid:2) e i Ue i ′ (cid:3) , e i Λ f j = (cid:2) e i Mf j (cid:3) , f j Λ f j ′ = h f j T f j ′ 00 0 i . (d) Hom Λ (Λ f j , Λ e i ) = 0 ∀ ( i, j ) ∈ I × J. Proof. The proof of (a), (b) and (c) is a straightforward calculation by usingmatrix operations. Finally, (d) follows from (c) since Hom Λ (Λ f j , Λ e i ) ≃ f j Λ e i as K -modules. (cid:3) Basic properties of the triangular algebra Λ . We start by proving the following fundamental lemma. Lemma 3.3. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebrasand M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) . Then, thelower triangular matrix K -algebra (Λ , { g r } r ∈ I ∨ J ) is basic and with enoughidempotents.Proof. It is clear that { g r } r ∈ I ∨ J is a family of orthogonal idempotents of Λ . Then, in order to show that (Λ , { g r } r ∈ I ∨ J ) is a w.e.i. K -algebra, we need toprove that P r ∈ I ∨ J g r Λ = Λ = P r ∈ I ∨ J Λ g r . Indeed, by Lemma 3.2 (a) andLemma 2.5, we get the equalities X r ∈ I ∨ J g r Λ = X i ∈ I (cid:2) e i M e i U (cid:3) + X j ∈ J (cid:2) f j T 00 0 (cid:3) = [ T M U ] = Λ . Analogously, we can show that Λ = P r ∈ I ∨ J Λ g r . Note that Lemma 3.2 (c) implies us that (Λ , { g r } r ∈ I ∨ J ) is Hom-finite. Letus prove now that it is also basic. Indeed, by Lemma 3.2 (c), we get the ringisomorphisms e i Λ e i ≃ e i U e i and f j Λ f j ≃ f j T f j ∀ i ∈ I, ∀ j ∈ L ; and thus TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 11 each idempotent g r is primitive. To finish the proof, we need to show that { Λ g r } r ∈ I ∨ J are pairwise non isomorphic Λ-modules. In order to do that, weneed to consider only the following two cases since, from Lemma 3.2 (d), it isclear that Λ e i Λ f j ∀ i ∈ I, ∀ j ∈ J. Case 1: Let i, i ′ ∈ I be such that i = i ′ . Suppose that Λ e i ≃ Λ e i ′ asΛ-modules. Then, by Lemma 3.2 (b), we know that Λ e i = (cid:2) Ue i (cid:3) , andthus there is an isomorphism U e i ≃ U e i ′ of U -modules, contradicting that( U, { e i } i ∈ I ) is basic.Case 2: Let j, j ′ ∈ J be such that j = j ′ . Suppose that Λ f j ≃ Λ f j ′ asΛ-modules. Then, by [13, Lemma 6.3], we get that f j Λ ≃ f j ′ Λ as Λ-modules.On the other hand, by Lemma 3.2 (a), we know that f j Λ = (cid:2) f j T 00 0 (cid:3) andthus there is an isomorphism f j T ≃ f j ′ T of T -modules, contradicting that( T, { e i } i ∈ I ) is basic (see Remark 2.4 (1)). (cid:3) Let A be an abelian category and X , Y be classes of objects in A . Weconsider the class X ⋆ Y whose objects are all the objects M ∈ A appearing inthe middle term of an exact sequence 0 → X → M → Y → , for some X ∈ X and Y ∈ Y . We recall the following result due to Ringel [14, Theorem] in therealm of finitely generated modules over Artin algebras. For the convenienceof the reader we include a proof. Lemma 3.4. Let A be an abelian category and X , Y be classes of objects in A which are closed under extensions and Ext A ( X , Y ) = 0 . Then, the class X ∗ Y is closed under extensions.Proof. Let 0 → L f −→ M g −→ N → A , with L, N ∈X ∗ Y . Then, we have exact sequences 0 → L α −→ L → L → → N β −→ N → N → , with L , N ∈ X and L , N ∈ Y . Thus, we have thepullback-pushout diagram 0 (cid:15) (cid:15) (cid:15) (cid:15) / / L f ′ / / M ′ g ′ / / h (cid:15) (cid:15) N β (cid:15) (cid:15) / / / / L f / / M (cid:15) (cid:15) g / / N (cid:15) (cid:15) / / N (cid:15) (cid:15) N (cid:15) (cid:15) and the following commutative diagram0 / / L α (cid:15) (cid:15) f ′ α / / M ′ γ / / N ′ β ′ (cid:15) (cid:15) / / / / L f ′ / / M ′ g ′ / / N / / . By snake lemma, Ker( β ′ ) = Coker( α ) = L . Then we have exact sequence η : 0 / / L / / N ′ β ′ / / N / / . Note that η splits since Ext A ( X , Y ) = 0 . Thus, we get the exact sequence0 / / N θ / / N ′ ψ / / L / / (cid:15) (cid:15) (cid:15) (cid:15) / / L / / M ′′ / / θ ′ (cid:15) (cid:15) N θ (cid:15) (cid:15) / / / / L f ′ α / / M ′ γ / / (cid:15) (cid:15) N ′ / / (cid:15) (cid:15) L (cid:15) (cid:15) L (cid:15) (cid:15) L , N ∈ X , we have that M ′′ ∈ X . Now, we consider the commutativediagram 0 / / M ′′ θ ′ / / M ′ h (cid:15) (cid:15) / / L / / (cid:15) (cid:15) / / M ′′ hθ ′ / / M / / Z / / TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 13 (cid:15) (cid:15) (cid:15) (cid:15) / / M ′′ θ ′ / / M ′ h (cid:15) (cid:15) / / L / / (cid:15) (cid:15) / / M ′′ hθ ′ / / M / / (cid:15) (cid:15) Z / / (cid:15) (cid:15) N (cid:15) (cid:15) N (cid:15) (cid:15) L , N ∈ Y , we have that Z ∈ Y . Therefore, the exact sequence0 / / M ′′ hθ ′ / / M / / Z / / M ∈ X ∗ Y , proving the result. (cid:3) Description of the modules in lower triangular matrix rings In order to give a description of the modules over a lower triangular matrix K -algebra, we need some results and notions from [10]. So in what follows,we collect them.Let U and T be ringoids and the bimodule M ∈ Mod ( U ⊗ T op ) . The triangular matrix ringoid Λ = [ T M U ] is defined as follows [10, Definition3.4]. The objects of Λ are matrices of the form [ T M U ] , with T ∈ T and U ∈ U . Given a pair of objects in [ T M U ] , (cid:2) T ′ M U ′ (cid:3) in Λ , the set of morphisms isHom Λ (cid:0) [ T M U ] , (cid:2) T ′ M U ′ (cid:3)(cid:1) := h Hom T ( T,T ′ ) 0 M ( U ′ ,T ) Hom U ( U,U ′ ) i . For a more detailed description of this matrix ringoid, we recommend thereader to see [10]. The bimodule M ∈ Mod ( U ⊗ T op ) induces a functor E : T op → Mod ( U ) , T M ( − , T ) . Moreover, by considering the Yonedafunctor Y : T op → Mod ( T ) , T Hom T ( T, − ) , there is a unique functor F : Mod ( T ) → Mod ( U ) with commutes with direct limits and such that F ◦ Y = E [10, section 5]. This functor F is called the tensor product and itis usually denoted by M ⊗ T − . Theorem 3.5. [10, Theorem 3.14, Proposition 5.3 ] Let U and T be ringoidsand M ∈ Mod ( U ⊗ T op ) . Then, there exists an equivalence of categories Mod (cid:16) [ T M U ] (cid:17) ≃ (cid:16) F (Mod( T )) , Mod( U ) (cid:17) . We recall that (cid:16) F (Mod( T )) , Mod( U ) (cid:17) is the so called comma category. Theobjects of this category are triples ( A, g, B ) with A ∈ Mod( T ) , B ∈ Mod( U )and g : M ⊗ T A → B a morphism of U -modules. A morphism between twoobjects ( A, g, B ) and ( A ′ , g ′ , B ′ ) is a pair of morphism ( α, β ) , where α : A → A ′ is a morphism of T -modules and β : B → B ′ is a morphism of U -modulesand such that the following diagram commutes M ⊗ T A M ⊗ T A ′ B B ′ . / / M ⊗ T α (cid:15) (cid:15) g (cid:15) (cid:15) g ′ / / β Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras and M bean U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) . By Lemma 3.3, weknow that the lower triangular K -algebra Λ = [ T M U ] is basic and with enoughidempotents { g r } r ∈ I ∨ J . We can construct the ringoids R ( U ), R ( T ) and thebifunctor M : R ( U ) ⊗ R ( T ) op → Ab , ( e i , f j ) e i M f j . Note that the ringoid R (Λ) is isomorphic to h R ( T ) 0 M R ( U ) i and thus it can be identified one with theother. Then, by Proposition 2.6 and Theorem 3.5, we have the equivalencesof categoriesMod (Λ) ≃ Mod (cid:16) h R ( T ) 0 M R ( U ) i (cid:17) ≃ (cid:16) F (Mod ( R ( T )) , Mod ( R ( U )) (cid:17) . It can be also shown that the following diagram commutesMod ( R ( T )) M ⊗ R ( T ) − / / (cid:15) (cid:15) Mod( R ( U )) (cid:15) (cid:15) Mod ( T ) M ⊗ T − / / Mod ( U ) , where the vertical arrows are the isomorphisms of categories given by Proposi-tion 2.6. Therefore, the category Mod (Λ) is equivalent to the comma category( M ⊗ T (Mod ( T )) , Mod ( U )) , and thus, each Λ-module L can be seen as triple( Y, ϕ, X ) , where Y ∈ Mod ( T ) , X ∈ Mod ( U ) and ϕ : M ⊗ T Y → X is amorphism of U -modules.For the lower triangular matrix K -algebra Λ = [ T M U ] , we have the canon-ical injective morphisms of K -algebras i U : U → [ T M U ] , u [ u ] , and i T : T → [ T M U ] , t [ t 00 0 ] ;and the canonical surjective morphisms of K -algebras p U : [ T M U ] → U, [ t m u ] u, and p T : [ T M U ] → T, [ t m u ] t. Then, by Proposition 2.2, we get the K -linear exact functors, TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 15 ( i U ) ∗ : Mod (Λ) → Mod ( U ) , ( i T ) ∗ : Mod (Λ) → Mod ( T ) , ( p U ) ∗ : Mod ( U ) → Mod (Λ) , ( p T ) ∗ : Mod ( T ) → Mod (Λ) . Proposition 3.6. For the lower triangular matrix K -algebra Λ = [ T M U ] , thefollowing statements hold true. (a) Let L ∈ Mod (Λ) . Then, its corresponding triple ( Y, ϕ, X ) in thecomma category ( M ⊗ T (Mod ( T )) , Mod ( U )) ≃ Mod (Λ) is the fol-lowing: Y = ( i T ) ∗ ( L ) , X = ( i U ) ∗ ( L ) and ϕ is obtained (by usingthe universal property of the tensor product) from the Z -bilinear and T -balanced map M × Y → X, ( m, y ) [ m ] y. (b) The sequence of Λ -modules / / ( Y , ϕ , X ) ( α ,α ) / / ( Y, ϕ, X ) ( β ,β ) / / ( Y , ϕ , X ) / / is exact in Mod (Λ) if, and only if, the following two sequences / / X α / / X β / / X / / , / / Y α / / Y β / / Y / / are exact, respectively, in Mod ( U ) and Mod ( T ) . (c) For the change of rings functors, described above, we have ( i T ) ∗ ◦ ( p T ) ∗ = 1 Mod ( T ) , ( i U ) ∗ ◦ ( p U ) ∗ = 1 Mod ( U ) and ( i U ) ∗ ◦ ( p T ) ∗ = 0 = ( i T ) ∗ ◦ ( p U ) ∗ . (d) ( p T ) ∗ (Mod ( T )) = { ( Y, , 0) : Y ∈ Mod ( T ) } . Moreover, if Y isclosed under extensions in Mod( T ) , then ( p T ) ∗ ( Y ) is closed underextensions in Mod (Λ) . (e) ( p U ) ∗ (Mod ( U )) = { (0 , , X ) : X ∈ Mod ( U ) } . Moreover, if X isclosed under extensions in Mod ( U ) , then ( p U ) ∗ ( X ) is closed underextensions in Mod (Λ) . (f) Ext (( p U ) ∗ (Mod ( U )) , ( p T ) ∗ (Mod ( T )) = 0 . (g) Let Y be closed under extensions in Mod( T ) and X be closed underextensions in Mod ( U ) . Then, the class ( p U ) ∗ ( X ) ⋆ ( p T ) ∗ ( Y ) is closedunder extensions in Mod (Λ) . Proof. (a) It follows from [10, Lemma 3.12, Lemma 3.13] and Proposition2.6.(b) It follows from [10, Proposition 5.3 and Proposition 5.3].(c) Let Y ∈ Mod ( T ) . Then, as an abelian group, we have( p T ) ∗ ( Y ) = X r ∈ I ∨ J p T ( g r ) Y = X j ∈ J f j Y = Y and as a Λ-module [ t m u ] · y = ty ∀ y ∈ Y. Moreover, as abelian group( i T ) ∗ (( p T ) ∗ ( Y )) = X j ∈ J f j · Y = X j ∈ J f j Y = Y. Furthermore, the structure of T -module on ( i T ) ∗ (( p T ) ∗ ( Y )) coincide with theone on Y, and thus ( i T ) ∗ ◦ ( p T ) ∗ = 1 Mod ( T ) . On the other hand,( i U ) ∗ (( p T ) ∗ ( Y )) = X i ∈ I e i · Y = 0;proving that ( i U ) ∗ ◦ ( p T ) ∗ = 0 . In a similar way, we can check the otherequalities of (c).(d) From (a) and (c), we get the equality in (d). Let Y be closed underextensions on Mod( T ) . Consider an exact sequence of Λ-modules0 / / ( Y , , ( α ,α ) / / ( Y, f, X ) ( β ,β ) / / ( Y , , / / , where Y , Y ∈ Y . Then by (b), and since Y is closed under extensions, weconclude that Y ∈ Y and X = 0 . Therefore ( Y, f, X ) ∈ ( p T ) ∗ ( Y ) . (e) It follows as in the proof of (d).(f) Consider the exact sequence of Λ-modules η : 0 / / ( Y, , ( α ,α ) / / ( Y ′ , ϕ ′ , X ′ ) ( β ,β ) / / (0 , , X ) / / . Then, by (b), we get the split exact sequences0 / / Y α / / Y ′ / / / / T )0 / / / / X ′ β / / X / / U ) . Thus, by (b) and the above split exact sequences, we obtain that η splits.Therefore, we get (f) from (d) and (e).(g) It follows from (d), (e), (f) and Lemma 3.4. (cid:3) Remark 3.7. For the lower triangular matrix K -algebra Λ = [ T M U ] , we havethe following. (1) Let ( Y, ϕ, X ) ∈ ( M ⊗ T (Mod ( T )) , Mod ( U )) ≃ Mod (Λ) . We can seethe Λ -module ( Y, ϕ, X ) as a matrix representation. Indeed, considerthe set of matrices [ Y X ] := { (cid:2) y x (cid:3) : x ∈ X, y ∈ Y } , with the structure of Λ -module given by [ t m u ] · (cid:2) y x (cid:3) := h ty ϕ ( m ⊗ y )+ ux i . Note that ( i T ) ∗ ([ Y X ]) ≃ Y as T -modules and ( i U ) ∗ ([ Y X ]) ≃ X as U -modules TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 17 (2) Let Y be a T -submodule of T and X be an U -submodule of M. Then,the map i Y,X : M ⊗ T Y → X, m ⊗ y my, is a morphism of U -modules and thus the triple ( Y, i Y,X , X ) is a Λ -module that can be seenas the matrix [ Y X ] . (3) Let X ∈ Mod ( U ) . Then, the set of matrices [ X ] becomes a Λ -modulevia the action [ t m u ] · [ x ] := [ ux ] . On the other hand, by (1), wehave the Λ -module [ X ] . Finally, an easy calculation shows that µ : [ X ] → [ X ] , [ x ] [ x ] , is an isomorphism of Λ -modules. (4) By Lemma 3.2, for any i ∈ I and j ∈ J, we get ( i T ) ∗ (Λ f j ) ≃ T f j , ( i T ) ∗ (Λ e i ) = 0 , ( i U ) ∗ (Λ f j ) ≃ Mf j and ( i U ) ∗ (Λ e i ) ≃ Ue i . Lemma 3.8. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) , ˜ A = { ˜ A i } i<α ∈ ℘ ( I ) and ˜ B = { ˜ B j } i<β ∈ ℘ ( J ) . Then, for ˜ C := ˜ A ∨ ˜ B ∈ ℘ ( I ∨ J ) (see 3.1), thecorresponding ˜ C -standard Λ -modules computed in the basic lower triangularmatrix K -algebra (Λ , { g r } r ∈ I ∨ J ) (see 3.3 ) are related with those of ( U, ˜ A ) and ( T, ˜ B ) as follows. (a) If t ∈ [0 , α ) × { } , then ˜ C ∆ r ( t ) ≃ ( p U ) ∗ ( ˜ A ∆ p ( r ) ( p ( t ))) = (cid:20) ˜ A ∆ p ( r ) ( p ( t )) (cid:21) . (b) If t ∈ [0 , β ) × { } , then ˜ C ∆ r ( t ) ≃ ( p T ) ∗ ( ˜ B ∆ p ( r ) ( p ( t ))) = (cid:20) ˜ B ∆ p ( r ) ( p ( t )) 00 0 (cid:21) . (c) F ( ˜ C ∆) = ( p U ) ∗ ( F ( ˜ A ∆)) ⋆ ( p T ) ∗ ( F ( ˜ B ∆)) = F (( p U ) ∗ ( ˜ A ∆)) ⋆ F (( p T ) ∗ ( ˜ B ∆)) . (d) F (( i T ) ∗ ( ˜ C ∆)) = F ( ˜ B ∆) and F (( i U ) ∗ ( ˜ C ∆)) = F ( ˜ A ∆) . (e) Let ( Y, ϕ, X ) ∈ Mod (Λ) . Then ( Y, ϕ, X ) ∈ F ( ˜ C ∆) ⇔ Y ∈ F ( ˜ B ∆) and X ∈ F ( ˜ A ∆) . Proof. We recall that ˜ C = { ˜ C t } t ∈ [0 ,α ) ∨ [0 ,β ) , were ˜ C t = ˜ A i × { } if t = ( i, , and ˜ C t = ˜ B j × { } if t = ( j, . We will make use of the equalities given inLemma 3.2. Let t ∈ [0 , α ) ∨ [0 , β ) and r ∈ ˜ C t . We have to compute ˜ C ∆ r ( t ) := ˜ C P r ( t )Tr ⊕ t ′ For the latest equality above, we used that X t ′ M f p ( k ) T f p ( r ) ⊆ M f p ( r ) , by doing the sum I + II, we get (b).(c) By Remark 2.7, we know that the classes F ( ˜ C ∆) , F ( ˜ A ∆) and F ( ˜ B ∆)are closed under extensions. Then, by Proposition 3.6 (g), we get that theclass Z := ( p U ) ∗ ( F ( ˜ A ∆)) ⋆ ( p T ) ∗ ( F ( ˜ B ∆)) is closed under extensions.From (a) and (b) we have that ˜ C ∆ ⊆ Z and thus F ( ˜ C ∆) ⊆ Z since Z isclosed under extensions. Now, we set W := F (( p U ) ∗ ( ˜ A ∆)) ⋆ F (( p T ) ∗ ( ˜ B ∆)) . Since ( p U ) ∗ and ( p T ) ∗ are exact functors, by (a) and (b), we get that( p U ) ∗ ( F ( ˜ A ∆)) ⊆ F (( p U ) ∗ ( ˜ A ∆)) ⊆ F ( ˜ C ∆) , ( p T ) ∗ ( F ( ˜ B ∆)) ⊆ F (( p T ) ∗ ( ˜ B ∆)) ⊆ F ( ˜ C ∆) . Therefore F ( ˜ C ∆) ⊆ Z ⊆ W ⊆ F ( ˜ C ∆) , proving (c).(d) It follows from (a), (b) and Proposition 3.6 (c).(e) Let Y ∈ F ( ˜ B ∆) and X ∈ F ( ˜ A ∆) . Thus, by (c), ( Y, ϕ, X ) ∈ F ( ˜ C ∆) . Let ( Y, ϕ, X ) ∈ F ( ˜ C ∆) . Then, by (c), there is an exact sequence 0 → (0 , , X ′ ) → ( Y, ϕ, X ) → ( Y ′ , , → , where X ′ ∈ F ( ˜ A ∆) and Y ′ ∈ F ( ˜ B ∆) . By using the exact sequence 0 → (0 , , X ) → ( Y, ϕ, X ) → ( Y, , → , weget from Proposition 3.6 (b) that X ≃ X ′ and Y ≃ Y ′ . Therefore Y ∈ F ( ˜ B ∆)and X ∈ F ( ˜ A ∆) . (cid:3) Theorem 3.9. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) , ˜ A = { ˜ A i } i<α ∈ ℘ ( I ) and ˜ B = { ˜ B j } i<β ∈ ℘ ( J ) . Then, for ˜ C := ˜ A ∨ ˜ B ∈ ℘ ( I ∨ J ) (see 3.1), andthe lower triangular matrix K -algebra Λ := [ T M U ] , the following statementsare equivalent. (a) (Λ , ˜ C ) is a left standardly stratified K -algebra. (b) { M f j } j ∈ J ⊆ F ( ˜ A ∆) and the pairs ( T, ˜ B ) and ( U, ˜ A ) are left stan-dardly stratified K -algebras.Proof. By Lemma 3.3, (Λ , { g r } r ∈ I ∨ J ) is a basic w.e.i. K algebra, where g r = e i if r = ( i, , and g r = f j if r = ( j, . Furthermore: (1) For j ∈ J, we have Λ f j = h T f j Mf j i ≃ ( T f j , ϕ, M f j ) ∈ Mod (Λ) , where ϕ := i T f j ,Mf j , see Remark 3.7 (2);(2) For i ∈ I, we have Λ e i = (cid:2) Ue i (cid:3) ≃ (0 , , U e i ) ∈ Mod (Λ) , see Remark3.7.(a) ⇒ (b): Let (Λ , ˜ C ) be a left standardly stratified K -algebra. Then, byProposition 2.10, we have that { Λ g r } r ∈ I ∨ J ⊆ F ( ˜ C ∆) . Let j ∈ J. Since ( i T ) ∗ : Mod (Λ) → Mod ( T ) is an exact functor, we getfrom (1), Remark 3.7 (1) and Lemma 3.8 (d) T f j ≃ ( i T ) ∗ (Λ f j ) ∈ ( i T ) ∗ ( F ( ˜ C ∆)) ⊆ F (( i T ) ∗ ( ˜ C ∆)) = F ( ˜ B ∆) . Then, from Proposition 2.10, it follows that ( T, ˜ B ) is a left standardly stratified K -algebra. On the other hand, using that ( i U ) ∗ : Mod (Λ) → Mod ( U ) is anexact functor, Remark 3.7 (1) and Lemma 3.8 (d), we get M f j ≃ ( i U ) ∗ (Λ f j ) ∈ ( i U ) ∗ ( F ( ˜ C ∆)) ⊆ F (( i U ) ∗ ( ˜ C ∆)) = F ( ˜ A ∆) . and thus { M f j } j ∈ J ⊆ F ( ˜ A ∆) . Let i ∈ I. Then, by the exactness of the functor ( i U ) ∗ : Mod (Λ) → Mod ( U ) , (2), Remark 3.7 (1) and Lemma 3.8 (d), U e i ≃ ( i U ) ∗ (Λ e i ) ∈ ( i U ) ∗ ( F ( ˜ C ∆)) ⊆ F (( i U ) ∗ ( ˜ C ∆)) = F ( ˜ A ∆) . and hence, from Proposition 2.10, we get that ( U, ˜ A ) is a left standardlystratified K -algebra.(b) ⇒ (a): Assume the hypothesis given in (b). In particular, by Proposi-tion 2.10, we get that { U e i } i ∈ I ⊆ F ( ˜ A ∆) and { T f j } j ∈ J ⊆ F ( ˜ B ∆) . In orderto show that (Λ , ˜ C ) is a left standardly stratified K -algebra, it is enough toprove that { Λ g r } r ∈ I ∨ J ⊆ F ( ˜ C ∆) . Let j ∈ J. Then, by (1) and Proposition 3.6 (b), there is an exact sequenceof Λ-modules η j : 0 → (0 , , M f j ) → ( T f j , ϕ, M f j ) → ( T f j , , → . On the other hand, by Proposition 3.6 (e),(0 , , M f j ) = ( p U ) ∗ ( M f j ) ∈ ( p U ) ∗ ( F ( ˜ A ∆));and by Proposition 3.6 (d),( T f j , , 0) = ( p T ) ∗ ( T f j ) ∈ ( p T ) ∗ ( F ( ˜ B ∆)) . Thus, from (1), Lemma 3.8 (c) and η j , we getΛ f j ≃ ( T f j , ϕ, M f j ) ∈ ( p U ) ∗ ( F ( ˜ A ∆)) ⋆ ( p T ) ∗ ( F ( ˜ B ∆)) = F ( ˜ C ∆) . Let i ∈ I. Then, by (2), Proposition 3.6 (e) and Lemma 3.8 (c), we getΛ e i ≃ ( p U ) ∗ ( U e i ) ∈ ( p U ) ∗ ( F ( ˜ A ∆)) ⊆ F ( ˜ C ∆);and thus the result follows. (cid:3) TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 21 Locally bounded K -algebras with enough idempotents Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) and lower triangular matrix K -algebra Λ = [ T M U ] which is basic and with enough idempotents { g r } r ∈ I ∨ J , see Lemma 3.3. In this section, we find conditions under which L = ( Y, X, ϕ ) ∈ mod (Λ) implies that Y ∈ mod ( T ) and X ∈ mod ( U ). We also study modulesof finite projective dimension in mod(Λ).Following [3], we start this section by recalling the following notions. Definition 4.1. [3] Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra. (a) M ∈ Mod ( R ) is endofinite if { e i M } i ∈ I ⊆ f.ℓ. (End R ( M ) op ) . (b) R is left (respectively, right) locally endofinite if Re i (respectively, e i R ) is endofinite, for every i ∈ I. We say that R is locally endofiniteif it is both left and right locally endofinite. Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra such that each e i is primitive. For M ∈ Mod( R ) , the support of M is the set Supp( M ) := { i ∈ I : e i M = 0 } . Itis said that R is left (respectively, right) support finite if the set Supp( Re i )(respectively Supp( e i R )) is finite for each i ∈ I. We say that R is supportfinite if it is both left and right support finite. Proposition 4.2. [3, Proposition 7] Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebrasuch that each e i is primitive. Then, the following statements are equivalent. (a) R is right locally endofinite and left support finite. (b) { Re i } i ∈ I ⊆ f.ℓ. ( R ) . Definition 4.3. [3, Definition 9] Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra. It issaid that R is locally bounded if the following conditions hold true. (a) each e i is primitive. (b) R is locally endofinite. (c) R is support finite. Remark 4.4. Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra. Note that ( R, { e i } i ∈ I ) islocally bounded if, and only if, its opposite K -algebra ( R op , { e i } i ∈ I ) is locallybounded. Proposition 4.5. [3, Corollary 8, Proposition 10] For a locally bounded w.e.i. K -algebra ( R, { e i } i ∈ I ) , the following statements hold true. (a) mod ( R ) = f.ℓ. ( R ) = fin . p . ( R ) . (b) mod ( R ) is an abelian full subcategory of Mod ( R ) which is closed un-der the formation of submodules. (c) Let M ∈ Mod ( R ) . Then, M ∈ mod ( R ) ⇔ Supp( M ) is finite and e i M ∈ mod( e i Re i ) ∀ i. Lemma 4.6. Let ( R, { e i } i ∈ I ) be a w.e.i. K -algebra and N ∈ Mod ( R ) suchthat e i N ∈ f.ℓ. ( K ) , for some i ∈ I. Then e i N ∈ f.ℓ. ( e i Re i ) . Proof. Since e i Re i is a K -module, we get that ϕ i : K → e i Re i , k k · e i , is amorphism of rings with unity. In particular, by Proposition 2.2, the change ofrings functor ( ϕ i ) ∗ : Mod ( e i Re i ) → Mod ( K ) is an exact K -functor such that( ϕ i ) ∗ ( X ) = X as abelian group ∀ X ∈ Mod ( e i Re i ) . Therefore any descending(ascending) chain of left e i Re i -submodules of N becomes stationary afterfinitely many steps and thus e i N ∈ f.ℓ. ( e i Re i ) . (cid:3) Proposition 4.7. Any support finite basic w.e.i. K -algebra is locally bounded.Proof. Let ( R, { e i } i ∈ I ) be a support finite basic w.e.i. K -algebra. We need toshow that R is locally endofinite. For each i ∈ I, we have the ring isomorphismEnd R ( Re i ) → e i Re i , f f ( e i ) , that will we be used as an identificationbetween these two rings. Then, by Lemma 4.6 and its dual, we get that e j Re i ∈ f.ℓ. ( e i R op e i ) ∩ f.ℓ. ( e j Re j ) since e j Re i ∈ f.ℓ. ( K ); proving the result. (cid:3) Remark 4.8. A natural source of support finite basic w.e.i. K -algebras isgiven by the Bongartz-Gabriel construction [6] . In order to do that, we startby taking a locally bounded quiver Q, a field K and an admissible ideal I ofthe path K -algebra K Q. Then, from the quotient path K -algebra K Q/I, wecan make a support finite basic w.e.i. K -algebra, see all the details in [13,Proposition 6.9] . Proposition 4.9. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) and the lowertriangular matrix K -algebra Λ = [ T M U ] which is basic and with enough idem-potents { g r } r ∈ I ∨ J (see Lemma 3.3). Then, the following statements are equiv-alent. (a) (Λ , { g r } r ∈ I ∨ J ) is support finite. (b) ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) are support finite, { M f j } j ∈ J ⊆ mod ( U ) and { e i M } i ∈ I ⊆ mod ( T op ) . (c) ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) are support finite and the sets Supp( M f j ) and Supp( e i M ) are finite ∀ i, j. Moreover, if one of the above equivalent conditions holds true, then U, T and Λ are locally bounded.Proof. By Lemma 3.2 (c), for i ∈ I and j ∈ J, we get the equalitiesSupp(Λ e i ) = Supp( U e i ) × { } , Supp( e i Λ) = (Supp( e i U ) ∪ Supp( e i M )) × { } , Supp(Λ f j ) = (Supp( T f j ) ∪ Supp( M f j )) × { } , Supp( f j Λ) = Supp( f j T ) × { } . On the other hand, by Lemma 4.6 and its dual, we get that e i M f j ∈ mod ( e i U e i ) ∩ mod ( f j T op f j ) ∀ i, j TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 23 since { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) . Then, the result follows from Remark 4.4,Proposition 4.5 and Proposition 4.7. (cid:3) Lemma 4.10. For a w.e.i. K -algebra ( R, { e i } i ∈ I ) , the following statementshold true. (a) Let M ∈ Mod ( R ) . Then M ∈ mod ( R ) if, and only if, there existsa finite subset I ′ ⊆ I and an epimorphism ` i ∈ I ′ ( Re i ) α i → M of R -modules, where each α i is a non-negative integer. (b) proj( R ) = add ( { Re i } i ∈ I ) . Proof. Note firstly that (b) follows from (a). We recall from [16, Section49] that ( R, { e i } i ∈ I ) is a ring with local units. That is, for a finitely many r , . . . , r k ∈ R there exists e = e ∈ R, which is a finite sum of elements in { e i } i ∈ I , such that { r j } kj =1 ⊆ eRe. Let M ∈ Mod ( R ) . We assert that for any m ∈ M there is e = e ∈ R, which is a finite sum of elements in { e i } i ∈ I , such that m = em. Indeed, for m ∈ M, there are { r j } nj =1 ⊆ R and { m j } nj =1 ⊆ M with m = P nj =1 r j m j . Since R has local units, there exists e = e ∈ R, which is a finite sum of elementsin { e i } i ∈ I , such that { r j } nj =1 ⊆ eRe. Then m = P nj =1 r j m j = P nj =1 er j m j = em ; proving our assertion.Let us prove (a). Suppose that M ∈ mod ( R ) . Then there is a finite gen-erating set { m j } kj =1 of the R -module M. By the assertion above, for each j, there exists ε j = ε j = P m j t =1 e j t such that m j = ε j m j . Note that the mapping k a j =1 Rε j → M, ( r j ε j ) kj =1 k X j =1 r j m j is well defined, since m j = ε j m j for each j, and it is an epimorphism of R -modules. Moreover, by using that Rε j = ⊕ m j t =1 Re j t for each j, we get thedesired epimorphism of R -modules. (cid:3) Proposition 4.11. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) and the lowertriangular matrix K -algebra Λ = [ T M U ] which is basic and with enough idem-potents { g r } r ∈ I ∨ J (see Lemma 3.3). Let (Λ , { g r } r ∈ I ∨ J ) be support finite and L := ( Y, ϕ, X ) ∈ mod (Λ) . Then the following statements hold true. (a) There is an epimorphism of Λ -modules π : P → L, where P = ` s ∈ S (Λ g s ) α s for some finite set S ⊆ I ∨ J and non-negative inte-gers α s . (b) For S I := S ∩ ( I × { } ) and S J := S ∩ ( J × { } ) , we have ( i T ) ∗ ( P ) ≃ a s ∈ S J ( T f p ( s ) ) α s ∈ proj ( T ) , ( i U ) ∗ ( P ) ≃ a s ∈ S J ( M f p ( s ) ) α s a a s ∈ S I ( U e p ( s ) ) α s ∈ mod ( U ) . (c) π : P → L induces two epimorphisms π : ( i T ) ∗ ( P ) → Y of T -modulesand π : ( i U ) ∗ ( P ) → X of U -modules. Moreover Y ∈ mod ( T ) and X ∈ mod ( U ) . (d) Let L ∈ proj(Λ) . Then, Y ∈ proj( T ) and X is a direct summand of ( i U ) ∗ ( P ) . Proof. By Proposition 4.9, we know that U, T and Λ are locally bounded.Moreover, { M f j } j ∈ J ⊆ mod ( U ) . Since L is finitely generated, we get (a) from Lemma 4.10 (a). Let now π : P → L be the epimorphism of Λ-modules from (a). Then, by Proposition3.6 (a), P ≃ (( i T ) ∗ ( P ) , ϕ ′ , ( i U ) ∗ ( P )) . In particular, the epimorphism π canbe written as π = ( π , π ) : (( i T ) ∗ ( P ) , ϕ ′ , ( i U ) ∗ ( P )) → ( Y, ϕ, X ) and thus, byProposition 3.6 (b), π : ( i T ) ∗ ( P ) → Y is an epimorphism of T -modules and π : ( i U ) ∗ ( P ) → X is an epimorphism of U -modules.Let us compute ( i T ) ∗ ( P ) and ( i U ) ∗ ( P ) . Indeed, from Remark 3.7 (4) andsince ( i T ) ∗ and ( i U ) ∗ are additive functors and P = ` s ∈ S (Λ g s ) α s , we get theisomorphisms stated in (b). The fact that ( i U ) ∗ ( P ) ∈ mod ( U ) follows fromthe inclusion { M f j } j ∈ J ⊆ mod ( U ) and Proposition 4.5 (b). Furthermore, Y and X are finitely generated since π and π are surjective. Finally, (d)follows from (b) and (c). (cid:3) Lemma 4.12. Let → X n → · · · → X → X → A → be an exact sequencein an abelian category A . Then pd( A ) ≤ n + max { pd( X i ) } ni =1 . Proof. By using that max { a, b + c } ≤ max { a, b } + c for any non-negativeintegers a, b and c, the proof can be carry on by induction on n. (cid:3) Proposition 4.13. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) and the lowertriangular matrix K -algebra Λ = [ T M U ] which is basic and with enough idem-potents { g r } r ∈ I ∨ J (see Lemma 3.3). If (Λ , { g r } r ∈ I ∨ J ) is support finite, thenthe following statements hold true. (a) For any X ∈ Mod ( U ) , ( p U ) ∗ ( X ) ∈ mod (Λ) ⇔ X ∈ mod ( U ) . (b) For any Y ∈ Mod ( T ) , ( p T ) ∗ ( Y ) ∈ mod (Λ) ⇔ Y ∈ mod ( T ) . (c) For any X ∈ mod ( U ) , there is a finite set J X ⊆ J such that pd(( p U ) ∗ ( X )) ≤ pd( X ) ≤ pd(( p U ) ∗ ( X )) + max { pd( M f j ) } j ∈ J X . (d) For any Y ∈ mod ( T ) , there is a finite set J Y ⊆ J such that pd( Y ) ≤ pd(( p T ) ∗ ( Y )) ≤ pd( Y ) + 1 + max { pd( M f j ) } j ∈ J Y . TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 25 Proof. Let (Λ , { g r } r ∈ I ∨ J ) be support finite. Consider E := add ( { M f j } j ∈ J ) . (a) Let X ∈ Mod ( U ) . If ( p U ) ∗ ( X ) ∈ mod (Λ) , by Proposition 4.11 we havethat X ∈ mod ( U ) . Suppose that X ∈ mod ( U ) . Then, there are finitely many x , x , · · · , x n in X such that X = P ni =1 U x i . On the other hand, by Remark3.7 (3), ( p U ) ∗ ( X ) can be seen as the matrix [ X ] and thus P ni =1 Λ (cid:2) x i (cid:3) = h P ni =1 Ux i i = [ X ] ; proving that ( p U ) ∗ ( X ) ∈ mod (Λ) . (b) It can be proved similarly as we did in (a).(c) Let X ∈ mod ( U ) . Assume that n := pd(( p U ) ∗ ( X )) < ∞ . Then, byProposition 4.11 and Proposition 3.6 (b), there is an exact sequence0 → E n a Q n → · · · → E a Q → E a Q → X → U ) , where E i ∈ E and Q i ∈ proj( U ) for all i. In particular, we canform a finite set J X ⊆ J such that { E i } ni =1 ⊆ add ( ` j ∈ J X M f j ) . Therefore,from Lemma 4.12, and the preceding exact sequence, we get that pd( X ) ≤ pd(( p U ) ∗ ( X )) + max { pd( M f j ) } j ∈ J X . Assume now that m := pd ( X ) < ∞ . Then, there is a projective resolution η : 0 → Q m → · · · → Q → Q → X → X. Since ( p U ) ∗ is anexact functor and ( p U ) ∗ ( U e i ) ≃ Λ e i ∈ proj(Λ) , by applying the functor ( p U ) ∗ to η, we get a projective resolution (of length m ) of ( p U ) ∗ ( X ) and hencepd(( p U ) ∗ ( X )) ≤ pd ( X ) . (d) Let Y ∈ mod ( T ) . Suppose there is a projective resolution (of length m ) of ( p T ) ∗ ( Y ) . Then, by Proposition 4.11 and Proposition 3.6 (b), there isa projective resolution (of length m ) of Y and thus pd( Y ) ≤ pd(( p T ) ∗ ( Y )) . Assume that n := pd( Y ) < ∞ . Then, there is a projective resolution0 → Q n → · · · → Q → Q → Y → Y. Thus, by applying the exactfunctor ( p T ) ∗ to this resolution, we get the exact sequence of Λ-modules θ : 0 → ( p T ) ∗ ( Q n ) → · · · → ( p T ) ∗ ( Q ) → ( p T ) ∗ ( Q ) → ( p T ) ∗ ( Y ) → . By Lemma 4.10, Q i ∈ proj( T ) = add ( { T f j } j ∈ J ) . Thus, we can form a finiteset J Y ⊆ J such that { ( p T ) ∗ ( Q i ) } ni =1 ⊆ add ( { ( p T ) ∗ ( T f j ) } j ∈ J Y . Therefore,pd (( p T ) ∗ ( Q i )) ≤ max { pd(( p T ) ∗ ( T f j )) } j ∈ J Y , for all i. Let j ∈ J. Note that Λ f j ≃ ( T f j , ϕ, M f j )) and thus we have the exactsequence 0 → ( p U ) ∗ ( M f j ) → Λ f i → ( p T ) ∗ ( T f j ) → p T ) ∗ ( T f j )) ≤ p U ) ∗ ( M f j )) ≤ M f j ) . Therefore max { pd (( p T ) ∗ ( Q i )) } ni =1 ≤ { pd( M f j ) } j ∈ J Y . Hence to finishthe proof, it is enough to apply Lemma 4.12 to the exact sequence θ. (cid:3) Theorem 4.14. For ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) basic w.e.i. K -algebras, M an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) and the lower triangu-lar matrix K -algebra Λ = [ T M U ] , which is basic and with enough idempotents { g r } r ∈ I ∨ J (see Lemma 3.3), such that (Λ , { g r } r ∈ I ∨ J ) is support finite, thefollowing statements hold true. (a) Let L = ( Y, ϕ, X ) ∈ Mod (Λ) and pd( M f j ) < ∞ ∀ j ∈ J. Then L ∈ P < ∞ Λ ⇔ Y ∈ P < ∞ T and X ∈ P < ∞ U . (b) Let m := max { pd( M f j ) } j ∈ J < ∞ , a := fin . dim . ( T ) , α := gl . dim . ( T ) ,b := fin . dim . ( U ) and β := gl . dim . ( U ) . Then max { β − m, α } ≤ gl . dim . (Λ) ≤ max { β, α + 1 + m } , max { b − m, a } ≤ fin . dim . (Λ) ≤ max { b, a + 1 + m } . Proof. (a) Consider the exact sequence 0 → ( p U ) ∗ ( X ) → L → ( p T ) ∗ ( Y ) → L ∈ mod (Λ) if, andonly if, Y ∈ mod ( T ) and X ∈ mod ( U ) . If Y ∈ P < ∞ T and X ∈ P < ∞ U , then by Proposition 4.13 and the above exactsequence, we conclude that L ∈ P < ∞ Λ . Let L ∈ P < ∞ Λ and n := pd ( L ) . Then by Proposition 4.11 and Proposition3.6 (b), there are two exact sequences θ Y : 0 → P n → · · · → P → P → Y → T -modules, where P i ∈ proj( T ) ∀ i ; and θ X : 0 → E n a Q n → · · · → E a Q → E a Q → X → U ) , where E i ∈ add ( { M f j } j ∈ J ) and Q i ∈ proj( U ) for all i. In partic-ular, we can form a finite set J X ⊆ J such that { E i } ni =1 ⊆ add ( ` j ∈ J X M f j ) . Now, from the exact sequence θ Y , we get that pd( Y ) ≤ n < ∞ . On theother hand, by applying Lemma 4.12 to the exact sequence θ X , it follows thatpd ( X ) ≤ n + max { pd( M f j ) } j ∈ J X < ∞ . (b) Let L = ( Y, ϕ, X ) ∈ mod (Λ) . Then, from the exact sequence 0 → ( p U ) ∗ ( X ) → L → ( p T ) ∗ ( Y ) → L ) ≤ max { pd(( p U ) ∗ ( X )) , pd(( p T ) ∗ ( Y )) }≤ max { pd( X ) , pd( Y ) + 1 + max { pd( M f j ) } j ∈ J Y }≤ max { gl . dim . ( U ) , gl . dim . ( T ) + 1 + m } , and thus gl . dim . (Λ) ≤ max { gl . dim . ( U ) , gl . dim . ( T ) + 1 + m } . Let Y ∈ mod ( T ) . Then, by Proposition 4.13 (d), pd( Y ) ≤ pd(( p T ) ∗ ( Y )) ≤ gl . dim . (Λ) and thus gl . dim . ( T ) ≤ gl . dim . (Λ) . Let X ∈ mod ( U ) . Then, by Proposition 4.13 (d),pd( X ) ≤ pd(( p U ) ∗ ( X )) + max { pd( M f j ) } j ∈ J X ≤ gl . dim . (Λ) + m and thus gl . dim . ( U ) ≤ gl . dim . (Λ) + m. Finally, the proof of the inequalitiesinvolving the finitistic dimension can be done, by using (a), as in the one wedid for the global dimension. (cid:3) TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 27 Theorem 4.15. Let ( U, { e i } i ∈ I ) and ( T, { f j } j ∈ J ) be basic w.e.i. K -algebras, M be an U - T -bimodule such that { e i M f j } ( i,j ) ∈ I × J ⊆ f.ℓ. ( K ) and pd( M f j ) < ∞ ∀ j ∈ J. Let Λ := [ T M U ] , which is basic and with enough idempotents { g r } r ∈ I ∨ J (see Lemma 3.3), be such that (Λ , { g r } r ∈ I ∨ J ) is support finite.Consider the partitions ˜ A = { ˜ A i } i<α ∈ ℘ ( I ) , ˜ B = { ˜ B j } i<β ∈ ℘ ( J ) , and ˜ C := ˜ A ∨ ˜ B ∈ ℘ ( I ∨ J ) (see 3.1). Then, the following statements are equiva-lent. (a) F ( ˜ C ∆) = P < ∞ Λ . (b) F ( ˜ A ∆) = P < ∞ U and F ( ˜ B ∆) = P < ∞ T . Moreover, if one of the above equivalent conditions holds true, then Λ , U and T are locally bounded and left standardly stratified K -algebras, fin . dim . (Λ) =pd( ˜ C ∆) , fin . dim . ( T ) = pd( ˜ B ∆) and fin . dim . ( U ) = pd( ˜ A ∆) . Proof. From Lemma 3.8 (a,b) and Theorem 4.14 (a), it can be shown that( ∗ ) ˜ C ∆ ⊆ P < ∞ Λ ⇔ ˜ A ∆ ⊆ P < ∞ U and ˜ B ∆ ⊆ P < ∞ T . (a) ⇒ (b): Since the classes P < ∞ U and P < ∞ T are closed under extensions,we get from (*), that F ( ˜ A ∆) ⊆ P < ∞ U and F ( ˜ B ∆) ⊆ P < ∞ T . Let X ∈ P < ∞ U . Then, by Theorem 4.14 (a), (0 , , X ) ∈ P < ∞ Λ = F ( ˜ C ∆) . Hence by Lemma 3.8 (e), we conclude that X ∈ F ( ˜ A ∆) and thus P < ∞ U ⊆F ( ˜ A ∆) . Similarly, it can be shown that P < ∞ T ⊆ F ( ˜ B ∆) . (b) ⇒ (a): Since the classes P < ∞ Λ is closed under extensions, we get from(*), that F ( ˜Λ ∆) ⊆ P < ∞ Λ . Let L = ( Y, ϕ, X ) ∈ P < ∞ Λ . Then, by Theorem 4.14(a), Y ∈ P < ∞ T = F ( ˜ B ∆) and X ∈ P < ∞ U = F ( ˜ A ∆) . Thus, by Lemma 3.8 (e), L ∈ F ( ˜ C ∆) . Assume now that one of the above equivalent conditions holds true. 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Vol. 25 No.1, 1-11, (2001).Eduardo Marcos:Instituto de Matem´aticas y Estadistica,Universidad de Sao Paulo,Sao Paulo, BRASIL. [email protected] Octavio Mendoza:Instituto de Matem´aticas,Universidad Nacional Aut´onoma de M´exico,Circuito Exterior, Ciudad Universitaria,M´exico D.F. 04510, M´EXICO. [email protected] Corina S´aenz:Departamento de Matem´aticas, Facultad de Ciencias,Universidad Nacional Aut´onoma de M´exico,Circuito Exterior, Ciudad Universitaria,M´exico D.F. 04510, M´EXICO. [email protected] TANDARDLY STRATIFIED LOWER TRIANGULAR RINGS 29