A NOTE ON STRONGLY SEPARABLE ALGEBRAS

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1 A NOTE ON STRONGLY SEPARABLE ALGEBRAS MARCELO AGUIAR Abstract. Let A be an algebra over a feld k. If M s an A bmodule, we let M A and M A denote respectvely the k spaces of nvarants and convarants of M, and ϕ M : M A M A be the natural map. In ths note we characterze those algebras A for whch ϕ M s a natural somorphsm as those separable algebras for whch the separablty dempotent can be chosen to be symmetrc, or as those fnte dmensonal algebras for whch the trace form s non-degenerate. These algebras are called strongly separable. We also prove that the cotensor product of C bcomodules has a rght adjont f and only f the coalgebra C s cosemsmple, and show that f C s strongly separable then the adjont s gven by maps of C bcomodules. 1. Introducton Let k be a feld, G a group such that G 0 n k and M a G space. Then the natural map between the spaces of G nvarants and G convarants M G = {m M / gm = m g G} M M G = M/ gm m / g G, m M s an somorphsm, snce the map M G M G, m 1 G gm s clearly a well-defned nverse. In ths note we consder the generalzaton of ths queston to arbtrary algebras A. In order to formulate the problem, one must consder A bmodules M (there s no essental dstncton between modules and bmodules for group algebras). The space of A nvarants of an A bmodule M s and the space of A convarants s g G M A = {m M / am = ma a A} M A = M/[A,M], where [A,M] = am ma / a A, m M. Here, and elsewhere, S denotes the k subspace generated by the set S. Whenever convenent, A bmodules wll be vewed as left A e modules, where A e = A A op. Let ϕ M : M A M M A be the obvous natural map. The purpose of ths note s to characterze those algebras A for whch ϕ s an somorphsm for every M. Date: March 14, Mathematcs Subject Classfcaton. Prmary:16H05 ; Secondary:16W30. Key words and phrases. Separable algebras, nvarants, convarants, coalgebras, Hopf algebras. Research done n 1996 whle the author was supported by Cornell Unversty under a Hutchnson Fellowshp. 1

2 2 M. AGUIAR A necessary condton s that A be projectve as A e module. For, M A = Hom A e(a,m) s a left exact functor of M, whle M A = A A e M s a rght exact one; hence, f these functors are naturally somorphc, they are both exact, so HH (M) = Ext A e(a,m) 0 and HH (M) = Tor Ae (A,M) 0, that s, A s projectve as left A e module and flat as rght A e module. A result of Vllamayor [V, corollary 2] establshes that ether of these condtons already mples the other. When they hold, the k algebra A s sad to be separable. We wll show, however, that separablty alone s not suffcent to guarantee that ϕ be an somorphsm. For nstance, we wll show (corollary 3.1) that A = M n (k) does not have ths property when chark dvdes n, even though t s separable. The addtonal condton that s needed s that the separablty dempotent can be chosen to be symmetrc (theorem 4.1). Such algebras have been studed before by Hattor [H], and called strongly separable. We further show that these algebras are characterzed by the property that the trace form s non-degenerate (theorem 3.1). In the last secton we show that for coalgebras that are dual to strongly separable algebras, the cotensor product of comodules has a rght adjont gven by maps of comodules (corollary 5.1). A more comprehensve study of separablty and strong separablty, specally n connecton to Hopf algebras, can be found n [KS]. 2. Strongly separable algebras. Defnton and examples As already mentoned, a k algebra A s called separable f A s projectve as A e module. Ths s equvalent to the exstence of an element e A e wth the propertes that (1) (2) (a 1)e = (1 a op )e a A µ A (e) = 1 where µ A : A e A s µ A (a b op ) = ab (e arses as the mage of 1 A under a splttng of µ A ). Such an element e s necessarly dempotent and s called a separablty dempotent for A. For ths basc materal the reader s referred to [P], chapter 10. In general e s not unque. For nstance f A = M n (k) then the elements e j = n =1 e j e j are separablty dempotents for each j = 1,...,n (e j denote the elementary matrces). Moreover, f chark does not dvde n then e = 1 n n j=1 e j s another separablty dempotent. Unlke the others, ths one has the addtonal property of beng symmetrc: (3) τ(e) = e where τ : A e A e s the antmorphsm τ(a b op ) = b a op. Defnton 2.1. A k algebra A s sad to be strongly separable f t possesses a symmetrc separablty dempotent e A e. In partcular such an algebra s separable. Explctly, e = u v must satsfy (1) au v = u v a a A,

3 (2) (3) STRONGLY SEPARABLE ALGEBRAS 3 u v = 1 u v = and v u. Remark 2.1. An obvous consequence of the last two equatons s (4) v u = 1. It turns out that condtons (1) and (4) suffce to mply the others. To prove ths we only need to show that (3) holds, for then (2) follows by symmetry. Ths s done as follows (4) u v = (1) u v v j u j = (1) v j u v u j = (4) v j v u u j = v j u j.,j,j,j j Ths shows that defnton 2.1 concdes wth Hattor s [H]. Perhaps the termnology symmetrcally separable would be more descrptve. Examples The dscusson above shows that f A = M n (k) and chark does not dvde n then A s strongly separable. We wll show (corollary 3.1) that f chark dvdes n then A s not strongly separable, even though t s separable. 2. If G s a fnte group, A = kg and chark does not dvde G, then A s strongly separable wth e = 1 g g 1. G g G If chark dvdes G then A s not even separable, snce t s not semsmple. 3. If X s a fnte set and A = k X, then A s strongly separable wth e = x X e x e x, where e x (y) = δ x,y x,y X. More generally, accordng to remark 2.1, any commutatve separable k algebra A s strongly separable, for n ths case (4) s just (2). Moreover, ths shows that any separablty dempotent for A s symmetrc. Ths, together wth the unqueness result n theorem 3.1 below, prove that n a commutatve separable algebra the separablty dempotent s unque (and symmetrc). 4. If chark = 0, then a k algebra A s strongly separable f and only f t s fnte dmensonal and semsmple (see corollary 3.1). We provde one more example of strongly separable algebras; t generalzes that of semsmple group algebras. For background on the relevant materal on Hopf algebras the reader s referred to [S1, chapter V], [M, chapter 2] or [Sc, chapter 3]. Part 1 of the next proposton s well-known (cf. Maschke s theorem for Hopf algebras, as

4 4 M. AGUIAR n [Sc, thm. 3.2]), except perhaps for the fact that fnte dmensonalty s not a necessary hypothess but rather a consequence. We have learned from W. Nchols that part 2 of the next proposton was obtaned earler by F. Kremer (unpublshed). Smlar results appear n [OS, secton 3]. Proposton 2.1. Let H be a Hopf algebra. 1. H s separable f and only f H s semsmple. In ths case, H s fnte dmensonal. 2. If H s semsmple and nvolutory (.e. f the antpode S satsfes S 2 = d H ), then H s strongly separable. Proof. 1. If H s separable, then t s semsmple and fnte dmensonal by general results: [P], corollares 10.4.b and Conversely, f H s semsmple then, frst of all, t s fnte dmensonal by a result of Sweedler [S2, p330]. Now, by Maschke s theorem for Hopf algebras [M, theorem 2.2.1], there s a left ntegral t H such that ǫ(t) 0. Also, snce H s fnte dmensonal, S s bjectve [M, theorem 2.1.3]. Let e = 1 ǫ(t) t2 S 1 (t 1 ) H H, where we have wrtten (t) = t 1 t 2 as usual. Let us show that e s a separablty dempotent for H. Frst, snce S 1 s an antpode for H cop, condton (2) for e holds. Second, for h H we have S(h) (t) = S(h 1 ) (ǫ(h 2 )t) = S(h 1 ) (h 2 t) = S(h 1 ) h 2 t 1 h 3 t 2 S(h)t 1 t 2 = S(h 1 )h 2 t 1 h 3 t 2 = t 1 ht 2 S 1 (t 1 )h t 2 = S 1 (t 1 ) ht 2 t 2 S 1 (t 1 )h = ht 2 S 1 (t 1 ), whch gves condton (1) for e. Thus e s a separablty dempotent for H. Ths completes the proof of If n addton S 2 = d H, then condton (4) holds, so by remark 2.1 e s a symmetrc separablty dempotent. Corollary 2.1. Let H be a fnte dmensonal Hopf algebra H over a feld of characterstc zero. The followng condtons are equvalent: 1. H s nvolutory, 2. H s semsmple, 3. H s cosemsmple, 4. H s separable, 5. H s strongly separable. Proof. Condtons 1,2 and 3 are equvalent by theorems of Larson and Radford [LR1, theorem 4] and [LR2, theorem 4.4]. Condtons 1 and 2 mply 5 by proposton 2.1 (or smply by corollary 3.1 below). The mplcatons are trval.

5 STRONGLY SEPARABLE ALGEBRAS 5 Remark 2.2. Recently t has been shown by Etngof and Gelak [EK] that any (fnte dmensonal) semsmple and cosemsmple Hopf algebra s nvolutory, regardless of the characterstc of the feld. It follows from proposton 2.1 that any such algebra s strongly separable. It s conjectured [K] that any (fnte dmensonal) semsmple Hopf algebra s nvolutory. Ths would mply that any semsmple Hopf algebra s strongly separable. 3. Strongly separable algebras and the non-degeneracy of the trace In ths secton we characterze strongly separable algebras n terms of the trace form. Recall that for any fnte dmensonal k algebra A the trace form Tr : A k s defned as Tr(a) = tr(l(a)), where l : A End k (A) s the left regular representaton and tr s the ordnary trace of a lnear endomorphsm. It gves rse to a symmetrc blnear form T : A A k va T(a,b) = Tr(ab). When A = M n (k), Tr(a) = ntr(a). In partcular Tr 0 f chark dvdes n. Smlarly, when A = kg, Tr( g G a gg) = G a e. The connecton between separablty and the non-degeneracy of T s well-known n the commutatve case (e.g. [J] lemma on page 621). There s a more general result as follows. Frst let us name one further condton (1) u av = u a v a A ; obvously (1) + (3) (1). Theorem 3.1. Let k be a feld and A a k algebra. Then the followng are equvalent: 1. A s strongly separable. 2. A s fnte dmensonal and T : A A k s non-degenerate. Moreover, f these condtons hold, then the symmetrc separablty dempotent s unque. Proof Let e = u v op A e be the symmetrc separablty dempotent. We wll show that x = Tr(xu )v x A, from where t mmedately follows that T s non-degenerate. Frst recall that by a theorem of Vllamayor and Zelnsk ([P, corollary 10.3], or [VZ, proposton 1.1]), any separable algebra s fnte dmensonal. Thus, the trace form s defned. We can assume that {u / I} s a k bass for A. For each j I wrte xu u j = h x jh u h, wth x jh k.

6 6 M. AGUIAR Then u u j v x (1),(1) = xu u j v =,h Hence u j v h x = x jh v for each j,h I. Thus Tr(xu )v = x jj v = j j x jh u h v = h u j v j x (2) = x u h x jh v as announced Let {u / I} be a k bass for A and {v / I} ts dual bass wth respect to T, so that (*) x = Tr(xu )v x A. We wll show that e = u v op A e s a symmetrc separablty dempotent. Snce T s a symmetrc form, e s symmetrc: ( ) u j v j = Tr(u j u )v v j = ( ) v Tr(u u j )v j = v u,,j,j,j j provng (3). Now, u v a ( ) =,j =,j u Tr(v au j )v j =,j Tr(au j v )u v j (3) =,j provng (1). Recall that (1) + (3) (1). Fnally, for each, j I wrte Tr(v au j )u v j = Tr(au j u )v v j ( ) = j au j v j, (**) u u j = h a jh u h wth a jh k. Then for each j I, u u j v (1) = u u j v =,h a jh u h v = h u h a jh v, from where (***) u j v h = a jh v j,h I Hence 1 ( ) = Tr(u )v ( ) =,j a jj v ( ) = j u j v j, provng (2) and thus completng the proof of the mplcaton.

7 STRONGLY SEPARABLE ALGEBRAS 7 Notce also that the unqueness of the symmetrc separablty dempotent has been settled: t s the element of A A correspondng to d A End k (A) under the somorphsm t : A A End k (A), t(a b)(c) = Tr(ca)b. Corollary 3.1. Let A be a k algebra. 1. If chark = 0 then A s strongly separable f and only f t s fnte dmensonal and semsmple. 2. If chark dvdes n and A = M n (k) then A s not strongly separable. Proof. 1. If A s separable then t s fnte dmensonal and semsmple by [P], cor.10.3 and 10.4.b. Conversely, assume that A s fnte dmensonal and semsmple. Let K be an algebrac closure of k. Then A K = M n1 (K)... M nr (K) by the Wedderburn-Artn theorem. It follows from the remarks precedng theorem 3.1 that the trace form for A K over K s non-degenerate, snce chark = 0. But the trace form s nvarant under extenson of scalars, so the same concluson holds for the trace form for A over k, and the theorem apples to conclude that A s strongly separable. 2. As mentoned before, Tr 0 n ths case, so the theorem apples. Corollary 3.2. If K s algebracally closed feld and A a strongly separable K algebra, then chark does not dvde dm K S for any smple A module S. Proof. Ths follows from the proof of the prevous corollary, snce the dmensons of the smple A modules are precsely the numbers n n that proof. Remark 3.1. The prevous result generalzes corollary 8 n [R], whch s the partcular case when A s a Hopf algebra n the hypothess of proposton Symmetrcally separable algebras and the somorphsm between nvarants and convarants Consder now A bmodules M. These can be regarded as left A e modules va (a b op )m = amb. We wll adopt ether pont of vew as convenent. Recall the defntons of M A, M A and ϕ M from the ntroducton. For nstance M A = {m M / am = ma a A} = {m M / (a 1)m = (1 a op )m}. Lemma 4.1. Let A be a separable k algebra, e A e any separablty dempotent and M an A bmodule. Then (a) M A = em, and (b) [A,M] = {m M / τ(e)m = 0} =: τ(e) r. Proof.

8 8 M. AGUIAR (a) : If m M A then em = u mv = u v m (2) = m so m em. : If m M and a A then aem = (a 1)em (1) = (1 a op )em = ema so em M A. (b) : τ(e)(am ma) = τ(e)(a 1 1 a op )m = τ((1 a op a 1)e)m (1) = 0 so am ma τ(e) r. : If τ(e)m = 0 then m = (1 1 τ(e))m (2) = ( 1 vop u op v u op )m = (1 vop v 1)(1 u op )m = (mu )v v (mu ) so m [A,M]. Theorem 4.1. Let A be a k algebra. Then the followng are equvalent: 1. A s strongly separable. 2. The natural map ϕ M : M A M A s an somorphsm for every A bmodule M. Moreover, f e s the symmetrc separablty dempotent, then the map s the nverse of ϕ M. M A M A, m em Proof Let e be the symmetrc separablty dempotent for A. Then, by the lemma, M A = em and M A = M/e r. On the other hand, M = em (1 e)m = em e r obvously. Hence ϕ M : M A = em M M/e r = M A s an somorphsm. Moreover, t s clear that ts nverse s as stated Let M = A A, an A bmodule under a(x y)b = ax yb (so M = A e ). Then M A = A A/ ax y x ya = A A A = A (under the swtch followed by the multplcaton map of A). Let e M A be such that ϕ M (e) = 1 1 M A, say e = u v A A. Snce e M A we have au v = u v a a A, that s (1). Snce e M A maps to 1 A (under ϕ M and the map descrbed above), we have v u = 1, that s (4). Accordng to remark 2.1, e s a symmetrc separablty dempotent for A. Examples Let A = kg, where G s a fnte group wth chark does not dvde G, so that A s strongly separable. Let M be a left G space; vew t as a kg bmodule wth trval G acton. Then M G = {m M / gm = m g G} and M G = M/ gm m / g G, m M are the usual spaces of G nvarants and convarants, and theorem 4.1 recovers the well-known fact that M G = M G under the present hypothess.

9 STRONGLY SEPARABLE ALGEBRAS 9 2. Let A be a strongly separable k algebra and M = A wth ts usual A bmodule structure, a m b = amb. Then M A = Z(A), the center of the algebra A, and M A = A/[A,A], where [A,A] s the k subspace of A generated by all commutators ab ba. Therefore theorem 4.1 mples that f A s strongly separable then Z(A) = A/[A,A]. Ths observaton already appears n [H]. Now suppose that A s actually a strongly separable balgebra (as for nstance n proposton 2.1). Then (A/[A,A]) = Cocom(A ), the space of cocommutatve elements of the coalgebra A. We deduce that n ths case Z(A) = Cocom(A ), n partcular dmz(a) = dmcocom(a ). A smlar result s [R, corollary 4]. 5. Symmetrcally separable algebras and the cotensor product of comodules Let A be a strongly separable k algebra, V a left A module and W a rght one. Let M = V W, wth ts obvous A bmodule structure. Then M A = V W/ av w v wa / v V, w W, a A = W A V under the swtch map. On the other hand, let A be the dual coalgebra of A (recall that A s necessarly fnte dmensonal) and vew V as a rght A comodule and W as a left A comodule as follows: s : V V A v v f ff av = f (a)v a A t : W A W w f w ff wa = f (a)w a A For background on ths materal see [S1]. Recall that f C s a k coalgebra and V and W are rght and left C comodules va s and t as above, then the cotensor product of V and W over C s defned as V C W = Ker(V W s d W d V t V C W). In the case when C = A and V, W and M = V W are as above, t follows mmedately that V A W = M A. Thus theorem 4.1 above says that for a strongly separable algebra A wth symmetrc separablty dempotent e = u j v j, there s a natural somorphsm V A W W A V, v w w A v, wth nverse W A V V A W, w A v j u j v wv j.

10 10 M. AGUIAR Elaboratng on ths observaton we wll now show that the cotensor product over strongly coseparable coalgebras has a rght adjont. Ponderng on when ths was true s what started us nto the consderatons of ths note. Corollary 5.1. Let C, D, and E be k coalgebras. Consder bcomodules as follows: U a C D bcomodule, V an E D bcomodule and W an E C bcomodule. Assume that C and E are fnte dmensonal, so that Hom D (U,V ) carres a natural structure of E C bcomodule as follows: s : Hom D (U,V ) Hom D (U,V ) C T j T j c j ff (d C T)α U (u) = c j T j (u) u U, where α U : U C U s the left C comodule structure on U, and t : Hom D (U,V ) E Hom D (U,V ) T e T ff α V T(u) = e T (u) u U, where α V : V E V s the left E comodule structure on V. Then, f C s strongly separable, there s a natural somorphsm Hom E D (W C U,V ) = Hom E C (W,Hom D (U,V )). Proof. Snce C s fnte dmensonal, there s an equvalence of categores {rght C comodules} = {left C modules} ([S1], chapter II). The comments precedng the corollary show that under the present hypothess we have W C U = U C W naturally. We combne these facts together wth the well-known adjuncton for tensor products of modules over algebras to obtan Hom E C (W,Hom D (U,V )) = Hom C E (W,Hom D (U,V )) = Hom D E (U C W,V ) The proof s complete. = Hom E D (U C W,V ) = Hom E D (W C U,V ). Let U be a C D bcomodule. The functor ( ) C U : {E C bcomodules} {E D bcomodules} may have a rght adjont even f C s not strongly separable. In fact, we wll close ths note by characterzng these coalgebras. In partcular, t follows from theorem 5.1 below that any group-lke coalgebra C = kx has ths property, whle C = k X s strongly separable only when X s fnte. However, t s only under these extra assumptons that we are able to fnd an explct descrpton for the rght adjont functor.

11 STRONGLY SEPARABLE ALGEBRAS 11 Defnton 5.1. A coalgebra C s called cosemsmple f every left C comodule s a drect sum of smple subcomodules, or, equvalently, f every short exact sequence of left C comodules splts. Theorem 5.1. The followng condtons are equvalent for any coalgebra C: 1. C s cosemsmple. 2. For any coalgebras D and E and C D bcomodule U, the functor ( ) C U : {E C bcomodules} {E D bcomodules} has a rght adjont. 3. For any left C comodule U, the functor has a rght adjont. ( ) C U : {rght C comodules} {k spaces} Proof Snce every short exact sequence of left C comodules splts, the functor Hom C (,U) : {left C comodules} {k spaces} s rght exact. By a result of Takeuch [T1, proposton A.2.1], ths functor s rght exact f and only f so s the functor It follows that the functor ( ) C U : {rght C comodules} {k spaces}. ( ) C U : {E C bcomodules} {E D bcomodules} s also rght exact. Snce t clearly preserves drect sums, t s cocontnuous (.e. preserves coequalzers and small coproducts). The result now follows from the specal adjont functor theorem [ML, corollary V.8], snce the category {E C bcomodules} s well-powered, small cocomplete and skeletally small Obvous Snce ( ) C U has a rght adjont, t s rght exact. Ths mples, as n the proof 1 2, that C s cosemsmple. Remark 5.1. On the other hand, t follows from another result of Takeuch [T2, proposton 1.10] that the functor ( ) C U : {rght C comodules} {k spaces} has a left adjont f and only f U s fnte dmensonal. The author thanks Stephen Chase for pontng out the relevance of separablty n relaton to the questons of ths note and Warren Nchols and the referees for useful comments and correctons. References [EK]: P.Etngof and S.Gelak On fnte dmensonal semsmple and cosemsmple Hopf algebras n postve characterstc, to appear n IMRN. [H]: A.Hattor, On strongly separable algebras, Osaka J. Math. 2 (1965), [J]: N.Jacobson, Basc Algebra 2, W.H. Freeman and Co (1989).

12 12 M. AGUIAR [K]: L.Kadson and A.A.Stoln, Separablty and Hopf Algebras, Göteborg Unversty Preprnt (1999). [KS]: I.Kaplansky, Balgebras, Unversty of Chcago Lecture Notes (1975). [LR1]: R.G.Larson and D.E.Radford, Semsmple cosemsmple Hopf algebras, Amer. J. Math 110 (1988), [LR2]: R.G.Larson and D.E.Radford, Fnte Dmensonal Cosemsmple Hopf Algebras n Characterstc 0 Are Semsmple, J. Algebra 117 (1988), [ML]: S.Mac Lane, Categores for the Workng Mathematcan, GTM 5 (1971). [M]: S.Montgomery, Hopf algebras and ther actons on rngs, CBMS n82 (1993). [0S]: U.Oberst and H.-J.Schneder, Über untergruppen endlcher algebrascher gruppen, Manuscrpta math. 8, (1973), [P]: R.S.Perce, Assocatve algebras, GTM 88 Sprnger Verlag (1982). [R]: D.E.Radford, The trace functon and Hopf algebras, J. of Algebra 163 (1994), (1988), Vol 2. [Sc]: H.-J.Schneder, Lectures on Hopf Algebras, Unversdad de Córdoba Trabajos de Matemátca, Sere B, No.31/95 (1995). [S1]: M.Sweedler, Hopf Algebras, W.A. Benjamn (1969). [S2]: M.Sweedler, Integrals for Hopf algebras, Annals of Math 89 (1969), [T1]: M.Takeuch, Formal Schemes Over Felds, Comm. Algebra 5 (1977) [T2]: M.Takeuch, Morta theorems for categores of comodules, J. Fac. Sc. Unv. Tokyo 24 (1977) [V]: O.Vllamayor, On weak dmenson of algebras, Pacfc J. Math. Vol 9 n3 (1959), [VZ]: O.Vllamayor and D.Zelnsky Galos theory for rngs wth fntely many dempotents, Nagoya Math. J. Vol 27 n2 (1966), Department of Mathematcs, Northwestern Unversty, Evanston, IL E-mal address: maguar@math.nwu.edu