Radical cube zero weakly symmetric algebras and support varieties

RADICAL CUBE ZERO WEAKL Y SYMMETRIC ALGEBRAS AND SUPPOR T V ARIETIES KARIN ERDMANN AND ØYVIND SOLBERG Abstract. One of our main results is a cl ass i fication all the weakly symmet- ric radical cube zero finite dimensional algebras o v er an algebraically closed field having a theo ry of support via the Hochsc hild cohomology ri ng satisfying Dade’s Lemma. Along the wa y we give a charact erization of when a finite di- mensional K oszul algebra has suc h a theory of supp or t in terms of the graded cen tre of the Koszul dual. Introduction The suppo r t v ariety of a module, if it exists, is a p ow erful in v ariant. F or gro up algebras of finite gr oups, or co co mm utative Hopf a lg ebras it is defined in ter ms of the maximal ideal spectrum o f the gr oup cohomolog y ring. F or a more general finite-dimensional algebra Λ, ins tead of g roup cohomolo g y one can take a subalgebra of the Hochschild cohomolo gy , pr ovided it has suitable finite gener ation prop erties (see Section 1 for details ). It w as sho wn in [8], that if these hold and Λ is self- injectiv e, then many of the standar d results from the theory of supp ort v arieties for finite groups g e ne r alize to this situation. The question is no w to understand whether o r not these finite genera tion prop- erties hold for g iven classes of a lgebras. Since Ho chsc hild cohomology is difficult to calcula te explicitly , one would rather not do this and hav e other w ays to detect finite g eneration. In this pap e r w e present a metho d, for K oszul algebras, which gives a criterion in terms of the Ko s zul dual, to show that the finite gene r ation condition holds. W e denote this condition by (Fg) . W e apply t his method to a g eneral weakly symmetric a lgebra with radical cub e zero, a nd also to quantum exterio r algebras. In [1], D. Benson characterize d the rate of growth of resolutions for weakly symmetric alg ebras with radical cub e zero. W e use his results, and we show that almost all such algebra s which are of tame representation t yp e satisfy the finite generation h yp othesis. There a re o nly tw o exceptional cases, wher e a deformatio n parameter q appears and whenever q is not a ro ot of unit y . It is clear from [1] that the algebras in Bens o n’s list which hav e wild t yp e, canno t satisfy the finite generation condition. Using [6] it follows eas ily that all w eakly s ymmetric (in fact all s elfinjectiv e) algebras of finite r epresentation t yp e ov er an algebraica lly clo sed field satisfy the finite generatio n condition. The precise a nswer is the following (see Theorem 1.5 for the definition o f the quivers involv ed). Date : Nov ember 17, 2018. 2000 Mathematics Subje ct Classific ation. 16P10, 16P20, 16E40, 16G20; Secondary: 16S37. Key wor ds and phr ases. W eakly symmetri c algebras, support v ari eties, Koszul algebras. The authors ac kno wledge supp ort fr om EPSRC gran t EP/D077656/1 and NFR Storforsk grant no. 167130. 1 2 ERDMANN AND SOLBERG Theorem. L et Λ b e a fi n ite dimensional symmetric algebr a over an alg ebr aic al ly close d field with r adic al cub e zer o and r adic al squar e non-zer o. Then Λ satisfies (Fg) if and only if Λ is of finite r epr esentation typ e, Λ is of typ e e D n for n ≥ 4 , e Z n for n > 0 , g D Z n , e E 6 , e E 7 , e E 8 , or Λ is of t yp e e Z 0 or e A n when q is a r o ot of unity. In addition we show that a quantum exter io r algebra satisfies (Fg) if and only if a ll deforma tion par ameters are ro o ts of unit y . This is genera lized in [2]. F ur- thermore, this result, together with [8 ] also generalizes almo st all of Theo rem 2.2 in [7]. As obse r ved in [3 ], ev en “nice” selfinjective alg ebras do not necessarily satisfy (Fg) , in spite of sharing many of the same repres en tation str uctural prop erties with algebras having (Fg) . Hence, whether or not a finite dimensional algebra has the prop erty (Fg) is a more subtle question than one fir st would exp ect. Ho wev er, the list of cla s ses o f algebra s satisfying (Fg) is quite extensive: (i) any blo ck of a g roup ring of a finite g roup [9, 1 1, 26], (ii) any blo ck of a finite dimensional co co mm utative Hopf a lgebra [10], (iii) in the co mm utative setting for a co mplete intersection [14], (iv) an y exterior algebra , (v) all finite dimensional selfinjectiv e algebra s ov er an algebraic ally closed field of finite representation t ype [6], (vi) quantum complete int ersections [2], (vii) all Go renstein Nak ayama algebras (announce d by H. Naga se), (viii) any finite dimensiona l pointed Hopf algebra, having ab elian group of gro up- like e lemen ts, under some mild restrictio ns on the gro up order [19]. A weakly symmetric alg ebra is, in particular , a finite dimensional selfinjective algebra. In [21] it is shown that a finite dimens io nal Ko s zul a lgebra Λ over a field k with deg ree zer o par t isomor phic to k , is selfinjective with finite complexit y if and only if the K oszul dual E (Λ) is an Artin-Schelter regular Koszul a lgebra. This was extended in [18] to finite dimensional Koszul algebra s Λ over a field k with Λ 0 ≃ k n for s o me p ositive integer n . Then it is natura l to say that a (no n- connected) Ko szul k -algebr a R = ⊕ i ≥ 0 R i is an Artin-Schelter re gular algebr a of dimension d , if (i) dim k R i < ∞ for all i ≥ 0, (ii) R 0 ≃ k n for so me p ositive integer n , (iii) gldim R = d , (iv) the Gelfand-Kirillov dimension of R is finite, (v) for all s imple gra ded R - modules S w e hav e Ext i R ( S, R ) ≃ ( (0) , i 6 = d, S ′ , i = d and some simple g raded R op -mo dule S ′ . As in [2 0] it fo llows that classifying all selfinjective Koszul a lgebras of finite com- plexity d and Lo ewy leng th m + 1 (up to isomorphis m) is the same as class ifying Artin-Schelter regular Ko szul algebras with Gelfand-Kir illov dimension d and globa l dimension m (up to isomorphism). Moreov er , if R is such a n Ar tin- Schelter regular Koszul algebr a , then Ext d R op (Ext d R ( S, R ) , R ) ≃ S and consequently E xt d R ( S, − ) is a “p er mutation” of the simple graded R - and R op -mo dules. The selfinjective alge- bra Λ b eing weakly symmetric, co rresp onds to the pro per t y that R = E (Λ), the Koszul dual of Λ, satisfies Ext d R ( S, R ) ≃ S op for all simple graded R -mo dules S . Hence, the w eakly symmetric a lgebras with radical cube zero found in [1] of finite complexity clas sify all the Artin-Schelter reg ular Koszul algebras of dimension 2, where Ex t 2 R ( − , R ) is the “ ide ntit y permutation”. Within this class of algebra s , we classify those that ar e finitely generated mo dules o ver their centres, such that RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 3 Ext 2 R ( − , R ) is the iden tit y per mutation. A similar classification is als o carried out for Artin-Sc he lter regular Koszul a lg ebras o f dimension 1 (see Propo sition 1.4). These algebras corr e s po nd to radica l square zero algebras, so after having stated this r esult, we exclude this class of algebr as from the discus s ion o f weakly symmetric radical cub e zero alge br as. An ear ly v ersion of this pap er w a s called Finite gener ation of the Ho chschild c ohomolo gy ring of some Koszul algebr as . All the results that existed in that early version are included in full in the current paper . The autho rs acknowledge the use of the Gr¨ obner bas is program GRB b y E. L . Green [1 2] in the exp erimental stages of this pap er. 1. Back ground and preliminar y resul ts Throughout let Λ be a finite dimensional algebra over an algebraica lly closed field k with Jac o bson radical r . Cohomologic a l suppo r t v a rieties for finite dimen- sional mo dules ov er Λ using the Ho chschild co homology ring HH ∗ (Λ) of Λ w ere int ro duced in [22] and further studied in [8]. It follows that Λ ha s a go o d theory of cohomolo gical supp ort v a rieties via HH ∗ (Λ) of Λ, if HH ∗ (Λ) is No e therian a nd Ext ∗ Λ (Λ / r , Λ / r ) is a finitely generated HH ∗ (Λ)-mo dule. Denote this condition b y (Fg) . The aim of this pap er is to ch ara c terize when a w eakly symmetric alg e bra Λ with r 3 = (0) satisfies (Fg) . F or the algebras Λ w e consider, it is well-kno wn that Λ ≃ kQ/ I for s ome finite quiver Q and so me ide a l I in k Q , up to Morita equiv a lence. F urthermore there is a homo morphism of graded rings ϕ M : HH ∗ (Λ) → Ext ∗ Λ ( M , M ) = ⊕ i ≥ 0 Ext i Λ ( M , M ) for a ll Λ-mo dules M , with Im ϕ M ⊆ Z gr (Ext ∗ Λ ( M , M )) (see [22, 27]). Here Z gr (Ext ∗ Λ ( M , M )) deno tes the gra ded centre of Ext ∗ Λ ( M , M ). The graded centre of a graded r ing R is gener a ted by { z ∈ R | r z = ( − 1) | r || z | z r, z ho mogeneous, ∀ n ≥ 0 , ∀ r ∈ R n } , where | x | denotes the degree of a homo g eneous element x in R . W eakly sy mmetric alg ebras a re selfinjective a lgebras where all indecomp osable pro jective mo dules P have the pr op erty that P / r P ≃ So c( P ). All selfinjective a lge- bras Λ of finite r epresentation t yp e ar e shown to b e perio dic algebra s [6], meaning that Ω n Λ ⊗ k Λ op (Λ) ≃ Λ for some n ≥ 1. It is ea sy to see that all p erio dic algebras Λ satisfy (Fg) . F urthermore, for s elfinjectiv e alge bras Λ with r 3 = (0) a nd r 2 6 = (0) we have the following result. Theorem 1.1 ([15, 17]) . L et Λ b e a selfinje ctive algebr a with r 3 = (0) and r 2 6 = (0) . Then Λ is Koszul if and only if Λ is of infinite r epr esentation t yp e. Hence in o ur study o f weakly symmetric alge br as Λ with r 3 = (0), w e can concentrate on infinite representation type and consequently Ko szul algebras. F o r Koszul algebras Λ the homomorphism of graded r ings from HH ∗ (Λ) to the E xt- algebra of the simple mo dules ha s an even nicer pro per t y than for general algebra s, as we discuss next. F or a quotien t of a path algebra Λ = k Q /I , given b y a quiver Q with relations I o ver a field k , it w as indep endently observed by Buc hw eitz and Green-Sna s hall- Solb erg, that when Λ is a K o szul algebra, the image of the na tural map from the Ho c hschild co homology ring HH ∗ (Λ) to the Koszul dual E (Λ) is equal to the gra ded 4 ERDMANN AND SOLBERG centre Z gr ( E (Λ )) of E (Λ). Here E (Λ) = ⊕ i ≥ 0 Ext i Λ (Λ 0 , Λ 0 ), wher e Λ 0 is the degree 0 part of Λ. This isomorphism w as obtained b y Buch w eitz as a part of a mor e general isomorphism b etw een the Ho chsc hild co homology r ing o f Λ and the g raded Ho c hschild cohomo logy r ing of the Koszul dual. This isomo rphism has since been generalized by Keller [16]. The statemen t from [4] re ads as follo ws. Theorem 1.2 ([4]) . L et Λ = k Q /I b e a Koszul algebr a. Then the image of t he natur al map ϕ Λ 0 : HH ∗ (Λ) → E (Λ) is the gr ade d c entr e Z gr ( E (Λ )) . This enables us to c haracter iz e when (Fg) holds for a finite dimensional Kosz ul algebra Λ ov e r an algebra ically closed field. Let Λ = k Q/ I b e a path algebra ov er an algebraic ally closed field k . In [8] the following conditions are crucial with r esp ect to having a go o d theory of cohomolo gical supp o rt v arieties ov er Λ: Fg1 : there is a commutativ e Noe ther ian graded subalgebr a H of HH ∗ (Λ) with H 0 = HH 0 (Λ), Fg2 : Ext ∗ Λ (Λ / r , Λ / r ) is a finitely gener ated H -module, where r deno tes the Ja - cobson radical of Λ. In [25, Prop os ition 5.7] it is shown that these conditions are satisfied if and only if the condition (Fg) ho lds for Λ. Suppose tha t (Fg) is satisfied f or Λ. Since Z gr ( E (Λ )) is an HH ∗ (Λ)-submo dule of E (Λ), th e Ho chsc hild co homology ring HH ∗ (Λ) is No e therian and E (Λ) a finitely gener ated HH ∗ (Λ)-mo dule, we infer that Z gr ( E (Λ )) is a finitely generated HH ∗ (Λ)-mo dule as well. Hence Z gr ( E (Λ )) is a No etherian algebra a nd E (Λ) is clearly finitely g enerated as a Z gr ( E (Λ ))-module. If Λ is a finite dimensional Kos zul algebr a, the con verse is also true. Supp ose that Λ is a finite dimensio nal Ko szul alg ebra with Z gr ( E (Λ )) a No etherian algebra and E (Λ) a finitely generated module ov er Z gr ( E (Λ )). Then Z gr ( E (Λ )) contains a com- m utative No etheria n even-degree gr aded subalgebra e H , such tha t Z gr ( E (Λ )) is a finitely generated module ov er e H . Let H be an inv erse ima ge of e H in HH ∗ (Λ). Then H is (can b e chosen to be) a commutativ e No etherian graded subalg ebra of HH ∗ (Λ). Therefore the conditions (Fg1) and (Fg2 ) are satisfied, and conseque ntly (Fg) holds true for Λ by [25, Prop osition 5 .7]. Hence we hav e the follo wing. Theorem 1.3. L et Λ = k Q/ I b e a finite dimensional algebr a ove r an algebr aic al ly close d field k , and let E (Λ) = Ext ∗ Λ (Λ / r , Λ / r ) . (a) If Λ satisfies (Fg) , then Z gr ( E (Λ )) is No etherian and E (Λ) is a finitely gener ate d Z gr ( E (Λ )) -mo dule. (b) When Λ is Koszul, then the c onverse implic ation also holds, that is, if Z gr ( E (Λ )) is No etherian and E (Λ) is a finitely gener ate d Z gr ( E (Λ )) - mo dule, t hen Λ satisfies (Fg) . In part (b) it is e no ugh to find a commutative No etheria n g r aded subring of Z gr ( E (Λ )) o ver which E (Λ) is a finitely genera ted module. Our main fo cus in this pap er is radical cube zero a lgebras. Ho wev er , let us illustrate the abov e result on radical square zero algebras. Let Λ = k Q/J 2 be a finite dimensional ra dical squar e zero algebra over a field k . Hence J is the ideal generated b y the arrows in Q . Since all quadratic monomial algebra s are Koszul [13], Λ is a Ko szul alg ebra and E (Λ) = k Q op . If Q do e s not hav e a n y orien ted cycles, the global dimension o f Λ is finite and there is no in teres ting theor y of suppo rt v ar ieties via the Ho chsc hild cohomology ring. So, as sume that Q has at RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 5 least one oriented cycle. Consequently k Q op is an infinite dimensional k -alg e bra. In addition we have that Z gr ( E (Λ )) = ( k , if Q is no t an oriented cycle ( e A n ), k [ T ] , otherwise , see for example [5]. The follo wing result is an immediate conseq ue nc e of the ab ov e. Prop ositio n 1.4. L et Λ = k Q /J 2 for some qu iver Q with at le ast one oriente d cycle and a field k . Then Λ satisfies (Fg) if and only if Λ is a r adic al squar e zer o Nakayama algebr a. Let Λ b e a weakly symmetr ic algebra with r 3 = (0) and r 2 6 = (0). Denote b y { S 1 , . . . , S n } all the non- isomorphic simple Λ-mo dules, and let E Λ be the n × n - matrix given by (dim k Ext 1 Λ ( S i , S j )) i,j . These algebras are classified in [1], a nd among other things the following is proved there. Theorem 1. 5 ([1]) . L et Λ b e a finite dimensional inde c omp osable b asic we akly symmetric algebr a ove r an algebr aic al ly close d field k with r 3 = (0) and r 2 6 = (0) . Then t he m atrix E Λ is a s ymm et ric matr ix , and the eigenvalue λ of E Λ with lar gest absolute value is p ositive. (a) If λ > 2 , then the dimensions of the mo dules in a minimal pr oje ctive r eso- lution of any fin itely gener ate d Λ - mo dule has exp onential gr owth. (b) If λ = 2 , then the dimensions of the mo dules in a minimal pr oje ctive r esolu- tion of any fi nitely gener ate d Λ -mo dule ar e either b ounde d or gr ow line arly. The matrix E Λ is the adjac ency m atrix of a Euclide an diagr am e A n for n ≥ 1 , e D n for n ≥ 4 , e E 6 , e E 7 , e E 8 , or e Z n : 0 1 n − 1 n for n ≥ 0 or g D Z n : 0 Q Q Q Q Q Q 2 3 n − 1 n 1 n n n n n n for n ≥ 2 . (c) If λ < 2 , then t he dimensions of the mo dules in a minimal pr oje ctive r eso- lution of any fin itely gener ate d Λ - mo dule is b ounde d. The trichotom y in Theorem 1.5 cor resp onds to the division in wild, tame and finite re pr esentation t yp e as p ointed out in [1]. By [8, Theore m 2.5 ] the complexity of any finitely gener a ted mo dule ov er an alg ebra satisfying (Fg) is b ounded above by the Kr ull dimension o f the Ho c hschild cohomo logy ring, hence finite. It follows from this tha t a weakly sy mmetric algebra with r adical cub e zero only can satisfy (Fg) in case (b) and (c) in the ab ov e theor em. W e remarked ab ov e that all the algebras in (c) sa tisfy (Fg) , so w e only need to consider the a lgebras g iven in (b). The above result gives the quiver of the algebra Λ, but since Λ is supp o sed to be weakly symmetric with r 3 = (0), it is ea sy to write down the pos s ible r elations. In these relations o ne ca n in tro duce s c alars from the fie ld. Most of the times the results are indep endent of these sca lars, except in the e Z 0 case and the e A n case, 6 ERDMANN AND SOLBERG where it suffices to intro duce one scalar q in one co mmutativit y relatio n. The ne x t sections a re devoted to discussing these cases. W e end this section with a remar k which is a go o d guide in doing a ctual com- putations. Let Λ be a finite dimensional selfinjectiv e K oszul a lgebra o ver an al- gebraica lly closed field k satisfying (Fg) . In finding non-nilp otent generators for Z gr ( E (Λ )) as an algebra, then [8 , P rop osition 4.4] and Theor em 1.2 says essentially that the degree of a non-nilpo tent generator for Z gr ( E (Λ )) is a m ultiplum of the Ω-p erio dicity of some Ω-p erio dic mo dule. In actual computations this usually g ives a go o d indication where to lo o k for non- nilpotent genera tors. 2. The ˜ A n -case In this sectio n w e characterize when a weakly symmetric algebr a o ver a field k with radical cub e zer o of t yp e ˜ A n satisfies (Fg) . F or a computation of the Ho c hschild cohomology ring o f a more general class of alg ebras containing the al- gebras we consider in this section consult [23, 24]. Let Q be the quiv er given by 0 a 0 / / 1 a 0 o o a 1 / / 2 a 1 o o a 2 / / a 2 o o a n − 2 / / n − 1 a n − 2 o o a n − 1 / / n a n − 1 o o a n / / 0 a n o o where the extreme v ertices are identified as the notation sugge s ts. Let k b e a field, and let I ′ be the ideal in k Q gener ated b y the elements { a i a i +1 } n i =0 , { a i +1 a i } n i =0 , and { a i a i + q i a i − 1 a i − 1 } n i =0 , for so me nonzero elemen ts q i in k . Here we compute the indices mo dulo n + 1. By c hanging basis the algebra Λ = k Q/ I ′ can be repre- sented b y the same quiver, but an ideal I generated by the elemen ts { a i a i +1 } n i =0 , { a i +1 a i } n i =0 , a nd { a i a i + a i − 1 a i − 1 } n i =1 ∪ { a 0 a 0 + q a n a n } for some nonzero elemen t q in k . That is, Λ ≃ k Q/ I . Next we apply Theorem 1.3 to character ize when Λ sa tisfies (Fg) . Prop ositio n 2.1. Le t Q and I in k Q b e as ab ove for a field k . Then Λ = k Q /I satisfies (Fg) if and only if q is a r o ot of u n ity. Pr o of. The graded cen tre of E (Λ) a nd E (Λ) op are the same and the cr iterion for chec king (Fg) can equiv alent ly b e per fo rmed using E (Λ) op . The algebra E = E (Λ) op is given by kQ / h{ a i a i − a i − 1 a i − 1 } n i =1 , q a 0 a 0 − a n a n i . Consider the length left-lexicogra phic ordering of the paths in k Q b y letting the vertices b e less than any arrow, and order the arrows like a 0 < · · · < a n < a 0 < · · · < a n . Then { a i a i − a i − 1 a i − 1 } n i =1 , q a 0 a 0 − a n a n is a Gr¨ obner basis for the idea l they genera te in k Q . In a ddition E is bigra ded via the degrees in a -arrows and a -arr ows. The graded centre inherits the bigrading from E , and the elemen ts in Z gr ( E ) ar e sums of bi-homo geneous elements. Let x = n X i =0 a i a i +1 · · · a n a 0 · · · a i − 1 and y = n X i =0 a i − 1 a i − 2 · · · a 0 a n · · · a i RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 7 in E . Any path in k Q viewed as an element in E can b e written uniquely as AAy r x s for some natural num ber s r and s , and so me path A in the a rrows { a i − 1 } and some path A in the arrows { a i } , eac h of whic h has length at most n . W e hav e that a i x = xa i , a i − 1 y = y a i − 1 , a i − 1 x = q xa i − 1 and a i y = q − 1 y a i for all i . Hence, if q is a d -th ro ot of unity , then { x 2 d , y 2 d } is in Z gr ( E ). It is immediate that E is a finitely generated mo dule ov er Z gr ( E ) when q is a ro ot of unit y . Suppo se that q is not a ro ot of unit y . Ass ume that z is an elemen t in Z gr ( E ) of homogeneous bi-degr ee ( r , s ) with r + s ≥ 1. Then z = z 0 + z 1 + · · · + z n with z i in e i E e i of bi-degre e ( r, s ) for i = 0 , 1 , . . . , n . If z i 6 = 0, then z i a i = a i z i +1 and a i − 1 z i = z i − 1 a i − 1 are non-zer o , he nc e both z i − 1 and z i +1 are non-zero. It follows that each z i is no n- zero for all i . By cons ide r ing some pow er of z w e can assume that each z i = α i e i y r ′ x s ′ e i for some p ositive int egers r ′ and s ′ and α i in k \ { 0 } . Since at least o ne of r ′ and s ′ is po s itive, it follows without lo ss of g enerality that q r ′ ( n +1) = 1. This is a co n tradiction. H ence, when q is no t a ro ot of unity , Z gr ( E ) is k . F urthermore (Fg) is not satisfied, and we hav e shown that Λ satisfies (Fg) if and only if q is a ro ot of unit y .  3. The e Z n -case This section is dev oted to proving that the weakly symmetric algebras over a field k with radical cub e ze r o of type e Z n all sa tisfy (Fg) for n > 0, ag ain using Theorem 1.3 . The case n = 0 is discussed in Section 9. The quiv er Q of the algebras of t y pe e Z n are giv en by 0 b : : a 0 / / 1 a 0 o o a 1 / / 2 a 1 o o a 2 / / a 2 o o a n − 2 / / n − 1 a n − 2 o o a n − 1 / / n a n − 1 o o c e e where we imp ose the following relatio ns when n > 0 { b 2 + a 0 a 0 , ba 0 , a 0 b, { a i a i +1 } n − 2 i =0 , { a i a i − 1 } n − 1 i =1 , { a i a i + a i − 1 a i − 1 } n − 1 i =1 , a n − 1 c, ca n − 1 , c 2 + q a n − 1 a n − 1 } . Let I denote the ideal ge nerated by this set of relations. One could deform this algebr a by int ro ducing non-zero co efficients in all the “commutativit y” r elations ab ov e. H ow ever, with a suitable basis change, we can remov e all these commutativit y co efficient s and only have one remaining , for in- stance the q as chosen ab ov e. In a dditio n, if k contains a square ro ot o f q , then we can r e place q by 1. Prop ositio n 3.1. L et Λ = k Q/I , wher e Q and I ar e as ab ove. Then Λ satisfies (Fg) . Pr o of. W e hav e that E (Λ) op = k Q / ( b 2 − a 0 a 0 , { a i a i − a i − 1 a i − 1 } n − 1 i =1 , q c 2 − a n − 1 a n − 1 ) . Let x = a n − 1 a n − 1 + P n − 1 i =0 a i a i . Then direct calculations show that x is in Z gr ( E (Λ )). F or d in { b, c } , denote b y d [ s ] th e sho r test closed path in Q , start- ing in vertex s inv olving d . Then let y = P n i =0 ( b [ i ] c [ i ] + c [ i ] b [ i ]). Straightf orward computations sho w that y is in Z gr ( E (Λ )). W e use the above to show that E (Λ) is a finitely generated mo dule ov er Z gr ( E (Λ )) (o r a ctually the subalgebra gene r ated by x and y ). An y path in Q , starting and ending in the sa me vertex and in v olving only the arrows a ? and a ? , is 8 ERDMANN AND SOLBERG as an element in E (Λ) a p ow er of x times the appropr iate idempotent. Then an y path p in Q , as an element in E (Λ), can be written as x m p ′ for some path p ′ , wher e p ′ = X l r X for X in { b [ i ] c [ i ] , c [ i ] b [ i ] } n i =0 and r X is a proper subpath of X from the left. Hence, any element in E (Λ) is a linear com bination of e le ments of the form { x m ( b [ i ] c [ i ]) l r B i , x m ( c [ i ] b [ i ]) l r C i } n i =0 . W e directly verify that b [ i ] 2 and c [ i ] 2 are in the span of x m e i for all i . Then by induction it is ea sy to see that ( b [ i ] c [ i ]) l and ( c [ i ] b [ i ]) l are in k h x, y i ( b [ i ] c [ i ] , c [ i ] b [ i ] , e i ) for all l . It follows fro m this that E (Λ) is a finitely generated Z gr ( E (Λ ))-module, and hence Λ satisfies (Fg) .  4. The g D Z n -case This section is dev oted to showing that a symmetric finite dimensional algebra Λ over a field k o f type g D Z n satisfies (Fg) . Let Q be the quiv er given by 0 a 0   = = = = = = = 2 a 0 ^ ^ = = = = = = = a 1          a 2 / / 3 a 2 o o a 3 / / a 3 o o a n − 3 / / n − 2 a n − 3 o o a n − 2 / / n − 1 a n − 2 o o a n − 1 / / n a n − 1 o o b e e 1 a 1 @ @        Assume that n > 2. Let I be the ideal in kQ generated by the elemen ts { a 0 a 1 , a 0 a 2 , a 1 a 0 , a 2 a 1 , a 0 a 0 − a 1 a 1 , a 2 a 0 , a 1 a 1 − a 2 a 2 , { a i a i +1 } n − 2 i =1 , { a i a i − 1 } n − 1 i =2 , { a i − 1 a i − 1 + a i a i } n − 1 i =3 , a n − 1 b, a n − 1 a n − 1 + q b 2 , ba n − 1 } for some q in k \ { 0 } . When n = 2, the ideal I we factor out is slightly different . This ideal is implicitly given in the end o f the proo f of the next result. Before proving these algebra s satisfies (Fg) , we discuss deformatio ns o f the al- gebra Λ. One could def orm t his algebra by in tro ducing no n-zero co efficients in all the commut ativity relations above. How ever with a suitable basis change, we can r emov e all these comm utativity co efficients and only hav e one remaining, for instance the q as chosen ab ov e. W e thank the referee for p ointing out that with a basis change given by a i 7→ a i , a i 7→ q a i and b 7→ b , the s calar q can b e r eplaced b y 1. Prop ositio n 4.1. L et Q , I and Λ b e as ab ove. Then Λ satisfies (Fg) . Pr o of. Supp ose n ≥ 3. Again w e apply Theorem 1.3. The opp osite E (Λ) op of the Koszul dual of Λ is given by k Q modulo the r elations gener a ted by { a 0 a 0 , a 1 a 1 , a 0 a 0 + a 1 a 1 + a 2 a 2 , { a i − 1 a i − 1 − a i a i } n − 1 i =3 , a n − 1 a n − 1 − b 2 } . Let α = a 0 a 0 , β = a 1 a 1 and γ = a 2 a 2 . Note that αβ + β α = γ 2 and αa 0 = a 0 α = β a 1 = a 1 β = 0. Let x = P n i =0 x i with x i =          ( a n − 1 a n − 1 ) 2 , for i = n, ( a i a i ) 2 , for 2 ≤ i ≤ n − 1 , a 0 β a 0 , for i = 0 , a 1 α a 1 , f or i = 1 . RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 9 Then direct computations sho w that x is in Z gr ( E (Λ )). F or η in { α, β , b } denote by η [ s ] the shortest path in Q starting in vertex s inv o lving η , whenever this makes sense . In particular, α [ s ] = a s − 1 a s − 2 · · · a 2 αa 2 a 3 · · · a s − 1 for 3 ≤ s ≤ n , and α [2] = α . W e leave it to the reader to write o ut the similar formulae for the other cases. Let y ′ = P n i =0 y ′ i with y ′ i =          b [0] , i = 0 , b [1] , i = 1 , b [ i ] α [ i ] − α [ i ] b [ i ] , 2 ≤ i ≤ n and i ev en, b [ i ] β [ i ] − α [ i ] b [ i ] , 2 ≤ i ≤ n and i o dd. W e want to show that y = ( y ′ ) 2 is in Z gr ( E (Λ )). In doing s o the following equa lities are useful to have, αγ = − αβ = γ β ( † ) β γ = − β α = γ α ( ‡ ) and b [ i ] α [ i ] − α [ i ] b [ i ] = − ( b [ i ] β [ i ] − β [ i ] b [ i ]) for all 2 ≤ i ≤ n . Note that the la st prop erty is equiv alent to having b [ i ] γ [ i ] = γ [ i ] b [ i ] for 2 ≤ i ≤ n . The most cum ber some calculatio ns inv o lve vertex n , and her e it is useful to note that α [ n ] γ [ n ] is equal to γ [ n ] α [ n ] if n is o dd, while it is equal to γ [ n ] β [ n ] for n even. Poin ting out that α [ n ] 2 = 0 and β [ n ] α [ n ] = 0 when n is even and o dd, resp ectively , w e leav e it to the reader to sho w that y is in Z gr ( E (Λ )). Next w e w ant to show that E (Λ) is a finitely gener ated module ov er the subal- gebra Z 2 generated by { x 2 , y 2 } . Let E 2 = e 2 E (Λ) op e 2 . Then E 2 is g enerated as an k -algebr a by { α, γ , b [2 ] } . Let µ b e any non-zero monomial in { α, γ , b [2] } . W e ha ve that αγ = − γ 2 − γ α and b [2] γ = γ b [2], so we can suppose that a ll the γ ’s in µ ca n be mov ed to the left (since γ 2 = x 2 ). F ur thermore, we hav e that b [2] 2 = γ 2 n − 3 and α 2 = 0, so that we can write µ = ± γ t µ 1 µ 2 · · · µ r with µ i in { α, b [2 ] } for all i with µ i 6 = µ i +1 . W e hav e that x 2 = γ 2 and y 2 = ( b [2] α ) 2 + ( αb [2]) 2 + αγ 2( n − 1) . Let A be the set of mo nomials in { α, γ , b [2] } with a t most six factors . It is then easy to see that the factor E 2 / Z 2 A is zero, and hence E 2 is a finitely generated Z 2 -mo dule. An y oriented cycle in Q no t going through the vertex 2, can b e written as a p ow er of x times one of a finite set of cyc le s. It follows fro m this that (Fg) is satisfied for Λ, when n ≥ 2. Let n = 2. Then E (Λ) op is given by k Q mo dulo the relations { a 0 a 0 , a 1 a 1 , a 0 a 0 + b 2 + a 1 a 1 } . Let α = a 0 a 0 , β = a 1 a 1 and γ = b 2 . Let x 0 = γ [0], x 1 = γ [1], and x 2 = γ 2 = − ( αγ + γ α ) = − ( β γ + γ β ). Let y 0 = bαb [0], y 1 = bβ b [1] and y 2 = ( bα − αb ) 2 = ( bβ − β b ) 2 . Then it is easy to see that x = x 0 + x 1 + x 2 and y = y 0 + y 1 + y 2 are in Z gr ( E (Λ )). Using that x 2 = − αb 2 − b 2 α a nd αy 2 = αbαbα , it is immediate that Λ sa tisfies (Fg) also in this ca s e.  The reduction to showing that e 2 E e 2 is a finitely generated e 2 Z e 2 -mo dule as in the pr o of ab ov e will be used later again. This example also pro vides us with a dditional information o n Betti n um ber s of per io dic mo dules. Considering the Λ-mo dule M with radical layers ( n n ) it is easy to see that M is Ω- per io dic with pe r io d 2 n − 1, and all the pro jective mo dules in an initial p erio dic minimal pro jective resolutio n are indecomp osable except pro jective 10 ERDMANN AND SOLBERG nu mber n − 1, which is a direct s um o f tw o indecomp osable pr o jectiv e mo dules. This gives an example of an Ω-p erio dic module with no n-constant Betti n umbers. 5. The e D n -case This section is dev oted to proving that the weakly symmetric algebras over a field k with radica l cube zero of type e D n all sa tisfy (Fg) . Let Q be the quiv er given by 0 a 0   = = = = = = = = n − 1 a n − 2 z z u u u u u u u u u 2 a 0 ^ ^ = = = = = = = = a 1          a 2 / / 3 a 2 o o a 3 / / a 3 o o a n − 4 / / n − 3 a n − 4 o o a n − 3 / / n − 2 a n − 3 o o a n − 2 : : u u u u u u u u u b $ $ I I I I I I I I I I 1 a 1 @ @        n b d d I I I I I I I I I I Assume that n > 4. Let I be the ideal in kQ generated by the elemen ts { a 0 a 1 , a 0 a 2 , a 1 a 0 , a 2 a 1 , a 0 a 0 − a 1 a 1 , a 2 a 0 , a 1 a 1 − a 2 a 2 , { a i a i +1 } n − 3 i =1 , { a i a i − 1 } n − 2 i =2 , { a i − 1 a i − 1 + a i a i } n − 3 i =3 , a n − 2 b, ba n − 2 , a n − 3 b, a n − 2 a n − 2 − bb, bb − a n − 3 a n − 3 } . When n = 4 then I is gener ated by { a 0 a 1 , a 0 a 2 , a 0 b, a 1 a 0 , a 1 a 2 , a 1 b, a 2 a 0 , a 2 a 1 , a 2 b, ba 0 , ba 1 , ba 2 , a 0 a 0 − a 1 a 1 , a 1 a 1 − bb, bb − a 2 a 2 } . Similarly as b efore, deformations via co efficie nts in the commutativit y relatio ns can be removed via a n appr opriate basis change. Giv en this we can s how the following. Prop ositio n 5.1. L et Q , I and Λ b e as ab ove. Then Λ satisfies (Fg) . Pr o of. W e apply a gain Theorem 1.3. The o ppo site algebr a E = E (Λ) op of the Koszul dual of Λ is given by k Q modulo the r elations gener a ted by { a 0 a 0 , a 1 a 1 , a 0 a 0 + a 1 a 1 + a 2 a 2 , { a i − 1 a i − 1 − a i a i } n − 3 i =3 , a n − 2 a n − 2 , bb , a n − 2 a n − 2 + bb + a n − 3 a n − 3 } when n > 4. F or n = 4 the rela tions are giv en by { a 0 a 0 , a 1 a 1 , a 2 a 2 , bb , a 0 a 0 + a 1 a 1 + a 2 a 2 + bb } . Let α = a 0 a 0 , β = a 1 a 1 and γ = a 2 a 2 . F u rthermor e we write δ = a n − 2 a n − 2 , ω = b b and η = a n − 3 a n − 3 . Note that αβ + β α = γ 2 and αa 0 = a 0 α = β a 1 = a 1 β = 0 and as well δ a n − 2 = a n − 2 δ = ω b = bω = 0. Let x = P i x i , whe r e x 0 = a 0 β a 0 , x n − 2 = η 2 , x 1 = a 1 α a 1 , x n − 1 = a n − 2 ω a n − 2 , x i = ( a i a i ) 2 , for 2 ≤ i < n − 2 , x n = bδ b. Then direct co mputations show that x is in Z gr ( E (Λ )). This is also true when n = 4. Note that in this c a se we hav e α + β + δ + ω = 0 . RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 11 When n = 4 we find another element of degre e 4 in the centre of E , namely w = P 4 i =0 w i , whe r e w 0 = a 0 δ a 0 , w 2 = ( α + δ ) 2 = αδ + δ α, w 4 = bβ b. w 1 = a 1 ω a 1 , w 3 = a 3 αa 3 , W e assume now that n > 4. Supp ose ρ is one of α, β , δ, ω . As be fo re, we write ρ [ s ] for the shor test close d path sta r ting at s which inv o lves α . Let y = P n i =0 y i , whe r e the y i are defined as follows. F or 2 ≤ r ≤ n − 2 , y r =          α [ r ] δ [ r ] + δ [ r ] α [ r ] , r , n even , α [ r ] δ [ r ] + ω [ r ] β [ r ] , r o dd , n even , α [ r ] δ [ r ] − ω [ r ] α [ r ] , r even , n o dd , β [ r ] ω [ r ] − ω [ r ] α [ r ] , r, n o dd . F urthermore, y 0 = δ [0] , y 1 = ω [1] , y n − 1 = α [ n − 1] , y n = β [ n ] . W e wan t to show that y is in the centre of E (Λ ). One w ay to pro ve this is to fir st establish the following identities. F or 2 ≤ r ≤ n − 2, (1) α [ r ]( δ [ r ] + ω [ r ]) = ( ( δ [ r ] + ω [ r ]) α [ r ] , n − r − 1 even , ( δ [ r ] + ω [ r ]) β [ r ] , n − r − 1 odd . Moreov er (2) δ [ r ]( α [ r ] + β [ r ]) = ( ( α [ r ] + β [ r ]) δ [ r ] , r − 1 ev en , ( α [ r ] + β [ r ]) ω [ r ] , r − 1 odd . Similar for mulae hold b y interc hanging α and β in (1), and by interc hanging δ and ω , in (2). Using these for m ulae one gets several identities for the y r . Assume 2 ≤ r ≤ n − 2. Then y r =          β [ r ] ω [ r ] + ω [ r ] β [ r ] , n, r even , δ [ r ] α [ r ] + β [ r ] ω [ r ] , r o dd , n even , δ [ r ] α [ r ] − β [ r ] δ [ r ] = ω [ r ] β [ r ] − α [ r ] ω [ r ] , n, r o dd , δ [ r ] α [ r ] − α [ r ] ω [ r ] = ω [ r ] β [ r ] − β [ r ] δ [ r ] , r even , n odd . These show in particular that the anti-homomorphism induced by a → a and a → a of E + fixes ea ch y r . This means that one only has to c heck that y comm utes with the ar rows a r , then it automatically comm utes with a r . F urthermore, one chec ks that α [ r ] δ [ r ] a r = a r β [ r + 1] ω [ r + 1], and similarly β [ r ] ω [ r ] a r = a r α [ r + 1] δ [ r + 1], for 2 ≤ r < n − 2 . Using all these details , it is not difficult to c heck that y comm utes with a ll a i , and with b . Next, we wan t to show that E (Λ) is a finitely generated mo dule o ver the subal- gebra generated by { x, y } . As in the previous section, it suffices to show that the lo cal a lgebra E 2 = e 2 E (Λ) e 2 is finitely g enerated as a mo dule o ver the subalgebra Z genera ted b y { x 2 , y 2 } . Recall x 2 = γ 2 ; and w e take y 2 = ( αδ [2] + δ [2] α, n ev en δ [2] α − αω [2] , n o dd . 12 ERDMANN AND SOLBERG Note that this is also correct when n = 4, so we can deal with ar bitrary n ≥ 4 at the s a me time. The algebra E 2 is generated b y { α, γ , δ [2] } , note that δ [2] + ω [2] = − γ n − 3 . W e hav e further iden tities, namely γ α = − γ 2 − αγ , δ [2] γ = γ ω [2 ] . One checks that if n is even, δ [2] 2 = 0, a nd that for n o dd, δ [2] ω [2] = 0. Let A b e the s et o f mo nomials in { α, δ [2] , γ } with a t mo st three factors. W e wan t to show that E 2 / Z A is zer o , hence that E 2 is finitely genera ted ov e r Z . Assume µ ∈ E 2 is a non-zero monomial in α, δ [2] and γ . W e can mo ve all even powers o f γ to the left, note that these lie in Z . F urthermore, any factor of α in µ can b e mov ed to the left, using γ α = − αγ + z for z ∈ Z , a nd also using that for n even, δ [2] α = − αδ [2] + y 2 and for n o dd, δ [2] α = y 2 + αω [2] = y 2 − αδ [2] − αz where z = γ n − 3 ∈ Z . Hence w e may a ssume none except p ossibly the first factor of µ is eq ua l to α . Next, consider submonomials of length thr ee in δ [2 ] , γ o f µ w he r e successive factors are different. If it is o f the form δ [2] γ δ [2 ] = δ [2] ω [2 ] γ then is zero if n is o dd, and if n is even, it is equa l to − δ [2] 2 − δ [2] γ n − 3 = z δ [2] γ with z ∈ Z . Otherwise, it is of the form γ δ [2] γ = γ 2 ω [2] = z δ [2] + z ′ γ j with z , z ′ ∈ Z and j = 0 or 1. Using these one shows by induction on the length of µ that µ belo ngs to Z A .  6. The e E 6 -case This section is de voted to showing that the weakly symmetric algebr as over a field k with radica l cube zero of type e E 6 satisfy (Fg) . Let Q be the quiv er 4 a 3   3 a 3 O O a 2   0 a 0 / / 1 a 1 / / a 0 o o 2 a 4 / / a 2 O O a 1 o o 5 a 4 o o a 5 / / 6 a 5 o o with r elations {{ a i a i +1 } 4 i =0 , { a i a i − 1 } 5 i =1 , { a i − 1 a i − 1 + a i a i } i =1 , 3 , 5 , a 1 a 4 , a 2 a 4 , a 4 a 2 , a 4 a 1 , a 1 a 1 − a 2 a 2 , a 2 a 2 − a 4 a 4 } . Let Λ = k Q/I , where I is the ideal generated by the relations given ab ove for a field k . As before, we could deform the algebra b y in tro ducing non-zero scalars in the commutativit y relations, but by a suitable basis c hange all the scalars can be remov ed. Then we hav e the following. Prop ositio n 6.1. L et Q , I and Λ b e as ab ove. Then Λ satisfies (Fg) . Pr o of. The opposite E (Λ) op of the Koszul dual of Λ is giv en by k Q modulo the relations g enerated by { a 0 a 0 , a 3 a 3 , a 5 a 5 , { a i − 1 a i − 1 − a i a i } i =1 , 3 , 5 , a 1 a 1 + a 2 a 2 + a 4 a 4 } . RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 13 Let α = a 1 a 1 , β = a 4 a 4 and γ = a 2 a 2 . Let x 0 = γ [0], x 1 = ( αγ + γ α )[1], and x 2 = α 2 γ + αγ α + γ α 2 . W e wan t to define x i for i = 3 , 4 , 5 , 6 b y symmetry . T o do so , w e need the following details. Using the r elations in E (Λ) it follows that a 1 α 2 = α 2 a 1 = 0, a nd therefore α 3 = β 3 = γ 3 = 0. Applying this we infer that α 2 γ + αγ α + γ α 2 = − ( αγ 2 + γ αγ + γ 2 α ) ( † ) γ 2 α = − γ 2 β γ αγ = − γ β γ αγ 2 = − β γ 2 and similar for m ulae. W e define now x 3 = − ( αγ + γ α )[3], x 4 = − α [4], x 5 = − ( αβ + β α )[5] and x 6 = − α [6]. Utilizing symmetr y direct computations then s how that x = P 6 i =0 x i is in Z gr ( E (Λ )). Computing ( † ) · γ − γ · ( † ) we obtain γ 2 α 2 − αγ αγ = α 2 γ 2 − γ αγ α. F urthermore, since a 0 α 2 = 0 = α 2 a 0 , we hav e that γ 2 α [1] + αγ 2 [1] = − γ αγ [1 ] − αγ α [1] . Using that a 0 αγ α [1] = a 0 αγ 2 [1] = 0 = αγ α [1] a 0 = γ 2 α [1] a 0 and y 2 = β 2 γ 2 − γ β γ β , and letting y 2 = γ 2 α 2 − αγ αγ , y 1 = − γ αγ [1] , y 0 = γ 2 [0] , we o btain by symmetry an element y = P 6 i =0 y i in Z gr ( E (Λ )). Next we show that E (Λ) is a finitely gener ated module ov er the subalgebra generated b y { x, y } . The algebra e 2 E (Λ) e 2 is generated b y { α, γ } as an algebra. Given a monomia l µ in α and γ we can use − αγ 2 = x 2 + γ αγ + γ 2 α to mov e the o ccurrence of γ 2 to the le ft in µ . Hence, except for a shor t initial part, w e can assume that µ is a word in { α, α 2 , γ } . If αγ α 2 o ccurs so mewhere in µ , the equality γ α 2 = − x 2 − α 2 γ − αγ α gives αγ α 2 = − αx 2 − α 2 γ α and the o c currence of α 2 is mov e d further to the left in cre a ting o ne new monomial and one expressio n − x 2 µ ′ , where µ ′ is a monomial of degree three les s than µ . Hence, except for a shor t initia l part, we can a s sume that µ is a w ord in { α, γ } . T he equality y 2 + γ 2 α 2 = αγ αγ implies that γ y 2 = γ αγ αγ and y 2 α = αγ αγ α . By induction we obtain that any monomial in α and γ can b e written as a linear co m bination of pro ducts o f powers of { x 2 , y 2 } , and a finite set o f monomia ls in α and γ . Hence e 2 E (Λ) e 2 is a finitely generated module ov er the algebr a gener a ted by { x 2 , y 2 } . As b efor e it follows from this that (Fg) holds for Λ.  7. The e E 7 -case This section is dev oted to proving that the weakly symmetric algebras over a field k with radica l cube zero of type e E 7 satisfy (Fg) . 14 ERDMANN AND SOLBERG Let Q be the quiv er 4 a 3   0 a 0 / / 1 a 1 / / a 0 o o 2 a 2 / / a 1 o o 3 a 4 / / a 3 O O a 2 o o 5 a 5 / / a 4 o o 6 a 6 / / a 5 o o 7 a 6 o o with r elations {{ a i a i +1 } 5 i =0 , { a i a i − 1 } 6 i =1 , { a i − 1 a i − 1 + a i a i } i =1 , 2 , 5 , 6 , a 2 a 4 , a 3 a 4 , a 4 a 3 , a 4 a 2 , a 2 a 2 − a 3 a 3 , a 3 a 3 − a 4 a 4 } . Let Λ = k Q/I , where I is the ideal generated by the relations given ab ove for a field k . As for the e E 6 -case, deforming the algebr a by in tro ducing non-ze r o scala r s in the commut ativity relatio ns do es not change the algebra up to is o morphism. Then we have the following. Prop ositio n 7.1. L et Q , I and Λ b e as ab ove. Then Λ satisfies (Fg) . Pr o of. The opposite E (Λ) op of the Koszul dual of Λ is giv en by k Q modulo the relations g enerated by { a 0 a 0 , a 3 a 3 , a 6 a 6 , { a i − 1 a i − 1 − a i a i } i =1 , 2 , 5 , 6 , a 2 a 2 + a 3 a 3 + a 4 a 4 } . Let α = a 2 a 2 , β = a 4 a 4 and γ = a 3 a 3 . One easily shows that α 4 = β 4 = γ 2 = 0. F urthermore using that α + β + γ = 0, β γ = − αγ and γ β = − γ α , w e get β 3 γ + β 2 γ β + β γ β 2 + γ β 3 = − [( β γ ) 2 + γ β 2 γ + ( γ β ) 2 ] ( † ) = − ( β γ + γ β ) 2 = − ( αγ + γ α ) 2 = − [( αγ ) 2 + γ α 2 γ + ( γ α ) 2 ] = α 3 γ + α 2 γ α + αγ α 2 + γ α 3 . Let x = P 7 i =0 x i be the elemen t of degree 8 defined as follows x 0 = γ [0] , x 7 = γ [7] , x 1 = ( αγ + γ α )[1] , x 6 = ( β γ + γ β )[6] , x 2 = ( α 2 γ + αγ α + γ α 2 )[2] , x 5 = ( β 2 γ + β γ β + γ β 2 )[5] , x 3 = α 3 γ + α 2 γ α + αγ α 2 + γ α 3 , x 4 = − ( αγ α )[4] . Using ( † ), symmetry in α and β and preforming str aightforw ard co mputatio ns, we infer that x is in Z gr ( E (Λ )). Define y = P 7 i =0 y i as the following deg ree 1 2 element in E (Λ) with y 0 = − γ αγ [0] , y 7 = − γ β γ [7] , y 1 = γ α 2 γ [1] , y 6 = γ β 2 γ [6] , y 2 = ( − α 2 γ α 2 − α 2 γ αγ + γ α 2 γ α )[2] , y 5 = ( − β 2 γ β 2 − β 2 γ β γ + γ β 2 γ β )[5] , y 3 = α 2 γ α 2 γ + γ α 2 γ α 2 , y 4 = α 2 γ α 2 [4] . Using the last equality in ( † ) and a 0 a 0 = 0, it follows that that y 0 and y 1 “commute” with a 0 and a 0 . Premultiplying the last equality in ( † ) with α 2 gives α 3 γ α 2 + α 2 γ α 3 = − [ α 2 ( αγ ) 2 + α 2 ( γ α ) 2 + α 2 γ α 2 γ ] . RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 15 In computing a 2 y 3 − y 2 a 2 we o btain a 2 y 3 − y 2 a 2 = a 2 [ α 2 γ α 2 γ + α 2 γ αγ α + α 2 γ α 3 ] . F urthermore, substitute for α 2 γ α 3 using the a bove expression, w e can then cancel four ter ms and a re left with a 2 y 3 − y 2 a 2 = a 2 [ − α 3 γ α 2 − α 2 ( αγ ) 2 ] = 0 , since a 2 α 3 = 0 . Similar ar guments give that y 2 and y 3 commute with a 2 and y 1 and y 2 commute with a 1 and a 1 . One easily c hecks that y 3 and y 4 commute with a 3 and a 3 . Utilizing that γ β ( γ β + β γ ) β γ = 0, we conclude by a direct s ubs titution that y 3 = β 2 γ β 2 γ + γ β 2 γ β 2 . By symmetry in α and β the elements { y 3 , y 5 , y 6 , y 7 } satisfy the required equa tions, so that y is an elemen t in Z gr ( E (Λ )). Now we show that E (Λ) is a finitely genera ted mo dule over the algebra generated by { x, y } . As be fo re, we show that E 3 = e 3 E (Λ) e 3 is a finitely generated mo dule ov er the subalgebra generated b y { x 3 , y 3 } . Let µ be a n y monomial in α and γ . Recall that x 3 = α 3 γ + α 2 γ α + αγ α 2 + γ α 3 . Using this equation w e can mov e the o ccurrence of α 3 to the left, so that it r emains to analyze monomials µ , where α 3 do es not o cc ur except for in a shor t initial pa rt. Reca ll that y 3 = α 2 γ α 2 γ + γ α 2 γ α 2 . This g ives that γ α 2 γ α 2 γ = γ y 3 . Hence w e only ne e d to deal with submonomials of the form γ αγ αγ , γ αγ α 2 γ and γ α 2 γ αγ . Since γ x 3 γ = 0, we obtain that γ αγ α 2 γ = − γ α 2 γ αγ . In this wa y we c a n mov e o ccur rences of α 2 to the left, and if we c r eate a submonomial of the form γ α 2 γ α 2 γ , we r e place it b y γ y 3 as ab ov e. Then it remains to a na lyze a monomia l µ , which except for a short initial and a s ho rt terminal par t, is o f the form γ ( αγ ) t for some po sitive in teger t . Recall that x 3 = − αγ αγ − γ αγ α − γ α 2 γ , s o that γ x 3 = − γ αγ αγ . Combining all the observ ations ab ov e, we ha ve sho wn that a ny mono mial in α and γ can b e written as a linear combination of p ow ers of { x 3 , y 3 } times some finite set o f monomials in { α, γ } . This shows that E (Λ) is a finitely gener ated mo dule ov er the subalgebra g e ne r ated by { x 3 , y 3 } , and therefore Λ satisfies (Fg) .  8. The e E 8 -case This section is de voted to showing that the weakly symmetric algebr as over a field k with radica l cube zero of type e E 8 satisfy (Fg) . Let Q be the quiv er 3 a 2   0 a 0 / / 1 a 1 / / a 0 o o 2 a 3 / / a 2 O O a 1 o o 4 a 4 / / a 3 o o 5 a 5 / / a 4 o o 6 a 6 / / a 5 o o 7 a 7 / / a 6 o o 8 a 7 o o with r elations {{ a i a i +1 } 6 i =0 , { a i a i − 1 } 7 i =1 , { a i − 1 a i − 1 + a i a i } i =1 , 4 , 5 , 6 , 7 , a 1 a 3 , a 2 a 3 , a 3 a 2 , a 3 a 1 , a 1 a 1 − a 2 a 2 , a 2 a 2 − a 3 a 3 } . Let Λ = k Q/I , where I is the ideal generated by the relations given ab ove for a field k . Defor ming the a lg ebra by intro ducing no n-zero scalars in the commutativit y relations does not change the algebra up to isomor phis m. Giv en this, w e hav e the following. 16 ERDMANN AND SOLBERG Prop ositio n 8.1. L et Q , I and Λ b e as ab ove. Then Λ satisfies (Fg) . Pr o of. The opposite E (Λ) op of the Koszul dual of Λ is giv en by k Q modulo the relations g enerated by { a 0 a 0 , a 2 a 2 , a 7 a 7 , { a i − 1 a i − 1 − a i a i } i =1 , 4 , 5 , 6 , 7 , a 1 a 1 + a 2 a 2 + a 3 a 3 } . Let α = a 1 a 1 , β = a 3 a 3 and γ = a 2 a 2 . As before let E 2 = e 2 E (Λ) op e 2 be the local algebra at vertex 2. The structure of the pro of is as b efore, first w e exhibit t w o elemen ts x and y in Z gr ( E (Λ )). Then we show tha t E (Λ) is a finitely generated module over the subalgebra generated b y { x, y } , thr o ugh a nalyzing the the lo cal algebr a E 2 . The algebra E 2 is genera ted by { α, γ } , and they satisfy α 3 = 0 and γ 2 = 0. F ur thermore we have β 6 = 0. This is equiv alent with ( † ) 0 = ( γ α 2 γ α 2 + α 2 γ α 2 γ ) + ( αγ αγ α 2 + α 2 γ αγ α ) + ( γ α 2 γ αγ + γ αγ α 2 γ ) + αγ α 2 γ α + ( γ α ) 3 + ( αγ ) 3 . This will b e used often. F urthermore, we hav e, from expanding − β 3 = ( α + γ ) 3 that ( ‡ ) − β 3 + γ β γ = − β 3 − γ αγ = ( γ α 2 + αγ α + α 2 γ ) . Denote this elemen t b y ρ . Note that ρ comm utes with α . In giving elements in Z gr ( E (Λ )), the following iden tit y is helpful. Let ζ = ρ 2 , then ζ has deg ree 1 2, and it lies in the centre of E 2 . T o this end, first note that it can b e written in differen t ways. ζ = γ α 2 γ α 2 + α 2 γ α 2 γ + α 2 γ αγ α + αγ α 2 γ α + αγ αγ α 2 ( △ ) = − ( γ α 2 γ αγ + γ αγ α 2 γ + ( γ α ) 3 + ( αγ ) 3 ) = ( γ α 2 + αγ α + α 2 γ ) 2 . By the second identit y in ( △ ) we hav e ζ γ = − ( γ α ) 3 γ = γ ζ ; and since ρ commutes with α , so do es ζ . Hence ζ is in the cen tre of E 2 . An elemen t of degree 12 i n Z gr ( E (Λ )) . Define x = P 8 i =0 x i , whe r e x 0 = γ α 2 γ [0] , x 3 = − ( αγ ) 2 α [3] , x 6 = P 2 i =0 β 2 − i γ β i [6] , x 1 = ( αγ α 2 γ + γ α 2 γ + ( αγ ) 2 α )[1] , x 4 = P 4 i =0 β 4 − i γ β i [4] , x 7 = ( β γ + γ β )[7] , x 2 = ζ , x 5 = P 3 i =0 β 3 − i γ β i [5] , x 8 = γ [8] . W e claim that x is in the centre o f E (Λ). On the branch o f the quiver sta rting with vertex 0 one uses the fir s t expressio n fo r ζ given in ( △ ). On the branch with vertex 3 one uses the second expr ession for ζ . F or the long bra nch, we need ζ in terms of β and γ . Namely , we hav e ζ = ( − β 3 + γ β γ ) 2 = ( β 5 γ + β 4 γ β + β 3 γ β 2 + · · · + γ β 5 ) . T o see this, write the RHS as β 3 [ β 2 γ + β γ β + γ β 2 ] + [ β 2 γ + β γ β + γ β 2 ] β 3 . Expanding α 3 = 0 gives β 2 γ + β γ β + γ β 2 = − β 3 − γ β γ , substitute this and use β 6 = 0. RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 17 An element o f degree 20 in Z gr ( E (Λ )) . Next we find an element y o f de g ree 20 in Z gr ( E (Λ )). Define ω = α 2 γ αγ + αγ αγ α + γ αγ α 2 . This elemen t comm utes with α . F urthermore, ω 2 commutes wit h γ , using the following which is easy to c heck. ( 2 ) γ ω + ω γ = − ζ . Define y = P 8 i =0 y i , where y 0 = γ αγ α 2 γ αγ [0] , y 5 = P 3 i =0 β 3 − i α 2 γ α 2 β i [5] , y 1 = ( α ( γ αγ α 2 γ αγ ) + ( γ αγ α 2 γ αγ ) α + ( αγ ) 4 α )[1] , y 6 = P 2 i =0 β 2 − i α 2 γ α 2 β i [6] , y 2 = ω 2 , y 7 = P 1 i =0 β 1 − i α 2 γ α 2 β i [7] , y 3 = ( α 2 γ αγ α 2 γ α + αγ α 2 γ α 2 γ α + αγ α 2 γ αγ α 2 )[3] , y 8 = α 2 γ α 2 [8] . y 4 = P 4 i =0 β 4 − i α 2 γ α 2 β i + 2( β 4 γ β 4 )[4] , W e claim that y is a n element in the cen tre of E (Λ). It is straightf orward to chec k that o n each branch, aw ay from the branch vertex, we have a i y = y a i and a i y = y a i , and s imilarly a 1 y = y a 1 and a 1 y = y a 1 . The remaining identit y at the long bra nch will follow directly if we show (A) ω 2 = 5 X i =1 β 5 − i α 2 γ α 2 β i + 2[ β 5 γ β 4 + β 4 γ β 5 ] . F urthermore, the r emaining identit y at vertex 3 will follow dir ectly from (B) ω 2 = γ [ α 2 γ αγ α 2 γ α + αγ α 2 γ α 2 γ α + αγ α 2 γ αγ α 2 ] + [( α 2 γ αγ α 2 γ α + αγ α 2 γ α 2 γ α + αγ α 2 γ αγ α 2 ] γ . W e use the following identit y ( D ) ( αγ ) 3 α 2 γ α + αγ α 2 ( γ α ) 3 + αγ αγ α 2 γ αγ α = 0 , which is obtained from ( † ) by premultiplying with αγ and p os tmultiplying with γ α . W e start with proving (A) . First we c a lculate (A)-I = 5 X i =1 β 5 − i α 2 γ α 2 β i . This can be written a s β 3 m + mβ 3 , where m = αγ α 2 γ α 2 + γ α 2 γ α 2 γ + α 2 γ α 2 γ α = αζ + γ α 2 γ α 2 γ − α 2 γ αγ α 2 . Recall fro m ( ‡ ) that β 3 = − ρ − γ αγ , where ρ 2 = ζ . Substituting this gives β 3 m + mβ 3 = ( − ρm − γ αγ m − mγ αγ − mρ ) = − ρm − ( γ α ) 2 ζ + ( γ αγ α 2 ) 2 − ( αγ ) 2 ζ + ( α 2 γ αγ ) 2 − mρ. Note that the tw o sq ua res o ccur in ω 2 . Next − ρm = − ραζ − ργ α 2 γ α 2 γ + ρα 2 γ αγ α 2 = ( αγ ) 4 α 2 + α 2 ( γ α ) 4 − ργ α 2 γ α 2 γ + ρα 2 γ αγ α 2 18 ERDMANN AND SOLBERG (and reversing each term gives an identit y for − mρ ). Note that the firs t t wo terms o ccur in ω 2 . In the expr ession for β 2 m + mβ 3 − ω 2 we can canc e l t wo of the terms immediately . Namely , we obtain that ρα 2 γ αγ α 2 + α 2 γ αγ α 2 ρ = 0 , by fir st s ubstituting ρα 2 = α 2 γ α 2 = α 2 ρ and then pre- and po st-mult iply ( † ) with α 2 . (A)-II . W e get from this that β 3 m + mβ 3 − ω 2 is equal to ( αγ αγ α ) 2 + ( αγ ) 4 α 2 + α 2 ( γ α ) 4 − ( γ α ) 2 ζ − ( αγ ) 2 ζ − ρ ( γ α 2 γ α 2 γ ) − ( γ α 2 γ α 2 γ ) ρ. (A)-II I . By definition and an obvious substitution β 5 γ β 4 + β 4 γ β 5 = β 4 [ β γ + γ β ] β 4 = β 4 ( α 2 − β 2 ) β 4 = β 4 α 2 β 4 = β 3 γ α 2 γ β 3 =( ρ + γ αγ ) γ α 2 γ ( γ αγ + ρ ) = ρ ( γ α 2 γ ) ρ. So we must show that ( ∗ ) (A)-I I = − 2 ργ α 2 γ ρ. Using that γ α 2 γ = γ ( ρ − αγ α ) = ( ρ − αγ α ) γ and ρ 2 = ζ , we infer that ρ ( γ α 2 γ αγ α ) − ζ ( γ α ) 2 = − ρ ( αγ ) 3 α a nd αγ αγ α 2 γ ρ − ( αγ ) 2 ζ = − ( αγ ) 3 αρ . Using the same identit y as a bove we get tha t α 2 ( αγ ) 4 − ρ ( αγ ) 3 α = − αγ α 2 ( γ α ) 3 . The identit y o btained from this b y reversing the o rder in each monomial holds similarly . W e add the appropria te equations and cancel, a nd we get α 2 ( γ α ) 4 − ζ ( γ α ) 2 = − αγ α 2 ( γ α ) 3 − ρ ( γ α 2 γ αγ α ) . The identit y o btained b y reversing the o rder in ea ch term also holds. W e substitute these into (A)-I I and obtain that it is equal to − ( αγ αγ α ) 2 − αγ α 2 ( γ α ) 3 − ( αγ ) 3 α 2 γ α − ρ ( γ α 2 γ αγ α ) − ( αγ αγ α 2 γ ) ρ − ρ ( γ α 2 γ α 2 γ ) − ( γ α 2 γ α 2 γ ) ρ. The sum of the first three terms is zero, b y ( D ). Now we note that ( ∗ ) ργ α 2 γ ρ = ργ α 2 γ α 2 γ + ργ α 2 γ αγ α, so we can replace tw o terms in (A)-I I by − ργ α 2 γ ρ . The identit y obtaine d from ( ∗ ) by reversing the or der also holds, and we can therefore replace the o ther tw o ter ms in (A)-I I, w hich inv o lve ρ b y − ργ α 2 γ ρ . So in to tal we get that (A)-II is equal to − 2 ργ α 2 γ ρ as required. This proves (A). W e prov e now (B), that is ( 3 ) γ αγ α 2 γ α 2 γ α + γ αγ α 2 γ αγ α 2 + αγ α 2 γ α 2 γ αγ + αγ α 2 γ αγ α 2 γ − αγ αγ α 2 γ αγ α − ( αγ ) 4 α 2 − α 2 ( γ α ) 4 = 0 . T ak e relation ( † ), and pre- and post-multiply it with αγ , this gives (B)-I αγ α 2 γ α 2 γ αγ + αγ α 2 γ αγ α 2 γ = − αγ αγ α 2 γ α 2 γ − ( αγ ) 5 , and we can replace terms 3 and 4 in ( 3 ) by (B)- I. Symmetrica lly we can replace terms 1 and 2 in ( 3 ) by (B)-I ∗ − γ α 2 γ α 2 γ αγ α − ( γ α ) 5 . RADICAL CUBE ZER O WEAKL Y SYMMETRIC ALGEBRAS 19 Next we claim that (B)-I is equal to (B)-I I ( αγ ) 3 α 2 γ α + ( αγ ) 4 α 2 + αγ αγ α 2 γ αγ α. Namely αγ αγ α 2 γ α 2 γ = αγ αγ α 2 ( ρ − αγ α ) = ( αγ ) 2 ( ρ − αγ α ) ρ − αγ αγ α 2 γ αγ α = αγ ) 2 ζ − ( αγ ) 3 αρ − αγ αγ α 2 γ αγ α = − ( αγ ) 5 − ( αγ ) 3 α 2 γ α − ( αγ ) 3 αγ α 2 − αγ αγ α 2 γ αγ α. Symmetrically w e can replace (B)-I ∗ by (B)-I I ∗ . αγ α 2 ( γ α ) 3 + α 2 ( γ α ) 4 + αγ αγ α 2 γ αγ α W e substitute (B)-II and (B)-I I ∗ int o ( 3 ) and cancel, this leav es us to sho w that ( αγ ) 3 α 2 γ α + αγ α 2 ( γ α ) 3 + αγ αγ α 2 γ αγ α = 0 . W e get this dire ctly by premultiplying the identit y ( † ) with αγ and p ostmultiplying with γ α . This prov es the identit y (B), and hence that y is in Z gr ( E (Λ )). Finite generation. W e wan t to show that E 2 is finitely gener ated ov er the subal- gebra g e ne r ated by the central elements ζ and ω 2 . Let φ b e a monomial in { α, γ } . W e define its length l ( φ ), to b e the total num b er of factors { α, γ } . As a first goal tow a rds finite gener ation, we w ant to express φ as a polyno mial in ζ , ω 2 , wher e the co efficients are p olyno mials in { α, γ } , a nd so that w e hav e only finitely many co efficients. T o this end, w e hav e already seen that (i) ω and ρ comm ute with α , (ii) ω γ + γ ω = − ζ , (iii) ω ρ + ρω = − 3 ζ α 2 , (iv) γ ρ = ργ + ( γ α ) 2 − ( αγ ) 2 . F urthermore, we have that ( γ α ) 3 k γ = ( − 1) k ζ k γ for k ≥ 1, since ζ γ = − ( γ α ) 3 γ . And, ( α 2 γ ) 2 k +1 = ρ 2 k α 2 γ = ζ k α 2 γ for k ≥ 1, since α 2 γ = ω − αγ α − γ α 2 . These observ ations are a main step tow ards our first goal. A t the next step w e g et expressions , which hav e ω a s a factor. Although this is not central, we can mov e factors of ω to the left, as a consequence o f the next lemma. Lemma. Assume φ = pω q wher e p, q ar e monomials in α, γ . Then φ = − ω pq + ζ X φ i , wher e φ i ar e monomials in α, γ of length l ( φ i ) < l ( p ) + l ( q ) . W e leav e t he pro o f to the rea der, only p ointing out that induction and the ident ity γ ω = − ( ω − ζ ) are used. The final crucia l step is the follo wing. Lemma. Assu m e φ is a monomial in { α, β } . Then φ is a line ar c ombination of elements of the form ζ a ω b φ 1 , wher e φ 1 is a monomial in { α, γ } of the form ( ∗ ) α i ( γ α ) r ( γ α 2 ) s γ α j with r ≤ 2 and s ≤ 2 and i , j ≤ 2 . 20 ERDMANN AND SOLBERG Pr o of. If suffices to expr ess φ as a co m bination of ele ments ζ a ω b φ 1 with φ 1 as in ( ∗ ), but for arbitrary r and s . Then the co efficients can b e reduced a ccording to the first Lemma. Suppose therefore that φ is not of the form as in ( ∗ ). Then we can wr ite φ = p ( α 2 γ αγ ) q, where p and q are monomia ls in { α, γ } . This is equal to ( ∗∗ ) p [ ω − αγ αγ α − γ αγ α 2 ]) q = pω q − pαγ αγ αq − pγ αγ α 2 q and pω q = ω pq + ζ P φ i with l ( φ i ) < l ( p ) + l ( q ). F or pq we use induction. The second term in ( ∗∗ ) has one facto r α 2 less, and in the third term of ( ∗∗ ), α 2 o ccurs further to the right. So if w e star t with the rightmost α 2 in φ a nd iterate the ab ove substitution, then we can mo ve α 2 completely to the right. Hence we get ter ms with co efficients either o f shor ter length, or with fewer α 2 and where to the r ight only submo no mials . . . γ αγ . . . occur, but w her e the leng th do es not increa se. The claim follows now by induction.  The claim that Λ satisfies (Fg) now follows immediately .  9. Quantum exterior al gebras This final section is devoted to characterizing when the quantum exter io r k - algebra Λ = k h x 1 , x 2 , . . . , x n i / ( { x i x j + q ij x j x i } i 1 . Prop ositio n 9.1. The quantum exterior k -algebr a Λ = k h x 1 , x 2 , . . . , x n i / ( { x i x j + q ij x j x i } i j, and x j x p i =      q − p ij x p i x j , i < j, x p +1 i , i = j, x j x p i , i > j. F rom these for m ulae we obtain that x p i is in Z gr ( E (Λ )) if and o nly if (i) 1 = ( − 1 ) p q − p ij for j > i , (ii) 1 = ( − 1 ) p for j = i , (iii) q − p j i = ( − 1 ) p for j < i . F or c har k 6 = 2, this is clearly equiv alent to (A) p is even and (B) { q ij } j >i are a ll p -th ro o ts of unit y . F or char k = 2, then this is equiv alent to just (B). Suppo se now that not all the elements { q ij } j >i are ro ots of unity for some i in { 1 , 2 , . . . , n − 1 } . Hence x p i is not in Z gr ( E (Λ )) for any p ≥ 1. On the other hand x l i is in E (Λ) f or any l ≥ 1. But since E (Λ) is a domain and Z gr ( E (Λ )) (and E (Λ)) is multi-graded by N n , the only w ay the elemen ts x l i can b e genera ted by the gr aded centre Z gr ( E (Λ )) is that it is one of the genera tors. Hence E (Λ) is infinitely gener ated as a mo dule ov er Z gr ( E (Λ )). Consequent ly (Fg ) is not satisfied for Λ = E ( E (Λ)). Suppo se now that all the elemen ts { q ij } j >i are ro o ts of unit y fo r all i in { 1 , 2 , . . . , n − 1 } . Supp ose that each q ij is a ro ot o f unit y of degree d ij . Let N be the lea st co mmon even mu ltiple o f a ll the d ij ’s. F rom the calculations ab ov e we infer that x N i is in Z gr ( E (Λ )) fo r all i in { 1 , 2 , . . . , n } . Then the set { x t 1 1 x t 2 2 · · · x t n n | t i < N , ∀ i ∈ { 1 , 2 , . . . , n }} is a generating set of E (Λ) as a module ov er the graded cen tre Z gr ( E (Λ )). Hence E (Λ) is a finitely g enerated mo dule ov er Z gr ( E (Λ )), and therefor e (Fg) is satisfied for Λ = E ( E (Λ)).  This result is generalized by Bergh and Opp ermann in [2, The o rem 5 .5 ]. Remark. The Nak ayama a utomorphism ν of Λ is given b y ν ( x i ) = ( − 1) n − 1 Q n j = i + 1 q ij Q i − 1 j =1 q ji x i for i = 1 , 2 , . . . , n . F or n = 3, w e can choose q 12 = q , q 13 = q − 1 and q 23 = q 12 , where q is not a ro ot of unity . 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