Hereditary triangular matrix comonads

HEREDITARY TRIANGULAR MATRIX COMONADS arXiv:1504.07594v1 [math.RA] 28 Apr 2015 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS Abstract. We recognise Harada’s generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less or equal than 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting. Introduction Every complete set of pairwise orthogonal idempotent elements e1 , . . . , en of a unital ring R allows to express the ring as a generalized n×n–matrix ring with entries in the bimodules ei Rej for 1 ≤ i, j ≤ n. The origin of this decomposition can be traced back to the seminal work of B. Peirce [24]. Beyond the role of matrix rings in the classification of semi-simple artinian rings (i.e., Wedderburn-Artin’s Theorem), these generalized matrix rings were used to investigate rings of low homological dimension, in the framework of a systematic program of studying non commutative rings of finite homological dimension promoted by Eilenberg, Ikeda, Jans, Kaplansky, Nagao, Nakayama, Rosenberg and Zelinsky among others, see [11, 10, 12, 18, 19]. In particular, S. U. Chase [8] and M. Harada [15] used generalized triangular matrix rings to investigate the structure of the semi-primary hereditary rings (i.e., semi-primary rings with homological dimension 1). Inspired by the study of homological properties of abelian categories of diagrams given in [22, §IX], Harada formulated in [16] versions of some of his results on hereditary triangular matrix rings from [15] in the framework of the so called abelian categories of generalized commutative diagrams. In this paper, we recognize these categories as the categories of modules (or algebras) over suitable monads. This allows, apart from giving a more conceptual treatment of these categories, to obtain most of the main results from [16] with sharper (and simpler, we think) proofs. Our methods, based on a sort of “Linear Algebra” 2010 Mathematics Subject Classification. 15A30; 18G20; 18C20; 16T15. Key words and phrases. Matrix Comonads; Categories of Comodules; Abelian Hereditary Categories; Global homological dimension. Research supported by the Spanish Ministerio de Economı́a y Competitividad and the European Union, grant MTM2013–41992-P. 1 2 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS for additive functors and natural transformations, also provide some new results, including a characterization of hereditary categories of generalized commutative diagrams. We work in the dual, although formally equivalent, situation than in [16]. Thus, we consider a comonad defined on a finite direct product of additive categories, and we investigate how to express in matrix form both its comultiplication and its counit. In this way, its category of comodules (or coalgebras) is expressed in a such a way that Harada’s categories of generalized diagrams become a particular case (namely, the categories of modules over a normal triangular monad). This is done in Section 1. In Section 2 we generalize [16, Theorem 2.3] in several directions. On one hand, we do not assume that the base categories are abelian and, on the other, our categories of comodules are more general than the dual of the categories considered in [16] (see Theorem 2.4). We derive our Theorem 2.4 from a simpler result, namely Theorem 2.1, in conjunction with the kind of linear algebra for functors developed in Section 1. In Section 3 we give full proofs of the dual form of some results from [16]. These results deal with the structure of the injective comodules over a hereditary normal triangular matrix comonad. In particular, they contain part of the dual of [16, Theorem 3.6]. We also prove a characterization of hereditary normal triangular matrix comonads (see Theorem 3.6). Section 4 illustrates our general results by giving a characterization of the bipartite coalgebras (see [21, 17]) that are right hereditary. Our methods allow us to work over a general commutative ring. 1. Matrix comonads and their categories of comodules. In this section we introduce the notion of matrix comonad and we describe its category of comodules. We develop a kind of linear algebra for matrix functors and their matrix natural transformations. 1.1. Basic notions and notation. All the categories in this paper are assumed to be additive, and all functors between them are additive. If A is an object of a category, then its identity morphism is also denoted by A. We shall use the standard notation for composition of functors and/or natural transformations, see [2]. If A, A′ are objects of an additive category A, then A ⊕ A′ denotes its direct sum, and similarly for any finite collection of objects. Recall that the direct sum of finitely many objects is both the product and the coproduct of the family in the category. The symbol ⊕ will be used also for the direct sum of morphisms. If F, G : A → B are functors, then its direct sum functor F ⊕ G : A → B is given by the composition F ⊕G:A ∆ / A×A F ×G / B×B ⊕ / B, where ∆ denotes the diagonal functor, and C × D denotes the product category of two categories C, D. Thus, (F ⊕ G)(f ) = F (f ) ⊕ G(f ) for any morphism f of A. The direct sum of finitely many functors is defined analogously. HEREDITARY TRIANGULAR MATRIX COMONADS 3 Let η : F → G ⊕ H be a natural transformation, where F, G, H : A → B are functors. By πG : G ⊕ H → G (resp. πH : G ⊕ H → H) we denote the natural transformation defined by the canonical projection. Therefore, η is uniquely determined by the natural transformations µ = πG η : F → G and ν = πH η : F → H. We will then use the notation η = µ ∔ ν. We will consider comonads (or cotriples) on A. We refer to [2] for details on (co)monads and their categores of (co)modules (or (co)algebras). 1.2. Matrix notation. Let A1 , . . . , An , B1 , . . . , Bm be additive categories and consider the product categories A = A1 × · · · × An , B = B1 × · · · × Bm . (1) For every j = 1, . . . , n, let πj : A → Aj (resp. ιj : Aj → A) denote the canonical projection (resp. injection) functors, and similarly for B. An additive functor F : A → B is determined by m × n functors Fji = πi F ιj : Aj → Bi , j = 1, . . . , n, i = 1, . . . , m, since, from the equalities 1A = ι1 π1 ⊕ · · · ⊕ ιn πn , 1B = ι1 π1 ⊕ · · · ⊕ ιm πm , we obtain F = M ιi πi F ιj πj = i,j M ιi Fji πj . i,j This means that given a morphism f = (f1 , . . . , fn ) in A, we have F f = (⊕j Fj1 fj , · · · , ⊕j Fjm fj ). This expression can be represented in  F11  F12 Ff =   ... F1m matrix form as   F21 · · · Fn1 f1 F22 · · · Fn2   f2   .  .. ..  . .   ..  F2m · · · Fnm fn Now, if η : F → G is a natural transformation, where G : A → B is a second functor, then we have the natural transformations η ij = πj ηιi : Fij → Gij , i = 1, . . . , n, j = 1, . . . , m (2) which completely determine η as follows. For each i = 1, . . . , n, and each A ∈ A, we consider the canonical morphism ξAi : ιi πi A → A given by the decomposition A = ι1 π1 A ⊕ · · · ⊕ ιn πn A. From the naturality of η, we get the following commutative diagrams F A = F ι1 π1 A ⊕ · · · ⊕ F ιn πn A O ηA / GA = Gι1 π1 A ⊕ · · · ⊕ Gιn πn A O i F ξA F ιi πi A i GξA ηιi πi A / Gιi πi A 4 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS which show that ηA = ⊕i ηιi πi A and, thus, for each j = 1, . . . , n, we have πj ηA = ⊕i πj ηιi πi A = ⊕i ηπiji A Since ηA is defined by the morphisms πj ηA , j = 1, . . . , m, we get that η is entirely determined by η ij , i = 1, . . . , n, j = 1, . . . , m. So we can represent our natural transformation in a matrix form       11 G11 G21 · · · Gn1 F11 F21 · · · Fn1 η η 21 · · · η n1  G  η 12 η 22 · · · η n2   F12 F22 · · · Fn2  G22 · · · Gn2   −→  .12 : . . η= ..  .. . . . . .  ..  .. ..  .. ..   .. .. .  . G1m G2m · · · Gnm F1m F2m · · · Fnm η 1m η 2m · · · η nm 1.3. Matrix operations. Let A F / B G / C be functors, where C = C1 × · · · × Cl , and A, B are finite products as in (1). Then the composite functor H = GF is represented in matrix form as the “usual matrix product” of matrices representing G and F . This means that   H11 H21 · · · Hn1  H12 H22 · · · Hn2  , H= ..  ..  ... .  . H1l H2l · · · Hnl where Hji = m M Gki Fjk , i = 1, . . . , l, j = 1, . . . , n. k=1 As for natural transformations concerns, let η : F → F ′ , µ : F ′ → F ′′ be natural transformations, where F, F ′ , F ′′ : A → B are functors. A straightforward argument shows that the matrix representing µη is the “Hadamard product” of the matrices that represent µ and η, namely    11  11 η η 21 · · · η n1 µ µ21 · · · µn1  µ12 µ22 · · · µn2   η 12 η 22 · · · η n2  · . µη =  .. ..  .. ..   ... . .  . .   .. µ1m µ2m · · · µnm η 1m η 2m · · · η nm  11 11 µ η µ21 η 21 · · · µn1 η n1  µ12 η 12 µ22 η 22 · · · µn2 η n2 = .. .. ..  . . . µ1m η 1m µ2m η 2m · · · µnm η nm    (3)  Finally, let us consider the composition of a functor and a natural transformation. So let η : F → F ′ with F, F ′ : A → B, and G : B → C. The natural transformation HEREDITARY TRIANGULAR MATRIX COMONADS 5 Gη : GF → GF ′ is easily shown to have a matrix representation   (Gη)11 (Gη)21 · · · (Gη)n1  (Gη)12 (Gη)22 · · · (Gη)n2  , Gη =  .. .. ..   . . . (Gη)1l (Gη)2l · · · (Gη)nl where ji (Gη) = m M Gki η jk , i = 1, . . . , l, j = 1, . . . , n. k=1 That is, the composition Gη leads to the “usual matrix product”. This also holds for the composition in the opposite order, that is, given a natural transformation γ : G → G′ , for functors G, G′ : B → C, and F : A → B, the matrix representation of the natural transformation γF : GF → G′ F is given by   (γF )11 (γF )21 · · · (γF )n1  (γF )12 (γF )22 · · · (γF )n2  , γF =  .. .. ..   . . . (γF )1l where (γF )ji = m M γ ki Fjk , (γF )2l · · · (γF )nl i = 1, . . . , l, j = 1, . . . , n. k=1 1.4. Matrix comonads. Fix a category A = A1 × · · · × An . Let F : A → A be an endofunctor, and δ : F → F 2 a natural transformation. In matrix form, δ = (δ ij ) : (Fij ) → (Fij )(Fij ), where δ ij : Fij → ⊕k Fkj Fik is a natural transformation for every i, j = 1, . . . , n, which is uniquely expressed as δ ij = δ i1j ∔ δ i2j ∔ · · · ∔ δ inj for some natural transformations δ ikj : Fij → Fkj Fik with k = 1, . . . , n. The coassociativity of F is given by the equation (δF )δ = (F δ)δ or, equivalently, (δF )ij δ ij = (F δ)ij δ ij , We know that (δF )ij = M k δ kj Fik 1 ≤ i, j ≤ n. and (F δ)ij = M (4) Flj δ il . l Observe that for each i, j, k = 1, . . . , n, the natural transformation δ kj Fik : Fkj Fik → L l Flj Fkl Fik is given by δ kj Fik = δ k1j Fik ∔ δ k2j Fik ∔ · · · ∔ δ knj Fik while, for each i, j, l = 1, . . . , n, the natural transformation Flj δ il : Flj Fil → is defined by Flj δ il = Flj δ i1l ∔ Flj δ i2l ∔ · · · ∔ Flj δ inl . L k Flj Fkl Fik 6 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS We then obtain that ij ij (δF ) δ = M kj δ Fik k and ij ij (F δ) δ = M Flj δ l il ! ∔δ k ! ∔δ  = ∔(δ klj Fik )δ ikj  = ∔(Flj δ ikl )δ ilj . ikj ilj l k,l k,l Both vectors should be equal, which is equivalent to (δ klj Fik )δ ikj = (Flj δ ikl )δ ilj , (1 ≤ i, j, k, l ≤ n). That is, all the diagrams Fij δilj / Flj Fil Flj δikl δikj  Fkj Fik /  Flj Fkl Fik δklj Fik are commutative. Next, we will discuss when a comultiplication δ : F → F 2 is counital, for a given counity ε : F → 1A . The general matrix form of ε is   ε1 0 · · · 0 F11 F21 · · · Fn1 2  0 ε · · · 0   F12 F22 · · · Fn2 ε= : ..  .. ..  ... ...   ... . . . 0 0 · · · εn F1n F2n · · · Fnn  1   0  −→  .   ..    0 ··· 0 1 ··· 0  .. ..  . .  0 0 ··· 1 (5) for some natural transformations εi : Fii → 1Ai for i = 1, . . . , n. The counitality conditions (εF )δ = 1F = (F ε)δ lead to matrix equalities equivalent to (εj Fij )δ ijj = 1Fij = (Fij εi )δ iij , (i, j = 1 . . . , n) that is, all the diagrams δijj / Fjj Fij Fij ■■ ■■■■■ ■■■■ j ■■■■ δiij ■■■■ ε Fij ■   / Fij Fij Fii i Fij ε commute. We have so far proved the following proposition. Proposition 1.1. Let A = A1 ×· · ·×An be the product category of finitely many categories A1 , . . . , An , and F : A → A any functor. There is a bijective correspondence between (1) Comonads (F, δ, ε); (2) Sets of natural transformations {δ ikj : Fij → Fkj Fik : i, j, k = 1, . . . , n}, and εi : Fii → 1Ai : i = 1, . . . , n such that HEREDITARY TRIANGULAR MATRIX COMONADS 7 (a) For all i, j, k, l = 1, . . . , n the diagram δilj Fij / (6) Flj Fil Flj δikl δikj  Fkj Fik /  Flj Fkl Fik δklj Fik conmutes, and (b) for all i, j = 1, . . . , n, the diagram δijj (7) / Fjj Fij Fij ■■ ■■■■■ ■■■■ j ■■■■ δiij ■■■■ ε Fij ■   / Fij Fij Fii i Fij ε commutes. Remark 1.2. Take i = j = k in diagrams (6) and (7), then (Fii , δ iii , εi ) is a comomad over Ai . Also diagrams (6) and (7) show that each functor Fij is in fact an Fjj − Fii bicomodule functor, in the sense of [3, Definition 4.7], see also [14]. Furthermore, if l = k in diagram (6), we get that each of the Fik ’s is a balanced bicomodule in a dual sense of [3, §3.2]. In this way, if the ’cotensor product’ functor Fkj Fkk Fik do exist, then the natural transformation δ ikj factors through Fkj Fkk Fik . Definition 1.3. A comonad (F, δ, ε) on A = A1 × · · · × An is called normal if Fii = 1Ai and εi = 1 for all i = 1, . . . , n. Thus, for a normal comonad, we have necessarily that δ ijj = δ iij = 1Fij for all i, j = 1, . . . , n. Remark 1.4. In the case n = 2, we deduce from Proposition 1.1 that a normal comonad (F, δ, ε) is given by natural transformations δ 121 : 1A1 → F21 F12 and δ 212 : 1A2 → F12 F21 such that F12 δ 121 = δ 212 F12 and F21 δ 212 = δ 121 F21 . Therefore, the normal comonads over A1 × A2 are in bijection with the wide (right) Morita contexts between A1 and A2 as defined in [5, 6]. Remark 1.5. It is possible to formulate Proposition 1.1 in dual form, thus given the structure of the monads on A1 × · · · × An . 1.5. Comodules over matrix comonads. Consider a comonad (F, δ, ε) over A = A1 × · · · × An as in Proposition 1.1. Next we want to describe its Eilenberg-Moore category AF of comodules (or coalgebras). Recall that an object of AF is a morphism dA : A → F A in A such that the following diagrams are commutative: A dA / FA δA dA  FA F dA /  F 2A dA / FA ❈❈ ❈❈ εA ❈ 1A ❈❈!  A❈ A (8) 8 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS We want to describe these objects in terms of the categories A1 , . . . , An . Each structure morphism dA is determined by the projections M Fkj Ak , j = 1, . . . , n πj (dA ) : Aj → k and each of them is given by 2j nj πj (dA ) = d1j A ∔ dA ∔ · · · ∔ dA with dkj A : Aj → Fkj Ak . Let us see what restrictions impose the commutativity of the diagrams (8) on the morphism dkj A . We start with the equation δA dA = F (dA )dA , which lead to the following system of equations πj (δA )πj (dA ) = πj (F (dA ))πj (dA ), j = 1, . . . , n. (9) On the other hand, πj δA = M δAljl , with δAljl : Flj Al → πj (δA )πj (dA ) = M l On the other hand, πj (F dA )πj (dA ) = M k δAljl Fkj Flk Al . k l Therefore, M ! ∔ dlj A l Fkj πk (dA )  = ∔ δAljl dlj A l ! ∔ dkj A k    lkj lkj lj = ∔ ∔ δAl dlj A = ∔ δAl dA . l k k,l (10) = ∔ Fkj πk (dA )dkj A = k ∔  k ∔ Fkj (dlk A) l  lk kj dkj A = ∔ Fkj (dA )dA . (11) k,l We get from (9), (10) and (11) that the commutativity of the first diagram in (8) is equivalent to the commutativity of all the diagrams Aj dkj A / (12) Fkj Ak dlj A Fkj dlk A  Flj Al lkj δA /  Fkj Flk Al l for j, k, l = 1, . . . , n. The second diagram in (8) leads to the equalities πj (εA dA ) = 1Aj for each j = 1, . . . , n. Therefore,   j kj 1Aj = πj (εA dA ) = πj (εA )πj (dA ) = (0 ⊕ · · · ⊕ 0 ⊕ εAj ⊕ 0 ⊕ · · · ⊕ 0) ∔ dA = εjAj djj A, k for all j = 1, . . . , n. HEREDITARY TRIANGULAR MATRIX COMONADS 9 The previous discussion gives the following description of the category AF of comodules over F . Proposition 1.6. Let A = A1 ×· · ·×An be the product category of finitely many categories A1 , . . . , An . If (F, δ, ε) is a comonad over A, then the category of F –comodules is described as follows: (1) Objects: They are pairs (A, dA ) where A = (A1 , . . . , An ) ∈ A = A1 × · · · × An and dA is a set of morphisms dA = {dkj A : Aj → Fkj Ak : 1 ≤ j, k ≤ n} such that the diagrams Aj dkj A / djj A Fkj Ak dlj A / Fjj Aj Aj ❊ ❊❊❊❊ ❊❊❊❊ j ❊❊❊❊ ❊❊❊❊ εAj ❊  Aj Fkj dlk A  Flj Al lkj δA /  Fkj Flk Al (13) l are commutative for all j, k, l = 1, . . . , n. (2) Morphisms: A morphism f : (A, dA ) → (B, dB ) is a set of morphisms f = {fj : Aj → Bj : j = 1, . . . , n} such that the diagrams dkj A Aj / (14) Fkj Ak Fkj fk fj  Bj dkj B /  Fkj Bk commute, for all j, k = 1, . . . , n. Remark 1.7. Obviously, it is possible to give the dual statements of the above results in the case of monads. So a monad T over A = A1 × · · · × An is given by a set of functors {Tij : Ai → Aj , i, j = 1, . . . , n} with two sets of natural transformations {µikj : Tkj Tik → Tij : i, j, k = 1 . . . , n}, {η i : 1Ai → Tii , i = 1, . . . , n} such that the diagrams Tlj Tkl Tik µklj Tik / Tkj Tik , Tlj µikl µikj  Tlj Til µilj /  Tij ηj Tij / Tjj Tij Tij ❍❍ ❍❍❍❍❍ ❍❍❍❍ ijj ❍❍❍❍ Tij ηi ❍❍❍❍ µ ❍   Tij Tii iij / Tij µ commute for all i, j, k ∈ {1, . . . , n}. The corresponding category of modules (or algebras) is described by dualizing the statements of Proposition 1.6. When T is a normal matrix monad (i.e., Tii = 1Ai and η i = 1 for all i = 1, . . . , n), A is abelian, and Tij = 0 for i > j, these are the kind of categories which were studied by M. Harada in [16], and refereed to as categories of generalized diagrams in abelian categories (see also [22]). 10 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS 2. Triangular matrix comonads. We have seen in Remark 1.7 that the abelian categories investigated by Harada in [16] are categories of modules over triangular normal matrix monads. In this section, we will take advantage of the fact that the existence of a (co)monad representing these categories to give a more systematic approach to their study. In fact, our main theorem in this section is more general in several directions than [16, Theorem 2.3]. At the same time, we find the proof presented here sharpest in some aspects than Harada’s one. We will say (see Definition 2.3) that a matrix comonad F = (Fij ) is triangular if Fij = 0 for all 1 ≤ j < i ≤ n. The understanding of the case n = 2 is the key to the study of the general case. 2.1. Triangular matrix comonads of order 2. Let R C C / V / S D D / (15) be functors. By Proposition 1.1, every comonad structure over the endofunctor   R 0 F = V S of C × D is given by a set of natural transformations  V ρV δS / 2 δR / / V R, V λ / SV, R R2 , S S , V R εR / εS 1C , S  1D / such that the following eight diagrams commute. δR R R2 / RδR δR  R2 V ρV δR R / / ρV R /  V R2 S2 / SδS  R3 V δR  δS δS  S2 / δS S λV V VR ρV VR S  S3 SV / δS V λV  SV SλV /  S2V R ❇❇ δR  R2 δR / R2 ❇❇❇❇ ❇❇❇❇ ❇❇❇❇ εR R ❇❇  /R RεR Vε (17) (18) SεS λV / SV V ❊❊❊ ❊❊❊❊ ❊❊❊❊ ρV ❊❊❊❊ εS V ❊❊❊   /V VR R δS / S2 ❆❆❆❆ ❆❆❆❆ δS ❆❆❆❆ εS S ❆❆❆   2 /S S S ❆❆❆ (16) V λV / SV ρV (19) SρV  VR λV R /  SV R By Remark 1.2, the commutativity of the diagrams (18) just says that (R, δ R , εR ) and (S, δ S , εS ) are comonads, while the commutative diagrams (19) say that (V, λV , ρV ) is an S − R–bicomodule functor. Theorem 2.1. Let      R 0  : C × D V S / C×D be a triangular matrix comonad of order 2, with the comonad structure given by a sextuple of natural transformations as in (17) satisfying the conditions (18) and (19). If the equalizer HEREDITARY TRIANGULAR MATRIX COMONADS ρV C of every pair of arrows of the form V C / V dC / 11 V RC , where (C, dC ) is any R–comodule, do exist in D, and S preserves all these equalizers, then there exists a functor T : CR → DS such that the following diagram commutes, V C / DO , L U  T CR DS / where L : C → CR is the free functor and U : DS → D is the forgetful functor. Moreover, there exists an equivalence of categories   ∼ (C × DS ) (C × D) 0 0  = R 1 R     T 1 V S Proof. The existence of T is proved in [3, Proposition 4.29] in a different context. We give a direct construction for the convenience of the reader. Given an R–comodule (C, dC ), define an object T C of D as the equalizer TC ιC ρV C VC / V dC / (20) V RC / The S–coaction λTC : T C → ST C, making T C an S–comodule, is given by the universal property of the equalizer at the bottom of the following diagram. TC ιC / ρV C VC λV C λT C  ST SC SιC  SV C / / V dC / V RC λV RC SρV C / SV dC /  SV RC Some straightforward computations show that this gives the object part of a functor T : CR → DS . We only make explicit here its definition on morphisms. Given a morphism of R–comodules f : (C, dC ) → (C ′ , dC ′ ), the morphism T f : T C → T C ′ is uniquely determined by the universal property of the equalizer in the bottom row of the serially commutative diagram TC ιC / VC ρV C / V dC / V RC Vf Tf  T C′ ιC ′ / V Rf  V C′ ρV C′ V dC ′ / /  V RC ′   , let us → (CR × DS ) 0 R 0   1    T 1 V S describe the objects of these categories of comodules. By Proposition 1.6, a comodule over In order to construct an equivalence E : (C × D)    12     LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS  R 0  consists of an object A = (C, D) ∈ C × D and a tern of morphisms V S dA = { C ρC RC , D / d ρD VC , D / / SD } such that the following diagrams commute. C ρC / RC ρC D RρC  RC R δC /  R2 C ρD SD / ρD SρD  SD / S δD / VC V ρC  VC ρV C /  V RC D ρD / SD  VC λV C /  SV C D ❈❈❈ / SD ❈❈❈❈ ❈❈❈❈ S ❈❈❈❈ εD ❈  εR C  (21) Sd d ρD RC ❈❈❈❈ ❈❈❈❈ ❈❈❈❈ ❈ / d  S 2C ρC C ❈❈❈ d D (22) C D   On the other hand, a comodule over  1 0 is just a pair of comodules ((C, ρC ), (D, ρD )) ∈ T 1 CR × DS connected with a morphism of S–comodules d′ : D → T C. This last condition is just the commutativity of the diagram D d′ / (23) TC λT C ρD  SD Sd′ /  ST C The functor E will send a tern dA = (ρC , d, ρD ) satisfying the conditions (21) and (22) to the pair of comodules (C, ρC ), (D, ρD ) ∈ CR × DS with the morphism d′ : D → T C is given by the universal property of the equalizer in the following diagram ιC / VC TC O ①< ① ① ① ′ ① d ①① ①① d ρV C V dC / / V RC D The existence and uniqueness of d′ is guaranteed by the commutativity of the third diagram of (21), while the fact that d′ becomes a morphism of S–comodules (namely, the commutativity of (23)) is given by the fourth diagram in (21). This gives the object part of the functor E. On the other direction, there is a functor E ′ defined on objects by sending a morphism of comodules d′ : D → T C, for (C, ρC ) ∈ CR , (D, ρD ) ∈ DS , to the morphism d = ιC d′ . The computation ρVC d = ρVC ιC d′ = V ρC ιC d′ = V ρC d shows that d makes commute the third diagram in (21), while the computation λVC d = λVC ιC d′ = SιC λTC d′ = SιC Sd′ ρD = (Sd)ρD HEREDITARY TRIANGULAR MATRIX COMONADS is just the commutativity of the last diagram of (21). It is not hard to see that E and E ′ are mutually inverse. 13  Remark 2.2. A standard argument shows that, if C and D are abelian categories, and V , R and S are left exact functors, then T is a left exact functor between abelian categories. 2.2. Triangular matrix comonads. Definition 2.3. A functor F : A1 × · · · × An → A1 × · · · × An which is endowed with a comonad structure (F, δ, ε) as in Proposition 1.1 is called a triangular matrix comonad of order n over A = A1 × · · · × An if Fji = 0 for all j > i, that is,   F11 0 0 ··· 0 0  F12 F22 0 · · · 0 0  . (24) F = ..  . . . .  .. .. .. .. .  F1n F2n F3n · · · Fn−1 n Fnn Given 1 ≤ m < n, consider the categories A≤m = A1 × · · · × Am , A>m = Am+1 × · · · × An . All these categories can be considered, in an obvious way, as full subcategories of A. For instance, an object A = (A1 , . . . , Am ) of A≤m is identified with the object (A1 , . . . , Am , 0, . . . , 0) of A. It is clear that there are canonical functors (the projection functors) π≤m : A −→ A≤m , π>m : A −→ A>m . By Proposition 1.1, the triangular matrix comonad (F, δ, ε) over A, gives rise to the triangular matrix comonads (F ≤m , δ≤m , ε≤m ) and (F >m , δ>m , ε>m ) over A≤m and A>m defined, respectively, by   F11 0 0 ··· 0  F12 F22 0 ··· 0  F ≤m =  ..  . . .  .. .. .. .  F1m F2m F3m · · · Fmm δ≤m δ 11  δ 12 =  ...  0 δ 22 .. . 0 0 .. . ··· ··· 0 0 .. . δ 1m δ 2m δ 3m · · · δ mm F >m   ,  Fm+1 m+1 0  Fm+1 m+2 Fm+2 m+2 = .. ..  . .  Fm+1 n Fm+2 n ε≤m 0 0 .. .  ε1 0 · · · 0  0 ε2 · · · 0  = ..   ... ... .  0 0 · · · εm  ··· ···  0 0  ..  .  Fm+3 n · · · Fnn (25) 14 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS δ>m  m+1  δ m+1 m+1 0 0 ··· 0 ε 0 ··· 0 m+1 m+2 m+2 m+2 m+2  δ  0 δ 0 ··· 0  ε ··· 0  , ε>m =  . = . . . . . ..   .. .. .. .. ..  .. . δ m+1 n δ m+2 n δ m+3 n · · · δ nn 0 0 · · · εn  The functor F m : A≤m → A>m given by  F1 m+1 F2 m+1 · · · Fm m+1  F1 m+2 F2 m+2 · · · Fm m+2 Fm =  .. .. ..  . . . F1 n F2 n · · · Fmn         (26) (27) is an F >m −F ≤m –bicomodule functor with the structure natural maps λm : F m → F m F ≤m and ρm : F m → F >m F m defined by λij m : Fij → m M Fkj Fik , i1j λij ∔ · · · ∔ δ imj m = δ Fkj Fik , im+1j λij ∔ · · · ∔ δ inj , m = δ k=1 and ρij m : Fij → n M k=m+1 for 1 ≤ i ≤ m, m + 1 ≤ j ≤ n. Therefore,   ≤m 0 , F = F m F F >m     as comonads. The following consequence of Theorem 2.1 is the basic tool for the study of triangular hereditary comonads in the next section. The particular case where Fii = 1Ai for all i = 1, . . . , n was already stated in its dual form in [16, Theorem 2.3] under the additional hypothesis that the categories Ai are abelian. Theorem 2.4. Let F be a triangular matrix comonad on A = A1 ×· · ·×An as in Definition 2.3. Given an integer m with 1 ≤ m < n, assume that all equalizers do exist in Ai for i = 1, . . . , m, and that the functors Fij preserve equalizers for 1 ≤ i ≤ j ≤ m. Then there exists a functor T m : A≤m F ≤m / A>m F >m (28) such that the category of F -comodules is isomorphic to the category of Gm -comodules, where Gm is the normal triangular matrix comonad defined by T m . That is, AF ∼ = ( A≤m F ≤m × A>m F >m )Gm , where G m   = 1m 0. T 1    Proof. Apply Theorem 2.1 with R = F ≤m , S = F >m , and V = F m .  HEREDITARY TRIANGULAR MATRIX COMONADS 15 Remark 2.5. If the categories Ai are abelian for i = 1, . . . , n, and all the functors Fij , for 1 ≤ i < j ≤ n are left exact, then the functor F is also left exact, as well as the functors F >m , F m , and F ≤m , for every 2 ≤ m ≤ n − 1. On the other hand, the functor T m : A≤m → A>m F >m , constructed as an equalizer (see the proof of Theorem 2.1) becomes F ≤m left exact, see Remark 2.2. 3. Hereditary categories of comodules. An abelian category with enough injectives is said to be hereditary if its global homological dimension is 0 or 1, that is, for every epimorphism E0 → E1 , if E0 is injective, then E1 is injective. Our aim is to characterize when the category of comodules AF of a normal triangular matrix comonad F = (F, δ, ε) over A = A1 × · · · × An is hereditary. We are denoting by δ : F → F 2 the comultiplication of F , and by ε : F → A its counit. The shape of the matrix of functors representing F is   1 0 0 ··· 0 0  F12 1 0 ··· 0 0  F = . (29) . . . . ..   .. .. .. .. .  F1n F2n F3n · · · Fn−1 n 1 We assume that the categories A1 , . . . , An are abelian with enough injectives, and so is A = A1 × · · · × An . If F : A → A is a left exact functor, then AF is also abelian and the forgetful functor AF → A is exact, see [9]. On the other hand, F is exact if and only if Fij is exact for all i, j = 1, . . . , n. First we analyze the case n = 2. The case n = 2. Let A = A1 × A2 , for A1 , A2 abelian categories, and let   1 0 F = : A1 × A2 −→ A1 × A2 F12 1 be a functor with F12 : A1 → A2 a left exact functor. Consider the unique structure of normal triangular comonad on the left exact functor F . According to Proposition 1.6, any F –comodule can be identified with a pair (X, d12 X ), where X = (X1 , X2 ) ∈ A1 × A2 and d12 : X → F X . By convenience, the free functor will be denoted by F : A → AF . 2 12 1 X By Inj.dim(X) we denote the injective dimension of an object X in some abelian category with enough injectives. Proposition 3.1. Assume that A1 and A2 have enough injectives and that F12 is a left exact functor that preserves injectives. Then (1) AF has enough injectives. (2) Each injective comodule in AF is, up to isomorphisms, of the form F (E), for some injective object E in A. In particular, the arrow d12 X is a split epimorphism of A2 12 for any injective F -comodule (X, dX ). (3) Given an F -comodule (X, d12 X ), we have Inj.dim((X, d12 X )) ≤ max{Inj.dim(X1 ), Inj.dim(X2 )} + 1 16 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS Proof. (1). Observe that F : A → AF preserves injectives, since it is right adjoint to the forgetful functor AF → A, which is exact [9, Proposition 5.3]. So, given an injective object E = (E1 , E2 ) in A, we have an injective F -comodule F (E) = (E1 , F12 E1 ⊕ E2 , π), where π : F12 E1 ⊕ E2 → F12 E1 is the canonical projection (this is given by the comonad structure of F ). Now, for every F -comodule (X, d12 X ), we can consider monomorphisms ιi : Xi → Ei in Ai , with Ei injective for i = 1, 2. So we have a monomorphism of F -comodules 12 (ι1 , F12 ι1 ◦ d12 X ∔ ι2 ) : (X1 , X2 , dX ) → (E1 , F12 E1 ⊕ E2 , π) (30) which shows that AF has enough injectives. (2). If we assume that the F -comodule (X, d12 X ) is injective, then the monomorphism (30) splits, so that there exists a morphism of comodules (α, β) : (E1 , F12 E1 ⊕ E2 ) → 12 (X1 , X2 , d12 X ) which splits (ι1 , F12 ι1 ◦ dX ∔ ι2 ). Thus X1 is isomorphic to a direct summand of the injective object E1 , so it is injective. Therefore, without loss of generality, we can suppose that X1 is injective and that E1 = X1 . In this way, we get that β ◦ (d12 X ∔ι2 ) = 1X2 , which shows that X2 is isomorphic to a direct summand of the injective object F12 E1 ⊕ E2 and so it is injective too. On the other hand, since (α, β) is a comodule map, we obtain 12 that π = d12 X ◦ β. Since π is a split epimorphism, we deduce that dX is a split epimorphism. This implies that there exists an isomorphism ω : X2 → F12 X1 ⊕ E2′ in A2 such that ′ π ′ ◦ ω = d12 X , where π is the obvious canonical projection. Thus ′ ′ ′ (1X1 , ω) : (X1 , X2 , d12 X ) → F (X1 , E2 ) = (X1 , F12 X1 ⊕ E2 , π ) is an isomorphism of F -comodules. (3). Put m = max{Inj.dim(π1 (X)), Inj.dim(π2 (X))}, and take, for an F -comodule (X, d12 X ), a resolution in AF 0 / (X, d12 X) f0 / (E 0 , d12 E0 ) / ... / (E m , d12 Em ) fm / (C, d12 C) / 0 12 with (E i , d12 E i ) injective for all 0 ≤ i ≤ m. We need to show that (C, dC ) is injective too. If we apply the projection functor πi , for i = 1, 2 to this resolution, by part (2), we get a resolution in Ai for πi (X) with πi (E k ) injective and d12 E k split epimorphism for m k = 0, . . . , m. Since Inj.dim(πi (X)) ≤ m, we deduce that πi (f ) is a split epimorphism m m 12 and πi (C) is injective for i = 1, 2. We know that d12 C ◦ π2 (f ) = F12 (π1 (f )) ◦ dE m . Hence, 12 d12  C is a split epimorphism and (C, dC ) is injective. By gl.dim(A) we denote the global homological dimension of an abelian category with enough injectives A. Corollary 3.2. The assumptions are that of Proposition 3.1. Then we have gl.dim(AF ) ≤ max{gl.dim(Ai ) : i = 1, 2} + 1 ≤ gl.dim(AF ) + 1 Next we give the desired characterization for hereditary categories of comodules over a triangular normal (2 × 2)-matrix comonad. HEREDITARY TRIANGULAR MATRIX COMONADS 17 Theorem 3.3. Let A = A1 × A2 for A1 , A2 abelian categories with enough injectives, and F12 : A1 → A2 is a left exact functor. Consider the unique normal comonad structure on the endofunctor   1 0 F = : A −→ A F12 1 The category of comodules AF is hereditary if, and only if, the following conditions are satisfied: (a) F12 preserves injectives. (b) A1 and A2 are hereditary. (c) F12 p is a split epimorphism for each epimorphism p : E1 → E1′ between injective objects in A1 . Proof. Suppose that AF is hereditary. Given an injective object E1 in A1 , we know that F (E1 , 0) = (E1 , F12 E1 , 1F12 E1 ) is an injective F -comodule. Now, since (0, 1F12 E1 ) : (E1 , F12 E1 ) → (0, F12 E1 ) is an epimorphism in the hereditary category AF , we have that (0, F12 E1 ) is an injective object in AF . Henceforth, it is clear that F12 E1 is injective in A1 , from which (a) is derived. The statement (b) follows from Corollary 3.2. For the proof of (c), given an epimorphism p : E1 → E1′ in A1 , we get an epimorphism (p, 1) : F (E1 , 0) = (E1 , F12 E1 , 1) → (E1′ , F12 E1 , F12 p) in AF . Since F (E1 , 0) is injective, then so is (E1′ , F12 E1 , F12 p). Therefore, F12 p is a split epimorphism, by the characterization of the injectives in Proposition 3.1. Conversely, consider an F –comodule (X, d12 X ) and a resolution in AF 0 / (X, d12 X) / (f1 ,f2 ) F (E1 , E2 ) / (C, d12 C) / 0 with (E1 , E2 ) ∈ A injective. Since F12 preserves injectives, we have that E1 and F12 E1 ⊕E2 are injective. As f1 and f2 are epimorphism, and Ai is hereditary for i = 1, 2, we deduce that C1 and C2 are injective. Using the fact that F12 (f1 ) is a split epimorphism, we obtain from the commutative diagram F12 E1 ⊕ E2 f2 / C2 d12 C π  F12 E1 F12 f1 /  F12 C1 12 that d12 C is a split epimorphism and so (C, dC ) is injective. This shows that the injective 12 dimension of (X, dX ) is less or equal than 1, which means that AF is a hereditary category.  The case n ≥ 3. Let A = A1 × · · · × An denote a product of categories and F : A → A a normal triangular matrix comonad. Propositions 3.4 and 3.5 below, which will be used to deduce our main result in this section (Theorem 3.6), contain part of the dual form of [16, Theorem 3.6]. 18 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS Proposition 3.4. Assume that Ai is an abelian category with enough injectives for every i ∈ {1, · · · , n}, and that each of the functors Fij : Ai → Aj , for 1 ≤ i < j ≤ n, is left exact. If the category of comodules AF is hereditary, then we have (1) Each injective object (X, dX ) of AF is, up to isomorphisms, of the form X = F (E1 , . . . , En ) for some injective object (E1 , . . . , En ) ∈ A. In particular each arrow dij X : Xj → Fij (Xi ) is a split epimorphism. (2) If i < j < k in {1, · · · , n}, then for every injective object Ei in Ai , we have δEijki : Fik Ei → Fkj Fik Ei is a split epimorphism. (3) For each m ∈ {1, . . . , n − 1}, T m p is a split epimorphism, for any epimorphism p : (E, dE ) → (E ′ , dE ′ ) between injective F ≤m -comodules. Proof. (1). Since F is left exact, then the forgetful functor AF → A is exact. Therefore, the free functor F : A → AF preserves injectives, with implies that F (E1 , . . . , En ) is an injective F –comodule for every injective (E1 , . . . , En ) ∈ A1 × · · · × An . Let us prove that every injective F –comodule is of this form by induction on n. For n = 1, there is nothing to prove, so let n > 1. By Theorem 2.4, we can identify AF with the category of comodules >1 1 1 (A1 × A>1 F >1 )G . By Theorem 3.3, we have that AF >1 is hereditary and that T preserves injectives. In this case, T 1 = F 1 , see equation (27). Therefore, using Proposition 3.1, we have that any injective object of AF is of the form (X, dX ) = (E1 , F 1 (E1 ) ⊕ (V, dV )) for some injective objects E1 ∈ A1 and (V, dV ) ∈ A>1 F >1 . By induction hypothesis we get that (V, dV ) = F >1 (E2 , . . . , En ) for some injectives Ej in Aj , j = 2, · · · , n. Hence (X, dX ) = F (E1 , E2 , · · · , En ) for some injective objects Ei in Ai , i = 1, · · · , n. By Theorem 3.3, 1n 1 (d12 X , . . . , dX ) : F E1 = (F12 E1 , . . . , F1n E1 ) → (E2 , . . . , En ) is a split epimorphism, which implies that d1j X is a split epimorphism for j = 2, . . . , n. Since ij by induction we know that each of the dV , 2 ≤ i < j ≤ n is a split epimorphism, we deduce that every dij X , for every i < j in {1, . . . , n}, is a split epimorphism. (2). It follows from (1), since any object of the form F (0, · · · , 0, Ei , 0 · · · , 0), for i = 1, · · · , n − 2 is injective in AF (recall that the structure morphisms are exactly ijk djk F (0,...,0,Ei ,0...,0) = δEi , i < j < k). m is the (3). By Theorem 2.4, we know that (A≤m × A>m F >m )Gm is hereditary, where G F ≤m m triangular (2 × 2)-matrix comonad constructed by using the functor T from (28). Now we conclude by using Remark 2.5 and Theorem 3.3.  Proposition 3.5. Let n ≥ 2 be a positive integer. Assume that Ai is an abelian category with enough injectives for all i ∈ {1, · · · , n}, and that Fij : Ai → Aj , is left exact for every 1 ≤ i < j ≤ n. If the category of comodules AF is a hereditary category, then we following statements hold. (1) Each of the functors Fij preserves injectives. (2) Fij p is a split epimorphism, for any epimorphism p : Ei → Ei′ between injective objects in Ai , for every 1 ≤ i < j ≤ n. HEREDITARY TRIANGULAR MATRIX COMONADS 19 Proof. (1) For n = 2, the claim follows from Theorem 3.3. Assume n ≥ 3, we will proceed by induction on n. By Theorem 2.4, the hereditary category AF is isomorphic (A1 × >1 1 1 1 A>1 F >1 )F (here, T = F ). From Theorem 3.3 we get that AF >1 is hereditary. By induction hypothesis, all the functors Fij with 2 ≤ i < j preserve injectives. Given an injective object E1 in A1 , we know from the proof of Theorem 2.4 that F 1 E1 = ((F12 E1 , . . . , F1n E1 ), dF 1 E1 ), is injective in A>1 F >1 . On the other hand, by Proposition 3.4.(1) there exists an injective object (E2 , . . . , En ) ∈ A2 × · · · × An such that F 1 E1 = F >1 (E2 , . . . , En ). Therefore, since each F2j for j = 3, . . . n preserves injectives, we obtain that F1k (E1 ) is injective for every k = 2, . . . , n. This completes the induction. (2) Use induction on n and Theorems 2.4, 3.3.  The following is our main result in this section. Theorem 3.6. Assume that Ai is an abelian category with enough injectives for all i ∈ {1, · · · , n}, and consider a normal triangular n × n-matrix comonad F = (Fij ) : A → A such that Fij : Ai → Aj , is left exact for every 1 ≤ i < j ≤ n. Then the category of comodules AF is hereditary if, and only if, the following conditions are fulfilled. (a) T n−1 and every Fij preserves injectives for 1 ≤ i < j ≤ n. (b) For each m ∈ {1, . . . , n − 1}, T m p is a split epimorphism, for every epimorphism p : (E, dE ) → (E ′ , dE ′ ) between injective F ≤m -comodules. (c) For every injective object Ei in Ai , its image δEijki for i < j < k is a split epimorphism. (d) Each of the categories Ai is hereditary. Proof. ⇐). We use induction on n. For n = 2 this implication is given by Theorem 3.3, since in this case we have T 1 = F 1 = F12 . Let n ≥ 3, and suppose the implication is true for any category of comodules over a normal triangular matrix comonad constructed by using a linearly ordered set of length n − 1. Without loss of generality we can suppose by Theorem 2.4 that AF = (AF≤n−1 ≤n−1 × An )Gn−1 , where as before Gn−1 is the triangular 2 × 2-matrix comonad associated to the functor T n−1 : AF≤n−1 ≤n−1 → An given in (28). Since the axioms (a)-(d) are satisfied for the triangular (n − 1) × (n − 1)-matrix comonad F ≤n−1 , by induction hypothesis we know that its category of comodules AF≤n−1 ≤n−1 is hereditary. Henceforth, by Remark 2.5, Theorem 3.3 can be applied as An is already assumed to be hereditary. Therefore, AF is hereditary since T n−1 preserves injectives. ⇒). Conditions (b) and (c) follow from Proposition 3.4. By Proposition 3.5.(1), we know that Fij preserves injectives for every 1 ≤ i < j ≤ n. Since, by Theorems 2.4 and 3.3, we get that T n−1 preserves injectives, we conclude (a). We use induction to prove (d). For n = 2, this implication is clear from Theorem 3.3. Suppose that (d) holds for any hereditary category of comodules over a triangular matrix comonad which was constructed by using a linearly ordered set of length n − 1. By Remark 2.5, we know that T m is a left exact functor for any m ∈ {1, · · · , n − 1}. Thus by Theorems 3.3 and 2.4, we know that AF≤n−1 ≤n−1 and An are hereditary. Hence Ai , i = 1, · · · , n are hereditary. This gives to us condition (d).  20 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS 4. Hereditary triangular matrix coalgebras. We illustrate our results by applying some of them to categories of comodules over coalgebras. We refer to [4] for basic information on coalgebras over commutative rings and their categories of comodules. 4.1. Triangular matrix coalgebras. Let C and D be coalgebras over a commutative ring K, and M be a C −D–bicomodule. Then the functors −⊗K C, −⊗K D give comonads over the category ModK of K–modules, and − ⊗K M : ModK → ModK becomes a (− ⊗K D)−(−⊗K C)–bicomodule functor (see Subsection 2.1). These functors define a triangular matrix comonad on ModK × ModK = ModK×K represented by the K × K–coalgebra   E = C 0 . M D    The comultiplication of this coalgebra over R = K × K is given by ∆  c m  X c(1) 0 = 0 d (c)      X  0 0 0 0 c 0 0 ⊗R + ⊗R (2) 0 d(2) 0 d(1) 0 0 0 (d)   X  X m 0 0 0 (−1) 0 + ⊗R + m(0) m(0) 0 0 0 (m) (m)    0 0 0 , (31) ⊗R 0 m(1) 0 and its counity is defined by     c 0 εC (c) 0 ε = , m d 0 εD (d) where we are using Heyneman-Sweedler’s notation. Assume that C and M are flat as K–modules. The category of right C–comodules is denoted by ComodC , and similarly for any other coalgebra over a commutative ring.   0 on By Theorem 2.1, we have the normal triangular matrix comonad G = 1 −C M 1 ComodC × ComodD , and the equivalence of categories (32) (ComodC × ComodD )G ∼ = (ModK×K )−⊗ E = ComodE . R 4.2. Base change ring by a Frobenius algebra. Let R be a commutative Frobenius algebra over a commutative ring K (see [20]), that is, the functor − ⊗K R : ModK → ModR is right adjoint to the forgetful functor from ModR to ModK (see [23] and [7, Remark 2.3.1]). The counit of this adjunction evaluated P at K gives the Frobenius functional ψ : R → K. If η denotes the unit, then ηR (1) = i ei ⊗ fi ∈ R ⊗K R is the Casimir element. Given any R–coalgebra E, we get the adjoint pairs of functors ComodE o / −⊗R E / ModR o −⊗K R ModK , where the unlabelled arrows denote the forgetful functors that are left adjoints to − ⊗R E and − ⊗K R. By composing these adjoint pairs we obtain the adjoint pair ComodE o / −⊗K E ModK , (33) HEREDITARY TRIANGULAR MATRIX COMONADS 21 where we are using the isomorphism E ∼ = R ⊗R E. In this way, we obtain a comonad − ⊗K E : ModK → ModK , which is determined by the structure of K–coalgebra on E with comultiplication P ˜ ∆ / E ⊗K E, / E x✤ (x),i x(1) ei ⊗ fi x(2) and counity ε̃ = ψ ◦ ε, where ε : E → R the counity of the R–coalgebra E. The adjoint pair (33) gives rise to the comparison functor V : ComodE → (ModK )−⊗E (see [2, Section 3.2]). If E is flat as an R–module, then ComodE is an abelian category and the faithful forgetful functor ComodE → ModR is exact [9]. This easily implies that the forgetful functor ComodE → ModK satisfies the hypotheses of Beck’s theorem (precisely, the dual of [2, Theorem 3.3.10]) and, hence, V is an equivalence of categories. We get thus that the categories of right comodules over the R–coalgebra (E, ∆, ε) and of right ˜ ε̃) are equivalent. comodules over the K–coalgebra (E, ∆, 4.3. Bipartite coalgebras. Let C, D be K–coalgebras, and M be a C − D–bicomodule.   0  C , the Assume that C, D, M are flat K–modules. In the case of the coalgebra E = M D base ring R = K ×K is a Frobenius K–algebra with Frobenius functional ψ : R → K given by ψ(a, b) = a + b for all (a, b) ∈ K × K, and Casimir element e = u ⊗ u + v ⊗ v ∈ R ⊗ R, where u = (1, 0), v = (0, 1). By 4.2 , E is a K–coalgebra. Explicitly, the comultiplication and counity are ˜ ∆  c m  X c(1) 0 = 0 d (c)     X 0 0 0 0 0 ⊗ + 0 d(1) 0 d(2) 0 (d)    X X m 0 0 0 (−1) 0 ⊗ + + 0 0 m(0) m(0) 0   0 c ⊗ (2) 0 0 (m) and (m)    0 0 0 ⊗ , (34) 0 0 m(1)   c m ε̃ = εC (c) + εD (d) 0 d We recover thus the construction of a bipartite K–coalgebra from [21, p. 91] (called triangular matrix coalgebra in [17]). In the following theorem we say that a K–coalgebra is right hereditary if the category ComodC is hereditary. Theorem 4.4. Let C, D be K–coalgebras, and M be a C–D–bicomodule. Assume that   C 0   is right hereditary C, D and M are flat as K–modules. The bipartite K–coalgebra  M D if and only if the following conditions hold. (1) UC M is an injective right D–comodule for every injective right C–comodule U; (2) C and D are right hereditary; (3) pC M is a split epimorphism for each epimorphism p : E1 → E1′ of injective right C–comodules E1 , E1′ . 22 LAIACHI EL KAOUTIT AND JOSÉ GÓMEZ-TORRECILLAS Proof. Since we assume M, C and D to be flat over K, it follows that the functor −C M : ComodC → ComodD is left exact. Now, the theorem follows from Theorem 3.3, the equivalence of categories (32), and the equivalence of categories given at the end of paragraph 4.2.  4.5. Generalized matrix coalgebras. Let Mij , i, j = 1, . . . , n, be a set of modules over a commutative ring K, and consider the endofunctors Fij = Mij ⊗K − : ModK → ModK . Then the matrix functor F : ModnK → ModnK has the structure of a comonad if and only if there exists a set of natural transformations δ ikj and εi as in Proposition 1.1. These ikj natural transformations are determined by linear maps φikj = δK : Mij → Mkj ⊗ Mik and i n ǫi = εK : Mii → K. We thus obtain a K -coalgebra   M11 M21 · · · Mn1  M12 M22 · · · Mn2  . F (K) =  .. ..   ... . .  M1n M2n · · · Mnn By Remark 1.2, each of the entries in the diagonal of this matrix is a K–coalgebra, and Mij is an Mjj − Mii –bicomodule. Also, every φikj factors trough the cotensor product Mkj Mkk Mik . Using the base change of ring from 4.2 with the Frobenius K–algebra R = K n we get the ‘comatrix’ K-coalgebra of [17, Section 2]. We prefer the name matrix coalgebras to avoid confusion with the notion of a comatrix coring (and, in particular, comatrix coalgebra) from [13], which is a different construction. As in 4.2, the comultiplication and counit of this matrix K–coalgebra can be computed explicitly. 4.6. Triangular matrix corings. More generally, we may consider corings C and D over different base rings A and B, respectively (see [4]). Given a C − D–bicomodule M, we get   0   C . Under suitable flatness conditions, a result the triangular matrix A × B-coring  M D similar to Theorem 4.4 may be formulated for corings. Remark 4.7. Let C be an R-coring which is flat as left R-module. Assume that R is a (possibly non commutative) Frobenius algebra over a commutative ring K. 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