Virtually indecomposable tensor categories

VIR TUALL Y INDECOMPOSABLE TENSOR C A TEGOR IES SHLOMO GELAKI Abstract. Let k be any field. J-P . Serre prov ed that the spectrum of the Grothendiec k ring of the k − r epresen tation category of a group is connected, and that the sa me holds in char act eristic zer o for the represen tation category of a Lie algebra ov er k [Se]. W e say that a tensor category C ov er k is vir- tual ly inde c omp osable if its Grothendiec k ri ng conta ins no non trivial cen tral idempotents. W e pr o ve that the following tensor categories are virtually i n- decomposable: T ensor categories wi th the Chev alley prop erty; representat ion categories of affine group sc hemes; representa tion categories of formal groups; represen tation categories of affine supergr oup sc hemes (in c haracteristic 6 = 2); represen tation cat egories of formal sup ergroups (in c haracteristic 6 = 2); sym- metric tensor categories of exponential growth in characte ristic zero. In par- ticular, we obtain an alternativ e pro of to Serre’s Theorem, deduce that the represen tation category of an y Lie algebra ov er k is virtually indecomposable also in p ositive char acteristic (t his a nswers a que stion of Serre [ Se]), and (using a theo rem of Deligne [ D ] i n the super case, and a theorem of Deligne-Milne [DM] in the even case) deduce that any (super)T annakian category is virtually indecomposable (this answe rs another question of Serre [Se]). 1. introduction The following theorem is due to J-P . Serr e. Theorem 1 .1. [Se, Co rollary 5.5 & Section 5.1.2; Ex. 3 ] L et k b e a field. 1) L et G b e any gr oup, let C := R ep ( G ) b e the c ate gory of finite-dimensional r epr esentations of G over k , and let Gr ( G ) b e its (c ommutative) Gr othendie ck ring. Then the sp e ctrum Sp e c ( Gr ( G )) of Gr ( G ) is c onne cte d. 2) Assume t hat k has characteris tic zero . L et g b e a Lie algebr a over k , let C := Re p ( g ) b e the c ate gory of finite-dimensional r epr esentations of g over k , and let Gr ( g ) b e its (c ommutative) Gr othendie ck ring. Then Sp e c ( Gr ( g )) is c onne cte d. The pro of of Theo rem 1.1 uses , a mong o ther things, the fact that the semisimple representations o f a group G are detected by their c hara c ter s, in c hara c ter istic zero, and by their Brauer characters, in p ositive characteristic. Recall that the category Rep( G ) is an exa mple of a T annakian ca tegory [DM] (see Section 2). Mo tiv ated by T he o rem 1 .1 and this fact, Se r re asked the following question. Question 1.2. [Se, Section 5 .1 .2; Ex. 4] Let C be a T annakian categor y ov er any field k . Is it true that Sp ec( Gr ( C )) is co nnected? In particula r, let g be a Lie algebra ov er any field k and let C := Rep( g ) b e the ca tegory of finite-dimensio nal representations o f g o ver k . Is it true that Spec( Gr ( g )) is co nnected? Date : Nov ember 4, 2018. Key wor ds and phr ases. tensor category , Grothendieck ring, Hopf (sup er)algebra, affine (su- per)group scheme, f ormal (super)group. 1 2 SHLOMO GELAK I Question 1.2 can b e extended to any tensor c ate gory ov er k , na mely to a k − linear lo cally finite ab elian categ ory with finite-dimensional Hom − spa ces, equipp ed with an ass o ciative tens o r pro duct and unit. (See e.g. [E], for the definition of a tensor category and its genera l theory .) Definition 1.3. Let k b e any field, and let C be any tensor category ov er k . Let R be any commutativ e ring. W e say that C is virtual ly inde c omp osable over R if its Gro thendieck r ing R ⊗ Z Gr ( C ) with R − co e fficien ts has no no nt rivial central idempo ten ts, and that C is s t r ongly virtual ly inde c omp osable over R if R ⊗ Z Gr ( C ) has no non trivial idempo ten ts. In the case R = Z we shall suppres s the phrase “ov er Z ” . Question 1.4. Is it true that an y tenso r category ov er a ny field is vir tually inde- comp osable? Strong ly virtually indec ompo sable? Our goal in this pap er is to provide a p ositive answer to Ques tio n 1.4 for a v ar iet y of tensor categ ories ov er any field k . More precis ely , w e pr ov e that the following tensor catego ries are vir tually indecomp os a ble: • T enso r categ ories with the Chev alley prop erty . • Representation catego ries o f affine group schemes. • Representation catego ries o f for mal gro ups. • Representation categ o ries of affine sup ergroup s chemes (in characteristic 6 = 2). • Representation catego ries o f for mal sup ergr oups (in characteris tic 6 = 2). • Symmetric tensor categ ories of e xpo nent ial g rowth in c haracter istic zero. In particular, we obta in b oth a n a lternative pro of to Theorem 1.1 and a p ositive answer to Question 1.2. Ac kno wl edgments. The author is grateful to J- P . Serr e for sending him his pro of of Theorem 1.1, for sugge s ting Question 1 .2, and for helpful commen ts. The author is indebted to P . Etingof for his help with the pr o ofs, and for his int erest in the paper and encourag ement . The a uthor thanks V. Ostr ik for telling him ab out the pap er [R]. The res e a rch w as partially supp orted by The Isra el Science F oundation (gr ant No. 317/ 09). 2. The main resul ts The following sta nda rd lemma s hows that w itho ut loss of gener ality we may (and shall) work ov er an alg ebraically clo sed field. Lemma 2. 1. If C is a lo c al ly finite ab elian c ate gory over a field k then the map Gr ( C ) → Gr ( C ⊗ k k ) is inje ctive. Pr o of. It is well known tha t C is equiv a lent to the ca teg ory of finite-dimensional A − como dules ov er k , where A is a co algebra ov er k . Let us denote Gr ( C ) b y Gr ( A ). W e need to sho w that the map Gr ( A ) → Gr ( A ⊗ k k ) is injective. Clear ly , we may assume that A is finite-dimensional, so C = Rep( A ∗ ). Then w e can pass to the quotient o f A ∗ by its radical and as sume that A ∗ is semisimple. So we can assume that A ∗ is simple, i.e., A ∗ = M at n ( D ), D a division algebra ov er k . But in this case the claim is ob vious since Gr ( A ) = Z .  VIR TUALL Y INDECOMPOS ABLE TENSOR CA TEGORIES 3 Corollary 2.2. A tensor c ate gory C over k is virtual ly inde c omp osable if C ⊗ k k is virtual ly inde c omp osable.  Therefore, throughout the pap er we shall w ork ov er an algebraically closed field k . 2.1. Based rings. In Section 3.1 we r ecall the definition of a unital ba sed ring, and then prov e in Section 3.2 the following theore m ab out them. Theorem 2.3. L et A b e any u n ital b ase d ring. Then A is virtual ly inde c omp osable. Recall that a k − linear ab elian rig id tenso r category C is said to have the Cheval- ley pr op erty if the tensor pro duct o f any tw o semisimple ob jects of C is also semi- simple. In other w ords, the sub categor y C ss of semisimple ob jects in C is a tensor sub c ategory . F or example, in characteristic zero, C = Rep( G ) and C = Rep( g ), where G is a n y group and g is an y Lie alg e bra, hav e the Chev a lley prop erty [C]. Of cour se, if C is semisimple (e.g., a fusion c ategory) then C has the Chev a lley prop erty . Now, if C has the Chev alley pr op erty then Gr ( C ) = Gr ( C ss ), so Gr ( C ) is a unital based ring . Hence, Theorem 2.3 implies the following co r ollary . Corollary 2. 4 . L et C b e a k − line ar ab elian rigid tensor c ate gory. If C has the Cheval ley pr op erty then C is virtual ly inde c omp osable.  Remark 2.5. In general it is not true that the repres en tation ca teg ories of gro ups and Lie algebras in p ositive characteristic have the Chev alley prop erty , and likewise for sup erg roups and L ie sup eralge br as in any characteristic. 2.2. The H o pf alg ebra case. In Section 4 we prov e the following inno cent lo oking result, whic h will turn out to pla y the k ey role in pro ving our results concerning (super )groups and (supe r)Lie alg e bras. Theorem 2.6. L et H b e a (not ne c essarily c ommu tative) Hopf algeb r a over a field k , and let Cor ep ( H ) denote the tensor c ate gory of finite-dimensional H − c omo dules over k . Supp ose that I is a Hopf ide al in H such t hat T n ≥ 1 I n = 0 . L et R b e any c ommutative ring and, if t he char acteristic of k is p > 0 , assume that T n ≥ 1 p n R = 0 . Then, if Cor ep ( H /I ) is virtual ly inde c omp osable over R t hen so is Cor ep ( H ) . Remark 2.7. In fact, Theorem 2.6 ho lds also, with the same pr o of, in the top o- logical cas e (i.e., when H is a topolo gical Hopf algebra; see below). 2.3. The group case. In Section 5 we use Theo rem 2.6 to pr ove increasingly strong res ults, culminating in the following theorem. Theorem 2. 8. L et k b e any field, and let G b e an affine gro up scheme over k . L et S b e the set of al l primes not e qual to the char acteristic of k and not dividing | G/G 0 | . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( G )) is c onn e cte d. Theorem 2.8 generalizes to formal groups. Recall that a formal gr oup G ov er a field k , whose subset of closed points (= reduced pa rt) is the affine proa lgebraic group G ov er k , is the following algebraic structure. W e hav e a structure alge - bra O ( G ) ov er k , which ha s an ideal I such that O ( G ) /I = O ( G ), and O ( G ) is complete and sepa rated in the to p olo gy defined by I (i.e., O ( G ) = lim ← − O ( G ) /I m ). Finally , w e ha ve a co commutativ e copr o duct ∆ : O ( G ) → O ( G ) b ⊗O ( G ), where the 4 SHLOMO GELAK I latter c o mpleted tens or pr o duct is lim ← − ( O ( G ) /I m ⊗ O ( G ) /I m ), defining a top olog ical Hopf algebr a structure o n O ( G ), suc h that I is a Hopf ideal, and the isomorphism O ( G ) /I → O ( G ) is a Hopf algebra iso mo rphism. Thu s, com bining Theo rems 2.6 and 2.8, we obtain the following result. Theorem 2.9. L et k b e any field, and let G b e a formal gr oup over k with r e duc e d p art G . L et S b e t he set of al l primes not e qual to the char acteristic of k and not dividi ng | G/G 0 | . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( G )) is c onn e cte d.  Therefore, as a n immediate cor ollary o f Theo rem 2.9 (the case G = 1), we deduce a p os itive a nswer to the seco nd par t of Serr e’s Questio n 1.2. Nevertheless, in Section 4.3 w e s ha ll also give a self con tained pro of o f this theor em in the p ositive characteristic ca se. Theorem 2. 10. L et g b e a Lie algebr a over any field k and let C := R ep ( g ) b e the c at e gory of finite-dimensional r epr esent ations of g over k . Then S p e c ( Gr ( g )) is c onne cte d.  Remark 2.11 . Note that the cas e G = 1 (for mal groups with one closed p oint) re- duces to Lie algebr as in ch ara cteristic zero, but in pos itiv e characteris tic it contains m uch mo re. Recall that a Hopf algebra H ov er a field k is called c o c onne cte d if every s imple H − como dule ov er k is tr ivial (see e.g . [EG] where, in particular, co connected Ho pf algebras ov er C a re cla s sified in Theorem 4.2). W e have the follo wing result which extends Cor ollary 2.10. Theorem 2.12. L et H b e a c o c onne cte d Hopf algebr a over any field k , and let S b e the set of al l primes not e qual to the char acteristic of k . Then Rep ( H ) is virtual ly inde c omp osable over Z [ S − 1 ] .  Pr o of. If H is co connected then H ∗ is a topolo gical Ho pf algebra with maximal ideal I := K er ( ǫ ), which is co mplete a nd se pa rated in the top ology defined by I (as the powers o f I are o rthogona l to the ter ms of the co radical filtration of H ). So the claim follows from the top ologica l version o f Theorem 2.6 (see Rema r k 2.7).  2.4. The s up ergroup case. In Section 6.1 w e r ecall the notion of a Hopf sup er- algebra, and in Section 6.2 we r ecall the notio ns o f an affine sup ergr oup scheme and a formal super group over k . W e then g eneralize in Section 6.3 the res ults from Section 5 to the super -case (assuming the characteristic of k 6 = 2). Let G b e an affine sup ergro up sc heme or, more genera lly , a forma l sup ergro up, and let u ∈ G b e an element of order 2 a c ting by parit y on the algebra of r egular functions O ( G ). Let Rep( G , u ) b e the category of repres ent ations of G on finite- dimensional super vector spaces ov e r k on which u acts b y parity , and let Gr ( G , u ) be its Gro thendiec k r ing. Theorem 2.13 . L et k b e any field of char acteristic 6 = 2 . L et G b e an affine sup er- gr oup scheme over k or, mor e gener al ly, a formal s up er gr oup over k . L et S b e t he set of al l primes 6 = 2 not e qual to the ch ar acteristic of k and not dividing |G / G 0 | . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( G , u )) is c onn e cte d. Remark 2.14. Note that the prime 2 must b e excluded (i.e., canno t b e inv erted). Indeed, alrea dy in the c ategory Sup erV ect of finite-dimensional sup ervector spa ces ov er k (see Section 6), the element 1 2 ( k 0 ⊕ k 1 ) is a nontrivial idemp otent. VIR TUALL Y INDECOMPOS ABLE TENSOR CA TEGORIES 5 Recall that a Lie su p er algebr a ov er a field k is a Lie algebra in Super V ect (see e.g, [B]). In other words, a L ie sup eralgebr a g = g 0 ⊕ g 1 is a s uper vector spa ce ov er k , eq uipp ed with a n o pe ration [ , ] : g ⊗ g → g satisfying the following axioms: [ x, y ] = − ( − 1) | x || y | [ y , x ] and [ x, [ y , z ]] = [[ x, y ] , z ] + ( − 1 ) | x || y | [ y , [ x, z ]], fo r homoge- neous elements x, y ∈ g and z ∈ g . The following result o n Lie sup eralgebr as is an immediate cor o llary of Theorem 2.13. Corollary 2.15. L et g b e a Lie su p er algebr a over a field k of char acteristic 6 = 2 . L et S b e the set of al l primes 6 = 2 not e qual to the char acteristic of k . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( g )) is c onne cte d.  By a theorem of Deligne [D] in c hara c ter istic zero , the c ategories Rep( G , u ) ex- haust all k − linear ab elian symmetric r igid tensor c ategories of ex po nen tial growth. Hence, we deduce the following corolla r y . Corollary 2.16 . If C is a k − line ar ab elian symmet ric rigid tensor c ate gory of exp onent ial gr owth over an algebr aic al ly close d field k of char acteristic zer o, then C is virtual ly inde c omp osable.  Recall that a (su p er)T annakian ca tegory over a field k is a k − linear ab elian sym- metric rigid tenso r category C , with End( 1 ) = k , where 1 denotes the unit ob ject, which admits a fib er functor to the ca tegory o f finite-dimensional (supe r )vector spaces (see [D ]). In the following prop os ition we de duce a p o sitive answer to the first par t of Serre’s Question 1.2. Prop ositio n 2.17. A (sup er)T annakian c ate gory C over any field k is virtual ly inde c omp osable. Pr o of. By (the sup er a nalog of ) a theor em of Deligne-Milne [DM] (which is in [D]), C is equiv alent to a ca tegory of the form Rep( G , u ), so the claim follows by Theorem 2.13.  3. The vir tua ll y indecompos ability of a unit al based ring In characteristic zero there is an alternative (“combinatorial”) pro of of (a sligh t generaliza tion of ) Theorem 1.1 in the framew ork of unital based rings. 3.1. Based ring s. Let A be a fr e e Z − algebra with a distinguishe d Z + − basis { b i } (not necessar ily of finite rank), which contains the unit element, suc h that b i b j = P k n k ij b k , wher e n k ij ∈ Z + . The map ( P i n i b i , P i m i b i ) 7→ P i n i m i defines a po sitive inner pro duct ( , ) : A × A → Z on A . W e call A a un ital b ase d ring if there is an inv olution i 7→ i ∗ such tha t the induced map x = P i n i b i 7→ x ∗ := P i n i b i ∗ satisfies ( xy , z ) = ( x, z y ∗ ) = ( y , x ∗ z ) for all x, y , z ∈ A . In pa rticular, it follows tha t the matrix of m ultiplication b y x ∗ is trans p os ed to the ma trix of multiplication by x , for an y x ∈ A . Example 3.1. If C is a k − linear semis imple rigid tenso r ca tegory , its Gro thendiec k ring Gr ( C ) is a unital ba sed r ing . A typical e x ample of such categ o ry is the categ ory C := Core p( H ) of finite-dimensional co mo dules of a cosemisimple Hopf algebra H . The dis ting uished Z + − basis of Gr ( C ) is formed by the isomorphism classes of the simple H − como dules , a nd the inv olution ∗ is given by taking the k − linear dua l of a como dule. 6 SHLOMO GELAK I 3.2. The pro of o f Theo rem 2.3. Let e 6 = 1 be a central idemp otent in A . W e hav e to show that e = 0. W e first note that e is a pro jection oper a tor on an inner pr o duct space, which is normal (i.e ee ∗ = e ∗ e ), so e is self-adjoint. I ndee d, ( e (1 − e ∗ ) , e (1 − e ∗ )) = ( e ∗ e (1 − e ∗ ) , 1 − e ∗ ) = ( ee ∗ (1 − e ∗ ) , 1 − e ∗ ) = 0. Thus by po sitivity o f the inner pro duct, e (1 − e ∗ ) = 0 , so e = ee ∗ , hence e = e ∗ . Then e is an orthogo nal pr o jector to a prop er subs pace of R ⊗ Z A , which do es not con tain 1. So 0 ≤ ( e, e ) = ( e 1 , e 1) < (1 , 1) = 1. But ( e, e ) is an integer, so ( e, e ) = 0, and hence e = 0.  Remark 3.2. It is interesting to mention here a class ic al re sult o f K aplansky which asserts that there is no nontrivial idempo tent in the integral gr oup ring o f any (not necessarily commutativ e) group (se e [K], [P]), i.e., the in tegral gro up ring of any group is stro ng ly virtually indecomp osable. Equiv alently , the tensor categ ory V ec G of G − gra de d vector spaces ov er k is strongly virtually indecomp os a ble for a ny group G . In fact, Pro po sition 3 in [R] extends the result of Kaplans k y to fusion ring s (= unital based rings of finite r ank). Eq uiv alently , any fusion catego ry is stro ngly virtually indec o mpo sable. 4. The pr oof of Theorem 2. 6. In this section w e let H be a Hopf alg ebra (not necess arily co mmutative) o ver k , and C := Cor ep( H ) b e the categ ory o f finite-dimensio nal r ight como dules o f H . Then C is a k − linear abelia n rig id tensor category in which every o b ject has a finite length. Let Gr ( C ) be the Gro thendieck ring of C ; it is the free Z − algebr a with a distinguished bas is formed by the classes [ X ] o f the simple ob jects X ∈ C . 4.1. Characters in Hopf algebras. Recall that any M ∈ C has a cano nical rational H ∗ − mo dule structur e . Definition 4.1. F o r an ob ject M ∈ C , the character ch ( M ) of M is the character of the H ∗ − mo dule M . In other words, the character ch ( M ) is the function H ∗ → k defined by ch ( M )( x ) := tr ( x | M ). Clearly , ch ( M ) ∈ H , ch ( M ) ch ( N ) = ch ( M ⊗ N ) a nd ch ( M ) + c h ( N ) = ch ( M + N ). Moreover, if M 1 , . . . , M n are the distinct comp osition factors of M , w ith mu l- tiplicities a 1 , . . . , a n , then ch ( M ) = P n i =1 a i ch ( M i ). In o ther w ords, the character of M and the character of its semisimplification ⊕ n i =1 a i M i coincide. W e there fo re hav e a w ell defined k − a lgebra homomorphism ch : k ⊗ Z Gr ( C ) → H , a ⊗ [ M ] 7→ a · ch ( M ) . Prop ositio n 4. 2. The char acter map ch is inje ctive. In other wor ds, if M , N ∈ C with ch ( M ) = ch ( N ) , then [ M ] = [ N ] in k ⊗ Z Gr ( C ) . Pr o of. It is eno ugh to show that if P m i =1 a i ch ( M i ) = 0 on H ∗ , fo r some finite nu mber of non- isomorphic ir r educible como dules M i ∈ C and some elements a i ∈ k , then a i = 0 for all i . Indeed, b y the density theorem, the map H ∗ → ⊕ i End k ( M i ) is surjective, so we can c ho ose a n element x ∈ H ∗ which maps to 0 on E nd k ( M j ) for j 6 = i , and to an element with trace 1 on End k ( M i ), which implies that a i = 0 for a ll i .  VIR TUALL Y INDECOMPOS ABLE TENSOR CA TEGORIES 7 Remark 4.3. Note that if the characteristic of k is zero then Pro po sition 4.2 implies that the character of M determines the comp osition fa c tors o f M together with their multiplicities, i.e., ch : Gr ( C ) → H is injective (so in particular , if M , N are semisimple then M , N a r e iso mo rphic). On the other ha nd, if the characteristic of k is p > 0 then Prop osition 4.2 implies only that the character of M determines the co mpo s ition factors o f M to gether with their mu lt iplicitie s mo dulo p . 4.2. The pro of of Theorem 2.6. Set C := Corep( H/ I ). The surjection of Hopf algebras H ։ H / I induces a tensor functor C → C , which in turn induces a ring homo mo rphism R ⊗ Z Gr ( C ) → R ⊗ Z Gr ( C ). Suppose E ∈ R ⊗ Z Gr ( C ) is a n idempo ten t which is not 0 or 1 , and let e be the image of E in R ⊗ Z Gr ( C ). By assumption, e is either 0 or 1. Without lo ss o f genera lit y we may assume that e = 0, replacing E b y 1 − E if needed. Now, if k has characteris tic p > 0, at least one of the coefficients of E is no t divisible b y p . Indee d, if E = pF then E n = p n F n = E , so E ∈ p n R ⊗ Z Gr ( C ) fo r all n , and hence it is zero , whic h is a con tradiction. Therefor e , the image E ′ of E in R ⊗ Z k ⊗ Z Gr ( C ) (which is ( R/pR ) ⊗ F p k ⊗ Z Gr ( C ) in p ositive character istic) is nonzero, and the image e ′ of e in R ⊗ Z k ⊗ Z Gr ( C ) is zero (as e = 0). Now, using the embedding ch : k ⊗ Z Gr ( C ) ֒ → H , we get a nonzero idempotent ch ( E ′ ) in R ⊗ Z H , whic h ha s zero imag e in R ⊗ Z H/ I (this image is ch ( e ′ )). This implies that ch ( E ′ ) ∈ R ⊗ Z I . But since ch ( E ′ ) is an idemp otent, c h ( E ′ ) n = ch ( E ′ ) for all n , so ch ( E ′ ) ∈ T n ≥ 1 ( R ⊗ Z I n ) = 0 , whic h is a contradiction.  4.3. A pro of of Theorem 2.10. Let g b e a Lie a lgebra over a field k of character- istic p > 0; we may assume without loss of generality that k is alg ebraically closed (see Co rollary 2.2). Let A := U ( g ) ∗ be the dual algebr a o f the universal env eloping algebra U ( g ) of g (it is a top olog ical Hopf algebra in the top ology defined by the maximal ideal I o f A ). Let E ∈ Gr ( g ) b e an idempotent which is not 0 or 1. W e can as s ume that T r E (1) = 0 mo dulo p by replacing E with 1 − E if needed. Now, at least one o f the co efficie nts of E is not divisible by p . Indeed, otherwise ( E /p ) n = E /p n , so E /p n ∈ Gr ( g ) for a ll n , but E /p n do es not have in teger co efficients for lar ge enough n . Consequently , the image E ′ of E in k ⊗ Z Gr ( g ) is no nze ro. Hence , using Pr op osition 4.2, w e g et a no nzero idemp otent ch ( E ′ ) in A . On the other hand, the a ugmentation map A → k maps ch ( E ′ ) to ze r o (since T r E (1) = 0 mo dulo p ). So c h ( E ′ ) is contained in I . But ch ( E ′ ) is an idempo ten t, so it is co nt ained in any power I n of I . But T n ≥ 1 I n = 0, s o ch ( E ′ ) is zero, whic h is a con tradiction.  5. The pr oof of Theorem 2. 8 The pr o of of Theo rem 2.8 will b e carried in several steps . 5.1. G is a reductiv e ab elian affine algebraic group ov er k . Recall that if G is a reductive abelia n affine a lgebraic group ov er k then G ∼ = G 0 × A , where G 0 = G n m is the n − dimensional torus ov er k a nd A is a finite ab elian gr oup of order prime to p (in ca se the characteristic of k is p > 0) (see e.g ., [Sp]). In par ticular, all finite-dimensional simple repre sent ations of G ov er k are 1 − dimensiona l, and their isomorphism t yp es are parameterized b y pair s ( z , χ ), where z ∈ Z n and χ ∈ b A . Thu s all idemp otents in Q ⊗ Z Gr ( G ) can b e easily descr ibe d in this case (they inv o lve a fac tor of 1 / | A | ), so the result follo ws in a straightforward manner. 8 SHLOMO GELAK I 5.2. G is an y ab elian affine alge braic group ov er k . Recall that if G is an ab elian affine algebraic group over k then G ∼ = G s × G u , wher e G s and G u are the subgroups of semisimple and unipo ten t element s o f G , r esp e ctively (see e.g., [Sp]). So Rep( G ) and Rep( G s ) hav e the s a me Gr othendieck ring s, and the c la im follows from 5.1 . 5.3. G is an y affine algebraic group ov er k . Let G b e any affine a lgebraic g roup ov er k , and let O ( G ) b e its co or dinate Hopf alg e br a; it is a finitely gener ated commu- tative reduced Hopf algebra over k . Supp ose e ∈ Gr (Rep( G )) = Gr (Co rep( O ( G ))) is an idempo ten t. Then ch ( e ) ∈ O ( G ) is a n idemp otent, so in particular a class function on G taking the v alues 0 , 1 (here we are using the trick with dividing by p , a s we did in the pro of o f Theor em 2.6, which explains why we ca nnot inv ert p ). Therefore it suffices to prove that ch ( e )( g ) = ch ( e )(1) for all g ∈ G . But for that purp ose w e ma y assume that G is the (Zariski closure o f the) cyclic group with generator g , so G is abelia n and the claim follo ws from 5.2. 5.4. G is an affine proalgebraic group o v er k . Let G b e an affine proalgebr aic group over k , and let O ( G ) b e its co ordina te Hopf alg ebra over k ; it is a commutativ e reduced Ho pf a lgebra over k (not necess a rily finitely gener ated). But it is w ell known that O ( G ) is the inductive limit of its finitely generated Hopf s ubalgebras (see e.g. [A]), so the claim follo ws in a stra ig ht forward manner from Part 5.3. 5.5. G i s an affine group scheme o v er k . Let G b e an affine group sch eme, and let O ( G ) b e the commutativ e Hopf algebra repres e n ting the gro up functor G (it is not necessarily reduced) (see e .g ., [W]). Since k is algebraic a lly closed, it is well known (essent ially by Hilbert Nullstellensa tz) that the nilradica l I of O ( G ) is a Hopf ideal. No w, the commutativ e reduced Hopf alg e br a O ( G ) /I ov er k represents an affine pr oalgebr a ic g roup ov er k , s o by 5 .4, Corep( O ( G ) /I ) is virtually indecomp osable. Finally , since T n ≥ 1 I n = 0, it follows from Theo rem 2.6 that Corep( O ( G )) = Rep( G ) is virtually indecompo sable, as claimed. This c oncludes the pro of of T he o rem 2.8.  6. The pr oof of Theorem 2.13 6.1. Hopf sup eralgebras. Rec a ll that a Hopf su p er algebr a ov er a field k is a Hopf algebra in the k − linear ab elian symmetric tensor ca tegory Sup erV ect of s uper vector spaces o ver k (s e e e.g, [B]). In other w ords, a Hopf sup eralgebr a H = H 0 ⊕ H 1 is an ordina ry Z 2 − graded asso ciative unital algebra over k (i.e., a sup er algebr a ), equipp e d with a coasso ciative mo rphism ∆ : H → H ⊗ H in Sup erV e c t, which is m ultiplicative in the sup er- s ense, and with a co unit and an tipo de satisfying the standard axioms. Her e mu ltiplicativity in the s uper -sense means that ∆ satisfies the r elation ∆( ab ) = X ( − 1) | a 2 || b 1 | a 1 b 1 ⊗ a 2 b 2 , where a, b ∈ H ar e ho mogeneous elements, | a | and | b | denote the degrees of a and b , ∆( a ) = P a 1 ⊗ a 2 and ∆( b ) = P b 1 ⊗ b 2 . A Ho pf sup eralg e bra H is sa id to be c ommu tative if ab = ( − 1) | a || b | ba for all homogeneous elements a, b ∈ H . Let J ( H ) := ( H 1 ) = H 2 1 ⊕ H 1 be the Hopf ideal of H gener ated b y the o dd elements H 1 . Then the quotient H := H /J ( H ) is an ordinary Hopf a lgebra. Note that if H is commutativ e then J ( H ) co nsists of nilp otent elements. VIR TUALL Y INDECOMPOS ABLE TENSOR CA TEGORIES 9 Let us recall the following useful cons tr uction, intro duce d in Section 3.1 in [AEG]. Let H b e any Hopf supera lg ebra ov er k , and le t e H := k [ Z 2 ] ⋉ H b e the semidirect pro duct Hopf sup era lgebra with comultiplication e ∆ and antipo de e S , where the generator g of Z 2 acts o n H b y g hg − 1 = ( − 1 ) | h | h . F or x ∈ e H , write e ∆( x ) = e ∆ 0 ( x ) + e ∆ 1 ( x ), where e ∆ 0 ( x ) ∈ e H ⊗ e H 0 and e ∆ 1 ( x ) ∈ e H ⊗ e H 1 . Then one can define a n or dinary Hopf alg ebra s tructure on the a lgebra H ′ := e H , with comultip lication a nd antipo de maps given by ∆( h ) := e ∆ 0 ( h ) − ( − 1) | h | ( g ⊗ 1) e ∆ 1 ( h ) a nd S ( h ) := g | h | e S ( h ), h ∈ H ′ (see Theor em 3.1.1 in [AEG]). Theorem 6. 1. L et H b e a Hopf sup er algebr a over a field k of char acteristic 6 = 2 , a nd let Cor ep ( H ) b e the t ensor c ate gory of fin ite- dimensional H− c omo dules in Super V ect over k . The tensor c ate gories Cor ep ( H ) and Cor ep ( H ′ ) ar e e quivalent. Pr o of. An H− como dule in Sup erV ect is a (contin uous) Z 2 − graded H ∗ − mo dule in the ca tegory V ect of vector s paces ov er k , which is the same as a cont inuous mo dule ov er the algebr a k [ Z 2 ] ⋉ H ∗ . But k [ Z 2 ] ⋉ H ∗ is isomorphic to ( H ′ ) ∗ as a n algebr a, so a H − como dule in Sup erV e c t is the sa me thing as a H ′ − como dule in V ect. Finally , it is easy to c heck that this eq uiv alence is in fact a tensor equiv alence.  W e note that as a consequence of Theor em 6.1, w e can define the character map ch : k ⊗ Z Gr (Cor ep( H )) → H ′ , a ⊗ [ M ] 7→ a · ch ( M ), a nd deduce fr om Prop ositio n 4.2 that it is an injective k − alg ebra homomorphism. 6.2. Affine sup ergroup sc hemes and formal sup ergroups. Recall that an affine sup er gr oup scheme G is the s pectr um of a (not necessar ily finitely gener a ted) commutativ e Hopf sup er a lgebra O ( G ) o ver k (see e.g ., [D]). In other words, it is a functor G fr o m the categor y of sup ercommutativ e algebr as to the categ ory of groups defined by A 7→ G ( A ) := Ho m( O ( G ) , A ), where Hom( O ( G ) , A ) is the group of algebr a maps O ( G ) → A in Sup erV ect . Recall that a formal sup er gr oup G ov er a field k , with reduced part G , is the following a lgebraic structure . W e hav e a sup eralg ebra O ( G ) over k , which has an ideal I such that O ( G ) is complete and separated in the top olog y defined by I (i.e., O ( G ) = lim ← − O ( G ) / I m ), and O ( G ) / I = O ( G ). Finally , we hav e a sup erco com- m utative copr o duct ∆ : O ( G ) → O ( G ) b ⊗O ( G ), where the la tter completed tensor pro duct is lim ← − ( O ( G ) / I m ⊗ O ( G ) / I m ), defining a topo logical Hopf algebra s tructure on O ( G ), s uc h tha t I is a Hopf ide a l, and the iso morphism O ( G ) / I → O ( G ) is a Hopf sup era lgebra isomorphis m. Let G b e a n affine sup ergr oup scheme. The ordinary commutativ e Hopf algebra O ( G ) is isomor phic to O ( G ) for some a ffine group s cheme G , which is referre d to as the even p art o f G . Let G b e an affine sup ergro up scheme ov er k , or, mor e generally , a formal su- per group ov er k , a nd let Rep( G ) denote the categ o ry of finite-dimensio na l algebraic representations o f G in Sup erV ect ov er k . Then Rep( G ) is a k − linear ab elian sym- metric rigid tensor category with End( 1 ) = k , where 1 denotes the unit ob ject, which admits a fib e r functor (= a sy mmetr ic tensor functor) to the full k − linea r ab elian symmetric rigid tensor sub catego ry of Super V ect whose o b jects are the finite-dimensional sup ervector s paces. Just lik e in the ev en case, Rep( G ) is equiv a- lent to Corep( O ( G )). 10 SHLOMO GELAK I 6.3. The pro of of Theorem 2. 13. Let G b e an a ffine sup ergr oup scheme ov er k , let H := O ( G ) b e the comm utative Hopf sup eralge br a repres e n ting the gro up functor G , a nd let H ′ be the ordina ry Hopf algebra asso ciated with H . Then the quotient H ′ / ( k [ Z 2 ] ⋉ J ( H )) is a commutativ e Hopf algebra repres ent ing the group functor Z 2 × G , where G is the even part of G . Therefore Corep( H ′ / ( k [ Z 2 ] ⋉ J ( H ))) is virtually indecomp osable by Theorem 2.8, and it follows fro m Theo rems 2 .6 and 6.1 that Corep( H ) is virtually indecomposa ble, as claimed. F or for mal sup erg roups G over k the pro of is completely paralle l using Theo r em 2.9 ab out formal gr oups ov er k .  References [A] E. Ab e, H opf algebras. T ranslated fr om the Japanese by Hisae Kinoshita and Hiroko T anak a. Cambridge T racts in Mathematics, 74 . Cambridge Universit y Press, Cambridge- New Y ork, 1980. xii+284 pp. [AEG] N. Andruskiewitsc h, P . Etingof and S. Gelaki, T riangular Hopf algebras with the Chev al- ley proper t y , Michigan Journal of Mathematics 49 (2001), 277–298. [B] H. Bosec k, Affine Lie Sup ergroups, Math. Nac hr. 143 , 303–327 (1989). [C] C. Chev alley , Th ´ eorie des Groupes de Lie, V ol. I II , Hermann, Paris, 1954. [D] P . Deligne, Cat´ egories tensorielles. (F renc h) [T ensor ca tegories] Dedicated to Y uri I. Manin on the occasion of his 65th birthday . Mosc. Math. J. 2 (2002) , no. 2, 227–248. [DM] P . Deligne and J. Milne, T annakian Cat egories, Lecture Notes i n Mathemat ics 900 , 101-228, 1982. [E] P . Etingof, T ensor Categories, http://www-math.mit.edu/ ∼ et ingof/tenscat1.pdf [EG] P . Etingof and S. Gelaki, Quasisymmetric and unip oten t tensor categories, Mathematic al R esea r ch L e tters , 15 (2008) , no. 5, 857–866. [K] I . Kaplansky , Fields and rings, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, Ill., 1969. [P] D . Passman, Idempoten ts in group rings, Pr o c . Amer. Math. So c. 28 371-374 (1971). [R] R. Rouquier, F ami lles et blo cs d’alg` ebres de Heck e, C. R. Ac ad. Sci. Paris t. 3 29 , S ´ erie I, 1037–1042, 1999. [Se] J-P . Serre, Lectures on N X ( p ), to app ear. [Sp] T. A. Spri nger, Linear al gebraic groups. Second edition. Progress in Mathematics, 9 . Birkha”user Boston, Inc., Boston, MA, (1998) . xiv+334 pp. [W] W . W aterhouse, Introduction to affine group schemes. Graduate T exts in Mathematics, 66 . Springer-V erlag, New Y or k-Berlin, 1979. xi+164 pp. Dep ar tment of Ma thema tics, Technion-Israel Institute of Technology , Haif a 32000, Israel E-mail addr ess : gelaki@math.t echnion.ac.il VIR TUALL Y INDE COMPOSABLE TENSOR CA TEGORIES SHLOMO GELAK I Abstract. Let k b e an y field. J-P . Ser re prov ed tha t the sp ec- trum of the Grothendieck ring of the k − r epresentation ca teg ory of a gro up is connec ted, and that the same holds in char acter- istic zer o for the representation ca tegory of a Lie a lgebra over k [Se]. W e say that a tensor categor y C ov er k is virtual ly inde- c omp osable if its Grothendieck ring contains no nontrivial central idempo ten ts. W e prov e that the following tensor categ ories a re virtually indecomp osable: T ensor categorie s with the Che v alley prop erty; representation categ ories of affine group schemes; repre- sentation ca tegories of formal groups ; re pr esentation categories of affine supergr oup schemes (in characteristic 6 = 2); repres e ntation categorie s of formal sup ergro ups (in c harac teristic 6 = 2); symmet- ric tenso r ca tegories of exponential g rowth in characteristic zero. In particular, we obta in an alternative pro o f to Serre’s Theorem, deduce that the represe n tation categor y of any Lie algebra ov er k is vir tually indecomposa ble also in p ositive char acteristic (this answers a question of Serre [Se]), a nd (using a theorem of Deligne [D] in the sup er ca se, and a theorem of Deligne-Milne [DM ] in the even case) deduce that any (super )T annakia n catego ry is vir tually indecomp osable (this answers another que s tion of Serre [Se]). 1. introduction The following t heorem is due to J-P . Serre. Theorem 1.1. [Se , Corollary 5.5 & Section 5.1.2; Ex. 3] L e t k b e a field. 1) L et G b e any gr oup, le t C := R ep ( G ) b e the c ate gory of finite- dimensional r epr esentations of G over k , and let Gr ( G ) b e its (c ommu- tative) Gr othendi e ck ring. Then the sp e ctrum Sp e c ( Gr ( G )) of Gr ( G ) is c onn e cte d. 2) Assume that k has c haracteristic zero . L et g b e a Lie algeb r a over k , l e t C := R e p ( g ) b e the c ate gory of finite-dimensional r epr esentations Date : Nov e m b er 4, 2018. Key wor ds and phr ases. tensor category , Grothendieck ring, Hopf (sup er)algebr a, affine (sup er)gr oup scheme, formal (sup er )g roup. 1 2 SHLOMO GELAKI of g over k , and let Gr ( g ) b e its (c ommutative) Gr othendie ck ring. Then Sp e c ( Gr ( g )) is c o n ne cte d. The pro of of Theorem 1.1 uses, among other things, the fact that the semisimple represen tations of a group G are detected by their c hara c- ters, in c ha r a cteristic zero, and by their Brauer c haracters, in p ositive c haracteristic. Recall that the category Rep( G ) is an example of a T annakian cat- egory [DM] (see Section 2). Motiv ated b y Theorem 1.1 and this fact, Serre ask ed the f ollo wing question. Question 1.2. [Se, Section 5 .1 .2; Ex. 4] Let C be a T annakian category o v er an y field k . Is it true that Sp ec( Gr ( C )) is connected? In particular, let g b e a Lie algebra ov er an y field k and let C := Rep( g ) b e the category of finite-dimensional represen tations of g ov er k . Is it true that Sp ec( Gr ( g )) is connected? Question 1 .2 can b e extended to an y tensor c ate gory ov er k , namely to a k − linear lo cally finite ab elian category with finite-dimens ional Hom − spaces, equipp ed with an asso ciative tensor pro duct and unit. (See e.g. [E], fo r the definition of a tensor category and its general theory .) Definition 1.3. Let k b e an y field, and let C b e an y tensor category o v er k . Let R b e any commutativ e ring. W e sa y that C is virtu- al ly inde c omp osable over R if its Grothendiec k ring R ⊗ Z Gr ( C ) with R − co efficien ts has no non trivial central idemp otents , and that C is str ongly virtual ly inde c omp osabl e over R if R ⊗ Z Gr ( C ) has no nontriv- ial idemp oten ts. In the case R = Z w e shall suppress the phrase “ov er Z ”. Question 1.4. Is it t rue that an y tensor catego ry ov er an y field is virtually indecomp osable? Stro ngly virtually inde comp osable? Our goal in this pap er is to pro vide a p o sitive answ er to Question 1.4 for a v ariet y of tensor categories o v er any field k . More precisely , w e pro v e that the following tensor categories are virtually indecomp osable: • T ensor categories with the Chev alley prop ert y . • Represen tat io n catego r ies of affine group sc hemes. • Represen tat io n catego r ies of formal groups. • Represen tat io n categories of affine sup ergroup sc hemes (in c har- acteristic 6 = 2). • Represen tat io n categories of formal sup ergroups (in c haracter- istic 6 = 2). VIR TUALL Y INDECOMPOSA BLE TENSOR CA TEGORIES 3 • Symmetric tensor categories of exp onen tial g ro wth in charac- teristic zero. In particular, w e obta in b oth a n alternativ e pro of to Theorem 1 .1 and a p ositiv e a nswe r to Question 1.2. Ac kno wledgmen ts. The author is grateful t o J-P . Serre for sending him his pro of of Theorem 1 .1, for suggesting Question 1.2, a nd for helpful commen ts. The author is indebted to P . Etingof for his help with the pro ofs, and for his in terest in the paper and encouragemen t. The author tha nks V. Ostrik for telling him ab out the pap er [R]. The researc h w as pa r tially supp orted by The Israel Science F ounda- tion (grant No. 317 /09). 2. The main res ul ts The following standard lemma sho ws that without loss of g eneralit y w e ma y (and shall) w ork o v er an algebraically closed field. Lemma 2.1. If C is a lo c al ly finite ab elian c ate gory over a field k then the m ap Gr ( C ) → Gr ( C ⊗ k k ) is inje ctive. Pr o of. It is we ll known that C is equiv alen t to the category of finite- dimensional A − comodules ov er k , where A is a coalgebra o v er k . Let us denote Gr ( C ) b y Gr ( A ). W e need to show that the map Gr ( A ) → Gr ( A ⊗ k k ) is injectiv e. Clearly , w e ma y assume that A is finite- dimensional, so C = Rep( A ∗ ). Then w e can pass to the quotien t of A ∗ b y its ra dical and assume that A ∗ is semisimple. So w e can assume that A ∗ is simple, i.e., A ∗ = M at n ( D ), D a division algebra ov er k . But in this case the claim is o bvious since Gr ( A ) = Z .  Corollary 2.2. A tens or c ate go ry C over k is virtual ly in de c om p osable if C ⊗ k k is virtual ly inde c omp osable.  Therefore, throughout the pap er we shall w ork o v er an algebraically closed field k . 2.1. Based rings. In Section 3.1 we recall the definition of a unital based ring, and then prov e in Section 3.2 the followin g t heorem abo ut them. Theorem 2.3. L et A b e any unital b ase d ring. Th en A is virtual ly inde c omp osable. Recall that a k − linear ab elian rigid tensor category C is said to ha v e the Chev a l ley pr op erty if the tensor pro duct of a n y tw o semisimple ob jects of C is also semisimple. In other w ords, the sub category C ss 4 SHLOMO GELAKI of semisimple ob jects in C is a tensor sub category . F or example, in c haracteristic zero, C = Rep( G ) and C = Rep( g ), where G is an y g r o up and g is a ny Lie algebra, ha v e the Chev alley prop ert y [C]. Of course, if C is semisimple (e.g., a fusion category) then C has the Chev alley prop ert y . No w, if C ha s t he Chev alley prop erty then G r ( C ) = Gr ( C ss ), so Gr ( C ) is a unital ba sed ring. Hence, Theorem 2.3 implies the follo wing corollary . Corollary 2.4. L e t C b e a k − line ar a b elian rigid tensor c ate gory. If C has the Ch e val ley pr op erty then C is virtual ly inde c o mp osable.  Remark 2.5. In general it is not true that the represen tation cat- egories of groups and Lie algebras in p ositiv e c haracteristic ha v e the Chev alley prop ert y , and lik ewise fo r sup ergroups and Lie sup eralg ebras in an y characteristic . 2.2. The Hopf algebra case. In Section 4 we pro v e the following inno cen t lo oking result, whic h will t urn out to play the k ey role in pro ving our results conc erning (super)groups and ( sup er)Lie algebras. Theorem 2.6. L et H b e a (not ne c es sarily c ommutative) Hopf algebr a over a field k , and let Cor ep ( H ) denote the tensor c ate go ry of finite- dimensional H − c omo d ules over k . Supp ose that I is a Hopf ide al in H such that T n ≥ 1 I n = 0 . L et R b e any c ommutative ring and, if the char acteristic o f k is p > 0 , assume that T n ≥ 1 p n R = 0 . Then, if Cor ep ( H /I ) is virtual ly inde c omp osabl e over R then so is Cor ep ( H ) . Remark 2.7. In fa ct, Theorem 2.6 holds also, with the same pro of, in the top olo g ical case (i.e., when H is a top ological Hopf algebra; see b elo w). 2.3. The group case. In Section 5 w e use Theorem 2.6 to prov e in- creasingly strong results, culm inating in the following t heorem. Theorem 2.8. L et k b e any field, and let G b e an affine gr oup schem e over k . L et S b e the set o f al l p rimes not e qual to the cha r acteristic of k and n ot dividing | G/G 0 | . Then S p e c ( Z [ S − 1 ] ⊗ Z Gr ( G )) is c onne c te d. Theorem 2.8 generalizes to for ma l groups. R ecall that a formal gr o up G o v er a field k , w hose subset of closed p o ints ( = reduc ed pa rt) is the affine proalgebraic group G o v er k , is the follow ing algebraic structure. W e hav e a structure algebra O ( G ) o v er k , whic h has an ideal I suc h that O ( G ) /I = O ( G ), and O ( G ) is complete a nd separated in the top ology defined b y I (i.e., O ( G ) = lim ← − O ( G ) /I m ). Finally , w e hav e a co comm utativ e copro duct ∆ : O ( G ) → O ( G ) b ⊗O ( G ), where the VIR TUALL Y INDECOMPOSA BLE TENSOR CA TEGORIES 5 latter completed tensor pro duct is lim ← − ( O ( G ) /I m ⊗O ( G ) /I m ), defining a top ological Hopf algebra structure on O ( G ), suc h t ha t I is a Hopf ideal, and the isomorphism O ( G ) /I → O ( G ) is a Hopf algebra isomorphism. Th us, com bining Theorems 2.6 a nd 2.8, w e obtain the follo wing re- sult. Theorem 2.9. L et k b e any field, and let G b e a formal gr oup over k with r e duc e d p art G . L et S b e the set of al l primes not e qual to the char acteristic of k and not dividing | G/G 0 | . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( G )) is c onne cte d.  Therefore, as an immediate corollary of Theorem 2 .9 (the case G = 1), we deduce a p o sitiv e answ er to the second part of Serre’s Q uestion 1.2. Nev ertheless, in Section 4.3 w e shall also giv e a self contained pro of of this theorem in the p ositiv e c haracteristic case . Theorem 2.10. L et g b e a Lie algebr a o v er a ny field k an d let C := R ep ( g ) b e the c ate gory of finite-dimensional r epr es e ntations of g ove r k . Th e n S p e c ( Gr ( g )) is c onne cte d.  Remark 2.11. Note that the case G = 1 (fo rmal groups with one closed p o in t) reduces to Lie algebras in characteristic zero, but in p o s- itiv e c haracteristic it con tains m uc h more. Recall t hat a Hopf algebra H ov er a field k is called c o c onn e cte d if ev ery simple H − como dule ov er k is trivial (see e.g. [EG] where, in particular, co connected Hopf algebras o v er C are classified in Theorem 4.2). W e hav e the follo wing result whic h extend s Corollary 2 .10. Theorem 2.12. L et H b e a c o c onne cte d Hopf algebr a o ver any fiel d k , and let S b e the set of al l primes not e qual to the char acteristic of k . Then R ep ( H ) is virtual ly inde c omp o sable over Z [ S − 1 ] .  Pr o of. If H is co connected then H ∗ is a top olo gical Hopf algebra with maximal ideal I := K er ( ǫ ), whic h is complete a nd separated in the top ology defined b y I (as the p ow ers of I a re orthogo nal to the terms of the coradical filtration of H ). So the claim follo ws from the top olo gical v ersion of Theorem 2.6 (see Remark 2 .7).  2.4. The sup ergroup case. In Section 6.1 w e recall the notion of a Hopf supera lg ebra, and in Section 6.2 w e recall the notions of an affine sup ergroup sc heme a nd a for ma l sup ergroup o v er k . W e then generalize in Section 6.3 the results from Section 5 to the sup er-case (assuming the c haracteristic of k 6 = 2). Let G b e an affine sup ergroup sc heme or, more generally , a fo rmal sup ergroup, and let u ∈ G be an elemen t of order 2 acting b y parit y o n 6 SHLOMO GELAKI the algebra of regular functions O ( G ). Let Rep( G , u ) b e the category of represen tations of G on finite-dimensional sup erv ector spaces ov er k on whic h u acts by parit y , and let Gr ( G , u ) b e it s Grothendiec k ring. Theorem 2.13. L et k b e any field of cha r acteristic 6 = 2 . L et G b e an affine sup er gr oup sc h eme over k or, mor e gener a l ly, a formal sup er- gr oup over k . L et S b e the s e t of al l primes 6 = 2 not e qual to the char- acteristic of k and not dividing |G / G 0 | . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( G , u )) is c onn e cte d. Remark 2.14. Note that the prime 2 m ust b e excluded (i.e., c an- not b e in v erted). Indeed, already in the category Sup erV ect of finite- dimensional sup erv ector spaces ov er k (see Section 6), the elemen t 1 2 ( k 0 ⊕ k 1 ) is a no n trivial idemp oten t. Recall that a Lie sup er algebr a o v er a field k is a Lie alg ebra in Sup erV ect (see e.g, [B]). In other w ords, a Lie sup eralgebra g = g 0 ⊕ g 1 is a sup erv ector space ov er k , equipp ed with an op eration [ , ] : g ⊗ g → g satisfying the follo wing axioms: [ x, y ] = − ( − 1) | x || y | [ y , x ] a nd [ x, [ y , z ]] = [[ x, y ] , z ] + ( − 1) | x || y | [ y , [ x, z ]], for homogeneous elemen ts x, y ∈ g a nd z ∈ g . The follow ing result on Lie sup eralgebras is an immediate corol- lary of Theorem 2.1 3 . Corollary 2.15. L et g b e a Lie sup er alg ebr a ov er a fie l d k of char- acteristic 6 = 2 . L et S b e the set of al l prim es 6 = 2 not e qual to the char acteristic of k . Then Sp e c ( Z [ S − 1 ] ⊗ Z Gr ( g )) is c onne c te d.  By a theorem of Deligne [D] in c haracteristic zero, the categories Rep( G , u ) exhaust all k − linear ab elian symmetric rigid tensor cate- gories of exp onential growth. Hence, w e deduce the followin g coro lla ry . Corollary 2.16. If C is a k − line ar ab elian symmetric rig id tensor c ate gory of exp onential g r owth over an algebr ai c al ly close d field k of char acteristic zer o, then C is virtual ly inde c omp osab le.  Recall that a (sup er) T annakia n category ov er a field k is a k − linear ab elian symmetric rigid tensor category C , with End( 1 ) = k , where 1 denotes the unit ob ject, whic h admits a fib er functor to the category of finite-dimensional (sup er)vec tor spaces (see [D]). In the follow ing prop osition we deduce a p o sitive answe r to the first part of Serre’s Question 1.2. Prop osition 2.17. A (sup er)T annakian c ate gory C over a n y fi e l d k is virtual ly inde c omp osable. Pr o of. By (the sup er analog of ) a theorem of Deligne-Milne [DM] (whic h is in [D]), C is equiv alen t to a catego ry of the form Rep( G , u ), so the claim follows by Theorem 2 .13.  VIR TUALL Y INDECOMPOSA BLE TENSOR CA TEGORIES 7 3. The vir tuall y indecomposability of a unit al b ased ring In c haracteristic zero there is an alternativ e (“combinatorial”) pro of of (a sligh t generalization of ) Theorem 1.1 in the framew ork of unital based rings. 3.1. Based rings. Let A b e a ring with a distinguished Z − basis { b i } , i ∈ I , ( no t necessarily of finite rank), whic h con tains t he unit ele- men t 1, such that b i b j = P k n k ij b k , where n k ij ∈ Z + . The bilinear map ( P i n i b i , P i m i b i ) 7→ P i n i m i defines a p ositiv e inner pro duct ( , ) : A × A → Z on A . W e call A a unital b ase d ring if there is an inv o- lution i 7→ i ∗ suc h that the induced map x = P i n i b i 7→ x ∗ := P i n i b i ∗ satisfies ( xy , z ) = ( x, z y ∗ ) = ( y , x ∗ z ) for all x, y , z ∈ A . In particular, it follo ws t ha t the matrix o f m ultiplication b y x ∗ is transp osed to the matrix of multiplic ation b y x , fo r an y x ∈ A . Example 3.1. If C is a k − linear semisimple rig id tensor categor y , its Grothendiec k ring Gr ( C ) is a unital based ring. A typical example of suc h catego r y is the category C := Corep( H ) of finite-dimensional co- mo dules of a cosemisimple Hopf algebra H . The distinguished Z − basis of Gr ( C ) consis ts of the isomorphism classes of simple H − como dules, and the inv olution ∗ is give n by taking the k − linear dual of a como dule. 3.2. The pro of of Theorem 2.3. Let e 6 = 1 b e a cen tral idemp oten t in A . W e ha v e to sho w that e = 0 . W e first note that e is a pro jection op erator on an inner pro duct space, whic h is normal (i.e ee ∗ = e ∗ e ), so e is self-adj o in t. Indeed, ( e (1 − e ∗ ) , e (1 − e ∗ )) = ( e ∗ e (1 − e ∗ ) , 1 − e ∗ ) = ( ee ∗ (1 − e ∗ ) , 1 − e ∗ ) = 0. Th us b y p ositivit y of the inner pro duct, e (1 − e ∗ ) = 0, so e = ee ∗ , hence e = e ∗ . Then e is an ortho g onal pro jector to a prop er subspace of R ⊗ Z A , whic h do es not contain 1. So 0 ≤ ( e, e ) = ( e 1 , e 1) < (1 , 1) = 1. But ( e, e ) is an in teger, s o ( e, e ) = 0, and hence e = 0 .  Remark 3.2. It is interes ting to men tio n here a classical result of Kaplansky whic h asserts that there is no non t rivial idempotent in the in tegral group ring of an y (not necessarily comm utativ e) group (see [K], [P ]), i.e., t he integral group ring of an y g roup is strongly virtually indecomp o sable. Equiv alen tly , the tensor category V ec G of G − graded v ector spaces o v er k is strong ly virtually indecomposable for an y gro up G . In fact, Proposition 3 in [R] extends the result of Ka pla nsky to fusion rings (= unital based ring s of finite ra nk). Equiv alently , an y fusion category is strongly virtually indecomp osable. 8 SHLOMO GELAKI 4. The proo f of Theorem 2.6. In this section we let H be a Hopf algebra (no t necessarily commuta- tiv e) o ve r k , and C := Corep( H ) b e the category of finite-dime nsional righ t como dules of H . Then C is a k − linear ab elian rigid tensor cat- egory in whic h eve ry ob j ect has a finite length. Let Gr ( C ) b e the Grothendiec k ring of C ; it is the free Z − algebra with a distinguished basis formed by the classes [ X ] of the simple o b jects X ∈ C . 4.1. Characters in Hopf algebras. Recall tha t any M ∈ C has a canonical rational H ∗ − mo dule structure. Definition 4.1. F or an ob ject M ∈ C , the c haracter ch ( M ) of M is the c haracter of the H ∗ − mo dule M . In ot her w ords, the character ch ( M ) is the function H ∗ → k defined b y ch ( M )( x ) := tr ( x | M ). Clearly , ch ( M ) ∈ H , ch ( M ) ch ( N ) = ch ( M ⊗ N ) and ch ( M ) + ch ( N ) = ch ( M + N ). Moreo v er, if M 1 , . . . , M n are the distinct com- p osition factors of M , with multiplicitie s a 1 , . . . , a n , then ch ( M ) = P n i =1 a i ch ( M i ). In other w ords, the c haracter of M and the c ha r acter of its semisimplification ⊕ n i =1 a i M i coincide. W e therefore ha v e a w ell defined k − algebra homomorphism ch : k ⊗ Z Gr ( C ) → H , a ⊗ [ M ] 7→ a · ch ( M ) . Prop osition 4.2. T h e char acter map ch is inje ctive. In other wor ds, if M , N ∈ C with ch ( M ) = ch ( N ) , then [ M ] = [ N ] in k ⊗ Z Gr ( C ) . Pr o of. It is enough t o sho w tha t if P m i =1 a i ch ( M i ) = 0 on H ∗ , for some finite n um b er of non-isomorphic irr educible como dules M i ∈ C and some elemen ts a i ∈ k , then a i = 0 for all i . Indeed, b y the de nsit y theorem, the map H ∗ → ⊕ i End k ( M i ) is sur- jectiv e, so we can c ho ose an elemen t x ∈ H ∗ whic h maps to 0 on End k ( M j ) fo r j 6 = i , and to an elemen t with trace 1 on End k ( M i ), whic h implies that a i = 0 for all i .  Remark 4.3. Note that if the c haracteristic of k is zero then Prop o- sition 4.2 implies that t he c haracter of M determines the comp osition factors of M together with their multiplicities , i.e., ch : Gr ( C ) → H is injectiv e (so in particular, if M , N are semisimple then M , N are isomorphic). On the other hand, if the c haracteristic of k is p > 0 then Prop osition 4 .2 implies o nly that the c haracter of M determines the comp osition factor s of M to gether with their multiplicities m o dulo p . 4.2. The pro of of Theorem 2.6. Set C := Corep( H / I ). The surjec- tion of Hopf algebras H ։ H /I induces a tensor functor C → C , whic h VIR TUALL Y INDECOMPOSA BLE TENSOR CA TEGORIES 9 in turn induces a ring homomorphism R ⊗ Z Gr ( C ) → R ⊗ Z Gr ( C ). Supp ose E ∈ R ⊗ Z Gr ( C ) is an idemp otent whic h is not 0 or 1, and let e b e the image o f E in R ⊗ Z Gr ( C ). By assumption, e is either 0 or 1. Without loss of generalit y w e ma y assume that e = 0, replacing E b y 1 − E if nee ded. No w, if k has characteristic p > 0, at least one of the co efficien t s of E is not divisible b y p . Indeed, if E = pF then E n = p n F n = E , so E ∈ p n R ⊗ Z Gr ( C ) for all n , and hence it is zero, whic h is a con tradiction. Therefore, the image E ′ of E in R ⊗ Z k ⊗ Z Gr ( C ) (whic h is ( R /pR ) ⊗ F p k ⊗ Z Gr ( C ) in p ositive characteristic ) is nonzero, and the image e ′ of e in R ⊗ Z k ⊗ Z Gr ( C ) is zero (as e = 0). No w, using the em b edding ch : k ⊗ Z Gr ( C ) ֒ → H , w e get a nonzero idemp oten t ch ( E ′ ) in R ⊗ Z H , whic h has zero imag e in R ⊗ Z H /I (this image is ch ( e ′ )). This implies that ch ( E ′ ) ∈ R ⊗ Z I . But since ch ( E ′ ) is an idemp otent, ch ( E ′ ) n = ch ( E ′ ) for all n , so ch ( E ′ ) ∈ T n ≥ 1 ( R ⊗ Z I n ) = 0, whic h is a contradiction.  4.3. A pro of of Theorem 2.10. Let g be a Lie algebra o v er a field k o f c haracteristic p > 0; w e may assume without loss of generality that k is algebraically closed (see Corollary 2.2). Let A := U ( g ) ∗ b e the dual algebra of the univ ersal en ve loping algebra U ( g ) of g (it is a top ological Hopf algebra in the top ology defined b y the maximal ideal I of A ). Let E ∈ Gr ( g ) b e an idemp o t ent whic h is not 0 or 1. W e can assume that T r E (1) = 0 mo dulo p by replacing E with 1 − E if needed. No w, at least one of the co efficien ts o f E is not divisible by p . Indeed, otherwise ( E /p ) n = E /p n , so E /p n ∈ Gr ( g ) for all n , but E /p n do es not hav e integer co efficien ts for large enough n . Consequen tly , the image E ′ of E in k ⊗ Z Gr ( g ) is nonzero. Hence, using Pro p osition 4.2, w e get a no nzero idemp oten t ch ( E ′ ) in A . On the other hand, the augmen tatio n map A → k maps ch ( E ′ ) to zero (since T r E (1) = 0 mo dulo p ). So ch ( E ′ ) is con tained in I . But ch ( E ′ ) is an idemp o ten t, so it is contained in an y p ow er I n of I . But T n ≥ 1 I n = 0, so ch ( E ′ ) is zero, whic h is a con tradiction.  5. The proo f of Theorem 2.8 The pro o f of Theorem 2.8 will b e carried in sev eral steps . 5.1. G is a reductive ab elian affine algebraic gr oup o ver k . Re- call t ha t if G is a reductiv e ab elian affine algebraic group ov er k then G ∼ = G 0 × A , where G 0 = G n m is the n − dimensional torus o v er k and A is a finite ab elian group of order prime to p (in case the character- istic of k is p > 0) (see e.g., [Sp]). In particular, all finite-dimensional 10 SHLOMO GELAKI simple represen tations o f G ov er k a r e 1 − dimensional, and their iso- morphism types are par a meterized b y pairs ( z , χ ), where z ∈ Z n and χ ∈ b A . Th us all idempotents in Q ⊗ Z Gr ( G ) can b e easily describ ed in this case (they in v olve a f a ctor of 1 / | A | ), so the r esult follow s in a straigh tforw ard manner. 5.2. G is an y ab elian affine algebraic group ov er k . Recall that if G is an a b elian affine alg ebraic group ov er k then G ∼ = G s × G u , where G s and G u are the subgroups of s emisimple and unip otent elemen ts of G , resp ectiv ely (se e e.g., [Sp ]). So Rep( G ) and Rep( G s ) ha v e the same Grothendiec k rings, a nd the claim follo ws f r om 5.1. 5.3. G is an y affine algebraic group ov er k . Let G b e an y affine algebraic group ov er k , and let O ( G ) b e its co ordinate Hopf algebra; it is a finitely generated comm utativ e reduced Hopf algebra ov er k . Supp ose e ∈ Gr ( R ep( G )) = Gr (Corep( O ( G ))) is an idempotent. Then ch ( e ) ∈ O ( G ) is an ide mp otent, so in particular a class function on G taking the v alues 0 , 1 (here w e are using the tric k with dividing b y p , as we did in t he pro of of Theorem 2.6, whic h explains wh y w e cannot in v ert p ) . Therefore it suffices to prov e tha t ch ( e )( g ) = ch ( e )(1) for a ll g ∈ G . But fo r that purp o se w e may assume that G is the (Za r iski closure of the) cyclic group with generator g , so G is ab elian and the claim follows from 5.2. 5.4. G is an affine proalgebraic group ov er k . Let G b e an affine proalgebraic group o v er k , and let O ( G ) b e its co ordinate Hopf algebra o v er k ; it is a comm utativ e reduced Hopf algebra o v er k (not necessarily finitely generated). But it is we ll know n that O ( G ) is the inductiv e limit o f its finitely generated Hopf subalgebras (see e.g. [A]) , so the claim follows in a straigh tfor ward manner from P art 5.3. 5.5. G is an affine group sc heme o v er k . Let G b e an a ffine group sc heme, and let O ( G ) b e the comm utativ e Hopf algebra represen ting the group functor G (it is not necess arily reduce d) (see e.g., [W]). Since k is algebraically closed, it is w ell kno wn (essen tially by Hilb ert Null- stellensatz) that t he nilradical I of O ( G ) is a Hopf ideal. No w, the comm utativ e reduced Hopf algebra O ( G ) /I ov er k repre sen ts an affine proalgebraic group ov er k , so b y 5.4, Corep( O ( G ) /I ) is virtually inde- comp osable. Finally , since T n ≥ 1 I n = 0, it f ollo ws from Theorem 2.6 that Corep( O ( G )) = Rep( G ) is v irtually indecomposable, as claimed. This concludes the proof of Theorem 2.8.  VIR TUALL Y INDECOMPOSA BLE TENSOR CA TEGORIES 11 6. The proof of Theorem 2.13 6.1. Hopf sup eralgebras. Recall that a Hopf sup er algebr a o v er a field k is a Hopf algebra in the k − linear abelian symmetric tens or category Sup erV ect of sup erv ector spaces o v er k (see e.g, [B]). In other w ords, a Hopf sup eralgebra H = H 0 ⊕ H 1 is an ordinary Z 2 − graded asso ciative unital algebra ov er k (i.e., a sup e r algebr a ), equ ipp ed with a coasso cia- tiv e morphism ∆ : H → H ⊗ H in Sup erV ect, whic h is multiplic ativ e in the super- sense, and with a counit and antipo de satisfying the stan- dard axioms. Here m ultiplicativit y in the sup er-sense means that ∆ satisfies the relation ∆( ab ) = X ( − 1) | a 2 || b 1 | a 1 b 1 ⊗ a 2 b 2 , where a, b ∈ H are homogeneous elemen ts, | a | and | b | denote the degrees of a and b , ∆( a ) = P a 1 ⊗ a 2 and ∆( b ) = P b 1 ⊗ b 2 . A Hopf sup eralg ebra H is said to b e c o mmutative if ab = ( − 1) | a || b | ba for all homogeneous elem en ts a, b ∈ H . Let J ( H ) := ( H 1 ) = H 2 1 ⊕ H 1 b e the Hopf ideal of H generated b y the o dd elemen ts H 1 . Then the quotien t H := H /J ( H ) is an ordinary Hopf algebra. Note that if H is commutativ e then J ( H ) consists of nilp oten t elemen ts. Let us recall the follow ing useful construction, in tro duced in Sec- tion 3.1 in [AEG]. Let H b e a ny Hopf sup eralgebra ov er k , and let e H := k [ Z 2 ] ⋉ H b e the semidirect pro duct Hopf sup eralgebra with co- m ultiplication e ∆ a nd an tip o de e S , where the gene rator g of Z 2 acts on H by g hg − 1 = ( − 1) | h | h . F or x ∈ e H , write e ∆( x ) = e ∆ 0 ( x ) + e ∆ 1 ( x ), where e ∆ 0 ( x ) ∈ e H ⊗ e H 0 and e ∆ 1 ( x ) ∈ e H ⊗ e H 1 . Then one can define an or dina ry Hopf algebra structure on the algebra H ′ := e H , with com ultiplication and antipo de maps giv en b y ∆( h ) := e ∆ 0 ( h ) − ( − 1) | h | ( g ⊗ 1) e ∆ 1 ( h ) and S ( h ) := g | h | e S ( h ), h ∈ H ′ (see Theorem 3 .1.1 in [AEG]). Theorem 6.1. L et H b e a Hopf sup er algebr a over a field k of char a cter- istic 6 = 2 , and le t Cor ep ( H ) b e the tensor c ate go ry of finite-dimens i o nal H− c omo dules in Sup erV ect over k . The tensor c ate gories C or ep ( H ) and Cor e p ( H ′ ) ar e e quivale n t. Pr o of. An H − como dule in Sup erV ect is a (con tin uo us) Z 2 − graded H ∗ − mo dule in t he catego ry V ect of vector spaces o v er k , whic h is the same as a contin uous mo dule ov er the algebra k [ Z 2 ] ⋉ H ∗ . But k [ Z 2 ] ⋉ H ∗ is isomorphic to ( H ′ ) ∗ as a n algebra, so a H− como dule in Sup erV ect is the same thing as a H ′ − como dule in V ect. Finally , it is easy to c hec k that this equiv alence is in fact a tens or equiv alence.  12 SHLOMO GELAKI W e note that as a consequence of Theorem 6 .1, w e can define the c haracter map c h : k ⊗ Z Gr (Corep( H )) → H ′ , a ⊗ [ M ] 7→ a · ch ( M ), and de duce from Prop osition 4.2 t hat it is an injectiv e k − algebra ho- momorphism. 6.2. Affine sup ergroup sc hemes and formal sup ergroups. Re- call that an affine sup er gr oup scheme G is the sp ectrum of a (not nec- essarily finitely generated) comm utative Hopf superalg ebra O ( G ) o v er k (see e.g., [D]). In ot her w ords, it is a functor G f rom the category of supercommutativ e algebras to the catego ry of groups defined b y A 7→ G ( A ) := Hom( O ( G ) , A ), where Hom( O ( G ) , A ) is the group o f algebra maps O ( G ) → A in Sup erV ect. Recall that a formal s up er gr oup G ov er a field k , with reduced part G , is the follo wing algebraic structure. W e hav e a sup eralgebra O ( G ) o v er k , whic h has a n ideal I suc h that O ( G ) is complete and sepa- rated in the top ology defined b y I (i.e., O ( G ) = lim ← − O ( G ) / I m ), and O ( G ) / I = O ( G ). Finally , w e ha v e a sup erco commutativ e copro duct ∆ : O ( G ) → O ( G ) b ⊗O ( G ), where the latter completed tensor pr o d- uct is lim ← − ( O ( G ) / I m ⊗ O ( G ) / I m ), defining a top ological Hopf alg ebra structure on O ( G ), suc h that I is a Hopf ideal, and the isomorphism O ( G ) / I → O ( G ) is a Hopf sup eralgebra isomorphism . Let G b e an affine sup ergroup sc heme. The ordinary comm ut a tiv e Hopf algebra O ( G ) is isomorphic to O ( G ) for some affine group sc heme G , whic h is referred t o as the eve n p art o f G . Let G b e an affine sup ergroup sc heme o v er k , or, more g enerally , a f ormal sup ergr o up o v er k , and let Rep( G ) denote the category of finite-dimensional alg ebraic represen tations of G in Sup erV ect o v er k . Then Rep( G ) is a k − linear ab elian symmetric rigid tensor category with End( 1 ) = k , where 1 denotes the unit ob ject, whic h admits a fib er functor (= a symmetric t ensor functor) to the full k − linear ab elian symmetric r ig id tensor sub category of Sup erV ect whose ob jects are the finite-dimensional sup erv ector spaces. Just like in the ev en case, Rep( G ) is equiv alen t to Corep ( O ( G )). 6.3. The pro of of Theorem 2.13. Let G b e an affine sup ergroup sc heme ov er k , let H := O ( G ) be the comm utativ e Hopf sup eralgebra represen ting the gr o up functor G , and let H ′ b e the ordinary Hopf algebra associated with H . T hen the quotien t H ′ / ( k [ Z 2 ] ⋉ J ( H )) is a comm utativ e Ho pf algebra represen ting the gro up functor Z 2 × G , where G is the ev en part of G . Therefore Corep( H ′ / ( k [ Z 2 ] ⋉ J ( H ))) is virtually indecomp o sable by Theorem 2.8, and it follo ws from Theorems 2.6 and 6.1 that Corep( H ) is virtually indecomposable, as claimed. VIR TUALL Y INDECOMPOSA BLE TENSOR CA TEGORIES 13 F or formal sup ergroups G o v er k the pro of is completely parallel using Theorem 2.9 about f o rmal groups ov er k .  Reference s [A] E. Abe , Hopf a lgebras. T r anslated from the Japanese by Hisae Kinoshita and Hiroko T a nak a. Cambridge T racts in Mathematics, 74 . Cam bridge Univ ersity P ress, Cambridge-New Y ork, 19 8 0. xii+284 pp. [AEG] N. Andr us kiewitsch, P . Etingof and S. Gela k i, T r iangular Hopf algebras with the Chev alley prop erty , Michigan Journal of Mathematics 49 (2001 ), 277–2 98. [B] H. Bo seck, Affine Lie Super groups, Math. Nachr. 143 , 30 3–327 (1989). [C] C. 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Serre, Lec tures on N X ( p ), to app ear. [Sp] T. A. Springer , Linear a lg ebraic gro ups . Second editio n. P rogr ess in Math- ematics, 9 . Birkha”user Bo ston, Inc., Boston, MA, (1998). xiv+33 4 pp. [W] W. W aterhouse , Intro duction to affine group schemes. Gra duate T exts in Mathematics, 66 . Springer- V erlag , New Y ork-Ber lin, 1979. xi+ 164 pp. Dep ar tment o f Ma thema tics, Technion-Israel Institute of Technol- ogy, Haif a 32000 , I srael E-mail addr ess : gelak i@math .technion.ac.il