On the graded center of the stable category of a finite $p$-group

ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GR OUP MARKUS LINCKELMANN, RADU ST ANCU Abstract. W e sho w that for any finite p - group P of rank at least 2 and any algebraically closed field k of charac teristic p the graded cen ter Z ∗ (mod ( kP )) of the stable module category of finite-dimensional kP -modules has infinite dimension i n each o dd degree, and if p = 2 also in eac h ev en degree. In particular, this pr o vides examples of symmetric algebras A for which Z 0 (mod ( A )) is not finite-dimensional, answering a question rais ed in [10]. 1. Introduction The graded center of a k -linear triang ula ted category ( C ; Σ) over a commutativ e ring k is the gr aded k -mo dule Z ∗ ( C ) = Z ∗ ( C ; Σ) which in degree n ∈ Z consists of a ll k -linear na tural transformatio ns ϕ : Id C → Σ n satisfying Σ ϕ = ( − 1) n ϕ Σ; this beco mes a graded co mm utative k -algebr a with m ultiplication essentially induced b y co mpo sition in C . W e r efer to [10] for more details. By [5 ], the stable categ ory mo d( A ) of finitely genera ted mo dules ov er a finite- dimensional self-injective algebra A is a tria ngulated category with s hift functor the inv erse of the Heller op era tor. Its gra de d center has been calculated for Br auer tree algebr a s [7 ] and in particular also uniserial algebras [8]. These ca lculations suggest that T ate coho mlogy ring s o f blo cks and the graded centers of their stable mo dule categ ories a re clos ely r elated. Since the T ate cohomolog y of a blo ck is an in v ar iant of the fusion system of the blo ck, a go o d understanding of g r aded centers might shed some light on the question to wha t extent the fusion s ystem o f a blo ck is determined by its stable mo dule categ o ry . These calculations also suggest that in order to determine the graded cen ter of the stable module categor y of a blo ck one will need to do this first for a defect g roup algebra of the blo ck, hence for finite p -gro up alg ebras. This is what mo tiv ates the prese n t pap er. The following result shows that Z 0 (mo d( A )) nee d not be finite-dimensional, ans w ering a question raised in [10]. Theorem 1.1. L et P b e a fin ite 2 -gr oup of ra nk at le ast 2 and k an algebr ai c al ly close d field of char acteristic 2 . Evaluation at the trivial k P -mo dule induc es a surje ctive homomorph ism of gr ade d k -algebr as Z ∗ (mo d ( k P )) − → ˆ H ∗ ( P ; k ) whose kernel I is a nilp otent homo gene ous ide al which is infinite-dimensional in e ach de gr e e; in p articular, Z 0 (mo d ( k P )) has infinite dimension. F or o dd p we hav e a slightly weaker statement: Theorem 1. 2. L et p b e an o dd prime, P a finite p -gr oup of r ank at le ast 2 and k an alge- br aic al ly close d field of char acteristic p . Evaluation at t he t rivial k P -mo dule induc es a surje ctive homomorph ism of gr ade d k -algebr as Z ∗ (mo d ( k P )) − → ˆ H ∗ ( P ; k ) whose kernel I is a nilp otent homo gene ous ide al which is infinite-dimensional in e ach o dd de gr e e. Date : No v em ber 25, 2021. 1 2 MARKUS LINCKELM ANN, RADU S T ANCU It is ea s y to see that the canonical ma p Z ∗ (mo d ( k P )) → ˆ H ∗ ( P ; k ) is surjective with nilp otent kernel I (see Lemma 2.7 below). The p oint of the a bove theorem is that this kernel tends to hav e infinite dimension in eac h deg ree (if p = 2) and at least in each o dd deg ree if p > 2. Note though that for p o dd there is no known example with Z 0 (mo d ( k P )) having infinite dimens io n. The pro of shows more pr ecisely that these dimensions have as low er b ound the car dina lit y of the field k . T echnically , the pro ofs of the a bove theo rems are based on the fact that for A a symmetric algebra s, an a lmost split sequence ending in a n indeco mpos able non pr o jective A -mo dule U deter mines an almost vanishing morphism ζ U : U → Ω( U ) which in turn provides elements of degree − 1 in the gra ded cent er of the stable category; see e.g. [10, Prop osition 1.4] or section 2 below. Using modules with appro priate p er io ds, these can then b e “shifted” to all other deg rees if the under lying characteris tic is 2 and all other o dd degrees if the character is tic is o dd. The elements of Z ∗ (mo d ( A )) obtained in this way will b e called almost vanishing ; see 2.5 for de ta ils. F or Kle in four groups, almost split sequences turn out to b e the only wa y to obtain elements in the gra ded center of its stable mo dule catego ry b eyond T ate cohomo logy . This can b e seen us ing the classificatio n of indecomp osa ble mo dules ov er Klein four gr oups and leads to a slightly more precise statement. Theorem 1. 3. L et P b e a Klein four gr oup and let k b e an algebr aic al ly close d field of char- acteristic 2 . Then t he evaluation at the t rivial k P -mo dule induc es a surje ctive homomorphism of gr ade d k - algebr as Z ∗ (mo d( k P )) − → ˆ H ∗ ( P ; k ) whose kernel I is a homo gene ous ide al which is infinite-dimensional in e ach de gr e e. Mo r e over, we have I 2 = { 0 } and al l elements in I ar e almost vanishing. This raises the question for which finite p -g r oups P is the graded ce nter Z ∗ (mo d ( k P )) gener- ated by T ate cohomo logy and almos t v anishing elements. Another interesting question under- lying s ome of the techical details below is the following. Giv en a p erio dic mo dule U of per io d n of a symmetric algebra A ov er a field k , any is o morphism α : U ∼ = Ω n ( U ) induces an algebra automorphism of the stable e ndo morphism a lg ebra End A ( U ) sending ϕ to α − 1 ◦ Ω( ϕ ) ◦ α . When is this an inner automorphism? Equiv alently , w hen ca n α be chosen in such a wa y that this automorphism is the identit y? D. J. Benson [3] observed that the answer is p ositive if α is induced by an element in T ate co homology (or the T ate analo gue o f Ho chsc hild cohomology ) bec ause then α is the ev aluation at U of a natural transforma tion from the identit y functor on mo d ( A ) to the functor Ω n . 2. Back ground ma terial Let A b e a finite-dimensional algebra over a field k . The s table mo dule category mo d ( A ) has as ob jects the finitely ge nerated A -mo dules and as mo r phisms, for any tw o finitely gen- erated A -mo dules U , V , the k -vector space Hom A ( U, V ) = Hom A ( U, V ) / Hom pr A ( U, V ), where Hom pr A ( U, V ) is the k -vector space of A -homomor phisms from U to V which factor through a pro jective A -mo dule, with comp ositio n of morphis ms induced by the us ual comp o sition o f A - homomorphisms in the catego ry o f finitely generated A -mo dules mo d( A ). The Heller op erator Ω A sends an A -mo dule U to the kernel o f a pro jective cover P U → U of U ; this induces a functor, s till deno ted Ω A , on mo d ( A ), which is uniq ue up to unique isomorphism of functors. If A is self-injective then Ω A is an equiv a le nce, and mod( A ) be comes a triang ulated catego ry with shift functor Σ A = Ω − 1 A , s ending an A -module to a cokernel of an injectiv e h ull of U , with exact triangles induced by short exa ct seq uences in mo d( A ). See e.g. [5] for details. If A is symmetric - tha t is, A is isomo rphic, as A - A -bimo dule to its k -dual Hom k ( A, k ) - then A is ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GROUP 3 in particular self- injective. The liter ature on finite-dimensional algebras tends to privilege Ω A from a notational p oint o f view , while the one o n triangulated ca tegories w ould rather use Σ A ; we so metimes use both. If A is clear from the context a nd no co nfusion ar ises, we write Ω and Σ instead of Ω A and Σ A , r esp ectively . If B is a subalgebra of A we hav e a canonical isomor phism Ω n A ( A ⊗ B V ) ∼ = A ⊗ B Ω n B ( V ) in mod ( A ), where V is a finitely generated B -mo dule and n a po sitive integer. If mor eov er A is pr o jective a s B -mo dule, we also hav e a canonica l isomor- phism Ω n B (Res A B ( U )) ∼ = Res A B (Ω n A ( U )) in mo d ( B ), where U is a finitely ge nerated A -module, and the c a nonical adjunction is omorphism Hom A ( A ⊗ B V , U ) ∼ = Hom B ( B , Res A B ( U )) induces an is omorphism Hom A ( A ⊗ B V , U ) ∼ = Hom B ( V , Res A B ( U )), where U is an A -mo dule and V is a B -mo dule. The following well-kno wn lemma states that the Heller transla tes commut e with the adjunction isomor phisms (w e sketch a pro of for the convenience of the re a der). Lemma 2.1. L et A b e a finite-dimensional algebr a over a field k and B a sub algebr a such that A is pr oje ctive as B -mo dule. L et n b e a p ositive int e ger. The fol lowing diagr am of c anonic al isomorphi sms is c ommut ative: Hom A ( A ⊗ B Ω n B ( V ) , Ω A ( U )) ≃ / / ≃   Hom B (Ω n B ( V ) , Res A B (Ω n A ( U ))) ≃   Hom A (Ω n A ( A ⊗ B V ) , Ω n A ( U )) Hom B (Ω n B ( V ) , Ω n B (Res A B ( U ))) Hom A ( A ⊗ B V , U ) ≃ / / Ω n A ≃ O O Hom B ( V , Res A B ( U )) Ω n B ≃ O O Pr o of. W e sketc h the argument in the case n = 1 ; the general case follo w s either by induction, or directly with shor t exact sequences in the t wo dia grams b elow replaced by exact sequences with n + 2 terms of which all but p os s ibly the first and la st a re pro j e c tiv e. The Heller translates Ω A and Ω B are defined b y the fo llowing tw o commutativ e diagra ms with exact rows, where P X denotes a pro jective c over of the mo dule X : 0 / / A ⊗ B Ω B ( V ) / / ≃   A ⊗ B P V ν / / δ   A ⊗ B V / / 0 0 / / Ω A ( A ⊗ B V )) / / Ω α   P A ⊗ B V / / γ   A ⊗ B V / / α   0 0 / / Ω A ( U ) / / P U µ / / U / / 0 0 / / Ω B ( V ) / / Ω β   P V ρ / / φ   V / / β   0 0 / / Ω B (Res A B ( U )) / / ≃   P Res A B ( U ) / / ψ   Res A B ( U ) / / 0 0 / / Res A B (Ω A U ) / / Res A B ( P U ) σ / / Res A B ( U ) / / 0 4 MARKUS LINCKELM ANN, RADU S T ANCU Here µ , ρ are chosen s urjective morphisms and σ = Res A B ( µ ), ν = Id A ⊗ ρ . W e hav e that µ ◦ γ ◦ δ = α ◦ ν . Since α , β corre s po nd to ea ch other through the adjunction ismorphism, the tw o diagrams imply tha t the imag es of Ω α and Ω β corres p ond to each o ther through the adjunction Hom B (Ω B ( V ) , Res A B (Ω A ( U ))) ∼ = Hom A ( A ⊗ B Ω B ( V ) , Ω A ( U )).  A morphis m α : U → V in mo d ( A ) is called almost vanishing if α 6 = 0 and if for a n y morphism ϕ : X → U in mod ( A ) whic h is no t a s plit epimor phism we hav e α ◦ ϕ = 0. Since endomor phism algebras of indecomp osa ble A -modules are (not necessar ily split) lo cal, it follows from [5, I.4.1] that this condition is equiv alent to its dual; that is, for any morphism ψ : V → Y in mo d( A ) which is not a s plit monomorphism we hav e ψ ◦ α = 0. In or der to follow almost v a nis hing mor- phisms through the standard adjunction isomor phisms w e collect a few elemen tar y sta temen ts on split epimorphisms. In statement s and pro ofs where we consider b oth A -homomor phisms and their cla sses in the stable category we ado pt the notational c o nv ention that if ϕ : U → V is a homomorphism of A -modules then ϕ denotes the class of ϕ in Hom A ( U, V ). As in an y tri- angulated catego r y , epimor phis ms in mo d ( A ) are s plit epimorphisms, and those are essen tia lly induced by split epimorphisms in mod ( A ): Lemma 2.2. L et A b e a finite-dimensional k -algebr a and ϕ : U → V an A -homomorph ism. Supp ose that V is inde c omp osable non pr oje ctive. Then ϕ is a split epimorphism in mo d( A ) if and only if its class ϕ : U → V is a split epimorphism in mod( A ) . Pr o of. Suppo se ϕ is a split epimor phis m in mo d( A ). Let σ : V → U b e an A -homomorphism such that ϕ ◦ σ = Id V . Thu s ϕ ◦ σ − Id V factors through a pr o jective module. Since V is indecomp osable non pro jective this implies tha t ϕ ◦ σ − Id V ∈ J (End A ( V )), hence ρ = ϕ ◦ σ is an automor phism o f V a nd σ ◦ ρ − 1 a section for ϕ . The conv erse is trivial.  Lemma 2. 3. Le t A b e a k -algebr a and B a sub algebr a of A . Supp ose that B has a c omplement in A as a B - B -bimo dule. L et ϕ : V → W b e a homomorphism of left B -mo dules and set ψ = Id A ⊗ ϕ : A ⊗ B V → A ⊗ B W . Then ψ is a split epimorph ism if and only of ϕ is a split epimorphi sm . Pr o of. Suppo se that ψ is a split epimor phis m; tha t is, there is an A -homo mo rphism σ : A ⊗ B W → A ⊗ B V sa tisfying ψ ◦ σ = Id A ⊗ B W . Le t C be a co mplement of B in A as a B - B - bimo dule; that is, A = B ⊕ C as a B - B - bimodule. Denote b y π : A ⊗ B V → V and τ : A ⊗ B W → W the canonical pro jections with kernel C ⊗ B V a nd C ⊗ B W , resp ectively , induced by the pro jection A → B with kernel C . E x plicitly , for v ∈ V and a ∈ A we hav e π ( a ⊗ v ) = a ⊗ v if a ∈ B and π ( a ⊗ v ) = 0 if a ∈ C ; simila rly for τ . Since ψ = Id A ⊗ ϕ we have ϕ ◦ π = τ ◦ ψ . Define ρ : W → V by ρ ( w ) = π ( σ (1 ⊗ w )), for any w ∈ W . The n ϕ ( ρ ( w )) = ϕ ( π ( σ (1 ⊗ w ))) = τ ( ψ ( σ (1 ⊗ w ))) = τ (1 ⊗ w ) = w , hence ϕ is a split epimorphism with section ρ . The co n verse is trivial.  Lemma 2. 4. L et A b e a finite-dimensional algebr a over an algebr aic al ly close d field k and B a sub algebr a su ch that A is pr oje ctive as left and right B -mo dule and s u ch that B has a c omplement in A as a B - B -bimo dule. L et V b e a finitely gener ate d inde c omp osable non pr oje ctive B -mo dule such that U = A ⊗ B V is inde c omp osable. Then U is non pr oje ctive. L et ζ V : V → Ω B ( V ) and ζ U : U → Ω A ( U ) b e almost vanishing homomorp hisms in mo d ( B ) and mo d( A ) , r esp e ctively. Supp ose that ther e is an isomorphism β : Ω n B ( V ) ∼ = V in mo d ( B ) such that Ω B ( β ) ◦ Ω n B ( ζ V ) = ( − 1) n ζ V ◦ β . Then Ω n A ( U ) ∼ = U and for any isomorphism α : Ω n A ( U ) ∼ = U in mo d ( A ) we have the Ω A ( α ) ◦ Ω n A ( ζ U ) = ( − 1) n ζ U ◦ α . ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GROUP 5 Pr o of. W e first p oint o ut where we use the hypothesis on k being algebr aically closed. If we can find some isomorphism α : Ω n A ( U ) ∼ = U in mo d ( A ) suc h tha t Ω A ( α ) ◦ Ω n A ( ζ U ) = ( − 1) n ζ U ◦ α then this holds for any isomo rphism b eca us e End A ( U ) is split lo cal. Moreov er, ζ U is unique up to a nonzer o scalar. Similarly fo r V and ζ V . By the assumptions, A = B ⊕ C for some B - B -submo dule C o f A whic h is finitely gener ated pr o jective as left and r ight B -mo dule. Thus Res A B sends any pro jectiv e A -module to a pro jective B -mo dule, a nd V is a direct summand of Res A B ( U ). Since V is non pr o jective this implies that U is non pro jective. By adjunction we hav e Hom A ( U, U ) ∼ = Hom B ( V , Res A B ( A ⊗ B V )) W e pr ov e that the image η of ζ U under this adjunction is a morphism which, when c o mpo sed with any pro jection of Res A B ( A ⊗ B V ) onto an indecomp osa ble direct summand is either zero or almost v anishing in mo d ( B ). This is equiv alent to showing that η precomp osed with any morphism ending a t V which is not a split epimo rphism in mo d( B ) yields zero. Let X b e a B - mo dule and ϕ ∈ Hom B ( X, V ) such that the ima ge ϕ in Hom B ( X, V ) is not a split epimorphism. Then by Lemma 2.2 and Lemma 2.3 the morphism ψ = Id A ⊗ ϕ in Hom A ( A ⊗ B X , U ) is not a split epimorphism and hence ζ M ◦ ψ is zero in Hom A ( A ⊗ B X , M ). Applying the adjunction to ζ M ◦ ψ we get that η ◦ ϕ is zero in Hom B ( X, Res A B ( A ⊗ B V ))). Th us a ll comp onents of η to the indecomp osable direct summands of Res A B ( A ⊗ B V )) are either zero or a scalar m ultiple of ζ V . The h y p othes is on ζ V implies that Ω n B ( η ) = ( − 1 ) n η , mo dulo appropr iate iden tifica tions. It follows from Lemma 2.1 that Ω n ( ζ U ) = ( − 1 ) n ζ U , again modulo appr opriate identifications, which pr ov es the result.  The existence o f an almost v anishing mo rphism α : U → V in mod ( A ) forces that U , V are indecomp osable non pro jective A -mo dules, and that the exa c t triangle in mod ( A ) of the form Ω( V ) / / E / / U α / / V is almost split in the sense of [5, I .4 .1.], hence induced by a n a lmost split exact s e quence of A -mo dules of the form 0 / / Ω( V ) / / E / / U / / 0 which in turn forces that V ∼ = Ω( U ) a s A is symmetric (cf. [2, 4.12.8 ]). This shows that if there is a n a lmost v anishing morphism U → Σ n ( U ) in mo d ( A ) for some integer n then Σ n ( U ) ∼ = Σ − 1 ( U ), a nd hence either n = − 1 or U is per io dic of p erio d dividing n + 1. As men tioned ea r lier, almost v anishing morphisms define elements in the graded center. It is conv enient to extend the ter minology of almost v a nishing mor phisms to elements in the graded center: Definition 2. 5. Let A b e a symmetric alg e bra ov er a fie ld k . F or any in tege r n we say that an element ϕ ∈ Z n (mo d ( A )) is almost vanishing if for any indeco mpo sable non pr o jective A -module U either ϕ ( U ) = 0 or ϕ ( U ) : U → Σ n ( U ) is almost v anishing. An element in Z ∗ (mo d ( A )) is called almost vanishing if all of its homogeneous comp onents a r e almost v a nishing. W e deno te by I A the set of all almost v anishing elements in Z ∗ (mo d ( A )). This definition makes sense for ar bitr ary tr iangulated categor ies. Note that the zero element of Z ∗ (mo d ( A )) is contained in I A ; this slight departure from the definition of almost v anishing morphisms has the following ob vious adv a n tage: 6 MARKUS LINCKELM ANN, RADU S T ANCU Lemma 2. 6. Le t A b e a symmetric algebr a over a field k . Then I A is an ide al in Z ∗ (mo d ( A )) whose squar e is zer o. Mor e over, I A annihilates every ide al c onsist ing of nilp otent elements in Z ∗ (mo d ( A )) . Pr o of. If ϕ ∈ Z n (mo d ( A )) is nilp otent, where n is an integer, then for any indecomp osable non pro jective A -mo dule U the morphism ϕ ( U ) : U → Σ n ( U ) is not an is o morphism. Since U , Σ n ( U ) are b oth indecomp osable, ϕ ( U ) is neither a split epimor phism nor a split mono morphism. Thu s ϕ ( U ) co mpos ed with an y almost v anishing morphism yields zero. The result follows.  In the statement s o f Theo rem 1.2 and Theor em 1.3, in order to show that Z ∗ (mo d( A )) is not finite-dimensional in a par ticular deg ree we actually show that I A is not finite-dimensional in that degree . W e will need the following lemma, whic h is well-kno wn and strictly analog ous to [10, Pr o po sition 1.3], fo r instance. Lemma 2.7. L et p b e a prime, P a fin ite p -gr oup and k a field of char acteristic p . Evaluation at the t rivial k P -mo dule k induc es a surje ctive gr ade d k -algebr a homomorphism Z ∗ ( k P ) → ˆ H ∗ ( P ; k ) whose kernel I is a nilp otent ide al. Pr o of. This is the same pro of as that of [10, Pr op osition 1.3 ], with H ∗ ( P ; k ), H H ∗ ( k P ), D b ( k P ) replaced b y T ate cohomology ˆ H ∗ ( P ; k ), the T ate ana logue of Ho chschild co homology ˆ H H ∗ ( k P ) and mo d ( k P ), resp ectively .  3. On some m odules of period 1 and 2 In or der to co nstruct elements in a given degree n of the graded center of the stable ca te- gory of a sy mmetric alg ebra A ov er a field k it is not sufficient to c o nstruct A - mo dules with per io d dividing n + 1 ; we a lso need to ha ve some control over the effect of the shift on endomor - phisms in or de r to verify the additional compa tibilit y condition with the shift functor of natur a l transformatio ns b elonging to Z n (mo d ( A )). This is the purp o se of the elemen tar y observ a tions in this s ection. W e use without further comment that if k has po sitive characteristic p and if x ∈ Z ( A ) such that x p − 1 6 = 0, x p = 0, then 1 + x is an inv ertible element in A o f o r der p and gener ates a subalgebra which ca n be iden tified with the tr uncated p olynomia l algebr a k [ x ] / ( x p ) or the gro up algebr a k h 1 + x i , whic hever is more conv enient. The a lgebra A may or may not be pro jective a s k h 1 + x i -mo dule; this will dep end on x . If A = k P is the gro up algebra of an ele men ta ry ab elian p - group P with minimal gener ating set { u 1 , u 2 , .., u r } , then any nonzer o k -linear combination x = P r i =1 λ i (1 − u i ) has the pr op erty that A is pro jective as k h 1 + x i -mo dule. T he subgr oups gener ated by 1 + x for those x a re ca lled cyclic shifted subgroups of A . Lemma 3. 1. L et A b e a finite-dimensional algebr a over a field k of prime char acteristic p and x an element in A such that x p = 0 , x p − 1 6 = 0 and such that A is pr oje ctive as a k h 1 + x i - mo dule. L et u ∈ A × such that ux = xu . Then ( ux ) p = 0 , ( ux ) p − 1 6 = 0 and A is pr oje ctive as a k h 1 + ux i -mo dule. Pr o of. Since x and u commute we clearly hav e ( ux ) p = 0 and ( ux ) p − 1 6 = 0 . The h y p othes is that A is pro jective as k h 1 + x i -module is eq uiv a le nt to dim k ( x p − 1 A ) = 1 p dim k ( A ). As u is inv ertible and commutes to x , we hav e ( ux ) p − 1 A = x p − 1 A , a nd hence A is pro jective a s k h 1 + ux i -mo dule.  ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GROUP 7 Lemma 3.2. L et A b e a finite-dimensional algebr a over a field k of prime char acteristic p > 0 . L et x ∈ A such that x p − 1 6 = 0 , x p = 0 and such that A is fr e e as right k h 1 + x i -mo dule. Set M = Ax p − 1 . (i) If p = 2 ther e is an isomorphism α : M ∼ = Ω( M ) in mo d ( A ) su ch t hat ϕ ◦ α = α ◦ Ω( ϕ ) for any ϕ ∈ End A ( M ) . (ii) If p > 2 ther e is an isomorphism β : M ∼ = Ω 2 ( M ) in mo d ( A ) such that ϕ ◦ β = β ◦ Ω 2 ( ϕ ) for any ϕ ∈ End A ( M ) . (iii) We have End A ( M ) ∼ = End A ( M ) . Pr o of. Since A is free as right k h 1 + x i -mo dule w e hav e an exact seq uence of A -mo dules of the form 0 / / M / / A ǫ / / A π / / M / / 0 where ǫ is given b y right mult iplication with x and π is given b y rig h t multiplication with x p − 1 . Th us for this choice of pro jective covers w e hav e Ω( M ) = Ax and Ω 2 ( M ) = M . Let ϕ ∈ End A ( M ). Then ϕ lifts to an endomorphism ψ ∈ End A ( A ) satisfying π ◦ ψ = ϕ ◦ π . Let y ∈ A such that ψ ( a ) = ay for all a ∈ A . Thus ϕ ( ax p − 1 ) = ay x p − 1 = ax p − 1 y for all a ∈ A . Also, since x is central we hav e ǫ ◦ ψ = ψ ◦ ǫ . Thus ψ restricts to the endo mo rphism ϕ of M , which shows (i) and (ii). Supp ose now that ϕ factor s thro ugh a pr o jective A -mo dule. Then ϕ factors through the inclusion M ⊆ A , or equiv alently , ther e is y ∈ A such that ϕ ( m ) = my for all m ∈ M a nd such that Ay ⊆ M . But then in par ticular y ∈ M = Ax p − 1 , hence y = ux p − 1 for some u ∈ A . Since a ny m ∈ M is of the form m = ax p − 1 for so me a ∈ A we ge t that ϕ ( m ) = ϕ ( ax p − 1 ) = ax p − 1 ux p − 1 = aux 2( p − 1) = 0, whence (iii).  The next obser v a tion sho ws that if the cen ter o f a finite p -gr oup P has rank at least 2 then the construction principle for modules of p erio d 1 and 2 over the group algebra A = k P in the t wo prev ious lemmas yields infinitely man y is omorphism classes whenever the field k is infinite. Prop ositio n 3. 3. L et p b e a prime, P a finite p -gr oup and k a field of char acteristic p . Set A = k P . Su pp ose that Z ( P ) has an elementary ab elian su b gr oup E = h s i × h t i of r ank 2 , for some elements s, t ∈ Z ( P ) of or der p . L et ( λ, µ ) , ( λ ′ , µ ′ ) ∈ k 2 −{ (0 , 0) } . Set x = λ ( s − 1 )+ µ ( t − 1) and x ′ = λ ′ ( s − 1) + µ ′ ( t − 1) . Then Ax is absolutely inde c omp osable as left A -m o dule, and the fol lowing s tatements ar e e quivalent. (i) Ax ∼ = Ax ′ as left A -mo dules. (ii) Ax = Ax ′ . (iii) The images of ( λ, µ ) and ( λ ′ , µ ′ ) in P 1 ( k ) ar e e qual. Pr o of. Since A = k P is split lo cal, any quotient of A a s A -mo dule is abso lutely indecompo s- able. Suppo s e (i) holds. Since A is symmetric, any isomorphism Ax ∼ = Ax ′ extends to an endomorphism of A , which is given by r ight multip lication with an elemen t y ∈ A . In partic- ular, Axy = Ax ′ . But x ∈ Z ( A ) implies Axy = Ay x ⊆ Ax . Thus Ax ′ ⊆ Ax . E x ch anging the roles of x and x ′ shows Ax = Ax ′ , whence (ii) holds. The conv e r se is trivial. Suppo s e now that (ii) ho lds. Arguing b y contradiction, supp ose that the images o f ( λ, µ ), ( λ ′ , µ ′ ) in P 1 ( k ) are different, or eq uiv a le ntly , that ( λ, µ ), ( λ ′ , µ ′ ) are linearly indep endent in k 2 . Since Ax = Ax ′ contains any k -linea r combination of x and x ′ it follows that Ax co nt ains any k -linear combination of s − 1 and t − 1. But then A ( s − 1) = Ax = A ( t − 1 ) b ecaus e a ll involv ed modules hav e the same dimension p − 1 p dim k ( A ) as A is pro jective as right k E - mo dule. Ho wever, clearly 8 MARKUS LINCKELM ANN, RADU S T ANCU k E ( s − 1) 6 = k E ( t − 1), so this is impo ssible. This shows that (ii) implies (iii). The conv er se is again trivial.  Corollary 3.4. L et p b e a prime, P a finite p -gr oup of r ank at le ast 2 and k an infin ite field of char acteristic p . If p = 2 then k P has infin itely many isomorp hism classes of absolutely inde c omp osable mo dules of dimension | P | 2 whose p erio d is 1 , and if p > 2 then kP has infin itely many isomorphism classes of absolutely inde c omp osable m o du les of dimension | P | p whose p erio d is 2 . Pr o of. Let E b e an elementary a belia n subgroup of ra nk 2 of P . By Pro po sition 3 .3 and Lemma 3.2, the res ult holds for k E instead of k P . Induction Ind P E is an exact functor, pres erving per io dicity , and b y Mackey’s for m ula, for any indecomp osa ble kP -mo dule M there ar e , up to iso morphism, only finitely many indecomp osable k E -mo dules N satisfying Ind P E ( N ) ∼ = M , which implies the statement.  4. Proof o f Theorem 1.1 and Theorem 1 .2 Let p be a pr ime and P a finite p -gr oup of rank a t least 2. Let k b e an algebr aically closed field of characteris tic p . By Lemma 2 .7, ev aluation at the tr ivial k P -mo dule induces a surjective graded algebr a homomor phism Z ∗ (mo d ( k P )) → ˆ H ∗ ( P ; k ) with nilp otent kernel I . Let E = h s i × h t i be an elementary abelian s ubgroup of Z ( P ) o f rank 2 for some elements s , t in Z ( P ) of order P and let ( λ, µ ) ∈ k 2 − { (0 , 0) } Set x = λ ( s − 1) + µ ( t − 1) and M = Ax p − 1 . By Le mma 3.2 the module M has p erio d 1 if p = 2 and perio d 2 if p is o dd. The a lmo st split exact sequence ending in M is of the form 0 / / Ω 2 ( M ) / / E / / M / / 0 hence repr esented by an almos t v anishing morphism ζ M : M → Ω( M ). By [1 0 , P rop osition 1.4], this defines a na tural transformatio n ϕ M : Id → Ω as follows. Set ϕ M ( M ) = ζ M , set ϕ M ( N ) = 0 if N is indeco mpo sable non pro jective and not iso morphic to M or Ω( M ), and in the case p o dd, s e t ϕ M (Ω( M )) = − Ω( ζ M ). Then, thanks to Lemma 3.2 ag ain, ϕ is indeed a natural transformation belong ing to Z − 1 (mo d( k P )). If p = 2 (resp. p > 2)then for an y integer n (resp. even integer n ) we hav e M ∼ = Σ n ( M ), and hence ϕ determines also an element in Z n − 1 (mo d ( k P )) for a ny integer n (r esp. even integer n ). By Lemma 3 .3, this implies that Z n − 1 (mo d ( k P )) is infinite dimens ional for a n y integer n if p = 2 and for a ny even integer if p > 2 . This prov es Theorem 1.1 and Theorem 1.2. Remark 4.1. D. J. Benson [3] pointed o ut that it is ea sy to co ns truct indecomp osa ble mo dules for finite p -group algebras with p erio d one for o dd p . O ne migh t therefore w onder whether the stronger statement of Theorem 1.1 holds for p o dd as well. 5. Proof of Theorem 1. 3 Let k be a n alge braically closed field of c ha racteristic 2. Denote by V 4 = h s, t i a Klein four group, with commuting in volutions s, t . By results of Ba ˇ sev [1 ], Heller and Reiner [6] (see als o [2, 4.3 .3] for an exp os itory a ccount), for any p ositive in teg er n a nd any λ ∈ k ∪ {∞} there is , up to iso morphism, a unique indeco mp os a ble k V 4 -mo dule M λ n of dimensio n 2 n , having a k -basis { a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n } on which the e le men ts x = s − 1 and y = t − 1 act by xa i = b i (1 ≤ i ≤ n ) , y a i = λb i + b i +1 (1 ≤ i < n ) , y a n = λb n , xb i = 0 = y b i (1 ≤ i ≤ n ) ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GROUP 9 provided that λ ∈ k , and by xa i = b i +1 (1 ≤ i < n ) , xa n = 0 , y a i = b i (1 ≤ i ≤ n ) , y a n = b n , xb i = 0 = y b i (1 ≤ i ≤ n ) if λ = ∞ . Then M λ n , with λ ∈ k ∪ {∞} , is a set of repr esentativ es of the isomorphism classe s of indecomp osable k V 4 -mo dules of dimensio n 2 n . This set can b e indexed b y the elements in P 1 ( k ) with M λ n corres p onding to the image of ( λ, 1) in P 1 ( k ) if λ ∈ k , and with M ∞ n corres p onding to the image of (0 , 1 ) in P 1 ( k ). The natural action of GL 2 ( k ) on P 1 ( k ) translates to a transitive action of GL 2 ( k ) o n the set of isomorphism classes of 2 n -dimensional indecomp osa ble k V 4 - mo dules. The g roup GL 2 ( k ) can be iden tified with a group of k -a lgebra automorphisms of k V 4 , with g =  α β γ δ  acting o n k V 4 by g ( x ) = αx + β y a nd g ( y ) = γ x + δ y . In this wa y , any k V 4 -mo dule M gives rise via restr ic tion along the automor phism g to a k V 4 -mo dule g M , which is equal to M as k -vector spa ce, with z ∈ k V 4 acting as g ( z ) on M . This defines another a ction of GL 2 ( k ) on the set of iso morphism classes of 2 n -dimensional indecomp osable k V 4 -mo dules, and, of course, these actions are compatible: one chec ks that for λ ∈ k we ha ve M λ n ∼ = g ( M 0 n ), where g =  1 0 λ 1  , a nd that M ∞ n ∼ = h ( M 0 n ), where h =  0 1 1 0  . The point of these remarks is that in order to ca lculate (stable) endomo rphism alge br as of the modules M λ n , identify almost v a nishing morphisms and ch eck their inv aria nce under the Heller op era tor, it suffices to consider the ca se λ = 0. W e collect some basic facts on endomorphisms of the mo dules M λ n , many of which ar e, of course, well-kno wn (such as the fact that M λ n has p e rio d 1). In wha t follo ws we set M n = M 0 n , for any positive in teger n . Lemma 5.1. L et n b e a p ositive inte ger and denote by { a 1 , a 2 , .., a n , b 1 , b 2 , .., b n } a k -b asis of M n such that xa i = b i for 1 ≤ i ≤ n , y a i = b i +1 for 1 ≤ i < n , y a n = 0 and xb i = 0 = y b i for 1 ≤ i ≤ n . L et ϕ ∈ End kV 4 ( M n ) . (i) We have Ω( M n ) ∼ = M n . (ii) If ϕ ∈ End pr kV 4 ( M n ) then Im( ϕ ) ⊆ so c( M n ) . (iii) If Im( ϕ ) = k · b 1 then ϕ 6∈ End pr kV 4 ( M n ) . (iv) F or 1 ≤ i ≤ n denote by ϕ n i the endomorph ism of M n sending a i to b 1 and al l other b asis elements of M n to zer o. Then the set { ϕ n i } 1 ≤ i ≤ n is line arly indep endent in End kV 4 ( M n ) . (v) We have dim k (End pr kV 4 ( M n )) = n 2 − n . Pr o of. F or 1 ≤ i ≤ n denote b y s i the elemen t in ( k V 4 ) n whose i -th compone nt is 1 and whose other compo nenets ar e 0. Then { s 1 , s 2 , .., s n } is a basis o f the free k V 4 -mo dule ( k V 4 ) n . There is an injective kV 4 -homomorphis m ι : M n − → ( k V 4 ) n such that ι ( a i ) = y s i + xs i +1 for 1 ≤ i < n and ι ( a n ) = y s n , a nd there is a sur jectiv e k V 4 - homomorphism π : ( k V 4 ) n − → M n such that π ( s i ) = a i for 1 ≤ i ≤ n . One chec ks that Im( ι ) = ker( π ), which shows that Ω( M n ) ∼ = M n , whence (i). If ϕ ∈ End pr kV 4 ( M n ), the inclusion Im( ϕ ) ⊂ rad( M n ) is a general fact, and we hav e rad( M n ) = so c( M n ), whic h shows (ii). Since ϕ factor s thro ugh a pro jective mo dule, it factors through bo th ι a nd π ; thus there is ψ ∈ End kV 4 (( k V 4 ) n ) such that ϕ = π ◦ ψ ◦ ι . 10 MARKUS LINCKELM ANN, RADU S T ANCU F or 1 ≤ i, j ≤ n let α i,j ∈ k a nd r i,j ∈ rad( k V 4 ) = xk V 4 + y k V 4 , viewed as element in the j -th comp onent of ( k V 4 ) n , suc h that ψ ( s i ) = n X j =1 α i,j s j + r i,j Since Im( ι ) ⊂ rad(( kV 4 ) n ) o ne easily sees that the w e ma y assume r i,j = 0. Two sho rt verifica- tions show that ϕ ( a n ) = n X j =2 α n,j − 1 b j and, for 1 ≤ i < n , ϕ ( a i ) = α i +1 , 1 b 1 + n X j =2 ( α i,j − 1 + α i +1 ,j ) b j which also shows ag ain that Im( ϕ ) ⊆ so c( M n ). Mo r eov er , this calculation shows that if Im( ϕ ) ⊆ k b 1 , then the equation for ϕ ( a n ) implies α n,j = 0 for 1 ≤ j < n , and the equations for the ϕ ( a i ) then imply inductively that α i,j = 0 for 1 ≤ j < i ≤ n , which in turn yields ϕ = 0. This prov es (iii). Statemen t (iv) is a n immediate conseq ue nc e of (iii). F or a ny pair ( i, j ), with 1 ≤ i, j ≤ n there is a unique endomorphism of M n sending a i to b j and a ll other basis elements to zero. Thu s the space of endomor phisms o f M n whose image is co ntained in so c( M n ) has dimension n 2 . It follows form (iv) that dim k (End pr kV 4 ( M n )) ≤ n 2 − n . Setting ϕ i,j = π ◦ ψ i,j ◦ ι , where ψ i,j is the unique endomorphism determined b y the c o e fficien ts α i,j = 1 and α r,s = 0 for ( r, s ) 6 = ( i, j ) an easy verification shows that the set { ϕ i,j | 1 ≤ i ≤ n, 1 ≤ j < n } is linearly indep e nden t in End pr kV 4 ( M n ), whence the equality in (v).  Lemma 5.2. F or any p ositive inte ger n denote by { α n 1 , α n 2 , .., α n n , β n 1 , β n 2 , .., β n n } a k -b asis of M n such that xa n i = b n i for 1 ≤ i ≤ n , y a n i = b n i +1 for 1 ≤ i ≤ n − 1 , y a n n = 0 , and su ch t hat xb n i = 0 = y b n i for 1 ≤ i ≤ n . F or any p ositive inte gers m < n ther e is a homomorphism ζ n m : M m → M n such that ζ ( a m i ) = a n n − m + i for 1 ≤ i ≤ m , and a homomorphism ξ n m : M n → M m such that ξ n m ( a n i ) = a m i for 1 ≤ i ≤ m and ξ n m ( a n i ) = 0 for m + 1 ≤ i ≤ n . Then the se quen c e 0 / / M m ζ n m / / M n ξ n n − m / / M n − m / / 0 is exact. In p articular ζ n m is a non-split monomorphism and ξ n m is a non-split epimorphism. Pr o of. Straightforward verification.  Lemma 5 . 3. Le t n b e a p ositive inte ger. The fol lowing hold. (i) The endomorphism ϕ n 1 in End kV 4 ( M n ) is an almost vanishing morphism. (ii) F or any isomorphism α : M n ∼ = Ω( M n ) in mo d ( k V 4 ) We have α ◦ ϕ n 1 = Ω( ϕ n 1 ) ◦ α . Pr o of. Since M n has p erio d 1 , any almost v a nishing morphism starting at M n is an endomo r- phism of M n . Let ψ : M n → M n be a repr esentativ e o f an almost v anishing endomorphism. W e us e the notation of the previo us lemma, except that we drop the super scripts of basis ele- men ts. First compo s e ψ with ζ n +1 n . Given that ψ is a lmost v anis hing and ζ n +1 n is no t an split monomorphism, ζ n +1 n ◦ ψ ha s to b e a zero morphism in mo d ( k V 4 ). But a n y k V 4 -homomorphis m betw een indecomp osable non pr o jective mo dules that facto rs through a pro jective module has its imag e contained in the so cle of the target mo dule. As ζ n +1 n is a monomorphism, it implies ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GROUP 11 that the image of ψ has to b e co n tained in so c( M n ). Using Lemma 5.1 (iv), ψ can b e chosen in the k -vector space generated b y { ϕ n i | 1 ≤ i ≤ n } . So there exist α i ∈ k , 1 ≤ i ≤ n suc h that ψ = P n i =1 α i ϕ n i . The aim is to prove that α i = 0 fo r all 2 ≤ i ≤ n . T o do that w e pre-co mp os e ψ with ζ n j where j = n − i + 1. Again the cla ss of this comp osition has to be a zer o morphism in mo d( k V 4 ). The pro of is by reverse induction on i , which amounts to pro ceeding by induction ov er j = n − i + 1. If j = 1 then ψ ◦ ζ n 1 ( a 1 ) = α n ϕ n n ( a n ) = α n b 1 , where the notation of ba sis elements of M n is a s at the b eginning of the pro of of Lemma 5 .1. Pr e-comp osing with ξ n 1 one gets an endomorphism ψ ◦ ζ n 1 ◦ ξ n 1 of M n that is equal to α n ϕ n 1 . Since the c la ss of this mo r phism has to be 0 in mo d ( k V 4 ) we get that α n = 0. Supp ose now that α i = 0 for all i > n − j + 1. Pre-co mpos e ψ with ζ n j ; as b efore the class o f ψ ◦ ζ n j in mo d( k V 4 ) is zero. By direct computation ψ ◦ ζ n j ( a 1 ) = α n − j +1 ϕ n n − j +1 ( a n − j +1 ) = α n − j +1 b 1 and ψ ◦ ζ n j ( a i ) = α n − j +1 ϕ n n − j + i ( a n − j +1 ) = 0. Pre-co mpos e now with ξ n j and get an endomor phism ψ ◦ ζ n j ◦ ξ n j of M n that is equal to α n − j +1 ϕ n 1 . The class of the latter morphism is zero in mo d( k V 4 ) if a nd o nly if α n − j +1 = 0. The induction pro cedure sto ps at j = n − 1 , or e q uiv a len tly , at i = 2 , whence (i). In order to prove (ii), note first that of the statement holds fo r some isomorphism α it holds for any s uc h isomor phism bec ause End kV 4 ( M n ) is split slo ca l and ϕ n 1 is almost v anishing. W e use the notation intro duced at the b eginning of the pro of of Lemma 5.1; in particula r, we have an injective ho momorphism ι : M n → ( k V 4 ) n and a surjective homomo r phism π : ( k V 4 ) n → M n . Define θ : ( k V 4 ) n → ( k V 4 ) n by θ ( s 1 ) = xs 1 and θ ( s j ) = 0 for 2 ≤ j 6 = n . A straightforward v erification shows that the following diagram comm utes: 0 / / M n ι / / ϕ n 1   ( k V 4 ) n π / / θ   M n / / ϕ n 1   0 0 / / M n ι / / ( k V 4 ) n π / / M n / / 0 This shows that for some - and hence an y - identification M n = Ω( M n ) w e hav e Ω( ϕ n 1 ) = ϕ n 1 , which c ompletes the pro of.  In conjunction with the remarks a t the beginning of this section, this shows that fo r any even dimensional indecompo sable non pro jective k V 4 -mo dule M λ n there exists, up to mult iplication by a sca lar, a uniq ue element of the graded cen ter in any degree that is an almost v a nishing endomorphism of M λ n and zero on all indecomp osa ble mo dules not isomorphic to M λ n . No w let η be a natural tra ns formation belo nging to the kernel I of the cano nical ev aluation map Z ∗ (mo d ( k V 4 )) → ˆ H ∗ ( V 4 ; k ). Then η is zero when ev aluated at the trivial mo dule, hence ze r o at any o dd dimensional inedcompo s able k V 4 -mo dule, as these are pr ecisely the Ω- o rbit of k , up to isomorphism. In order to co nclude the pr o of of Theor em 1.3 we need to show that for any even-dimensional indecompo s able k V 4 -mo dule M the ev aluation η ( M ) is either zer o or almost v a nishing. Tha nks to the remarks at the b eginning of this section, it suffices to show this for the mo dules M n , for n a po sitive integer, and this is what the following lemma do es: Lemma 5 . 4. Le t η ∈ I and n a p ositive inte ger. Then η ( M n ) is a sc alar multiple of ϕ n 1 . Pr o of. The pro of is by induction on n . F or n = 1 the asser tion is trivial. Supp ose that n > 1 and that η M l is a multiple of ϕ l 1 for all 1 ≤ l < n . B y Lemma 5.1, there a r e co efficient s α i ∈ k such that η ( M n ) = P n i =1 α i ϕ n i . W e pr ov e tha t, fo r a n y for an y k such that n ≥ k > 1, if 12 MARKUS LINCKELM ANN, RADU S T ANCU η ( M n ) = P k i =1 α i ϕ n i then α k = 0. The following diag ram in mod( k V 4 ) is commut ative: M n ξ n n − k +1 / / η ( M n )   M n − k +1 η ( M n − k +1 )   M n ξ n n − k +1 / / M n − k +1 By induction η ( M n − k +1 ) is an almos t v anishing mo r phism or zero, so on one hand, η ( M n − k +1 ) ◦ ξ n n − k +1 is zero in mo d ( k V 4 ). On the other hand, a n easy verification shows that ξ n n − k +1 ◦ η ( M n ) ◦ ζ n n − k +1 = α k ϕ n − k +1 1 This forces α k = 0, whence the result.  Combining the a bove lemmas yields a pr o of of Theorem 1.3. 6. Rank 1 F or completeness w e include some rema r ks on the graded cen ter of the stable catego ry of a finite p -g roup P of r ank 1. If k is a field of o dd characteris tic p then P is cyclic, the alge bra k P has finite representation type, and the structure of mo d ( k P ) is calculated in [8] and [7]. If p = 2 and Q is a non cyclic finite 2-gro up of rank 1 then Q is a gener a lised quaternion group. Theorem 6.1. L et k b e an algebr aic al ly close d field of char acteristic 2 and let Q b e a gener alise d quaternion 2 -gr ou p. The kernel I of the c anonic al m ap Z ∗ (mo d ( k Q )) − → ˆ H ∗ ( Q ; k ) given by evaluation at t he t rivial k Q -mo dule is infinite dimensional in o dd de gr e es. The idea of the pro of of Theorem 6.1 is to reduce this to the cas e where Q = Q 8 is quater nion of order 8, and then to show that the 2-dimensiona l mo dules over the Klein four quotient group V 4 of Q 8 yield infinitely many isomorphism classes of k Q 8 -mo dules of p erio d 2 with the prop erty that Ω 2 acts triv ially on the almost v anishing morphisms starting a t these mo dules ; this implies that the a lmo st v anishing morphisms determine infinitely many linearly independent elements in each o dd degree o f the gr aded center of the stable mo dule ca tegory . In what follows, k is a (no t ne c e ssarily algebraica lly closed) field o f characteristic 2 and Q 8 a q uaternion gr oup o f order 8, with gener ators s , t of o rder 4 sa tis fying s 2 = t 2 and tst − 1 = s − 1 . Set z = s 2 ; this is the unique (and hence central) in volution of Q 8 . W e start with a few elementary observ ations. Lemma 6 . 2. W e have k Q 8 ( z − 1) ⊆ Z ( k Q 8 ) . Pr o of. W e hav e s ( z − 1) = s 3 + s ; this is the conjugacy class sum of s , hence contained in Z ( k Q 8 ). The sa me argument applies to all elements of order 4 in Q 8 , whence the result.  Lemma 6.3. L et ( λ, µ ) ∈ k 2 and set x = λ ( s − 1 ) + µ ( t − 1) . S upp ose that λ 2 + λµ + µ 2 6 = 0 . Then 1 + x is invertible of or der 4 in ( k Q 8 ) × , and k Q 8 is pr oje ctive as left or right k h 1 + x i -mo dule. Pr o of. W e have x 2 = ( λ 2 + µ 2 )( z − 1 ) + λµ ( s − 1 )( t − 1 ) + λµ ( t − 1 )( s − 1 ). A sho r t calcula tio n shows that ( s − 1 )( t − 1) + ( t − 1)( s − 1) = st ( z − 1). This is a n e le men t in Z ( k Q 8 ), by L e mma 6.2. Thus x 2 = u ( z − 1), where u = ( λ 2 + µ 2 ) · 1 + λµ · ( st ). The hypotheses on λ a nd µ imply that u is in vertible. Thus h 1 + x i has order 4 and Lemma 3.1 implies that k Q 8 is pro jective as k h 1 + x 2 i mo dule. Since h 1 + x 2 i is the unique subgr oup of order 2 of h 1 + x i it follows that k Q 8 is pro jective a s left or right k h 1 + x i -mo dule.  ON THE GRADED CENTER OF THE ST ABLE CA TEGOR Y OF A FINITE p -GROUP 13 The hypothesis λ 2 + λµ + µ 2 6 = 0 in the pr evious lemma means that at leat one of λ , µ is nonzero, and if they ar e both no nzero then λ µ is not a cub e ro ot of unity . Prop ositio n 6. 4. L et ( λ, µ ) ∈ k 2 and set x = λ ( s − 1) + µ ( t − 1) . Supp ose that λ 2 + λµ + µ 2 6 = 0 . Set M = k Q 8 ( z − 1) x and N = k Q 8 x . Then M , N ar e absolutely inde c omp osable, and the fol lowing hold. (i) We have isomorphisms α : M ∼ = Ω 2 ( M ) and β : Ω( M ) ∼ = N in mo d ( k Q 8 ) . (ii) We have M ⊆ N h z i = k Q 8 ( z − 1) . (iii) F or any ϕ ∈ Hom kQ 8 ( M , N ) ther e is an element c ∈ k Q 8 such that ϕ ( m ) = cm for al l m ∈ M ; in p articular, Im( ϕ ) ⊆ M . (iv) S et γ = Ω( β ) ◦ Ω( α ) ◦ β − 1 : N ∼ = Ω 2 ( N ) . Then, for any ϕ ∈ Hom kQ 8 ( M , N ) we have Ω 2 ( ϕ ) ◦ α = γ ◦ ϕ . Pr o of. The mo dules M , N a re cy c lic mo dules ov er the split lo cal a lgebra k Q 8 , hence absolutely indecomp osable. Since x 2 = u ( z − 1) for some u ∈ ( kQ 8 ) × we have x 2 ( z − 1) = 0. Thus N = k Q 8 x is con tained in the kernel of the surjective homomor phism k Q 8 → M given by right m ultiplication with ( z − 1) x , hence equal to this kernel b ecause b oth hav e dimensio n 6. The same argument shows that the endomor phism k Q 8 → k Q 8 given by r ight multiplication with x ha s kernel equal to M . This prov es that for this choice of pro jective covers of M and N we get Ω( M ) = N and Ω 2 ( M ) = M , whence (i). The inclusion M ⊆ N is trivial. Since z − 1 annihilates M , w e ha ve M ⊆ N h z i ⊆ k Q 8 ( z − 1), where the second inclusion uses the fa ct that k Q 8 is pro jective as right k h z i -mo dule. Cle arly also Im( ϕ ) ⊆ N h x i . Now k Q 8 is pro jective as a righ t k h 1 + x i -mo dule b y 6.3, th us kQ 8 is free of rank 2 as a k h 1 + x i -mo dule. It follows that k Q 8 x , when viewed as r ight k h 1 + x i -module, is isomorphic to the direct sum o f tw o c o pies of the unique (up to isomorphism) 3-dimensiona l k h 1 + x i -mo dule. The restr iction of this 3- dimensional mo dule to h 1 + x 2 i is of the form k ⊕ k h 1 + x 2 i , so the s pace annihilated b y x 2 has dimension 2 + 2 = 4. Since x 2 = u ( z − 1) for some inv er tible element u it follows that the subspace of N annihilated by z − 1 has also dimension 4, hence is equal to k Q 8 ( z − 1). This prov es (ii). Let ϕ ∈ Ho m kQ 8 ( M , N ). It follows fr o m (ii) that Im( ϕ ) ⊆ k Q 8 ( z − 1). Thus we may view ϕ as a homomorphism from the submo dule M o f k Q 8 ( z − 1 ) to k Q 8 ( z − 1 ). Both mo dules are annihilated by z − 1, so we may view this as a homomorphism of k V 4 -mo dules, where V 4 = Q 8 / h z i . But since M is a submo dule of k Q 8 ( z − 1) ∼ = k V 4 as k V 4 -mo dules and since k V 4 is commut ative it follows that ϕ is actually induce d b y left multiplication with an element in k V 4 . Pulling this ba ck to k Q 8 shows tha t ϕ , as a homomorphism of k Q 8 -mo dules, is induced by left m ultiplication o n M with some elemen t c ∈ k Q 8 , comp osed with the inclusion M ⊆ N . This prov es (iii). In order to prov e (iv), we cho ose notation such that α , β ar e e q ualities (that is, we identify N = Ω( M ) and M = Ω 2 ( M ) in the obvious wa y describ ed at the b eginning of this pro of ), and consider a commut ative diag ram of k Q 8 -mo dules of the form 0 / / M ι / / ϕ ′   k Q 8 x / / v   k Q 8 x ( z − 1) / / w   M / / ϕ   0 0 / / N κ / / k Q 8 x ( z − 1) / / k Q 8 x / / N / / 0 where ι , κ are the inclusions, v , w a re suitable elemen ts in k Q 8 and wher e an ar row la b elled by an element in k Q 8 means the homomor phism induced by right multiplication with that 14 MARKUS LINCKELM ANN, RADU S T ANCU element. W e need to show ϕ ′ = ϕ . F o r a ∈ k Q 8 , the co mm uta tivity of the right squa r e in this diagram means that ϕ ( ax ( z − 1 )) = aw x . The commutativit y o f the middle square is equiv ale n t to axw = av x ( z − 1), hence xw = v x ( z − 1 ). The c ommu tativity of the left square implies that ϕ ′ ( ax ( z − 1)) = ax ( z − 1) v = av x ( z − 1 ) = axw , whe r e we used that x ( z − 1) ∈ Z ( k Q 8 ) b y Lemma 6.2. In order to show that ϕ = ϕ ′ it suffices to show that w can b e c ho sen in Z ( k Q 8 ). By (iii) there is an element c ∈ k Q 8 such that ϕ ′ ( x ( z − 1)) = cx ( z − 1 ) = xc ( z − 1 ), and hence we ma y take w = c ( z − 1), which is a n element in Z ( k Q 8 ) by Lemma 6.2.  Pr o of of The or em 6.1. W e use the notation a nd a ssumptions of Theo rem 6 .1; in particula r , k is now algebraica lly clos ed. W e consider fir st the case wher e Q = Q 8 . It fo llows from Prop ositio n 3 .3 applied to the Klein four group V 4 = Q 8 / h z i and fro m Pro po s ition 6.4 (i) that if ( λ, µ ) ∈ k 2 runs ov er a set of r epresentativ e s of ele ments in P 1 ( k ) sa tisfying λ 2 + λµ + µ 2 6 = 0 then M λ,µ = k Q 8 ( z − 1)( λ ( s − 1 ) + µ ( t − 1 )) runs ov er pairwise noniso morphic indecompo sable k Q 8 - mo dules of pe rio d 2. It follows fro m Prop osition 6.4 (iv) that the almost v a nishing morphisms starting at these mo dules determine elements in the gr aded c e n ter of the stable mo dule category of k Q 8 in any o dd degre e , whence the result in the case Q = Q 8 . F or Q a gener alised quaternio n 2-gro up w e may identify Q 8 to a subg roup of Q , and then the induction functor Ind Q Q 8 sends the indecompo sable mo dules of p e rio d 2 considered in Prop ositio n 6.4 to indecomp osable k Q - mo dules of perio d 2. Mo reov er , it follows from Pr op osition 6.4 (iv) in conjunction with Lemma 2.4 that the almost v anishing morphisms star ting at these mo dules a re aga in in v ar iant under Ω 2 kQ , hence define elements in the graded cen ter of the stable catego ry of kQ - mo dules .  References [1] V. A. Ba ˇ sev, R epr ese nt ations of the gr oup Z 2 × Z 2 in a field of char acteri stic 2, (Russi an), Dokl. Ak ad. Nauk. SSSR 141 (1961) , 1015–1018. [2] D. J. Benson, R epr esent ati ons and Cohomolo gy , V ol. I: Cohomolo gy of gr oups and mo dules. Cambridge Studies in Adv anced Mathematics 30, Cam bri dge U niv ersity Press (1991) . [3] D. J. Benson, Pr iv ate comm unication. [4] K. Erdmann Blo cks whose defe ct gr oups ar e Klei n four gr oups: a co rr e ction , J. Al gebra 76 (1982), 505–518. [5] D. Happ el, T riangulate d c ate g ories in the r epr esentation the ory of finite-dimensional algebr as , London Math. Soc. 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