Equivalence of categories, Gruson-Jensen duality, and applications

EQUIV ALENCES OF CA TEGOR IES, GR USON-JENSEN DUAL ITY AN D APPLICA TIONS SEPTIMIU CRIVEI AND MIODRAG CRI STIAN IOV ANOV Abstract. F or coalgebras C o ver a field, we study when the categories C M of left C -comodules and M C of righ t C -comodules are symm etric categories, i n the sense that there is a dualit y b et ween the categories of finitely present ed unitary left R -mo dules and finitely presente d unitary left L -mo dules, where R and L are th e functor rings associated to the finitely accessible cate gories C M and M C . 1. Intr oduction Let R b e a r ing with iden tity a nd let M R denote the categor y of unitary right R -mo dules. Consider the functor ring s asso cia ted to the catego ries M R and M R op , s ay S and A op resp ectively . Then there is a duality b et ween the catego ries of finitely presented unitary left S -mo dules and finitely presented unitary right A -mo dules. This is a refor m ulation of the dua lit y proved b y Gruson and Jensen [14, Theor em 5.6 ]. The r esult was ex tended by Dung and Garc ´ ıa [9, Theo rem 2.9] to the ca s e o f unitar y mo dules over a ring with enough idemp otents, and fur thermore, by Crivei and Garc ´ ıa [5, C o rollar y 5.13] to the case of unitary and tor sionfree mo dules over an idemp otent r ing, provided the corresp onding mo dule categories are lo cally finitely genera ted. The latter was the firs t Grus on-Jensen duality established for categ ories not having enough pro jectives. Gruson-Jens en duality can also b e reformulated using the notion of symmetric ca tegories. The ide a of considering such a concept appea red firs t in the work of Herzo g [16], ac cording to the account given by Prest in [21], a nd was later on used by Dung and Garc ´ ıa f or finitely accessible catego ries [9], a nd by Criv ei and Garc ´ ıa for exactly definable categor ies [5]. In this language, the ab ov e Gruson-Jensen dualities say that the categ o ries M R and M R op are symmetric to each o ther whenever they are : categor ies o f unitar y mo dules over a ring w ith identit y , catego ries of unitary mo dules ov er a ring with enough idemp otents, or more generally , categor ies o f unitar y and torsionfree mo dules ov er an idempotent r ing, provided they are lo cally finitely generated. Every finitely a ccessible Grothendieck categor y has a s ymmetric category , but this fails in genera l to b e finitely acces sible Gro thendiec k; see the e xample from [5, p. 395 3 ]. Therefore, an interesting problem is to find further examples of finitely a c cessible Gr othendieck categor ies having their symmetr ic categorie s ag a in finitely acce s sible Grothendieck. In the pr e sent pap er we cons ide r categories o f como dules over a coalgebra C ov er a field. W e first recall gener al re s ults a nd g ive an interpretation o f Gruson-J ensen duality in terms of F reyd categ ories and homoto p y categor ies of certain categorie s of chain complexes. The categ o ries of le ft C -co mo dules and rig h t C -co mo dules are lo ca lly finite Gr othendieck [8], and so finitely acce ssible Grothendieck. The existence of a left-right setting simila r to the case of mo dules sugge sts that this w ould b e a g o o d framework 2000 M athematics Subject Classific ation. 18C35, 18A32, 16T15. Key wor ds and phr ases. Symmetric categories, duality , como dule, coalgebra. 1 2 SEPTIMIU CRIVEI AND MIODRAG CRISTIAN IOV ANOV for a Gruson-Jensen dualit y to hold. Let us fir st note that the ca tegories o f left C -co mo dules and r ight C -como dules ar e equiv a lent to categories of mo dules ov er rings with iden tity if C is a finite dimensional coalgebr a, and mo re ge ne r ally , they are eq uiv alent to categories of mo dules o ver rings with enough idempo ten ts if C is a left and right semip erfect co algebra [8]. So the catego ries of left C -como dules and of right C -co mo dules a re symmetric, or equiv alently , the Gruso n-Jensen duality takes pla ce in these cas es, bec ause it can be re duce d to the aforementioned mo dule-theor etic contexts. On the o ther hand, we shall construct several examples of co a lgebras with very go o d finitar y prop erties for which the Gruson- Jensen duality do es not hold. Although a coa lg ebra has v ery go o d built-in finiteness pr op erties, which might a t first glance lea d o ne to belie ve that such a duality w ould b e in place, it turns out that there are situations where some strong conditions ar e fulfilled but the Gr uson-Jensen duality do es not hold. W e show that for a coalgebr a which is only left or only right semip erfect, the Gruson-J ensen dualit y can fail. W e also sho w that another s et of strong conditions on a coalgebra C is not enough to have such a duality: we give a n example where C ∗ is left a nd right a lmost no etheria n (meaning that cofinite left idea ls and cofinite right ideals are finitely generated) and mo reov er C ∗ is even no etherian on one side, but the Gruson-Je ns en dua lit y betw een the functor r ing s of M C and C M do es no t hold. 2. Finitel y accessible ca tegories an d functor rings Throughout the paper all categ ories will b e additive a nd all modules will b e unitary . Let us recall some terminology o n finitely acces sible categorie s . An ob ject P of a catego ry C with direct limits is ca lled finitely pr esente d if the functor Hom C ( P, − ) comm utes with direct limits. A categor y C is called fi n itely ac c essible (or lo c al ly fi nitely pr esente d in the terminology of [4]) if C has direc t limits, the class fp( C ) of finitely pres e n ted o b jects of C is skeletally sma ll, and every ob ject of C is a dir ect limit of finitely presented o b jects [21]. A Grothendieck categor y is finitely accessible if and o nly if it has a fa mily of finitely presented generato r s. A finitely a ccessible ca teg ory is called lo c al ly c oher ent if e very finitely presented ob ject is coher ent , that is, finitely generated subo b jects of finitely presented ob jects a re finitely presented. Such a catego ry is necessa rily Grothendieck and has a family of finitely presented ge nerators [21]. Now let C be a finitely a c cessible categ o ry and let ( U i ) i ∈ I be a family of repr esentativ es of the isomo rphism classes of finitely presented o b jects o f C . W e asso ciate a ring R to the family ( U i ) i ∈ I in the fo llowing wa y (e.g., see [9], [13]): R = M i ∈ I M j ∈ J Hom C ( U i , U j ) as ab elian group, and the multiplication is given by the rule : if f ∈ Hom C ( U i , U j ) a nd g ∈ Hom C ( U k , U l ), then f g = f ◦ g if i = l and z ero otherwise . Then R is a ring with enough idemp otents [1 2], say R = L i ∈ I e i R = L i ∈ I Re i . The idemp otents e i are the elements of R which are the identit y on U i and zero elsew he r e, and they form a complete family of pairwise orthog onal idemp otents. The ring R constructed ab ov e is called the functor ring of C . The family ( Re i ) i ∈ I is a family o f finitely g enerated pro jective g enerators of the categor y R M of (unitary) le ft R -mo dules. A (unitar y ) left R -mo dule X is finitely presented if and only if there is an exact sequence L i ∈ F 1 Re i → L j ∈ F 2 Re i → X → 0 for some SYMMETR Y FOR COMODULE CA TEGORIES 3 finite se ts F 1 and F 2 of indices in I . No w denote U = L i ∈ I U i . Since Hom C ( U i , U ) = Re i , it follows that a left R -mo dule X is finitely pres e n ted if and only if there is a n exact sequence Hom C ( N , U ) → Hom C ( M , U ) → X → 0 . It is straightforward to sho w that the Y oneda functor Hom C ( − , U ) : fp( C ) → R M is a contrav ariant full, faithful a nd left exact functor, which r e flects monomor phisms to epimorphisms and epimor phisms to split monomorphisms. It induces a duality betw een finitely presented ob jects in C and finitely ge nerated pro jective ob jects in R M . W e recall a co uple of pro p er ties of finitely ac c e ssible categories which will b e needed later on. Prop ositio n 2.1. [21, Theorem 6.1] A finitely ac c essible c ate gory has pr o ducts if and only if the c ate gory of left mo dules over its functor ring is lo c al ly c oher ent. Prop ositio n 2 .2. [23, Pro po sition 2.2] A fin itely ac c essible c ate gory C is lo c al ly c oher ent if and only if fp( C ) is ab elian (with only finite c opr o ducts). The rea der is referred to [4] and [21] for more informatio n on finitely ac c essible categories . 3. Associa ted Freyd ca tegories Let C b e an additive categ o ry . F ollowing [2], we shall as so ciate to C tw o additive c a tegories w hich a re defined as follows. In the morphism categ ory Mor( C ) of C denote an o b ject u : M → N by ( M , u, N ) and a morphism b y ( f , g ) : ( M ′ , u ′ , N ′ ) → ( M , u, N ), where f : M ′ → M a nd g : N ′ → N are such tha t uf = g u ′ . Consider in Mor( C ) the full sub catego ries X a nd Y consis ting of all split monomorphisms and all split epimorphisms r esp ectively . Then the s table categories B ( C ) = Mor( C ) / X and A ( C ) = Mor( C ) / Y are called the F r eyd c ate gories asso ciated to C . An o b ject ( M , u, N ) from Mor( C ) will b e denoted by [ M , u, N ] and { M , u, N } when viewed as an ob ject in B ( C ) and A ( C ) resp ectively . Also, a mo rphism ( f , g ) from Mor( C ) will b e denoted b y [ f , g ] and { f , g } when view ed in B ( C ) a nd A ( C ) resp ectively . The category B ( C ) may b e viewed alternatively as Mor( C ) modulo the co ngruence gene r ated by the s ubgroup of Hom C ( M , N ) consisting o f all mor phisms ( f , g ) : ( M ′ , u ′ , N ′ ) → ( M , u, N ) for which there exis ts a morphism α : N ′ → M such that αu ′ = f , i.e. by commuting square mor phisms factoring as in the following diagram with the lower left tr iangle and the big square co mm uting (only): M u / / = N M ′ f O O u ′ / / N ′ α a a C C C C C C C C g O O Indeed, if ( f , g ) : ( M ′ , u ′ , N ′ ) → ( M , u, N ) is a morphism in Mor( C ), then [ f , g ] = 0 in B ( C ) if and only if there is a morphism α : N ′ → M such that αu ′ = f . In particular [ M , u, N ] = 0 if and only if f is a split monomo rphism. Ther efore, B ( C ) is one of the homoto p y ca teg ories in tro duced b y F reyd in [1 1]. W e no te that the subgr oup of Hom C ( M , N ) deter mining the a bove congruence is the sum of tw o gro ups , namely the group o f all morphisms ( f , g ) : ( M ′ , u ′ , N ′ ) → ( M , u, N ) for which there exists a morphis m α : N ′ → M such that αu ′ = f a nd uα = g and the gro up o f all mor phisms ( f , g ) : ( M ′ , u ′ , N ′ ) → ( M , u, N ) with f = 0 . This fo llows since if f = αu ′ then ( f , g ) = ( αu ′ , uα ) + (0 , g − uα ). It is also worth 4 SEPTIMIU CRIVEI AND MIODRAG CRISTIAN IOV ANOV to note that the equiv alence relation corr esp o nding to the firs t of these g roups is just the usua l homotopy equiv ale nc e of chain c omplexes, restr icted to b ounded chain complexes of length 2. Similarly , A ( C ) is the other homotopy categ ory intro duced by F reyd in [1 1], since it may b e viewed alternatively as Mor( C ) mo dulo the cong ruence gener ated by the subgroup of Hom C ( M , N ) (for ea ch pair of ob jects ( M , N )) consisting of all morphisms ( f , g ) : ( M ′ , u ′ , N ′ ) → ( M , u, N ) for which there exists a mor phism β : N ′ → M such that u β = g , i.e. by commuting sq ua re morphisms factoring a s in the following diagram with the upp er r ight tria ngle and the big square commuting (only): M u / / = N M ′ f O O u ′ / / N ′ β a a C C C C C C C C g O O The F reyd catego ries are related as follows. W e shall denote e q uiv alences of categories by “ ≈ ”. Prop ositio n 3.1. [2, Pro po sition 3.6 ] Le t C and D b e two c at e gories. Then: (i) C ≈ D if and only if A ( C ) ≈ A ( D ) if and only if B ( C ) ≈ B ( D ) . (ii) A ( C op ) ≈ B ( C ) op and B ( C op ) ≈ A ( C ) op . Prop ositio n 3.2. L et C b e an ab elian c ate gory. (i) The c ate gory B ( C ) is an ab elian c ate gory, e quivalent to the c ate gory of exact chain c omplexes of typ e M → N → P → 0 of obje cts in C up to the usual homotopy e quivalenc e. (ii) The c ate gory A ( C ) is an ab elian c ate gory, e quivalent to the c ate gory of exact chain c omplexes of typ e 0 → K → M → N of obje cts in C up to the usual homotopy e qu ivalenc e. Pr o of. (i) The catego ry B ( C ) is ab elian b y [2, P rop osition 4.5]. The seco nd pa r t of (i) follows b y consid- ering the functor given o n o b jects b y [ M , u, N ] 7− → ( M u → N → Coker ( u ) → 0) and observ ing that any morphism [ f , g ] : [ M ′ , u ′ , N ′ ] → [ M , u, N ] in B ( C ) can b e extended to a morphism h : Coker ( u ′ ) → Co ker ( u ). If [ f , g ] = 0, then ther e is α : N ′ → M such that αu ′ = f . Since ( uα − g ) u ′ = uαu ′ − g u ′ = u f − g u ′ = 0 , we see tha t K er ( p ′ ) = Im ( u ′ ) ⊆ Ke r ( uα − g ) so then there is α ′ : Co ker ( u ) → N s uch that α ′ p ′ + uα = g : M u / / = N p / / = Coker ( u ) / / 0 M ′ f O O u ′ / / N ′ α A A A A ` ` A A A A g = O O p ′ / / + + Coker ( u ′ ) α ′ d d h O O / / 0 Moreov er, we have that hp ′ = pg = pα ′ p ′ + puα = pα ′ p ′ , a nd so h = pα ′ ( p ′ is an epimorphis m), so ( f , g , h ) is a null-homotopic morphism of chain complexes. The inverse functor is obvious. (ii) is a nalogous to (i). W e note that the res ult of [2] that A ( C ) and B ( C ) are ab elian also follows by the ab ov e equiv alences.  SYMMETR Y FOR COMODULE CA TEGORIES 5 The full sub catego ries o f finitely presented left o r right mo dules over the functor ring o f a lo cally co- herent category C a re closely r elated to the F reyd categ ories asso cia ted to C (for a genera l c a se, see [2, Corollar y 3.9 ]). Corollary 3 . 3. L et C b e a lo c al ly c oher ent c ate gory and let R b e its functor ring. Then fp( M R ) ≈ A (fp( C )) and fp ( R M ) ≈ B (fp( C )) op . Pr o of. W e sketc h the second pa rt. Note that fp ( C ) is abelia n by Pro po sition 2.2. A finitely pres en ted left R -mo dule is the cokernel of a morphism be t ween finitely generated left R - mo dules . Using the duality betw een the categor ie s of finitely gener ated pro jective left R -mo dules and finitely g enerated pr o jective right R -mo dules, the equiv alence b etw een the category finitely g enerated pro jective rig h t R -mo dules and fp( C ), and Pro p os ition 3.2, we obtain a duality b etw een fp( R M ) and B (fp( C )). The dualit y is explicitly given on ob jects as follows. Le t ( U i ) i ∈ I be a family of representativ es of the isomor phism classes o f finite dimensional left C -co modules and deno te U = L i ∈ I U i . W e hav e see n that a n ob ject X of R M is finitely presented if and only if there is a finite presentation Hom C ( N , U ) → Hom C ( M , U ) → X → 0 for some ob jects M and N in fp ( C ). This is induced b y an o b ject [ M , u, N ] of B (fp( C )).  Now w e can give an interpretation of sy mmetry of finitely acc e ssible categor ies in terms of F reyd cate- gories. Definition 3.4. Let C and D be tw o finitely acces s ible catego r ies. W e c a ll C and D symmetric ca tegories if there is a duality betw een the categories B (fp ( C )) and B (fp( D )). Let us note that C and D a re symmetric in the ab ov e sense if and only if they are symmetr ic in the sense of Dung and Ga rc ´ ıa [9]. T o this end, let us deno te by R and L the functor rings of C and D re spe c tiv ely . Then b y P rop osition 3 .3 w e ha ve B (fp( C )) ≈ B (fp( D )) op if a nd only if (fp( R M )) op ≈ fp( L M ) if and only if C and D are symmetric in the sense of [9 , Definition 2 .8]. 4. Co algebras and co m o dules Now we recall se veral facts on coalgebra s and como dules, mainly following [8]. Let C b e a co algebra ov er a field k . Denote by C ∗ = Hom k ( C, k ) the dua l algebra of C ov er k . Then C ∗ is a to po logical vector s pa ce endow ed with the weak- ∗ topo logy , in whic h the closed s ubspaces are annihilators in C ∗ of subspa c e s of C . In this topolo gy , C ∗ has a basis of neig h b ourho o ds of 0 consis ting of ideals of finite co dimens io n. The coalgebr a C is called left F -no etherian if every closed a nd co finite le ft ideal o f C ∗ (in the sense that C ∗ /I is finite dimensio nal) is finitely genera ted in M C ∗ (see [2 2] and [6]). In pa rticular, every right semiper fect coalgebr a is left F -no etherian [6, Theo rem 2.1 2]. The coalg ebra C is called left str ongly r eflexive o r left almost n o etherian (o r C ∗ is left almost n o etherian , see [22] and [6]) if every cofinite left ideal I of C ∗ is finitely gener ated [6]. Clearly , every left almost no etherian co algebra is left F -no etherian. A right C ∗ -mo dule M is ca lle d r ational if for every x ∈ M , there a re x 1 , . . . , x n ∈ M a nd c 1 , . . . , c n ∈ C such that xc ∗ = n X i =1 x i c ∗ ( c i ) 6 SEPTIMIU CRIVEI AND MIODRAG CRISTIAN IOV ANOV for every c ∗ ∈ C ∗ . The cla ss Rat( M C ∗ ) of rational right C ∗ -mo dules is c lo sed under submodules, direct sums, direct pro ducts and ho momorphic images. In fa c t, Rat( M C ∗ ) = σ [ C C ∗ ], where σ [ C C ∗ ] is the full sub c ategory of the category M C ∗ of rig h t C ∗ -mo dules consisting o f the mo dules s ubgenerated by C . Denote by C M the ca tegory o f left C -como dules. Then there is an iso morphism o f categor ies C M ∼ = Rat( M C ∗ ). W e shall frequently make the identification b etw een left C -co mo dules and ra tional rig h t C ∗ -mo dules. The categor y of rational r ight C ∗ -mo dules, and so the categ ory of left C -como dules , is a Grothendieck category , which has a family of finite dimensional generator s, namely the rationa l r ight C ∗ -mo dules of the for m C ∗ /I with I a closed co finite (t wo-sided) idea l of C ∗ . It is easy to note that a C -como dule is finitely pr esented if and only if it is finitely gener ated if and o nly if it is finite dimensional. W e denote by fd( C M ) the c la ss o f finite dimensiona l left C -como dules. Similar consideratio ns may be made for the categor ies M C of rig h t C -co mo dules and Rat( C ∗ M ) of rational left C ∗ -mo dules. Note that the functor ( − ) ∗ defines a duality b etw een the categor ies fd( C M ) and fd( M C ) [8]. W e also refer to [8] and [3] fo r basics o n c oalgebra s and their como dules. W e note a n interesting characteriza tion of the categ ory Mo r (fd( M C )). O ne can pr ove witho ut muc h difficult y that this catego r y is eq uiv alent (in fact, isomorphic) to the category of finite dimensio nal right como dules over the upper triangula r matrix coalg ebra M 2 ∆ ( C ) =   C C 0 C   with comultiplication and counit given b y (we use the Sw eedler no tation with the summation symbol omitted)   x y 0 z   7− →   x 1 0 0 0   ⊗   x 2 0 0 0   +   y 1 0 0 0   ⊗   0 y 2 0 0   + +   0 y 1 0 0   ⊗   0 0 0 y 2   +   0 0 0 z 1   ⊗   0 0 0 z 2   ,   x y 0 z   7− → ε C ( x ) ε C ( z ) . The dua l algebr a of this coalgebr a is prec isely the algebr a M ∆ 2 ( C ∗ ) o f upp er triang ular matrices with ent ries in C ∗ . The stated equiv a lence ass o ciates to a n y M u − → N the M 2 ∆ ( C )-como dule M ⊕ N with coaction given b y:   m n   7− →   n 0 0   ⊗   n 1 0 0 0   +   f ( m 0 ) 0   ⊗   0 m 1 0 0   +   0 n 0   ⊗   0 0 0 m 1   . The inv erse of this equiv alence g o es as follows. Deno te E =   0 0 0 ε   , N =   0 ε 0 0   , F =   ε 0 0 0   as elements o f M ∆ 2 ( C ∗ ). Then, for a rig ht M 2 ∆ ( C )-como dule T , we have T = F · T ⊕ E · T and the inv ers e functor asso cia tes the ob ject N : E · T → F · T in Mor (fd ( M C )), with the mo rphism N giv en by left m ultiplication b y N ∈ M ∆ 2 ( C ∗ ) = ( M 2 ∆ ( C )) ∗ . The co rresp ondence on morphisms is similar. SYMMETR Y FOR COMODULE CA TEGORIES 7 5. Positive answers In wha t follows we shall discuss symmetry for c o mo dule ca tegories, or equiv a lent ly , the ex is tence of a Gruson-Jens en dua lit y in the case of co mo dule catego ries. As noted in the introduction, the ca se of a (left and right) semip erfect coalg ebra ca n b e solved by r educing it to a result for mo dule categories ov er rings with enough idempo ten ts. W e now give some details. Theorem 5.1. L et C b e a (left and right) semip erfe ct c o algebr a. Then the c ate gories C M and M C ar e symmetric. Mor e pr e cisely, ther e is a duality b etwe en fp( R M ) and fp( M R ) . Mor e over, if C is a right semip erfe ct c o algebr a and t he c ate gories C M and M C ar e symmetric, then C is (left and right) semip erfe ct. Pr o of. Denote by R a nd L the functor rings of C M and M C resp ectively . Since C is (left and rig h t) semip erfect, it is known and not difficult to show (see, fo r exa mple, [8, Chapter 3 ]) that Rat( C ∗ C ∗ ) = Rat( C ∗ C ∗ ) is an idemp otent ideal of C ∗ denoted simply Ra t( C ∗ ). Mor eov er, Rat( C ∗ ) is a ring with enough idempo ten ts, and the catego ry of rational left C ∗ -mo dules (right C -como dules) is the same as that o f left Rat( C ∗ )-mo dules (see also [3]). Similar ly , the categor y of left C -co mo dules is the same as that of right Rat( C ∗ )-mo dules. The r efore, applying [9, Theorem 2.9] (which solves the ca se of unitary mo dules ov er a ring with eno ugh idemp otents), there is a dua lity b etw een fp( R M ) and fp( M L op ). But R ≃ L op since R = M M ,N ∈ C M C Hom( M , N ) ≃ M M ,N ∈ C M C Hom( N ∗ , M ∗ ) = M P,Q ∈ M C Hom C ( P, Q ) = L op and, b ecause ( − ) ∗ is a contrav ariant functor, the m ultiplication of R (comp osition) is reverted. F o r the la s t part, no te that for a right semip erfect co algebra the category M C has a generating family of small pro jective ob jects (see [8 , Chapter 3]), and so, by well known results of ca tegory theory (e.g. see [15]) it is equiv ale nt to the category of unital left A -mo dules for a ring A with enough idemp otents. In this case, its symmetr ic ca tegory C M must be equiv alent to the catego ry of r ight unital A -mo dules by [9, Theorem 2.9 ], and so it has a ge ne r ating set of pro jectiv e o b jects. There fo re, C is also left semip erfect.  Prop ositio n 5.2. The c ate gories C M and M C ar e symmetric if and only if A (fd ( C M )) ≈ B (fd( C M )) if and only if A (fd( M C )) ≈ B (fd( M C )) . Pr o of. Using the duality b etw een fd( C M ) and fd( M C ) and Pr o po sition 3.1, we hav e B (fd( M C )) ≈ B (fd( C M ) op ) ≈ A (fd( C M )) op . Now it follows that C M and M C are symmetric if a nd only if there is a duality b et ween B (fd( M C )) and B (fd( C M )) if a nd o nly if A (fd ( C M )) ≈ B (fd( C M )). The last assertion follows in a similar wa y .  6. Nega tive answers In this section w e will study the simple o b jects of the categ o ry fp( R M ) of finitely presented left R -mo dules ov er the functor ring R of a loc ally finite ca teg ory and use the conclus io ns to give examples of situations where the Gr uson-Jensen duality does not happ en. Prop ositio n 6. 1. L et C b e a fi n itely ac c essible c at e gory with pr o ducts having functor ring R . Then a finitely pr esent e d left R - mo dule X is simple in fp( R M ) if and only if it is simple in R M . 8 SEPTIMIU CRIVEI AND MIODRAG CRISTIAN IOV ANOV Pr o of. By Pr op o sition 2.1, fp( R M ) is lo cally co herent. Then by P rop osition 2.2, fp( R M ) is ab elian (with only finite co pro ducts). It is eas y to deduce that a morphism b e t ween finitely presented left R -mo dules is a monomor phism (epimo r phism) in fp( R M ) if and only if it is a monomor phism (epimo rphism) in R M , and kernels and co kernels are computed in R M . Then the subo b jects o f an ob ject in fp ( R M ) are the finitely pr esented submo dules. Therefore, a simple finitely presented left R -mo dule will be a simple ob ject in fp( R M ). Conv ersely , assume that X is simple in fp( R M ). Deno te U = L i ∈ I U i , where ( U i ) i ∈ I is a family of representatives of the is o morphism cla sses o f finitely presented ob jects of C . If X is not a simple left R -mo dule, then ther e is some x ∈ X such that 0 6 = Rx 6 = X . But since X is unitar y , e i x = x fo r so me idempo ten t e i corres p onding to a split inclusion U i ֒ → U , and so in fact Rx is finitely genera ted and is a quotient of Re i . Since fp( R M ) is lo ca lly coherent, Rx has to b e finitely presented, a nd this co n tra dic ts the assumption tha t X contains no non-trivial finitely presented submo dules.  A finitely accessible ca tegory will b e called lo c al ly finite if every finitely pres en ted o b ject has finite length. In particular , it is lo cally coher en t, and so Gr o thendieck. The fo llowing result is imp ortant as it computes the s imple ob jects o f the abe lian categ ory B (fp ( C )) for a lo cally finite catego ry C , and it will be the k ey ingredient tha t we will use in or der to constr uct examples of non-s ymmetric como dule categ o ries. Prop ositio n 6.2. L et C b e a lo c al ly fi nite c ate gory with functor ring R . An obje ct X in fp ( R M ) is simple if and only if ther e is an obje ct u : M → N in B (fp( C )) , which is the c orr esp onding obje ct of X (up to e qu ivalenc e) thr ough the duality of Cor ol lary 3.3 , and such that M → N is an epimorphism with simple kernel and M is inde c omp osable inje ctive. Pr o of. Denote U = L i ∈ I U i , where ( U i ) i ∈ I is a family of repr esentativ es of the iso morphism classes of finitely prese nted ob jects of C . Consider a finite presentation Hom( N , U ) Hom( u,U ) / / Hom( M , U ) ϕ / / / / X / / 0 of the simple ob ject X for some o b jects M , N of fp( C ). Then u : M → N is an ob ject in B (fp( C )). Note that we may a ssume that M is indecomp osa ble. Indeed, write M = L i ∈ F M i as a finite direc t sum of indecomp osables. Then since ϕ 6 = 0 at least one of its restrictio ns to Hom( M i , U ) is nonzero, and since X is simple, this r estriction will b e surjective. Mor e ov er, the kernel of this r estricted mor phism (as a morphism of left R -mo dules) will be finitely presented, so in par ticular finitely gener ated. Let σ M be the (split) inclusion of M in to U . The fact that X is simple transla tes equiv alently to the following: for any α ∈ Hom( M , U ), either α factor s throug h u : M → N (so α = αu for some α ∈ R ) or there are β ∈ R and γ ∈ Hom( N , U ) such that σ M = β α + γ u (i.e. the ge ner ator σ M of Hom( M , U ) is generated by α mo dulo Hom( N , U ), for a n y α 6 = 0 mo dulo Hom( N , U )). M α    s σ M L L L L L L L L L L u / / N γ   M α   ?>=< 89:; 1 M A A A A A A A A u / / + + N γ   U β / / U = M ⊕ U ′ M P β / / M SYMMETR Y FOR COMODULE CA TEGORIES 9 Now, since α has finite image in U , and M is a direct summand with complement U ′ M , and β and γ also hav e finite images, this co ndition is equiv alent to the o ne g iven by the right diagra m a bove, wher e P c an be any ob ject of finite length of C . That is, w he ne ver α do es not fa c tor thro ugh u there are β and γ such that 1 M = β α + γ u . W e hav e tw o cases: • (1) Im ( u ) = N . Let K = Ker ( u ). W e note tha t a quotient α : M → M /L s plits throug h u if and o nly if K ⊆ L . When this is not true, eq uiv alently , when L ∩ K 6 = K , we see that L ∩ K ⊆ Ker ( β α ) ∩ K er ( γ u ), and since β α + γ u = 1 M , we get L ∩ K = 0 . This shows that K is a simple sub ob ject of M (pick L any sub o b ject of K for this purpo se); if K = 0, then we would hav e X = 0, a contradiction. Now choo se an arbitrar y inclusio n morphis m α : M → P and find such β , γ as ab ov e (be cause α do es not factor throug h u ). The equality 1 M = β α + γ u s hows that the morphism h = ( α + u ) : M → P ⊕ N is split by ( β + γ ). So we hav e h ( M ) ⊕ M ′ = N ⊕ P = ( L k ∈ K N k ) ⊕ ( L j ∈ J P j ), with each N k and P j indecomp osable. Now, s ince all these ob jects ar e of finite length, by (an equiv alent form of the) Krull-Remak-Schmidt-Azumay a theorem, we find that h ( M ) is a complemen t of a dir ect sum of so me s ubfamily of { N k , P j } k ∈ K,j ∈ J (see e.g. [20, Lemma 9.2.2] or [1]). Since h ( M ) is indeco mpo s able, and it cannot b e isomor phic to a ny of the N k ’s bec ause length( M ) ≥ length( N k ) ( u is an epimorphism with no nzero kernel) w e g et h ( M ) ⊕ N ⊕ ( L j 6 = j 0 P j ) = N ⊕ P , for j 0 ∈ J for which M ≃ h ( M ) ≃ P j 0 . In particula r , h ( M ) ∩ ( N ⊕ ( L j 6 = j 0 P j )) = 0, and ther efore, if p is the pr o jection onto P j 0 we o btain h ( M ) ∩ Ker ( p ) = 0 , so ph is a mono morphism. Thus, a s M ≃ P j 0 , ph is a n is omorphism. B ut note tha t ph = p ( α + u ) = pα + p u = pα , since Im ( u ) ⊆ N ⊆ Ke r ( p ). This shows that M splits off in P , and therefore it is injective in the category fp( C ). Using, similar arguments as those in [8, Chapter 2, Section 4], it follows that M is injective in C . • (2) Im ( u ) 6 = N . W e first no te that u must b e a monomorphism. Consider α : M − → P = Im ( α ) the corestrictio n of u . If this s plits through u , then it is easy to see that Im ( u ) is a direct summand in N . This situation reduces to the previous one, since if u : M → Im ( u ) ⊕ T , then w e can ha ve an exact sequence Hom(Im ( u ) , U ) → Hom( M , U ) → X → 0 (i.e. then M → N and M → Im ( u ) represent the same o b ject of Mor (fp( C ))). As b efore, finding β a nd γ with β α + γ u = 1 M will yield that K e r ( u ) = 0, since Ke r ( u ) = Ker ( α ). But now we ha ve a context similar to that of the pro of of (1), with the ro les of α and u reversed: u is a monomorphism and α is an epimorphism. As b efore, we get that u splits; but this situation is not pos sible, since in this case, u defines the zero ob ject of B (fp( C )) ≃ (fp( R M )) op (see Prop ositio n 3.3), i.e. X = 0. Finally , let u : M → N an epimorphism with simple k ernel S a nd M finite dimensional indecompo sable injectiv e. W e chec k that the finitely presented ob ject X corr espo nding to u : M → N is simple, and for this we chec k the equiv a lent condition given by the ab ov e right diagra m. Let α : M → P b e a morphism. If Ker ( α ) ⊇ S , then o bviously α factor s through u . Otherwis e , Ker ( α ) ∩ S = 0, a nd since M is indecomp osa ble injective and S is simple, S is the so cle of M a nd is essential in M . Thus we have Ker ( α ) = 0 (since K er ( α ) ∩ S = 0), so α is injectiv e. But then, since M is an injectiv e ob ject, α splits off and we can find β with β α = u , and so w e can ta ke γ = 0.  Next we recall the following definition and results from [7, 1 7, 19]. Definition 6. 3. A right C - como dule M is calle d chain (or uniserial ) if the lattice of its rig ht subco - mo dules is a chain. A co algebra C is called right serial if its indecomp osable injectiv e r ig ht como dules 10 SEPTIMIU CRIVEI AND MIODRAG CRISTIAN IOV ANOV are uniserial, equiv alently , C is a direct sum of uniserial right como dules. C is called serial if it is b oth left and right seria l. It is shown in [7] that a coa lgebra is s erial (see also [10, Coro llary 25.3 .4 ] and the pr o of of [17, Prop ositio n 4.4]) if and only if any finite dimensio nal right (equiv a len tly , any finite dimensio nal left) C -como dule is a direct sum of chain como dules . Recall that when X is a lo cally finite par tially or dered set a nd F is a field (so for each x ≤ y there are only finitely many z such that x ≤ z ≤ y ), C X = F { ( x, y ) ∈ X × X | x ≤ y } , the vector s pace with basis { ( x, y ) ∈ X × X | x ≤ y } b ecomes an F - coalgebra when endow ed with the co m ultiplication ∆(( x, y )) = P x ≤ z ≤ y ( x, z ) ⊗ ( z , y ) and co unit ε (( x, y )) = δ ( x,y ) . Let C N be the coalgebr a asso ciated in this wa y to the se t o f natural num b ers. Then w e hav e: Prop ositio n 6.4. C N = L n ∈ N F { ( n, p ) | p ≥ n } is a de c omp osition of C N into inde c omp osable inje ctive right c omo dules, and C N = L n ∈ N F { ( k , n ) | k ≤ n } is a de c omp osition of C N as a dir e ct sum of inde c omp osable inje ctive left c omo dules. Mor e over, al l these ar e chain c omo dules. Conse qu en tly, C N is a serial c o algebr a. It is right semip erfe ct, and not left semip erfe ct. Pr o of. It is easy to see that E r ( n ) = F { ( n, p ) | p ≥ n } a re rig h t sub como dules and E l ( n ) = F { ( k , n ) | k ≤ n } are left sub como dules. Also note that the cora dic a l of this coalgebr a is co commutativ e, and consists of the span of all grouplike elements F { ( n, n ) | n ∈ N } . Therefor e, the so cle (the simple part) o f ea ch E r ( n ) (and E l ( n )) is precisely F { ( n, n ) } , so it is simple. This shows that they a re indecomp osa ble injective (since they are summands of C N ). W e a lso se e that E r ( n ) /F { ( n, n ) } ≃ E r ( n + 1) and E l ( n ) /F { ( n, n ) } ≃ E l ( n − 1). Then, using a n inductiv e pro cess, we show that the Lo ewy filtratio n of each E r ( n ) has simple quo tien t at each step, so E r ( n ) are c hain como dules by [17]. Similarly , E l ( n ) are c hain como dules. This coalg ebra is rig h t semip erfect s ince the injective indecomp osa ble left como dules E l ( n ) are finite dimensional, and not left semiper fect since there are infinite dimensional injectiv e indecomp osable r ight como dules E r ( n ) (in fact, all these a re infinite dimensiona l).  Example 6.5. The coalgebr a C N do es not have Gr uson-Jensen duality . This follows immediately by applying Theorem 5.1, since this co algebra is right but not left s emiper fect. It also follows b y applying the r esults of this section, which provide a little more information a bo ut the categor ies of mo dules o ver the functor rings. Indeed, note that there is a simple injective left C N -como dule, namely F { (0 , 0) } . By Prop ositio n 6.2, there will be some s imple mo dules in the category o f finitely presented left mo dules ov er the functor ring R o f C M . How ever, ag ain by Pro p os ition 6.2, there will b e no simple mo dules in the category of finitely pr esented left mo dules over the functor ring L ≃ R op of M C , be cause there is no finite dimensional injective right C N -como dule, so there c an b e no epimorphis m Q → Q/ T with Q injective a nd T a simple right sub como dule o f Q (which would be the ob ject fro m B (fp( C )) corr esp onding to a simple finitely prese nted left L -mo dule). This illustrates the res ult of Theorem 5.1 b y example. In what follows, we show that other p ossible re pla cements of the semip erfect hypo thesis of Theo rem 5.1 to very strong finitary pro per ties are still not enough for this dualit y to hold. W e recall the follo wing g eneral type of construction, dual to the generalized uppe r triangula r ma trix ring (see also [1 8, Section 4]). Let C a nd D be tw o coalg e br as and M b e a C - D bicomo dule. W r ite c → c 1 ⊗ c 2 , SYMMETR Y FOR COMODULE CA TEGORIES 11 d → d 1 ⊗ d 2 for the co m ultiplications of C and D re s pec tively , and m 7→ m − 1 ⊗ m 0 and m → m 0 ⊗ m 1 the left and right coactio ns of M (b ecause of the bicomo dule condition, there is no danger of confusion even if b oth coa c tions ar e present ). Then o n the vector space H = C ⊕ M ⊕ D =   C M 0 D   we can int ro duce the coalgebr a structure g iven by:   c m 0 d   7− →   c 1 0 0 0   ⊗   c 2 0 0 0   +   m − 1 0 0 0   ⊗   0 m 0 0 0   + +   0 m 0 0 0   ⊗   0 0 0 m 1   +   0 0 0 d 1   ⊗   0 0 0 d 2     c m 0 d   7− → ε C ( c ) ε D ( d ) Using these, it is easy to note that we hav e:   C M 0 D   =   C 0 0 0   ⊕   0 M 0 D   as left co mo dules ,   C M 0 D   =   C M 0 0   ⊕   0 0 0 D   as rig ht co mo dules . Also, one can see that H ∗ =   C ∗ M ∗ 0 D ∗   , the us ua l upper triangula r matr ix ring with M ∗ a C ∗ - D ∗ bimo dule. Example 6.6. Let C b e the divided p ow er coalgebr a ov er a field F , which is the finite dual of the algebra of for ma l p ower series F [[ X ]]. It ha s a basis ( c n ) n ≥ 0 , and co multiplication ∆( c n ) = P i + j = n c i ⊗ c j and counit ε ( c n ) = δ 0 n . Let D = F a s a co algebra . Then ε : C → D is a morphism o f coalg e br as, and so M = C b ecomes a C - D bicomo dule (in fact the rig h t D -como dule s tr ucture is nothing else but the vector space structure of C ). Let H =   C M 0 D   =   C C 0 F   be the c o algebra defined above. More specifica lly , this c o algebra has a basis { ( c n ) n ; ( x n ) n ; t } with com ultiplication ∆( c n ) = P i + j = n c i ⊗ c j , ∆( x n ) = P i + j = n c i ⊗ x j + x n ⊗ t , ∆( t ) = t ⊗ t and co unit g iven by ε ( c n ) = δ 0 n , ε ( t ) = 1, ε ( x n ) = 0. Then: • The deco mpo sition of H as indecompo s able injective left como dules is H = F { ( c n ) n } ⊕ F { ( x n ) n ; t } . • The deco mpo sition of H as indecompo s able injective r ight como dules is H = F { ( c n ) n ; ( x n ) n } ⊕ F t . Denote by R and L the functor r ings o f H M and M H resp ectively . As b efore, by Prop osition 6.2, this shows tha t there ar e no simple mo dules in fp( R M ), b ecause there ar e no finite dimens io nal injective como dules in H M , but there are some simple mo dules in fp ( L M ) (corresp onding to the simple injective right H -como dule F t ), so the tw o ca tegories are not dual to each other. Note that this coa lgebra has some other very nice “finitary” prop erties , and still the Gruson-Jense n duality do es not ho ld. W e can see that the second term of the c o radical filtration o f H is H 1 = F { c 0 , c 1 , x 0 , x 1 , t } . Then, since dim( H 1 ) < ∞ , by [6, Theor em 2.8] we get that H is left a nd right almost no etherian, and 12 SEPTIMIU CRIVEI AND MIODRAG CRISTIAN IOV ANOV therefore, it is a lso left and right F - no etherian. Moreov er, H is left artinia n (i.e. artinian as a left H -co mo dule) and H ∗ is left no e ther ian (but no t right noether ian). This can b e see n b y lo oking at the dual ring H ∗ =   C ∗ C ∗ 0 F   of H . Then, by w ell k nown facts a bo ut ma tr ix rings of this type, H ∗ is left no etherian b ecaus e C ∗ = F [[ X ]] and F are noether ia n and C ∗ is left finitely generated, but it is no t right noether ian b ecause C ∗ is no t finitely genera ted over F . The fact that H is left artinian can b e easily seen either b e cause H ∗ is left noetheria n, or becaus e each indecomp osable injectiv e left H -como dule has only finite dimensio nal sub como dules. Remark 6.7 . We n ote that a left and right semip erfe ct c o algebr a is left and right F -no etherian, and the left (and right) F -no etherian is a c ate goric al c ondition for a c o algebr a: it m e ans t hat every finite dimensional r ational C ∗ -mo du le is finitely pr esente d as left C ∗ -mo du le, that is, the c ate gories of fp( M C ) and f .d. fp C ∗ M (finitely pr esent e d C ∗ -mo du les which ar e finite dimensional) c oincide. Be c ause of t his, one c ould t hen exp e ct that the r esult on the G-J duality for C ∗ might offer some insight to such a duality b etwe en the functor rings of fp( M C ) and fp( C M ) , but the ab ove example shows t hat this do es n ot hold. In fact, it is se en t hat even st ro nger c onditions, such as C ∗ almost no etherian on b oth s ides and even no et herian on one side ar e not enough to have the G-J duality. The ab ove example is also motivate d by the fol lowing fact: a c o algebr a C which is right semip erfe ct and left artinian (i.e. C is artinian as a left C -c omo dule, e quivalently, C ∗ is left no etherian) is ne c essarily fi n ite dimensional. Inde e d, if C is left artinian, then C = n L i =1 E ( S i ) a finite sum of artinian inde c omp osable inje ctives in C M ; but sinc e C is right semip erfe ct, these E ( S i ) ar e finite dimensional, and so C is finite dimensional. Thus, the c onditions of C b eing left artinian and C b eing right semip erfe ct c an b e thought as t wo r amific ations of the fin ite dimensional c o algebr as whose “interse ction ” is the class of finite dimensional c o algebr as. In view o f the ab ov e and of E xample 6.6, it is then natural to ask the following question: Question If C is a left and right art inian c o algebr a, ar e the c ate gories C M and M C symmetric? This w ould in fact provide a first example of a Gruson-Jensen dualit y (symmetry) where the catego ries of como dules in question a re of quite a different na ture than the catego r ies of unitary mo dules over a ring with eno ugh idempotents. Examples of Gruso n-Jensen duality for categor ies other than categ ories of unitary mo dules over a ring with enoug h idemp otents are scarce, apparently the only other known generic example of this t yp e b eing the cas e of the categor ies of unitary and torsionfree mo dules ov er an idemp otent ring, provided they are lo cally finitely genera ted (see [5]). Let us p oint out that the main obs ta cle in establis hing a Gr uson-Jensen duality for such ca tegories is the lack o f enough pro jective ob jects. Ackno wledgment The authors wish to thank the referee for c a reful comments, and for sugges ting the second part of Theorem 5.1. W e also wis h to thank Stefaan Caene p eel for useful disc us sions on the sub ject and the hos pita lit y of V rije Universiteit Brussels in the summer o f 2009. The seco nd author also wishes to thank the first for his hospitality during his 20 10 visit in the Department of Mathematics of “Bab e¸ s-Bolyai” Univ ersity of Cluj-Nap o ca. SYMMETR Y FOR COMODULE CA TEGORIES 13 The first author acknowledges the suppo rt of the Romanian grant P N- I I-ID-P CE-20 0 8-2 pro ject ID 2271. F o r the s econd author, this work w as supp orted by the strategic grant POSDRU/89/1.5 / S/5885 2, Pr o ject “Postdoc to ral programe for training s cient ific resear c hers ” cofinanced by the Eur op ean So cial F und within the Sectoria l Op erational Pr o gram Human Resources Development 200 7-201 3. References [1] F.W. Anderson and K.R. F uller, Ri ngs and Categories of Mo dules, Graduate T exts in Mathematics, Springer, Berl in- Heidelber g- New Y ork, 1974. [2] A. 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F acul ty of Ma thema tics and Computer S cience, “Babes ¸-Bol y ai” University, Str. Miha il K og ˘ alniceanu 1, 400084 Cluj-Napoca, Romania E-mail addr ess : crivei@math. ubbcluj.ro University of Bucharest, F ac. Ma tema tica & Informa tica, S tr. Academiei 1 4, Bucharest 010014 , Romania, &, University of Southern California, 36 2 0 South Vermont A ve. KAP 108, Los Angeles, CA 90089, USA E-mail addr ess : yovanov@gmai l.com, iovanov@usc.ed u