On Extensions of Rational Modules

ON EXTENSIO NS OF RA TIONAL MODULES MIODRAG CRISTIAN IOV ANOV Abstra ct. W e inv estiga te when the catego ries of all ratio nal A -mo du les and of finite dimensional rational modu les are close d und er extensions insi de the category of C ∗ - mod u les, where C ∗ is the co finite top ological completion of A . W e give a complete chara cterization of these tw o properties, in terms of a top ologica l and a homological condition. W e also give connections to other important notions in coalg ebra theory suc h as c oreflexive coalgebras. In particular, we are able to generalize many prev iously kn own partial results and answ er some q uestions in this direction, and obtain large classes of coalgebras for whic h rational m o du les are closed un der extensions as w ell as v ario us examples where this is not t rue. Introduction a nd Preliminaries Let C b e an ab elian category and A b e a fu ll sub category of C . W e say that A is closed if it is closed under sub ob jects, quotien ts and direct sums (copro d ucts). With any closed sub category A of C , there is an asso ciated trace f u nctor (or preradical) T : C → A which is righ t adjoin t to the inclusion fun ctor i : A → C ; that is T ( M )=the sum of all sub ob jects of M which b elong to A . Classical examples include the torsion grou p of an ab elian group or the or more generally the torsion part of an R -mo dule for a comm u tativ e domain R , or the singular torsion, whic h is, for a left R -mo dule M , defined as Z ( M ) = { x ∈ M | ann R ( x ) is an essen tial ideal } (see [G72 ]). Another imp ortan t example is that of rational mo dules: giv en an algebra A , we call a mo dule rational if it is a sum (colimit ) of its fi nite dimensional subm o dules. Th e category of rational A -mo du les is equiv alent to the category of right C -comodules M C , wh er e C = R ( A ) = A 0 is the coalgebra of representati v e fu n ctions on A or the finite d u al algebra of A ([DNR , Chapter 1]). The dual of th e coalge bra is again an algebra, and th e catego ry of rational A -mo d ules is a closed sub category of left C ∗ -mo dules C ∗ M = C ∗ − Mo d . In fact, there is a morphism A → C ∗ , and C ∗ can b e though t as a completion of A with resp ect to the linear top ology h a ving a basis of neigh b orho o ds of 0 consisting of fi nite ideals of A (see [T af72]). The ab o v e d escrib ed situation h as ro ots in algebraic geometry . Let G b e an affine algebraic group scheme ov er an algebraically closed field K of p ositiv e c haracteristic p and let A b e the Hopf algebra rep r esen ting G as a fun ctor fr om Commutative Algebr as to Gr oups ( A is the ”algebra of fun ctions” of G ). If M b e the augmen tatio n ideal of A , one d efi nes M n := A { x p n | x ∈ M } ⊆ A . Then there is a sequence of fin ite d imensional Hopf algebras A/ M n , with canonical p ro jections A → A/ M n → A/ M s for n ≥ s . Geometrically , these corresp ond to the n’th p o w er of the F rob enius morph ism of the scheme. T he d u al family of finite dim en sional Hopf algebras ( A/ M n ) ∗ together with the morphisms ( A/ M s ) ∗ ֒ → Key wor ds and phr ases. T orsion Theory , Splitting, Coalgebra, R ational Mo dule. 2000 Mathematics Subje ct Classific ation . Primary 16W30; Secondary 16S90, 16Lxx, 16Nxx, 18E40. 1 2 MIODRAG CR ISTIAN IO V ANOV ( A/ M n ) ∗ forms an in d uctiv e family of Hopf algebras, and the algebra B = lim − → n ( A/ M n ) ∗ is called the hyp er algebr a of G . The catego ry of finite dimensional G -repr esen tati ons is equiv alen t to the category of finite dimensional A -como dules. Note that B embedd s in A ∗ canonically as algebras, and so eve ry A -como dule is a B -mo dule, and h ence, the category of (rational) G -mod ules or A -comod ules is a closed sub categ ory of the category of B - mo dules (mo d ules o v er the hyper algebra of G ). W e refer th e r eader to [FP87, Su78] and the classical text [J] f or fu rther details. Giv en a closed su b category A of C , one is often int erested also in the situation when A is also closed under extensions, i.e. if 0 → M ′ → M → M ′′ → 0 is a sequ en ce in C with M ′ , M ′′ in A , then M is in C . In this case, A is u sually called a Serr e sub category of C . The pr op ert y of b eing closed u nder extensions has also b een called lo calizing, for an obvio us reason: in this situation, on e can form the quotien t category C / A (which is a lo calization of C ), see [G]. The p r op ert y of A b eing closed un d er extensions is easily seen to b e equiv alent to T b eing a radical, i.e. to the prop erty T ( M /T ( M )) = 0 for all ob jects M ∈ C . Examples of localizing categories are torsion mo dules ov er a comm utativ e domain, or f or the singular torsion m o dules o v er a rin g R (un d er an additional assump tion that the singular id eal of R is 0; [G72]), b ut, in general, a sub category n eed not b e lo calizing. F or example, in R − M od one could consider th e su b-category of semisimple mo dules in the situation when R is n ot semisimple. It is a n atural question to ask when the ab o v e mentioned rational sub category of of C ∗ -mo dules is closed und er extensions, or equiv alen tly , when is the functor Rat : C ∗ M − → M C = Rat ( C ∗ ) a radical; such a coalge bra is said to h a v e a (left) torsion Rat fu nctor. This qu estion wa s inv estigated by man y authors [C03, CNO, GTN, L74, L75, Rad73, HR74, Sh76, TT05]; it is known, for example, that if C is a righ t semip erfect coalgebra, then it has a (left) torsion Rat functor [L74, GTN ], or if C is su ch that C ∗ is left F -No etherian (meaning that left ideals whic h are closed in the fi nite top ology of C ∗ are fi n itely generated) then it also has a torsion Rat fu nctor ([Rad73, CNO]). On the other hand, if s ome aditional conditions on C are satisfied, then one can giv e an equiv alen t caracterization of this p rop erty: if the coradical C 0 of C is fi nite d imensional (suc h a coalgebra is called almost connected) then C has a torsion (left or righ t) Rat fu nctor if an d on ly if ev ery cofinite left (or, equiv alently in this case, right) ideal of C ∗ is fin itely generated (see [C03, CNO], and [HR74] f or an equiv alent form ulation), wh ich is furth er equiv alent to all the terms of th e coradical fi ltration b eing finite dimensional. In fact, as noted in [CNO], all the known classes of examples of coalge bras for wh ic h rational left C ∗ -mo dules are closed und er extensions turned out to b e F -Noetherian. This motiv ated the authors in [CNO] to raise the question of whether the conv erse is true, i.e. if a coalge bra with torsion (left) rat fu nctor is necesarily (left) F -Noetherian. Ho w ev er, the construction in [TT 05] show ed that th is is n ot true. Hence, the pr oblem of completely caracterizing this prop ert y remained op en, and is p erh aps one of the m ain problems in coalgebra theory , and is imp ortant from a more general categ orical p ersp ectiv e as p oin ted out ab o v e. In this pap er, we prop ose a solution to this pr oblem. Ther e are seve ral main p oint s of our treatmen t. W e provide a complete c haracteriz ation for wh en a coalg ebra has a left rational torsion fun ctor in Theorem 3.7, in terms of a top ological condition and a homologica l one. In fact, w e fir st characte rize the situation when the finite d imensional rational (left, or righ t) mo d ules are closed u nder extensions (ins id e C ∗ M = C ∗ -Mod ); this is equiv alen t to a top ological condition, namely , the set closed cofinite (equiv al en tly , op en) ideals of C ∗ is stable u nder p ro ducts. Theorem 3.7 states ON EX TENSIONS OF RA TIONAL MODULES 3 Theorem A c o algebr a has a left r ational torsion functor if and only if the op en c ofinite ide als ar e close d under pr o ducts and Ex t 1 C ∗ ( C ∗ C 0 , C ∗ C ) = 0 W e also analyze and give conditions equiv alen t to the h omological pr op ert y in the ab o v e statemen t. S econd, we also pr o vide connections of this p r op ert y with other imp ortan t coalge bra n otions, suc h as coreflexive coalgebras. In fact, if the set of simple como d ules is a non-measurab le s et, then the ab o v e top ological condition is equ iv alen t to the coalgebra b eing coreflexiv e, w h ic h is a concept d ev eloped b y seve ral authors [Rad73, Rad74, HR74, T af72] (we note that no examples of n on -measur able set is known, so this equiv alence is in place in any ”reasonable” example). Third, this general result allo w s us to giv e general classes of examples of coalgebras having a r ational torsion fun ctor. In particular, these generalize all the previously kn o wn examples, as w ell as sev eral results c haracterizing coreflexiv e, F -n o etherian or almost no etherian coalgebras (i.e. coalgebras C such that any cofinite id eal in C ∗ is finitely generated). In particular, we sho w th at: Theorem If the set of c ofinite close d ide als is close d u nder pr o ducts and one of the fol- lowing c onditio ns is satisfie d, then left r ational C ∗ -mo dules ar e close d under extensions in C ∗ M : -left i nj e ctive inde c omp osables have finite c or adic al filtr ation; -right inje ctive i nde c omp osables have finite c or adic al filtr ation; -for e ach right inje ctive inde c omp osable, ther e is some n such that the n ’th qu otient of the L o ewy series (c or adic al filtr ation) L n +1 E /L n E is finite dimensional One other generaliz ation of results on coreflexiv e coalgebras, F -no etherian and almost no etherian coalgebras [C03, HR74] and also on coalge bras with rational torsion functor [CNO] is Theorem 4.8; this states that Theorem If in the Ext Gabriel quiver of the left c omo dules of a c o algebr a C , vertic es have finite left de g r e e, then the fol lowing ar e e qui v alent: - E ( T ) has finite dimensional terms in its c or adic al filtr ation of al l right c omo dules T -in E ( T ) ∗ c ofinite submo dules ar e finitely gene r ate d - C is (left) F -no etherian - C is lo c al ly finite (se e Se ction 4) - C has a left r ational torsion func tor F ourth, we also giv e some simple equiv alent c haract erization of the prop erty of C b eing left F -no etherian. With this, we can easily construct simple examples of coalgebras whic h hav e a left rational torsion functor but are not F -notherian, also answ ering the aforemen tioned question of [CNO]. Moreo ver, the examples sh o w that, while a righ t s emip erfect coalge bra is necesarily left F -no etherian, it do es n ot hav e to b e right F -n o etherian, and also that the F -no etherian pr op ert y is not a left-righ t symmetric prop erty . W e close with a few op en qu estions and future p ossible d irections of researc h . In p articular, w e ask whether p otentia lly the homological E xt condition migh t b e eliminated from the main r esult (or, m ore generally , w h ether E xt ( C 0 , C ) is alw ays 0); also, all th e known examples of coalgebras with left rational functor, also h av e a right r ational fun ctor, so it is natural to ask wh ether this is alw a ys the case. Based on our general examples, we suggest a p ossible wa y to attempt a counte r-example, and also to appr oac h th e other questions. Basic prop erties and notat ions. W e refer to th e monographs [DNR] and [M] for definitions and prop erties of coalge bras and rational como du les. W e recall here a few basic facts and n otatio ns; most defin itions and notions used are recalled and refered throughout 4 MIODRAG CR ISTIAN IO V ANOV the p ap er. If M is a left C -como dule, we consider the finite top ology on M ∗ with a basis of neigh b ourh o o ds of 0 consisting of su b mo dules (subspaces) of M ∗ of the type X ⊥ M = { f ∈ M ∗ | f | X = 0 } , for finite dimensional su b como dules (subsp aces) X of M . W e write X ⊥ = X ⊥ M when there is no danger of confusion. The submo d ules X ⊥ for X ⊂ M are precisely the closed su bmo dules of M . F or a s u bspace V of M ∗ w e write V ⊥ = { x ∈ M | f ( x ) = 0 , ∀ f ∈ V } . W e ha v e X ⊥ ⊥ = X for X ⊂ M and Y ⊥ ⊥ is the closure of Y ⊆ M ∗ . W e will frequently u s e the isomorphisms of C ∗ -mo dules ( M /X ) ∗ ∼ = X ⊥ , M ∗ /X ⊥ ∼ = X ∗ and X ⊥ / Y ⊥ ∼ = ( Y /X ) ∗ for sub comodu les X ⊆ Y of M . Recall, for example, from [IO] that any finitely generated subm o dule N of M ∗ is closed in this top ology , i.e. N = X ⊥ , for some su b como dule X of M . Also, n ote that if X ⊥ is a closed cofinite left ideal of C ∗ and I is a left ideal s uc h that X ⊥ ⊂ I , then I is closed too: let f 1 , . . . , f n ∈ I b e such th at I = P i C ∗ f i + X ⊥ ; then n P i =1 C ∗ f i = Z ⊥ is closed, so w e get th at I = Z ⊥ + X ⊥ = ( Z ∩ X ) ⊥ . Also, note that if C ∗ /I is fi n ite dimensional rational then I is closed, since th er e is W ⊥ closed cofin ite ideal su ch that W ⊥ ( C ∗ /I ) = 0, which sho ws that W ⊥ ⊆ I . W e will denote by J the Jacobson radical of C ∗ , J = J ac ( C ∗ ); w e ha v e that J = C ⊥ 0 , where C 0 is the coradical of C (see [DNR, Chapter 3]). If C 0 ⊆ C 1 ⊆ · · · ⊆ C n is the coradical filtration of C , then we also h a v e that ( J n ) ⊥ = C n , ( J n ) ⊥ ⊥ = C ⊥ n . F or a C -como dule M w e write L 0 M ⊆ L 1 M ⊆ · · · ⊆ L n M ⊆ . . . for its Lo ewy series (coradical fi ltration), so L n +1 M /L n M is the so cle (semisimp le su b(co)mod ule of M /L n M ). F or como du les, one has M = S n L n M . One also h as that a right comod ule M is semisimple if and only if J M = 0, and L n M = M if and only if J n +1 M = 0 (e.g. [I09, Lemma 2.2]). Recall that a coalgebra ( C , ∆ , ε ) is called lo cally fi nite ([HR74]) if for every t w o finite dimensional subs p aces X , Y of C , we ha v e that the ”w edge” X ∧ Y is finite dimensional, where X ∧ Y = ∆ − 1 ( X ⊗ C + C ⊗ Y ). Note that by the fund amen tal theorem of coalgebras, this is equiv ale n t to asking the condition for all fi nite sub coalg ebras X , Y of C . In d eed, if it is true for th e wedge of fi nite sub coalgebras, and X, Y are fi nite dimen s ional subs paces of C , there are finite d im en sional sub coalge bras X ′ , Y ′ suc h that X ⊆ X ′ and Y ⊆ Y ′ and then X ∧ Y ⊆ X ′ ∧ Y ′ whic h is fi nite dimensional. W e also r ecall that a left (or right) C -comod ule X is called quasi-finite [T ak77 ] if and only if Hom( S, X ) for eve ry simple left (right ) como d u le S , or, equiv alent ly , Hom ( N , X ) is fin ite dimensional for ev ery fin ite dimensional como dule N . F or a righ t como du le ( M , ρ ) subsp ace X ⊆ C w e denote by cf ( X ) the coalgebra of co efficien ts of X , the smallest sub coalg ebra W of C f or whic h ρ ( M ) ⊆ M ⊗ W . 1. Extensions of finite dimensional ra tional mo dules One of the imp ortant notions connected to the ”rational extension p r oblem” w ill prov e to b e that of lo cally finite coalg ebras. W e thus fir st note a f ew in teresting caracterizations of lo cally finite coalge bras. Lemma 1.1. If X is a right sub c omo dule of C , W a finite dimensional sub c o algebr a of C , then ( X ∧ W ) /X = P f ∈ Hom( P, C /X ) f ( P ) , wher e P is a finite dimensional gener ator of W ∗ M = M W . In p articular, if S is a simple right c omo dule and W = cf ( S ) , then X ∧ W = P f ∈ Hom( S,C /X ) f ( S ) . ON EX TENSIONS OF RA TIONAL MODULES 5 Pro of. Let Y ⊂ C b e the sub space suc h that Y /X = P f ∈ Hom( P, C /X ) f ( S ), and denote π : C → C /X the canonical pro jection. T hen for y ∈ Y , write y = P i y i suc h that eac h ( C ∗ y i + X ) /X is a quotien t of P ; since P ∈ M W w e ha v e π ( y i 1 ) ⊗ y i 2 ∈ Y /X ⊗ W . Therefore, ∆ ( y i ) = y i 1 ⊗ y i 2 ∈ π − 1 ( Y /X ⊗ W ) = Y ⊗ W + X ⊗ C . Ind eed, if w e write ∆( y i ) = P j u j ⊗ a ij + P k v k ⊗ b ik , wher e { u j } is a fi xed b asis of X and { v k } is a fi xed b asis of Y mo du lo X , then we get π ( v k ) ⊗ b ik ∈ Y /X ⊗ W so b ik ∈ W sin ce π ( v k ) is a basis of Y /X . Therefore, y i ∈ ∆ − 1 ( X ⊗ C + Y ⊗ W ) ⊆ X ∧ W , so Y ⊆ X ∧ W . Conv ersely , let y ∈ X ∧ W , so y 1 ⊗ y 2 = ∆ ( y ) ∈ X ⊗ C + C ⊗ W . Then π ( y 1 ) ⊗ y 2 ∈ C /X ⊗ W , so C ∗ π ( y ) is canceled b y W ⊥ . Th erefore, it has an indu ced W ∗ ∼ = C ∗ /W ⊥ -mo dule s tructure. Hence, there is an epimorph ism ( P ) n → C ∗ π ( y ) → 0 as it is finite dimensional, and this shows that π ( y ) ∈ Y .  Lemma 1.2. Th e fol lowing assertions ar e e qui v alent: (i) C is lo c al ly finite; (ii) F or every finite dimensional right sub c om o dule X of C , C /X is quasifinite. (iii) F or every simple right sub c o mo dule T of C , C /T is quasifinite. (iv) F or every two si mple sub c o algebr as U, W of C , U ∧ W i s finite dimensional. (v) The left c om o dule versions of (ii) and (iii). Pro of. (i) ⇒ (ii) F or simple S and W = cf ( S ), sin ce P f ∈ Hom( S,C /X ) f ( S ) = ( X ∧ W ) /X is finite d imensional, it f ollo ws that Hom( S, C /X ) is finite d imensional to o. (ii) ⇒ (i) Use (ii) for a su b coalgebra X , w h ic h is then also a right sub como dule; tak e W also a finite d imensional sub coalgebra of C and since P = W ∗ is a finite d im en sional co- mo dule wic h generates M W , it follo ws that ( X ∧ W ) /X = P f ∈ Hom( W ∗ ,C /X ) f ( W ∗ ) is finite dimensional sin ce Hom( W ∗ , C /X ) is finite dimensional. (ii) ⇒ (iii) and (i) ⇒ (iv) are obvious. (iii) ⇒ (ii) is prov ed b y induction on l eng th ( X ). F or simple X it is obvious. Assu me the statement is true for X w ith l eng t h ( X ) ≤ n ; consider X with l eng th ( X ) = n + 1, and let Y ⊂ X w ith X/ Y simple. Since C / Y is quasifinite, we can wr ite the so cle of C / Y as s ( C / Y ) = L i ∈ I S n i i , with S i simple nonisomorp h ic, n i finite, and s u c h that X/ Y = S 0 (0 ∈ I ). Then there is an embedd ing C / Y ֒ → L i 6 =0 E ( S i ) n i ⊕ E ( S 0 ) n 0 with fi nite n i ’s, whic h can b e extended to an essential em b edding C /X ∼ = ( C / Y ) / ( X/ Y ) ֒ → H = E ( S 0 ) /S 0 ⊕ E ( S 0 ) n 0 − 1 ⊕ L i 6 =0 E ( S i ) n i . No w , since for eac h simp le S , Hom( S, E ( S 0 ) /S 0 ) is finite dimensional, then Hom( S, H ) = Hom( S, E ( S 0 ) /S 0 ) ⊕ Hom( S, E ( S i ) n i − δ i, 0 ) is obvi- ously finite dimen s ional. It follo ws that Hom( S, C /X ) is finite dimensional, and the pro of is fin ished. (iv) ⇒ (iii) If T , L are a simple r ight su b como dules of C , let U = cf ( T ), W = cf ( L ); then T ∧ W ⊆ U ∧ W is finite dimensional. Therefore, by the previous Lemma, w e get P f ∈ Hom( L,C /T ) f ( L ) = ( T ∧ W ) /T is fin ite dimen s ional, and so Hom( L, C /T ) is finite dimensional.  W e prov e now the connection mentio ned at the b eginn in g of this section: Prop osition 1.3. L e t C b e a c o algebr a such that left r ational C ∗ -mo dules ar e close d under extensions. Then C is lo c al ly finite. 6 MIODRAG CR ISTIAN IO V ANOV Pro of. Assume C is not lo cally fin ite. Then there are simple left como dules S, L such that Hom ( L, C /S ) is infi nite dimensional. Let X ⊆ C b e s uc h that X/S ∼ = L n ∈ N L , and let X n /S = L k ∈ N \{ n } L . Then X/X n ∼ = L and T n X n = S . Let I = P n X ⊥ n ⊆ S ⊥ ; note that I ⊥ = T n X ⊥ n ⊥ = T n X n = S , and X ⊥ ⊆ X ⊥ n . Note that T ⊥ /X ⊥ ∼ = ( X/T ) ∗ ∼ = Q n L ∗ , whic h is a rational mo dule, since it is cancele d b y cf ( L ) ⊥ ; moreo v er, it b ecomes a cf ( L ) ∗ - mo dule, with cf ( L ) a finite d imensional simple coalgebra, so it is semisimple. T herefore, ev ery quotien t of T ⊥ /X ⊥ is rational and semisimp le. Obviously , I 6 = T ⊥ , sin ce I /X ⊥ is coun table dimensional while T ⊥ /X ⊥ is uncount able d imensional. Therefore, since T ⊥ /X ⊥ is semisimple rational, we can find I ⊂ K ( T ⊥ suc h that T ⊥ /K is rational simp le. S ince C ∗ /T ⊥ is rational, the h yp othesis sho ws that C ∗ /K is rational. Therefore, K is closed by the remarks in the in tro duction: K = Y ⊥ . But then Y = K ⊥ ⊂ I ⊥ = S , so K = Y ⊥ = S ⊥ , a con tradiction.  W e note that in [GNT] a como dule M such that ev ery qu otien t of M is quasifin ite w as called str ictly quasi-finite. Lemma 1.2 p r o vides an in teresting connection b et w een the notion of lo cally finite coalgebra and a slightl y we ak er v ersion of ”strictly quasifin ite coal- gebra”, where qu otien ts only b y finite d imensional como du les are quasifinite. In p articular, w e h a v e Corollary 1.4. A strictly quasifinite c o algebr a is lo c al ly finite. W e note ho w ev er that the conv erse of this corollary do es not hold. W e use an example b elonging to a family whic h pro vides man y examples an d counterexamples in coalge bras (see [I3 ]). Th ese are coalgebras which are sub coalgebras of the full qu iv er coalgebra of a quiv er and hav e a basis of paths; suc h coalgebras w ere called path sub coalg ebras in [DIN], and hav e also b een called m on omial in literature. Example 1.5. Consider the fol lowing qui ver: a b 1 o o x 1 / / y 1 b 2 O O x 2   y 2 . . . b n   x n _ _ y n . . . and let C b e the K -c o algebr a with b asis { a, b n , x n , y n , p n | n ∈ N } wher e p n is the p at h p n = x n y n , as a sub c o algebr a of the ful l p at h c o algebr a of the ab ove qu iver. We have thus formulas for ∆ and ε : ∆( a ) = a ⊗ a ∆( b n ) = b n ⊗ b n ∆( x n ) = a ⊗ x n + x n ⊗ b n ∆( y n ) = b n ⊗ y n + y n ⊗ a ∆( p n ) = a ⊗ p n + x n ⊗ y n + p n ⊗ b n ε ( a ) = ε ( b n ) = 1 ; ε ( x n ) = ε ( y n ) = ε ( p n ) = 0 . As in [DIN] , we have C 0 = K { a, b n | n ∈ N } (this is the K -sp an), C 1 = K { a, b n , x n , y n | n ∈ N } , C = C 2 . We note then that C /C 1 ∼ = ON EX TENSIONS OF RA TIONAL MODULES 7 L N K { a } as left c omo dules, sinc e in C /C 1 , the c omultiplic ation maps p n − → a ⊗ p n . This shows that C is not strictly quasifinite. However, we note that C is lo c al ly finite. L et X, Y b e finite subsp ac es of C ; then ther e ar e finite dimensional sub c o algebr as U, W which have b ases of p aths and such that X ⊆ U and Y ⊆ W . F or example, one c an take U to b e the sp an of al l p aths which o c cu r in elements i n X and their r esp e ctive subp aths. L et V n = K { a, b k , x k , y k , p k | k ≤ n } . Then V n is a sub c o al gebr a of C and S n V n = C . Sinc e U, W ar e finite dimensional, it fol lows that ther e is some n such that U, W ⊆ V n . But now it is e asy to se e that V n ∧ V n = V n , so X ∧ Y ⊆ U ∧ W ⊆ V n ∧ V n = V n . This shows that C is lo c al ly finite. Prop osition 1.6. Assume the finite dimensional left r ational C ∗ -mo dules ar e close d under extensions. L et 0 → M → P → N → 0 b e an exact se quenc e of A -mo dules su c h that M , N ar e r ational, N is finite dimensional and P is cyclic. Then M /J M is finite dimensional. Pro of. Note that if w e quotien t out b y J M , we get an exact sequence 0 → M /J M → P /J M → N → 0 with the same prop erties, so we m a y in fact assu me that M is semisim- ple, and sh ow that then M is finite dimensional. Let us firs t sho w th at in M there are only finitely many isomorp h ism t y p es of s imple comod ules. Indeed, assume otherwise, and th en again after taking another qu otien t w e ma y assu me we ha v e an exact sequen ce 0 → L n ∈ N S n → P ′ → N → 0, with S n noniso- morphic simple como dules. Also, P ’ is cyclic so there is an epimorphism π : C ∗ → P ′ . Let I = π − 1 ( L n S n ); then since ob viously C ∗ /I ∼ = N is finite dimensional r ational, we ha v e that I is closed: I = X ⊥ , with X a fi n ite dimensional left sub como dule of C . Also, if M n = L k 6 = n S k , then we ha v e an exact sequence 0 → S n → P ′ / M n → N → 0, and as the finite d imensional rational left C ∗ -mo dules are closed un der extensions, we see that P ′ / M n is rational. T herefore, π − 1 ( M n ) = X ⊥ n , X ⊂ X n ⊆ C . Note that T k ∈ F M k 6⊆ M n for an y finite set F ⊂ N with n / ∈ F . This s ho ws that ( P k ∈ F X k ) ⊥ = T k ∈ F X ⊥ k 6⊆ M n = X ⊥ n , so X n 6⊆ P k ∈ F X k . Also, ( X n /X ) ∗ ∼ = X ⊥ /X ⊥ n ∼ = S n , so X n /X is simp le. Let Y = P n ∈ N X n ; the previous considerations sho w that Y /X ∼ = L n S ∗ n . Also, k er( π ) = π − 1 (0) = π − 1 ( \ n ∈ N M n ) = \ n ∈ N π − 1 ( M n ) = \ n ∈ N X ⊥ n ⊇ Y ⊥ . W e thus ha v e an epimorph ism I / Y ⊥ → I / k er π ∼ = L n ∈ N S n , so an epimorph ism X ⊥ / Y ⊥ → L n S n . But X ⊥ / Y ⊥ = ( Y /X ) ∗ = ( L n ∈ N S ∗ n ) ∗ = Q n ∈ N S n , so we ha v e ob tained an epimorph ism of left C ∗ -mo dules. Y n ∈ N S n → M n ∈ N S n → 0 But now, since L n S ∗ n is a direct sum of nonisomorp hic simp le rational left como dules, we ha v e an em b edding L n ∈ N S ∗ n ֒ → C 0 ⊂ C , and therefore, d ualizing w e get an epimorphism C ∗ → Q n ∈ N S n . Combining with the ab ov e, we obtain an epimorphism C ∗ → L n ∈ N S n ; b ut L n ∈ N S n is not fi nitely generated, and this is a con tradiction. 8 MIODRAG CR ISTIAN IO V ANOV No w, let us sho w that it is not p ossible to h a v e in fi nitely man y copies of the same comod ule S as summands of M . Assume otherwise, and k eep the ab o v e notations, only no w w e will ha v e S n ∼ = S , for some simple righ t como dule S . As ab o v e, w e obtain Y /X ∼ = L n ∈ N S , and since X is fin ite d imensional, this con tradict s the h yp othesis that C is lo cally finite, which follo w s by Prop osition 1.3. This end s our p r o of.  Prop osition 1.7. A ssume the same hyp otheses (and notations) of the pr evious pr op o- sitions hold. Then the se quenc e M ⊇ J M ⊇ J 2 M ⊇ · · · ⊇ J n M ⊇ . . . eventual ly terminates: J n M = J n +1 M = . . . , and M /J n M is finite dimensional. Pro of. By th e previous p rop osition, w e hav e M /J M is finite dimen s ional. Therefore, since the fin ite d imensional rational are closed under extensions, P /J M is r ational to o since it fi ts into the exact sequen ce 0 → M /J M → P /J M → P / M → 0. W e can again apply th e p revious p rop osition for th e sequence 0 → J M → P → P /J M → 0 and obtain that M /J 2 M is finite dimensional, and inductive ly , w e obtain M /J n M is fi n ite dimensional and also that P /J n M is rational. L et again π : C ∗ → P b e an epimorphism. As b efore, I = π − 1 ( M ) is closed, s o I = X ⊥ , with X a left sub como d ule of C . Similarly , since C ∗ /π − 1 ( J n M ) ∼ = P /J n M is rational, so again π − 1 ( J n M ) = X ⊥ n , for a left finite dimensional sub como d ule X n ⊂ C . Obviously X ⊥ n ⊇ X ⊥ n +1 , so X n ⊆ X n +1 , and it su ffices to sh o w th at X n = X n +1 = . . . from some n onw ard . Let Y = S n ∈ N X n ; since ker( π ) ⊆ X ⊥ n , k er( π ) ⊆ T n ∈ N X ⊥ n = ( P n ∈ N X n ) ⊥ = Y ⊥ . T herefore, there is an epimorph ism ind uced by π , making the d iagram commutat iv e: X ⊥ π / / p   M p { { w w w w w w w w w X ⊥ / Y ⊥ W e now sh o w that the so cle of Y /X is X 1 /X . F or this, let Z/X b e a simple como dule with Z ⊂ Y ; this means th at ( Z/X ) · J = 0 or, equiv alently J ( Z/X ) ∗ = J · ( X ⊥ / Z ⊥ ) = 0. But J · ( Y /X ) ∗ = J p ( M ) = p ( π ( π − 1 ( J M ))) = pπ ( X ⊥ 1 ) = p ( X ⊥ 1 ) = X ⊥ 1 / Y ⊥ . Therefore J · ( X ⊥ / Z ⊥ ) = J · ( X ⊥ / Y ⊥ Z ⊥ / Y ⊥ ) = J · ( X ⊥ / Y ⊥ ) + Z ⊥ / Y ⊥ Z ⊥ / Y ⊥ = X ⊥ 1 / Y ⊥ + Z ⊥ / Y ⊥ Z ⊥ / Y ⊥ = 0 This sho ws that X ⊥ 1 / Y ⊥ ⊆ Z ⊥ / Y ⊥ and so Z ⊆ X 1 . No w, since the so cle of Y /X is X 1 /X and is fin ite d imensional, there is an em b edding Y /X ֒ → C n for some n and so, b y d ualit y , w e get an epimorph ism ( C ∗ ) n → ( Y /X ) ∗ → 0, so ( Y /X ) ∗ is fi nitely generated. S ince it is r ational (a qu otien t of M ), it has to b e finite dimensional. Therefore, the sequence X n ⊆ X n +1 ⊆ X n +2 ⊆ . . . m ust terminate. This end s the pr o of.  W e n ote that by [CNO, Lemma 2.10] we h a v e that the set of closed cofinite left id eals of C ∗ is closed un der pr o ducts if C is lo cally finite and X ⊥ Y ⊥ = ( X ∧ Y ) ⊥ for all left fin ite dimensional sub comodu les X , Y of C . W e n ote th at with the same pro of, w e ha v e that the set of tw o-sided closed cofinite ideals of C ∗ is closed u nder pro du cts if and only if C is lo cally fin ite and U ⊥ W ⊥ = ( U ∧ W ) ⊥ for all finite dimens ional sub coalg ebras U, W of C . W e see that these t w o conditions are in f act equiv ale n t: ON EX TENSIONS OF RA TIONAL MODULES 9 Prop osition 1.8. The fol lowing assertions ar e e quivalent for a c o algebr a C : (i) The finite dimensional r ational C - c omo dules ar e close d under extensions; (ii) The set of left close d c ofinite ide als (e qu ivalently, op en ide als) of C ∗ is c lose d under pr o ducts (of ide als), i. e. X ⊥ Y ⊥ is close d c ofinite whenever X, Y ar e finite dimensional left sub c omo dules of C . (iii) The set of two-side d c ofinite close d ide als of C ∗ is close d under pr o ducts. (iv) The right hand side version of (ii ). Pro of. (iii) ⇒ (ii) If X , Y are finite dimensional left sub como du les of C so that X ⊥ , Y ⊥ are closed cofinite left id eals in C ∗ , let U = cf ( X ), W = cf ( Y ) which are finite dimensional sub coalgebras of C ; then U ⊥ W ⊥ ⊆ X ⊥ Y ⊥ . But by h yp othesis U ⊥ W ⊥ = ( U ⊥ W ⊥ ) ⊥ ⊥ = ( U ∧ W ) ⊥ (here we can also use [DNR, L emm a 2.5.7]) is closed and cofinite, and since X ⊥ Y ⊥ ⊇ ( U ∧ W ) ⊥ it f ollo ws that X ⊥ Y ⊥ is closed and cofinite. (ii) ⇒ (iii) is obvious. (iii) ⇒ (i) Let 0 → M ′ → M → M ′′ → 0 b e an exact sequen ce of C ∗ -mo dules with M ′ , M ′′ finite d imensional rational. Let U = cf ( M ′ ), W = cf ( M ′′ ). Th en U ⊥ · M ′ = 0, W ⊥ · M ′′ (in fact, U = an n C ∗ ( M ′ ) ⊥ b y [DNR, Prop osition 2.5.3]), and then, for x ∈ M it follo ws that W ⊥ x ∈ M ′ and thus U ⊥ W ⊥ x = 0, so U ⊥ W ⊥ ⊂ ann C ∗ ( M ). Bu t by the hyp othesis of (iii), as b efore we h a v e U ⊥ W ⊥ = ( U ∧ W ) ⊥ and this is closed cofinite. Hence since eac h x ∈ M , is canceled by a closed cofinite ideal ( U ∧ W ) ⊥ , it follo ws that C ∗ x is rational, so M is rational. (i) ⇒ (iii) Let U, W b e finite dimensional s ub coalgebras of C . Consider the exact sequ en ce (1) 0 → W ⊥ /U ⊥ W ⊥ → C ∗ /U ⊥ W ⊥ → C ∗ /W ⊥ → 0 Note that M ′ = W ⊥ /U ⊥ W ⊥ is rational since it is cancele d b y the cofinite closed ideal U ⊥ (and we can argue as ab ov e in (iii) ⇒ (i)). Moreo v er, there is some n su c h that U ⊆ C n − 1 , and then U ⊂ ( J n ) ⊥ = C n − 1 (use again [DNR, Lemma 2.5.7]). Then J n ⊆ C ⊥ n − 1 ⊆ U ⊥ so J n · M ′ = 0. W e no w note that we are un d er the assump tions of Prop osition 1.7: C ∗ /U ⊥ W ⊥ is cyclic, C ∗ /W ⊥ ∼ = W ∗ is fin ite dimensional rational and M ′ = W ⊥ /U ⊥ W ⊥ is rational. Therefore, it follo ws that J k M ′ = J k +1 M ′ = . . . from some k . But also for large k , J k M ′ = 0. Moreo v er, M ′ /J k M ′ is also fin ite dimensional, and so it follo w s that M ′ is fin ite dimensional. It follo ws that the exact sequence in (1) is a sequ ence of fin ite dimensional C ∗ -mo dules, w ith rational ”ends”, and b y the h yp othesis of (i) it follo w s that M = C ∗ /U ⊥ W ⊥ is finite dimensional r ational. Therefore, if H = cf ( M ), then H is fin ite dimensional and w e ha v e H ⊥ M = 0, and so H ⊥ ⊆ U ⊥ V ⊥ . It then follo ws by the in itial remarks that U ⊥ V ⊥ is closed and cofinite. Equiv alence with (iv) follo ws by the symmetry of (i) and (iii).  2. Connection to coref lexive coalgebras W e note now a connection with another imp ortan t notion in coal gebra theory , that of coreflexiv e coalge bras. Recall fr om [T af72] that a coalgebra is coreflexive if the natural map C → ( C ∗ ) o is sur j ectiv e (so an isomorphism). By the results of [HR74], there is a tigh t connection of th is to rational mo dules: C is coreflexiv e if and only if ev ery finite dimensional left C ∗ -mo dule is rational; by symmetry , this is equiv ale n t to eve ry finite dimensional r igh t C ∗ -mo dule b eing rational. Therefore we ha v e: Prop osition 2.1. A c o algebr a C is c or eflexive if and only if C 0 is c or eflexive and the finite dimensional r ational left (or, e q uivalently, right) C ∗ -mo dules ar e close d under extensions. 10 MIODRAG CR ISTIAN IO V ANOV Pro of. If C is coreflexiv e, then C 0 is coreflexiv e as a sub coalg ebra of C (by [HR74, 3.1.4]); ob viously th e fin ite dim en sional C ∗ -mo dules are closed under extensions since if (2) 0 → M ′ → M → M ′′ → 0 is an exact sequence with finite rational M ′ , M ′′ , then M is fi nite dimensional so it is rational by the hyp othesis. Conv ers ely , we pro ceed as in the pr o of of [DIN , Theorem 7.1] b y in d uction on the length (or dimension) of the fi nite d imensional mo d ule M to sho w that finite d imensional C ∗ -mo dules are rational: it is true for the the sim p le ones since C 0 is coreflexiv e, and in general, tak e a sequence as in (2) for some prop er sub comod ule M ′ of M and apply the induction hypothesis on th e C ∗ -mo dules M ′ , M ′′ of length smaller than M to get that they are rational. T herefore, M is r ational as an extension of M ′′ b y M ′ .  Applying prop osition 1.8 we get: Corollary 2.2. C is c or e flexive if and only if C 0 is c or eflexive and the top olo gy of close d c ofinite (op en) ide als of C ∗ is close d under pr o ducts. W e also note the follo wing: by the results of [Rad73] (see [Rad73, 3.12]), if K is infi n ite, w e ha v e that a cosemisimple coalgebra C = ` i ∈ I C i with C i simple coalgebras is coreflexiv e if and only if ` i ∈ I K = K ( I ) is coreflexiv e. On the other hand , by [HR74, Section 3.7], the coalgebra K ( I ) is coreflexiv e f or eve ry set I whose cardinalit y is nonmeasur able (or if card( I ) < card( K )). A cardinal X is called nonmeasur able if eve ry Ulam ultrafilter on X (that is, an ultrafilter wh ic h is closed u nder countable intersectio ns) is prin cipal (i.e. equal to the set of all subsets of X conta ining s ome x 0 ∈ X ). Th e class of nonmeasurable sets is closed under ”us u al” constructions, su c h as sub sets, unions, pro du cts, p o w er set, and con tains the countable set, and in fact there is no kno wn example of set w ic h is measurable (i.e. not nonmeasur ab le). Hence, it is reasonable to esp ect that C 0 is coreflexiv e, f or an y coalge bra C o v er an infinite field K (for fi nite fields, [Rad73, 3.12] provi des a simp le test). Hence, the ab ov e corollary sho ws that in general, if the set of simple como dules of C is an y set that we m igh t ”reasonably” exp ect, to test coreflexivit y is equiv alen t to testing that the finite top ology of id eals of C ∗ is closed under pr o ducts, equiv ale n tly , C is lo cally finite and U ⊥ W ⊥ = ( U ∧ W ) ⊥ . 3. Ra tional Extens ions and the homological Ext Lemma 3.1. Assu me that C is a c o algebr a such that if 0 → M ′ → M → M ′′ → 0 is any exact se q u enc e of left C ∗ -mo dules with M ′ , M ′′ r ational and M ′′ cyclic, then M is r atio nal. Then the left r ational C ∗ -mo dules ar e close d under extensions. Pro of. Let 0 → M ′ → M π → M ′′ → 0 b e an exact s equ ence with M ′ , M ′′ rational. W rite M = M ′ + P i C ∗ x i , so that the images π ( x i ) of x i in M ′′ generate M ′′ . Sin ce M ′′ is rational, C ∗ π ( x i ) is fin ite dimen sional, and so w e ha v e exact sequen ces 0 → M ′ → M ′ + C ∗ x i → C ∗ π ( x i ) → 0, wh ic h sho ws that M ′ + C ∗ x i is rational. Therefore, M = P i ( M ′ + C ∗ x i ) is rational.  Prop osition 3.2. L et 0 → M ′ → M → M ′′ → 0 b e an exact se quenc e with M ′ , M ′′ r ational, and such that M ′ has finite L o ewy length (finite c or adic al filtr ation). If finite dimensional r atio nal left C ∗ -mo dules ar e close d under extensions, then M is r at ional. ON EX TENSIONS OF RA TIONAL MODULES 11 Pro of. The previous Lemma sho ws th at we ma y assum e M ′′ is cyclic rational and finite dimensional. By [I09, Lemma 2.2] w e ha v e that J n M ′ = 0 for some n since M ′ has finite Lo ewy length. Prop osition 1.7 no w sho ws that M ′ = M ′ /J n M ′ is fi nite d imensional. Therefore, M is finite dimensional s ince fin ite dimensional rationals are assumed closed under extensions.  Prop osition 3.3. Assume R at ( C ∗ M ) is not lo c alizing, bu t finite dimensional r ational mo dules ar e close d under extensions. Then ther e is an exact se quenc e 0 → M → P → S → 0 and such that (i) P is cyclic and non-r ational. (ii) J M = M and M has simple so cle (and infinite L o ewy length). (iii) S is simple finite dimensional r ational. Pro of. By Lemma 3.1 we can fin d suc h a sequ en ce with P cyclic and S fi nite dimens ional. Let u s consider suc h a sequence with S of min imal length (or dimension). T hen we s ee that S is simple. Ind eed, if S is not, th en let x ∈ P \ M suc h that L = ( C ∗ x + M ) / M is simple, L ( S . If C ∗ x is rational, then we get an exact sequen ce 0 → M + C ∗ x/C ∗ x → P /C ∗ x → P / ( M + C ∗ x ) → 0, which has the prop er ties: P / ( M + C ∗ x ) has p ositiv e length smaller than S , P /C ∗ x is cyclic and not rational (if it were r ational, P w ould b e rational since then P /C ∗ x and C ∗ x w ould b e finite dimen sional r ational). This cont radicts the minimalit y c hoice. Therefore, C ∗ x is not rational, and we get an exact sequence 0 → M ∩ C ∗ x → C ∗ x → C ∗ x/ ( C ∗ x ∩ M ) ∼ = L → 0, with M ∩ C ∗ x rational and L simple. Since the initial S was of minimal p ossible length of all sequences with this feature, S is simple. No w, note th at M m ust hav e infi n ite Lo ewy length, since otherwise P would b e rational b y Prop osition 3.2. Also, by Pr op s ition 1.7, we can find n suc h that J n M = J n +1 M and M /J n M is fi nite d imensional. Since M is rational, w e can fin d a finite d im en sional sub como d u le N of M s u c h that J n M + N = M . Let M ′ = M / N , P ′ = P / N , s o th at we ha v e an exact sequence 0 → M ′ → P ′ → S → 0. Note that J M ′ = J ( J n M + N ) / N = ( J n +1 M + J N + N ) / N = ( J n M + N ) / N = M ′ . Let L b e a simple sub como du le of M ′ and X b e a maximal su b como du le of M ′ suc h that L ∩ X = 0 (suc h a sub comod ule can b e found , for example, by Zorn ’s Lemma). It is then n ot d ifficult to see that L = L + X/X ֒ → M ′ /X is an essen tial su b como dule of M ′′ = M ′ /X . Moreo v er, J M ′′ = J ( M ′ /X ) = ( J M ′ + X ) /X = M ′ /X = M ′′ . This also sho ws that M ′′ has infin ite Lo ewy length, since M ′′ 6 = 0 (for example, again by [I09, Lemm a 2.2]). T hen, if P ′′ = P ′ /X , w e ha v e P ′′ / M ′′ ∼ = P ′ / M ′ ∼ = P / M = S , and so the exact sequence 0 → M ′′ → P ′′ → S → 0 has the required prop er ties: J M ′′ = M ′′ , M ′′ has simp le so cle an d in finite Lo ewy length, S is simple and P ′′ is cyclic and nonrational (if P ′′ is rational, it is finite dimens ional and in th is case so is M ′′ , whic h is not p ossible).  Prop osition 3.4. L et 0 → M → P → S → 0 b e a short exact se qu enc e of left C ∗ - mo dules with M , S r ational mo dules, and S a simple mo dule. Then ther e is an extension 0 → E ( M ) → P → S → 0 , wher e E ( M ) i s the inje ctive hul l of M as right. Mor e over if P is not r ational, then P is not r ational e i ther, and the last se quenc e is not split. 12 MIODRAG CR ISTIAN IO V ANOV Pro of. W e tak e P to b e th e push out in the category of left C ∗ -mo dules of the follo wing diagram: E ( M )  q " " M -  < < y y y y y y y y y  q # # F F F F F F F F F P P -  < < By th e p rop erties (or the construction) of the push-out, w e h a v e E ( M ) + P = P , E ( M ) ∩ P = M ; this sho ws that P /E ( M ) = ( E ( M ) + P ) /E ( M ) = P /P ∩ E ( M ) = P / M ∼ = S and this giv es us th e requ ired extension. If P is not rational, then P is not rational either, as it con tains P . O bviously , the sequence 0 → E ( M ) → P → S → 0 is n ot split, sin ce otherwise, P w ould b e rational.  W e note a few equiv alen t inte rpretations of the condition in the ab o v e p rop osition. Prop osition 3.5. L et S b e a left r at ional C ∗ -mo dule, T = S ∗ , and E an inje ctive right C -c omo dule. The fol lowing assertions ar e e quivalent: (i) Ext 1 ( S, E ) = 0 ; (i’) Any se quenc e 0 → E → P → S → 0 splits. (i”) E is inje ctive in the lo c alizing sub c ate gory (i.e. close d under extensions) of C ∗ M gener ate d by R at ( C ∗ M ) . (ii) F or any f ∈ Hom C ∗ ( T ⊥ , E ) , Im( f ) is finite dimensional. (iii) If M = T ⊥ E ( T ) is the maximal submo dule of E ( T ) ∗ (which is unique by [I, Lemma 1.4] ), any f ∈ Hom C ∗ ( M , E ) has finite dimensional image. Pro of. The equiv alence of (i)-(i”) is obvious. If we write C = E ( T ) ⊕ H , we note that C ∗ = E ( T ) ∗ ⊕ H ∗ so then T ⊥ = M ⊕ H ∗ . Since H ∗ is cyclic (as a d irect s u mmand of C ∗ ), we get that morp hisms f ∈ Hom C ∗ ( H ∗ , E ) ha v e finite images. This shows the equiv alence of (ii) and (iii). (i) ⇒ (ii) The cond ition yields an exact sequence 0 → Hom C ∗ ( S, E ) → Hom C ∗ ( C ∗ , E ) → Hom C ∗ ( T ⊥ , E ) → Ext 1 ( S, E ) = 0, therefore any f in the follo wing diagram extends to some g : 0 / / T ⊥ f   / / C ∗ g } } E Therefore, sin ce Im( g ) is cyclic r ational, it is finite d im en sional, and Im( f ) ⊆ Im( g ). (ii) ⇒ (i) Let f ∈ Hom C ∗ ( T ⊥ , C ∗ ), K = k er( f ), F = Im( f ); we ha v e the comm utativ e diagram: 0 / / T ⊥ f     σ / / C ∗ g     0 / / F u   σ / / C ∗ /σ ( K ) v z z E ON EX TENSIONS OF RA TIONAL MODULES 13 where f = u ◦ f , g is the canonical p ro jection an d σ is induced by σ . Since C ∗ /T ⊥ = T ∗ and T ⊥ /K are finite d imensional, the second ro w of the diagram consists of finite dimensional comod ules; since E is injectiv e, the diagram extend s with some v such that v ◦ σ = u (e.g. b y [DNR, Theorem 2.4.17]). Hence, g = v ◦ g extends f . Th is giv es an exact sequence 0 → Hom C ∗ ( S, E ) → Hom C ∗ ( C ∗ , E ) σ ∗ → Hom C ∗ ( T ⊥ , E ) 0 → Ext 1 ( S, E ) 0 → Ext 1 ( S, C ∗ ) = 0 Since σ ∗ is surjective , it is s tand ard to see that the ab o v e sequencew e yields Ext 1 ( S, E ) = 0.  Corollary 3.6. L et C b e a c o algebr a and assume finite dimensional left r ational mo dules ar e close d under extensions. Then the fol lowing ar e e qu i valent: (i) R at ( C ∗ M ) is close d under extensions (i.e. Rat i s a torsion functor). (ii) Ext 1 ( S, E ) = 0 f or every simple right C -c omo dule S and every inje ctive inde c om p osable right C - c omo dule E . (iii) Ext 1 ( S, C ) = 0 , for al l simple right C -c omo dules S . (iii)’ E x t 1 ( C 0 , C ) = 0 as left C ∗ -mo dules. (iv) F or e v ery simple lef t C -c omo dule T , every inje ctive inde c omp osable rig ht c omo dule E and any f ∈ Hom C ∗ ( T ⊥ , E ) , Im ( f ) is finite dimensional. (v) Ther e i s no exact se quenc e of left C ∗ -mo dules 0 → M → P → S → 0 with M r ational with simple so cle and J M = M , S simple r ational and P cyclic . Pro of. (v) ⇒ (i) follo ws from Prop osition 3.3 (i) ⇒ (ii) and (iii) follo ws sin ce if 0 → E → P → S → 0 is an exact sequence, (i) implies that P is rational and so the sequence splits since E is an injectiv e rational como dule. (ii) ⇒ (v) follo w s by Prop osition 3.4. (iv) ⇔ (ii) is cont ained in Prop osition 3.5 (iii) ⇒ (ii) is obvious sin ce C = E ⊕ H for some H . (iii) ⇔ (iii) ’ is obvious.  W e can now pro ceed with th e main charact erization of the ”rationals closed und er exten- sion” pr op ert y . Theorem 3.7. L et C b e a c o algebr a. Then the left r ational C ∗ -mo dules ar e close d under extensions if and only if the fol lowing two c onditio ns hold: (i) pr o ducts of close d c ofinite ide als ar e close d; (e quivalently, C is lo c al ly finite and for any two finite dimensional sub c o algebr as V , W of C , V ⊥ W ⊥ = ( V ∧ W ) ⊥ .) (ii) Ext 1 ( S, E ) = 0 , for eve ry simple right C -c omo dule S and inje ctiv e right C -c omo dule E (e qui valently, Ext 1 ( S, C ) = 0 , for al l such S , or Ext 1 ( C 0 , C ) = 0 as left C ∗ -mo dules). W e n ote that Ext 1 ( S, E ) (or Ext 1 ( C 0 , C )) can b e computed from the long exact sequence of homology: 0 → Hom C ∗ ( S, E ) → Hom C ∗ ( C ∗ , E ) → Hom C ∗ ( T ⊥ , E ) → Ext 1 ( S, E ) → Ext 1 ( S, C ∗ ) = 0 where T = S ∗ . So Ext 1 ( S, E ) = Hom( T ⊥ ,E ) ֒ → Hom( C ∗ ,E ) with ֒ → Hom( C ∗ , E ) representing the im age of Hom( C ∗ , E ) in Hom ( T ⊥ , E ); similarly , Ext 1 ( C 0 , C ) = Hom( C ⊥ 0 ,C ) ֒ → Hom( C ∗ ,C ) . 4. A gen eral suf ficient condition an d appl ica tions W e giv e a quite general suffi cien t condition u nder we h av e th at th e rational f unctor is a torsion fu n ctor. In f act, these will b e situations in wh ic h closure of finite dimensional 14 MIODRAG CR ISTIAN IO V ANOV rationals under extensions is enough to ha v e closure und er extensions of all r ational mo d- ules. A r igh t comod ule M is called finitely cogenerated if it em b edds in a finite d ir ect sum of copies of C . First, let u s n ote: Prop osition 4.1. Assume any inje ctive inde c omp osable right c omo dule E has finite L o ewy length. If op en (i.e. close d c ofinite) ide als of C ∗ ar e close d under pr o ducts, then R at ( C ∗ M ) is close d under extensions. Pro of. If rational mo du les are not closed und er extensions, there is an exact sequence 0 → M → P → S → 0 as in P rop osition 3.3, with M with simple so cle and of infinite Lo ewy length; b ut then E ( M ) is indecomp osable of infinite Lo ewy length, a con tradiction.  Theorem 4.2. L et C b e a c o algebr a such that ide als of C ∗ ar e close d under pr o ducts. Assume that for e ach lef t inde c omp osable inje ctive c omo dule E ( T ) ther e is 0 6 = X ⊆ E ( T ) such that E ( T ) /X is finitely c o gener ate d and X has finite L o e wy length. Then left r ational C ∗ -mo dules ar e close d under extensions. Pro of. Assume the con trary , and consid er an exact sequence of left C ∗ -mo dules as p ro- vided b y Prop osition 3.3: 0 → M → P π → S → 0 Let T = S ∗ ; since P is cyclic, M = J M ⊆ J P 6 = P , s o M = J P is the Jacobson r adical of P . If E ( T ) ∗ p → S is the canonical p ro jection, there is p : E ( T ) ∗ → P with π p = p . As Im ( p ) 6⊆ M = ke r π , Im ( p ) + J P = P so Im ( p ) = P since P is finitely generated (b y Nak a y ama lemma). If H = T ⊥ E ( T ) = { f ∈ E ( T ) ∗ | f ( x ) = 0 , ∀ x ∈ T } is the max- imal sub mo dule of E ( T ) ∗ (whic h is unique b y [I, Lemm a 1.4]), then H = J E ( T ) ∗ so p ( H ) = p ( J E ( T ) ∗ ) = J p ( E ( T ) ∗ ) = J P = M . Since E ( T ) /X is finitely cogenerated b y E ( T ) /X ֒ → C n , dualizing w e get that the sub m o d- ule Y = X ⊥ E ( T ) = ( E ( T ) /X ) ∗ ֒ → E ( T ) ∗ is finitely generated, so p ( Y ) is fin ite d imensional. If k is the Lo ewy length of X , then X J k +1 = 0 and J k +1 X ∗ = 0. Also, X ∗ = E ( T ) ∗ / Y , and J X ∗ = ( J E ( T ) ∗ + Y ) / Y = H / Y , so J k ( H / Y ) = 0. No w, since there is an epimor- phism H / Y − → p ( H ) /p ( Y ), w e get that M /p ( Y ) h as finite Lo ewy length. S ince p ( Y ) is finite d im en sional, it follo ws that M has finite Lo ewy length, whic h cont radicts the initial c hoice give n by Prop osition 3.3. This ends the p ro of.  R emark 4.3 . It is n ot hard to see that the h yp othesis of E ( T ) /X b eing finitely cogen- erated is actually equiv alen t to the fact that X ⊥ E ( T ) ∼ = ( E ( T ) /X ) ∗ ⊆ E ( T ) ∗ is finitely generated. Indeed, if X ⊥ E ( T ) ∗ is generated b y f 1 , . . . , f n , then one can see that ψ : E ( T ) ∋ x 7− → ( f i ( x 0 ) x − 1 ) i =1 ,...,n ∈ C n is a morph ism of left C -como dules (right C ∗ -mo dules), and k er( ψ ) = X , since one sho ws easily that ψ ( x ) = 0 if and only if f ( x ) = 0 for all f ∈ X ⊥ E ( T ) (so ker( ψ ) = ( X ⊥ E ( T ) ) ⊥ = X ). Hence, we hav e the follo wing Corollary 4.4. Supp ose the op en ide als of C ∗ ar e close d under pr o ducts. If for al l simple left C -c omo dules T , the maximal ide al T ⊥ of C ∗ is finitely gener ate d (e quivalently, T ⊥ E ( T ) is finitely gener ate d), then left r atio nal C ∗ -mo dules ar e close d under extensions. W e r ecall f r om [CNO] that a coalgebra is called left F -Noetherian if ev ery closed cofinite left ideal of C ∗ is finitely generated. The f ollo win g C orollary is pro v ed in [CNO], but also follo w s as a particular case of the ab o v e r esult: ON EX TENSIONS OF RA TIONAL MODULES 15 Corollary 4.5. If C is left F -No etheria n, then left r ational C ∗ -mo dules ar e close d under extensions. The n ext corollary f ollo ws dir ectly from th e ab ov e resu lts, but it giv es s ome particularly nice and easy conditions to c hec k in order to get that Rat ( C ∗ M ) is closed under extensions. Corollary 4.6. L et C b e a c o algebr a such that the set of op en ide als of C ∗ is close d under pr o ducts. If any of the fol lowing c onditions is true, then the left r atio nal C ∗ -mo dules ar e close d under extensions. (i) F or e ach simple rig ht C -c omo dule S , its inje ctive hul l E ( S ) has finite L o ewy length (in p articular, when it is finite dimensional). (ii) F or e ach simple left C -c omo dule T , either • i ts inje ctive hul l E ( T ) has finite L o ewy length (in p art icular, if it is finite dimensional), or • L n +1 E ( T ) /L n E ( T ) is finite dimensional for some n (in p art icular, this is true when E ( T ) is artinian), or • Ext 1 ( L, T ) 6 = 0 for only finitely many simple left c omo dules L (which, in this c ase, i s e quivalent to: L 1 E ( T ) /L 0 E ( T ) is finite dimensional). W e n o w giv e some applications of the ab o v e results. In particular, we note how man y results in [CNO] can b e obtained as a corollary . W e first need the follo wing easy but useful Lemma: Lemma 4.7. L et M b e a left C -c omo dule su c h that M is finitely c o gener ate d. Then J M ∗ = M ⊥ 0 = ( M 0 ) ⊥ M . Conse quently, if M is a left c omo dule such that al l M n = L n M ar e finite dimensional, then J n +1 M ∗ = M ⊥ n and every c ofinite submo dule of M ∗ is finitely gener ate d (and close d). Pro of. The fact that J M ⊆ M ⊥ 0 is straigh tforw ard (it follo w s, for example, by [I09, Lemma 2.2]). Let σ : M ֒ → C n b e an embedd ing of left como dules. Let f ∈ M ⊥ 0 ⊆ M ∗ , and let α : M → C the morphism of left C -como dules defined by α ( m ) = f ( m 0 ) m − 1 . Then α | M 0 = 0 since f ∈ M ⊥ 0 , and s o it factors as α = g ◦ p as b ello w ; the follo wing diagram is obvio usly comm utativ e with η inj ectiv e since C n 0 ∩ M = M 0 ( p and π are th e canonical pro jections). M ֒ → σ / / p   C n π   0 / / M / M 0 g   ֒ → η / / ( C /C 0 ) n h y y C By the injectivit y of C , the morphism g extends to some h , and so we h a v e α = g p = hη p = ( hπ ) σ . But using the standard pro jections p i and injections σ i of C n , this implies that α = n P i =1 ( hπ σ i )( p i σ ) = P i u i ◦ α i with u i = hπ σ i : C → C and α i = p i σ : M → C . 16 MIODRAG CR ISTIAN IO V ANOV Therefore, if f i = ε ◦ u i and h i = ε ◦ α i , w e h a v e n X i =1 f i h i ( m ) = X i f i ( m − 1 ) h i ( m 0 ) = X i ε ( u i ( m − 1 )) ε ( α i ( m 0 )) = X i εu i ( α i ( m ) 1 ) ε ( α i ( m ) 2 ) − since α i is a comod ule map = X i ε ( u i α i ( m )) = ε ( α ( m )) = f ( m ) F or the last part, use indu ction on n : if J n M ∗ = M ⊥ n − 1 , then J n +1 M ∗ = J M ⊥ n − 1 ; apply the first part to M / M n − 1 and get that J ( M / M n − 1 ) ∗ = ( M n / M n − 1 ) ⊥ M / M n − 1 whic h corresp onds to M ⊥ n through the isomorphism ( M / M n − 1 ) ∗ ∼ = M ⊥ n − 1 ; therefore J M ⊥ n − 1 = M ⊥ n . Finally , if I is a cofinite su bmo dule of M ∗ , then J n · ( M ∗ /I ) = 0 for some n , and so J n M ∗ ⊆ I , i.e. M ⊥ n ⊆ I . Sin ce I conta ins a cofinite closed id eal, it is closed; also, since M / M n is finitely cogenerate d, we hav e that M ⊥ n ∼ = ( M / M n ) ∗ is finitely generated, so I is finitely generated to o.  W e fir st n ote a generalization of [CNO, Th eorem 2.8], part of w hic h wa s prov ed first in [HR74, Theorem 4.6]. In fact, the f ollo win g also generalizes [CNO , Theorem 2.11], and, in p articular, reco vers the caracterizati on of the comm u tativ e case. It also genega lizes some results of [C03]. Recall a coalgebra C is called left s trongly reflexive, or C ∗ is called almost No etherian if ev ery cofinite left ideal of C ∗ is finitely generated (see [HR74 ]). W e ma y thus call a p seudo compact left C ∗ -mo dule M ∗ (i.e. M is left C -como dule) almost No etherian if every cofinite sub mo dule is fin itely generated. Theorem 4.8. L et C b e a c o algebr a such that for e ach simple left C -c om o dule T , we have Ext C, 1 ( L, T ) 6 = 0 for only finitely many simple left C - c omo dules L , i.e. Hom( L, E ( T ) /T ) 6 = 0 for only finitely many simple lef t c omo dules L . Then the fol lowing assertions ar e e quiv- alent: (i) R ational left C ∗ -mo dules ar e close d under extensions (i.e. C has a left R at torsion functor). (ii) L n E ( T ) is finite dimensional for al l n and al l simple left C -c om o dules T . (iii) L 1 E ( T ) is finite dimensional for al l simple left C -c omo dules T . (iv) E ( T ) ∗ is an almost no etheria n C ∗ -mo dule for al l simple left c omo dules T , i.e. every c ofinite submo dule of E ( T ) ∗ is finitely gener ate d. (v) C is lo c al ly finite. (vi) C ∗ is left F -No e therian. Pro of. (i) ⇒ (iii) F ollo ws since in this case C is lo cally finite; therefore, Hom( L, E ( T ) /T ) is fin ite dimensional for all L ; bu t it is also 0 for all bu t finitely many left como du les L ’s. This sh ows that the so cle of E ( T ) /T is fi nite dimensional. (iii) ⇒ (ii) F ollo ws by applying Lemma 1.2 in d uctiv ely . (ii) ⇒ (iv) F ollo ws f r om Lemma 4.7 (iv) ⇒ (vi) F ollo w s since eac h closed cofinite left ideal X ⊥ ⊂ C ∗ can b e decomp osed as X ⊥ = X ⊥ L i ∈ F E ( T i ) ⊕ L i ∈ H \ F E ( T i ) ∗ , where C = L i ∈ H E ( T i ) is a d ecomp osition of C into indecomp osables suc h that X ⊆ L i ∈ F E ( T i ), F -finite. Then one find s fin ite d imensional X i ⊆ E ( T i ) suc h that X ⊆ L i ∈ F X i ; w e get L i ∈ F X ⊥ i E ( T i ) ⊆ X ⊥ L i ∈ F E ( T i ) ⊆ L i ∈ F E ( T i ) ∗ , so ON EX TENSIONS OF RA TIONAL MODULES 17 X ⊥ L i ∈ F E ( T i ) is finitely generated. (vi) ⇒ (i) is kno wn (from [CNO] or ab o v e considerations: Corollary 4.5, or firs t obtain (ii) and apply Corollary 4.6). (iii) ⇔ (v) is a dir ect consequence of Lemma 1.2  The h yp othesis of the previous prop osition sa ys that th e (left) Gabriel quiv er of the coalg e- bra C has only finitely man y arro ws going into any vertex T . In particular, the hyp othesis is tru e if C is almost connected, i.e. if it has only finitely many t yp es of isomorphism of simple como du les. In this particular situation, w e r eco ver the ab o v e men tioned results of [CNO, Th eorem 2.9]. Another app lication is in the case of semip erfect coalgebras. It is prov ed in [L74, Lemma 2.3] (see also [GTN, Theorem 3.3] and [CNO, Theorem 2.12]) that if C is righ t s emip erfect, then C is F -No etherian and has a left Rat torsion functor. W e note an alternate pr o of of this as a a consequence of the ab o v e results, and also a strenghtening: Corollary 4.9. L et C b e a right semip erfe ct c o algebr a. Then C ∗ is left F - N o etherian and C has a left and right torsion functor, i.e. r ational left mo dules and r at ional right mo dules ar e close d under extensions. Pro of. Th e fact that C ∗ is F -No etherian follo ws directly f rom Theorem 4.8; therefore, finite dimensional rationals (left or righ t) are closed under extensions. Since C is r igh t semip erfect, left in d ecomp osable injectiv e como dules are fin ite dim en sional, so Corollary 4.6 (ii) implies that left r ationals are closed u n der extensions and Corollary 4.6 (i) (its left-righ t symmetric v ersion), implies that right rationals are closed u nder extensions to o.  W e w ill see in Examp le 5.2 that for a righ t semip erfect coalgebra C , its dual C ∗ need not b e righ t F -No etherian. Another d irect consequence of T h eorem 3.7 (but whic h can b e easily obtained also dir ectly , see also [L75]) is Corollary 4.10. L et C = L i ∈ I C i b e a dir e ct sum of c o algebr as. C has a lef t r ational torsion functor i f and only if al l C i have left r atio nal torsion functors. 5. F -Noetherian coalgebras and tors ion ra tional fun ctor W e giv e connections b et w een F -No etherian and prop eries inv estigated b efore, more sp ecif- ically , lo cally finiteness. F or a simple left comod ule S and a comod ule M , let [ M ; S ] denote the m ultiplicit y of S in the so cle of M . It can b e finite or in finite. Note that a como dule M is quasifinite if [ M ; S ] < ∞ for all simples S . The follo wing p rop osition giv es a cryterion to test the F -No etherian prop ert y: Prop osition 5.1. The fol lowing assertions ar e e quivalent: (i) C ∗ is left F -No e therian. (ii) C /X is finitely c o gener ate d for e ach finite dimensional left sub c omo dule X ⊆ C , that is, ther e is a monomorphism C /X ֒ → C n for some n . (iii) su p { [ C /X ; S ] [ C ; S ] | S simple left como dule } < ∞ for e ach finite dimensional left sub c omo dule X of C . Pro of. (i) ⇔ (ii) W e pro ceed similar to Remark 4.3 . I f C /X ֒ → C n is a monomorp hism, dualizing we get an epimorphism ( C ∗ ) n → M ∗ → 0. Con v ersely , let f 1 , . . . , f n generate the right C ∗ -mo dule X ⊥ , and let X i = ( C ∗ f i ) ⊥ . Then X ⊥ = n P i =1 X ⊥ i = ( n T i =1 X i ) ⊥ so 18 MIODRAG CR ISTIAN IO V ANOV X = n T i =1 X i . T he map ϕ i : C → C , ϕ ( c ) = f i · c is a morphism of r igh t C ∗ -mo dules (so of left C -como dules), and ke r( ϕ i ) = { c | c 1 f i ( c 2 ) = 0 } = { c | c ∗ ( c 1 f i ( c 2 )) = 0 ∀ c ∗ ∈ C ∗ } = ( C ∗ f i ) ⊥ = X i . This shows that C /X i em b edds in C . Th erefore w e can get an emb edding C /X = C / n \ i =1 X i ֒ → n M i =1 C /X i ֒ → C n (i) ⇔ (iii) If C /X em b edds in C n then the m ultiplicit y of S in C /X is smaller than in C n , so [ C /X ; S ] ≤ n [ C ; S ]. Con v ersely , if [ C /X ; S ] ≤ n [ C ; S ] for some fixed n (dep ending on X ) and all S , w e see th at th e so cle of C /X em b edds in C n 0 : s ( C /X ) = L S S [ C /X ; S ] ֒ → L S S n [ C ; S ] = C n 0 . Hence s ( C /X ) em b edds in C n , so C /X emb edds in C n since s ( C /X ) is essen tial in C /X and C n is inj ective . Th is fi n ishes the pr o of.  It was conjectured in [CNO, Remark 2.13] that if C has a left torsion Rat-functor, i.e. the left rational are closed under extensions, then C is F -No etherian. Th e conjecture was motiv ated b y a series of results on the Rat fu nctor whic h gav e evid en ce for it. Ho wev er, it w as pro v ed in [TT05] by using a certain more complicated semigroup coalge bra construc- tion that this conjecture is f alse. This simple charact erization of F -Noetherian, together with a r esult of Radf ord f rom [Rad74], yield an easy wa y to give counterexamples to this conjecture Example 5.2. Consider the fol lowing qui ver Γ : a b 1 o o x 11 b 2 . . . b n W W x 21 W W x 21 0 0 x n 1 ... 0 0 x nn 0 0 . . . L et C b e the p ath sub c o algebr a of the ful l p ath c o al gebr a of Γ , which has as a K -b asis the set { a, b n , x ni | n ∈ N , 1 ≤ i ≤ n } . We se e that this c o algebr a has C = C 1 . Ar guing exactly as we did in Example 1.5, we se e that this c o algebr a is lo c al ly finite (or apply L emma 1.2 and note that the sp ac e of ( u, v ) skew primitives is finite dimensional for any two gr ouplike elements u, v ). As p ointe d out b efor e, if K is infinite then C 0 ∼ = K ( N ) is c or eflexive. Then, by [Rad74, 3.3] , sinc e C 0 is c or eflexive, C is lo c al ly finite and C = C 1 , we get that C is c or eflexive (note: this is c al le d r eflexive in [Rad74] ). By the r esults of the pr evious se ctions (for e xample, Pr op osition 2.1 or Cor ol lary 4.6), C wil l have a torsion R at-functor (for b oth lef t and right C ∗ -mo dules). Bu t note that [ C / K { a } ; K { b n } ] = n as right c omo dules, and C is p ointe d so [ C ; K { b n } ] = 1 , so c ondition (iii) of Pr op osition 5.1 is not verifie d for the right c omo dule K { a } . Ther efor e, by (the right version) of Pr op osition 5.1, we have that C ∗ is not right F -N o etherian. We note that at e ach vertex u , ther e ar e only finitely many arr ows (and p aths) into u , so by wel l known char acterizations of the inje ctive inde c omp osable obje cts over p ath c o al gebr as (se e [Sim09] , [DIN ] ) it fol lows that this left inje ctiv e inde c omp osable ar e finite dimensional. Ther efor e, C is right semip erfe ct. However, infinitely many arr ows go out of a , so the inje ctive hul l of the right c om o dule K { a } is infinite dimensional. Henc e, C i s not left semip erfe ct. This example shows also ON EX TENSIONS OF RA TIONAL MODULES 19 that if C is a left (right) semip erfe ct c o algebr a then C ∗ is not ne c essarily left (right) F - No etherian, and also, that C ∗ c an b e only one-side d F - No etherian, so it makes sense to distinguish b etwe en the two notions. Closing Remarks W e hav e seen that in general, the rational mo d ules are closed und er extensions if and only if the finite d imensional rationals are so, and a homological condition holds. W e n ote that in all th e particular classes of examples that w e ha ve for w h ic h the r ationals are closed un der extension (e.g. see Corollary 4.6, Th eorem 4.8, and their consequences), the E xt-condition from Th eorem 3.7 is automatically satisfied. Th at is, if finite rationals are closed under extensions, then all left C ∗ -mo dules are so. The known examples for which the Rat fun ctor is not torsion, are in fact examples w here the fin ite rational mo d ules are not closed und er extensions. One can see that the counter-exa mple to the ab o v e mentioned question [CNO, 2.13] build in [TT05, Example 12] is also of this t yp e. Ind eed, one can see that the coalgebra C = K S in that example has coradical filtration of length 2 (i.e. C = C 2 ), and so by Corollary 4.6, the finite rationals are not closed under extensions; in fact, this coalgebra C is n ot coreflexiv e, so the example is of the same nature as th e ab o v e Example 5.2. This leads one to ask th e follo w ing questions: Question 1 If the finite dimensional rational (left, or equiv alen tly , righ t) C ∗ -mo dules are closed under extensions, d o es it follo w that R at ( C ∗ M ) (and, b y symmetry , also Rat ( M C ∗ )) is closed un der extensions? Question 2 If R at ( C ∗ M ) is closed un der extensions, d o es it also follo w that R at ( M C ∗ ) is closed un der extensions? Question 3 If E is an in jectiv e indecomp osable left comodu le, and S is a simple left C -comod ule, do es it follo w that Ext 1 ( S, E ) = 0? Note that an afirmativ e answer to Q1 implies afirmativ e an answ er to Question 2, and afirmativ e answ er to Q3 implies afirmativ e Q1. W e b eliev e th at a counterexa mple to Q1, Q2 and Q3 migh t b e constructed by usin g the c haracte rizations of Th eorem 4.8, more precisely , a coalgebra for whic h for all s imple left como du les S , Ext C, 1 ( L, S ) 6 = 0 for only finitely m an y simple left como du les L , but for wh ic h this condition is not tr u e for righ t simple como dules. Su ch a coalgebra migh t b e obtained by consid ering path (sub)coalgebras of quiver coalgebras for quiv ers Γ in whic h at eac h v ertex there are only finitely many arro ws going in, but for some vertic es th ere are infinitely arrows going out. In this case, C w ould h av e the follo wing pr op erties: finite dimensional rational mo dules are closed u nder extensions, and C ∗ is left F -No etherian, so R at ( C ∗ M ) is closed u n der extensions; but one might exp ect that Rat ( M C ∗ ) is not closed under extensions, whic h w ould answer Q1,Q2 and Q3 in th e n egativ e. A cknowledgment The author wishes to th an k E.F riedlander for interesting insights and conv ersations on the theory of r ational mo d ules. This w ork was su pp orted by the s trategic grant POS - DR U/8 9/1.5/ S/58852 , Pr o j ect “Po stdo ctoral programe for trainin g scient ific researc hers” cofinanced b y the Europ ean So cial F und within the S ectorial Op erational Program Human Resources Dev elopmen t 2007-201 3. 20 MIODRAG CR ISTIAN IO V ANOV Referen ces [A] E. A b e, H opf Algebr as , Cambridge Un iv. Press, 1977. [AF] D. A nderson, K. F uller, Rings and Cate gories of M o dules , Grad. T exts in Math., Springer, Berlin- Heidelb erg-New Y ork, 1974. [CMo97] W. Chin and S. Montgomery , Basic Co algebr as , in Mo dular Interfaces (R iverside, CA, 1995) AMS/IP S tudies in Adv anced Math. vol.4, Providence RI , (1997) 41-47. [C03] J. Cuadra, Extensions of r ational mo dules. , Int. J. Math. Math. S ci. 2003, no. 69, 4363-4371. [C05] J. Cuadra, W hen do es the r ational submo dule split off ? , Ann. U niv. F errarra -S ez. VI I- S c. Mat. V ol. LI ( 2005), 291-298. [CGT] J. Cuadra, J. Gomez-T orrecillas , Serial Co al gebr as , J. Pure Ap p. Algebra 189 (2004), 89-107. [CNO] J. Cuadra, C. N ˘ ast˘ asescu, F. V an Oy staeyen, Gr ade d almost no etherian rings and applic ations to c o algebr as , J. Algebra 256 (2002) 97110. [DIN] S. D˘ asc˘ alescu, M.C.Iov anov, C. N˘ ast˘ asescu, Path sub c o algebr as, fini teness pr op erties and quantum gr oups , arXiv:1012.4335 (37p). [DNR] S. D˘ asc˘ alescu, C. N˘ ast˘ asescu, S ¸ . Raianu, Hopf Algebr as: an intr o duction . V ol. 235. Lecture Notes in Pure and A pplied Math. V ol.235, Marcel D ekker, New Y ork, 2001. [F] C. F aith, Algeb r a I I: Ring The ory . V ol 191, Sp ringer-V erlag, Berlin-Heidelb erg-New Y ork, 1976. [FP87] E.M.F riedlander, B.P arshall, Rat ional actions asso ciate d to the adjoint r epr esenta tion , Ann.Sci. E.N.S. 4, t ome 20, no. 2 (1987), 215–226. [G72] K. Go odearl, Singular splitting and the torsion pr op ert ies . Mem. A mer. Math. So c 124 (1972), 1-86. [G] P . Gabriel, Des c ate gories ab eliennes , Bul. Soc. Math. F rance 90 (1962), p.323-448. [GTN] J. G´ omez-T orrecillas, C. N ˘ ast˘ asescu, Quasi-c o- F r ob enius c o algebr as , J. Algebra 174 (1995), 909-923. [GMN] J. G´ omez-T orrecillas, C. Manu, C. N˘ ast˘ asescu, Quasi-c o-F r ob enius c o algebr as II , Comm. Algebra V ol 31, N o. 10, pp. 5169-5177, 2003. [GNT] J. G´ omez-T orrecill as, C. N ˘ ast˘ asescu, B. T orrecillas, L o c ali zation in c o algebr as. Applic ations to finiteness c onditions , J. Algebra A ppl. 6 (2007), no. 2, 233–24 3. eprint arXiv:math/0403248, http://a rxiv.org/abs/math/040 3248 . [HR74] R.G.Heyn eman, D.R.Radford, R eflexivity and c o algebr as of finite typ e , J. Algebra 28 ( 1974), 215– 274. [I09] M.C. I o v an ov, The Gener ating Condition for Co algebr as , Bull. Lond . Math. Soc. 41 (2009), no. 3, 483-494. [I] M.C. Iov anov, Co-F r ob enius Co al gebr as , J. A lgebra 303 ( 2006), n o. 1, 146–153; eprint arXiv:math/0604251. [I3] M.C.Io v anov, Abstr act Inte gr als in A lgebr a , preprint, arXiv:0810.374 0 v2. [IO] M.C. Iov ano v , When do es the r ational torsion split off for finitely gener ate d mo dules , Algebr. Rep re- sent. Theory 12 (2009), no. 2-5, 287–309. [J] J.C. Jantzen, R epr esentations of Algeb r aic Gr oups , Mathematical Surveys and Monogra phs 107, American Mathematical So ciet y , (2003), 576p. [L74] B.I-Peng Lin, Semip erfe ct c o algebr as , J. Algebra 30 (1977), 357 - 373. [L75] B.I. Pe ng Lin, Pr o ducts of torsion the ories and appli c ations to c o algebr as , Osak a J. Math. 12 (1975) 433439. [LS] C. Lomp, A. Sant’ana, Chain Co algebr as and Distributivity , J. Pure Appl. Algebra 211 (2007), no. 3, 581–595 . eprint arXiv:math.RA/0610135. [M] S. Montgomery , Hopf Algebr as and their A ctions on Rings , A MS, CBMS n o. 82, 1993. [Mon95] S. Montgome ry , Inde c omp osa ble c o algebr as, simple c omo dules, and p oi nte d Hopf algebr as , Proc. Amer. Math. So c. 123 (1995) 23432351. [NT] C. N˘ ast˘ asescu, B. T orrecillas, The splitting pr oblem for c o algebr as , J. Algebra 281 (2004), 144-149. [NTZ] N˘ ast˘ asescu, B. T orrecilla s, Y . Zhang, Her e ditary Co algebr as , Comm. Algebra 24 (1996), 1521-1528. [Rad73] D. Radford, Cor eflexive c o algebr as , J. Algebra 26 (1973), 512–535. [Rad82] D. E. Radford, On the struct ur e of p ointe d c o algebr as , J. Algebra 77 (1982), 114. [Rad74] D.E. Radford, On the Structur e of Ide al s of the Dual A lgebr a of a Co algebr a , T rans. Amer. Math. Soc. V ol. 198 (Oct 1974), 123-137. [Sim09] D. Simson, Incidenc e c o algebr as of i nterval ly finite p osets , their i nte gr al quadr atic forms and c o- mo dul e c ate gories , Colloq. Math. 115 (2009), 259-295. [Sh76] T. Shudo, A note on c o algebr as and r ational mo dules , H iroshima Math. J. 6 (1976) 297304. [Su78] J.B.Sulliv an, R epr esent ations of the Hyp er algebr a of an Algebr aic Gr oup , Amer. J. Math., V ol . 100, No. 3 (Jun., 1978), pp. 643-652 ON EX TENSIONS OF RA TIONAL MODULES 21 [Sw] M.E. Sweedler, Hopf Algebr as , Benjamin, New Y ork, 1969. [T af72 ] E. T aft, R eflexivity for Algebr as and Co algebr as , Amer. J. Math., V ol. 94, No. 4 (Oct., 1972), p p. 1111-1130 . [T ak77] M. T akeuc hi, Morita the or ems f or c ate gories of c omo dules , J. F ac. Sci. U niv. T okyo Sect. IA Math. 24 (1977), n o. 3, 629-644. [TT05] M.L.T eply , B.T orrecillas, Co algebr as with a r adic al r ational functor , J. Algebra 290 (2005), 491502. University of Southern California, 3620 South Vermont A ve. KAP 108, Los A ngeles, CA 90089, USA &, U niversity of B ucharest, F a c. Ma tema tica & Informa tica, Str. Ac ademiei 14, Bucharest 010014, Ro mania, E-mail addr ess : yovanov@gma il.com, iovano v@usc.edu