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THE HOCHSCHILD HOMOLOGY OF A(1). 6 ANDREWSALCH 1 0 2 Abstract. WecomputetheHochschildhomologyofA(1),thesubalgebraofthe2-primary SteenrodalgebrageneratedbythefirsttwoSteenrodsquares,Sq1,Sq2. Thecomputation n isaccomplishedusingseveralMay-typespectralsequences. u J 9 2 Contents ] T 1. Introduction. 1 A 2. ConstructionofMay-typespectralsequencesforHochschildhomology. 4 . 3. TheMayandabelianizingfiltrations. 9 h t 4. RunningtheHH-Mayandabelianizingspectralsequences. 11 a 4.1. Input. 11 m 4.2. d -differentials. 14 1 [ 4.3. d2-differentials. 16 2 References 19 v 7 4 5 2 1. Introduction. 0 The2-primarySteenrodalgebraA,thatis,thealgebraofstablenaturalendomorphisms . 1 ofthemod2cohomologyfunctorontopologicalspaces,hasgeneratorsSq1,Sq2,Sq3,..., 0 the Steenrod squares. The subalgebra of A generated by the first two Steenrod squares, 5 Sq1 and Sq2, is called A(1), and A(1) is an eight-dimensional, graded, noncommutative 1 : (not even graded-commutative), co-commutative Hopf algebra over F2. The homologi- v calalgebraof A(1)-moduleseffectivelydetermines,viatheAdamsspectralsequence,the i X 2-completehomotopytheoryof spaces and spectra smashed with the connective real K- r theoryspectrumko.Theseideasareallclassical;anexcellentreferencefortheSteenrodal- a gebraisSteenrod’sbook[12],andanexcellentreferenceforA(1)-modulesandtheAdams spectralsequenceisthethirdchapterofRavenel’sbook[11]. Asastudentofhomotopytheory,whenIfirstlearnedthedefinitionofHochschildho- mologyofalgebras,myfirstreactionwastotrytocomputetheHochschildhomologyof A(1). I knowat least three otherhomotopytheoristswho havetold me that theyhad the same reaction when learning aboutHochschildhomology! Computing HH (A(1),A(1)), ∗ however,isanontrivialtask,anditseemsthatthiscomputationhasneverbeensuccessfully done1. Date:January2015. 1Bo¨kstedt, inhisextremely influential unpublished paperontopological Hochschild homology, computes the Hochschild homology ofπ∗(HFp∧HFp), i.e., the Hochschild homology ofthe linear dual ofthe entire Steenrodalgebra,butthisisverystraightforward,sincethedualoftheSteenrodalgebraispolynomialatp=2 andpolynomialtensoredwithexterioratp>2. Forthesamereason,itisalsoeasytocomputetheHochschild 1 2 ANDREWSALCH In this paper we compute HH (A(1),A(1)) by using two different filtrations on A(1) ∗ andstudyingthespectralsequencesinHochschildhomologyarisingfromthesefiltrations. ThesespectralsequencesaretheanaloguesinHochschildhomologyofJ.P.May’sspectral sequence for computingExt over the Steenrod algebra (see [7]), so we think of these as “May-type”spectralsequences. The problem of computing HH (A(1),A(1)) is made rather difficult by the fact that ∗ A(1)isnoncommutativeandsoHH (A(1),A(1))doesnotinheritaproductfromtheshuf- ∗ fle product on the cyclic bar complex, and as a consequence, the May-type spectral se- quenceconvergingto HH (A(1),A(1))thatonewouldconstructinthemostna¨ıvewayis ∗ not multiplicative, i.e., it does not have a productsatisfying a Leibniz rule. This makes thecomputationofdifferentialsinthatspectralsequencebasicallyintractable.Instead,we takethelineardualofthestandardHochschildchaincomplexonA(1),andweusetheco- commutativecoproductonA(1)togivethecohomologyofthislineardualcochaincomplex aproductstructurearisingfromthecoproductonA(1)andthelineardualoftheAlexander- Whitney map. In Proposition 4.3 we set up multiplicative spectralsequencescomputing thecohomologyofthelineardualcochaincomplexofthestandardHochschildchaincom- plex of A(1). By an easy universal coefficient theorem argument (Proposition 2.5), this cohomologyistheF -lineardualofthedesiredHochschildhomologyHH (A(1),A(1)). 2 ∗ We then compute the differentials in these spectral sequences. In the end there are nonzero d and d differentials, and no nonzero differentials on any later terms of the 1 2 ∼ spectralsequences. In4.2.1and4.3.1wepresentchartsofthe E andE = E -pagesof 2 3 ∞ thetherelevantspectralsequences. OurchartsaredrawnusingtheusualAdamsspectral sequenceconventions,describedbelow.Thisisthemostconvenientformatif,forexample, onewantstousethisHochschildhomologyastheinputforanAdamsspectralsequence, and it also makes it easier to see the natural map from this Hochschild homologyto the classical Adams spectral sequence computing π (ko) , the 2-complete homotopygroups ∗ 2 of the connective real K-theory spectrum ko, in Probposition 4.4 and in the charts 4.3.1 and4.3.2. Inparticular,thechart4.3.1isachartofthe(F -lineardualofthe)Hochschildhomology 2 of A(1), and gives our most detailed description of HH (A(1),A(1)). We reproduce that ∗ charthere: 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 homologyofthelineardualofA(1). ButthelineardualofA(1)isacompletelydifferentringthanA(1),sothis shedsnolightontheHochschildhomologyofA(1)itself! THEHOCHSCHILDHOMOLOGYOFA(1). 3 Theverticalaxisishomologicaldegree,so therow srowsabovethe bottomofthechart istheassociatedgradedF -vectorspaceofafiltrationonHH (A(1),A(1)).Thehorizontal 2 s axisis,followingthetraditioninhomotopytheory,theAdamsdegree,i.e.,thetopological degree(comingfromthetopologicalgradingonA(1))minusthehomologicaldegree.The horizontal lines in the chart describe comultiplications by certain elements in the linear dual Hopf algebra homF2(A(1),F2) of A(1), and the nonhorizontallines describe certain operations in the linear dual of HH (A(1),A(1)), described in Convention 4.6. The en- ∗ tirepatterndescribedbythischartisrepeatedeveryfourverticaldegreesandeveryeight horizontaldegrees:thereisaperiodicityclass(notpictured)inbidegree(4,8). Informationabout the F -vector space dimension of HH (A(1),A(1))in each grading 2 ∗ degreeisprovidedbyTheorem4.10,whichwereproducebelow(wedonotdescribeany ringstructureonHH (A(1),A(1))becauseA(1)isnoncommutativeandsothereisnoring ∗ structureonitsHochschildhomologyinducedbyshuffleproductoncyclicchains): Theorem. TheF -vectorspacedimensionofHH (A(1),A(1))is: 2 n 2n+5 if2|n dimF2 HHn(A(1),A(1))= 22nn++46 iiffnn≡≡31 mmoodd 44. HencethePoincare´seriesofthegradedF-vectorspaceHH (A(1),A(1))is 2 ∗ 5+8s+9s2+10s3+ 8s4 1−s. 1−s4 IfweadditionallykeeptrackoftheextragradingonHH (A(1),A(1))comingfromthe ∗ topological grading on A(1), then the Poincare´ series of the bigraded F -vector space 2 HH (A(1),A(1))is ∗,∗ (1+u) 1+u2+su(1+u2+u5)+s2u2(1+2u5+u7)+s3u3(1+u4+u5+u6+u9) s4u4(1+u4+u5+u9) 1 + +u6+su2+su8+s2u4 1−su ! !1−s4u12 wheresindexesthehomologicalgradinganduindexesthetopologicalgrading. Our computation of HH (A(1),A(1)) can be used as the input for other spectral se- ∗ quences in order to make further computations. For example, one could use it as input for the Connes spectral sequence, as in 9.8.6 of [13], computing the cyclic homology HC (A(1),A(1)).Thisisprobablyoflimitedutility,however,sinceA(1)isanalgebraover ∗ afieldofcharacteristic2,sothecyclichomologyof A(1)isprobablynotagoodapprox- imationto the algebraic K-theoryof A(1). Insteadone oughtto computethe topological cyclichomologyofA(1).Forthis,onecoulduseourcomputationofHH (A(1),A(1))asin- ∗ putforthePirashvili-Waldhausenspectralsequence,asin[10],computingthetopological Hochschild homologyTHH (A(1),A(1)), and then one would need to run the necessary ∗ homotopy fixed-point spectral sequences to compute π∗(THH(A(1),A(1))Cpn) and then TR (A(1)) and finally TC (A(1)), which, using McCarthy’s theorem (see [3]), gives the ∗ ∗ 2-completealgebraic K-groups K (A(1)) (the algebraic K-groupscompletedaway from ∗ 2 2oughtto bemucheasier: anappropriabtenoncommutativeversionofGabberrigidity,if true, wouldimplythattheyvanishinpositivedegrees. See IV.2.10of[14]). See e.g.[5] fora surveyoftrace methodcomputationsof thiskind. Those computationsare entirely outsidethescopeofthepresentpaper. 4 ANDREWSALCH We remark that our methods also admit basically obvious extensions to methods for computingHH (A(n),A(n))forarbitraryn,butoneseesthatforn > 1,carryingoutsuch ∗ computationswouldbeadauntingtask. OurHH-MayspectralsequenceofProposition4.3 surjects on to the classical May spectral sequence computingExt∗ (F ,F ), and for the A(1) 2 2 samereasons,then> 1analogueofourHH-Mayspectralsequencemapsnaturallytothe classicalMayspectralsequencecomputingExt∗ (F ,F ). Wesuspectthatthismapisstill A(n) 2 2 surjectiveforn > 1, althoughwe havemadenoattemptto verifythis. Consequentlythe computationofHH (A(n),A(n))usingourmethodsisofatleastthesamelevelofdifficulty ∗ asthecomputationofExt∗ (F ,F ). Forn=2thisisalreadyquitenontrivial. A(n) 2 2 Itisworthpointingoutthat,in[2],J.-L.Brylinskiconstructsaspectralsequencerelated tothepresentpaper’sabelianizingspectralsequence. Brylinski’sspectralsequencecom- putestheHochschildhomologyofanoncommutativealgebraoverafieldofcharacteristic zero,forthepurposesofstudyingPoissonmanifolds,andremarksthat“[e]xamplesshow thatthisspectralsequencetendstodegenerateatE2,”inparticular,thatBrylinski’sspectral sequencecollapsesatthe E2-termforthealgebraofdifferentialoperatorsonanalgebraic orcomplex-analyticmanifold. Inthepresentpaper’scomputationof HH (A(1),A(1))we ∗ insteadgetcollapseonetermlater,atE ratherthanE . 3 2 Remark 1.1. Given a field k and a graded k-algebra A, there are two notions of the HochschildhomologyHH (A,A)whichareincommoncirculation: ∗ • onecanforgetthegrading,andsimplyconsidertheHochschildhomologyofthe underlyingungradedk-algebraof A. Thisis the rightthingto doin manyappli- cationsofHochschildhomologyinclassicalalgebra;forexample,ifonewantsto usetracemethodstocomputethealgebraicK-groupsof(theunderlyingungraded algebraof)A. (Thecomputationsin[4]areanexcellentexampleofthis.) • Alternatively, one can instead compute the “graded algebra Hochschild homology,” which incorporatesa sign conventioninto the cyclic bar complex. This is the right thing to do in many applications of Hochschild homologyin algebraic topology;forexample,if onewantsto use Bo¨kstedt’sspectralsequenceto compute topological Hochschild homology of a ring spectrum. (The computations in[8]areanexcellentexampleofthis.) SinceA(1)hascharacteristic2,signconventionsareirrelevant,andbothnotionsofHochschild homologycoincide.Thismakesthecomputationsinthispaperequallyapplicableinclassi- calalgebraasinalgebraictopology.Anyfutureodd-primaryanaloguesofthesecomputa- tions,however,wouldrequirethatonecarryout“ungraded”HH computationsseparately ∗ fromthe“graded”HH -computations. ∗ 2. ConstructionofMay-typespectralsequencesforHochschildhomology. Proposition2.1. (MayspectralsequenceforHochschildhomology.)Letkbeafield,A analgebra,and (2.0.1) F0A⊇ F1A⊇ F2A⊇... afiltrationofAwhichismultiplicative,thatis,if x∈ FmAandy∈ FnA,thenxy∈ Fm+nA. Thenthereexistsaspectralsequence Es,t ∼= HH (E A,E A)⇒ HH (A,A) 1 s,t 0 0 s ds,t :Es,t → Es−1,t+r. r r r ThebigradingsubscriptsHH areasfollows: sistheusualhomologicaldegree,whiletis s,t theMaydegree,definedandcomputedasfollows:givenahomologyclassx∈HH (E0A,E0A), s THEHOCHSCHILDHOMOLOGYOFA(1). 5 itsMaydegreeisthetotaldegree(inthegradingonE0AinducedbythefiltrationonA)of anyhomogeneouscyclerepresentativeforxinthestandardHochschildchaincomplex. Thisspectralsequenceenjoysthefollowingadditionalproperties: • If the filtration 2.0.1 is finite, i.e., FnA = 0 for some n ∈ N, then the spectral sequenceconvergesstrongly. • IfAisalsoagradedk-algebraandthefiltrationlayersFnAaregenerated(astwo- sidedideals)byhomogeneouselements, thenthisspectralsequenceisaspectral sequenceofgradedk-vectorspaces,i.e.,thedifferentialpreservesthegrading. • IfAiscommutative,thensoisE A,andtheinputforthisspectralsequencehasa 0 ringstructuregivenbytheusualshuffleproductontheHochschildhomologyofa commutativering(seee.g.[13]),andthespectralsequenceismultiplicative,i.e., the differentialsin the spectral sequenceobey the gradedLeibniz rule. Further- more, the product in the spectral sequence converges to the product induced on theassociatedgradedbytheusualshuffleproductonHH (A,A)(thisistheusual ∗ situationinspectralsequencesofdifferentialgradedalgebras). • Thedifferentialinthespectralsequenceis(likeanyotherspectralsequenceofa filteredchaincomplex)computedonaclassx∈HH (E A,E A)bycomputinga ∗,∗ 0 0 homogeneouscyclerepresentativeyfor xinthestandardHochschildchaincom- plexforE A,liftingytoahomogeneouschainy˜inthestandardHochschildchain 0 complexfor A,applyingtheHochschilddifferentiald toy˜,thentakingtheimage ofdy˜inthestandardHochschildchaincomplexforE A. 0 Proof. LetCH (A,A)denotethestandardHochschildchaincomplexofA,andletFnCH (A,A) • • denotethe sub-chain-complexofCH (A,A)consistingof allchainsoftotalfiltrationde- • gree≤ n. OurMayspectralsequenceisnowsimplythe spectralsequenceofthefiltered chaincomplex (2.0.2) CH (A,A)=F0CH (A,A)⊇ F1CH (A,A)⊇ F2CH (A,A)⊇.... • • • • If A is commutative,then filtration 2.0.1being multiplicativeimpliesthat filtration 2.0.2 is a multiplicative filtration of the differential graded algebra CH (A,A), with product • given by the shuffle product. It is standard (see. e.g. [6]) that the spectral sequence of amultiplicatively-filteredDGAismultiplicative. The producton the spectral sequencebeing givenby the shuffle productis due to the naturalityoftheconstructionofCH (A,A)inthechoiceofk-algebraA: ifAiscommuta- • tive,thenthemultiplicationmapA⊗ A→ Aisamorphismofk-algebras,hencewegeta k mapofchaincomplexes CH (A⊗ A,A⊗ A)→CH (A,A), • k k • whichwecomposewiththeEilenberg-Zilber(i.e.,“shuffle”)isomorphism ∼ = CH (A,A)⊗ CH (A,A)−→CH (A⊗ A,A⊗ A). • k • • k k Theotherclaimsarealsoallstandardaboutaspectralsequenceofafilteredchaincom- plex,exceptperhapsthestrongconvergenceclaim,whichweeasilyproveasfollows:sup- pose the filtration 2.0.1 satisfies FnA = 0 for some n ∈ N. Then the group of i-cycles CHi(A,A) ∼= A⊗ki+1 hasnononzeroelementsoffiltrationgreaterthan(n+1)i. Sothefil- tration in E of HH(A,A) is a finite filtration, and the column in the spectral sequence ∞ i converging to HH(A,A) is zero after the E -page. So the spectral sequence con- i (n+1)i+2 vergesstrongly. (cid:3) 6 ANDREWSALCH Definition2.2. Let k be a fieldand A a coalgebraover k with comultiplicationmap ∆ : A → A⊗ A. By the cyclic cobar construction on A we mean the cosimplicial k-vector k space // // oo // oo // A oo // A⊗k A oo //// A⊗k A⊗k A oooo //... // withcofacemapsd0,d1,...,dn : A⊗kn →A⊗kn+1givenby a ⊗...⊗a ⊗∆(a)⊗a ⊗...⊗a 0 i−1 i i+1 n−1 di(a0⊗...⊗an−1)= iτiff(ii∆<=(ann0,,)⊗a1⊗...⊗an−1) whereτisthecyclicpermutationtowardtheleft,i.e., τ(a ⊗...⊗a )=a ⊗...⊗a ⊗a . 0 n 1 n 0 Thecodegeneracymapsareconstructedfrom thecounit(augmentation)mapon A inthe usualway. BythecycliccobarcomplexofA,denotedcoCH•(A,A),wemeanthealternatingsign cochain complex of the cyclic cobar construction on A. We write coHH∗(A,A) for its cohomology,whichwecalldualHochschildcohomology. Remark 2.3. Clearly, if B is a finite-dimensional co-commutative Hopf algebra over a fieldk,andifAisthek-lineardualofB,thenthecycliccobarcomplexonAisisomorphic to the k-linear dual of the cyclic bar complex on A, and the cyclic cobar complex then inheritsaproductfromthecoproductonthecyclicbarcomplexofB. It is worthwhile to be very explicit about the coproducton the cyclic bar complex of a co-commutative Hopf algebra, since this point is not as well documented in the liter- ature as it could be (although it does appear in some places, for example, in [1]). A Hochschild n-chain is an element of A⊗ A⊗ ...⊗ A, with n+1 tensor factors of A; k k k because the tensor factor on the far left plays a special role, we will adopt the common notationa [a ⊗...⊗a ]insteadofa ⊗a ⊗...⊗a fora(termina)Hochschildn-chain. 0 1 n 0 1 n Thenthecoproductonthecyclicbarcomplexisgivenby ℓ n ∆(a [a ⊗...⊗a ])= a′ a ⊗...⊗a ⊗a′′ a ⊗...⊗a 0 1 n 0,i 1 j 0,i j+1 n Xi=1 Xj=0 h i h i where ℓ and {a′ ,a′′} are given by the determined by the coproduct in A and the 0,i 0,i i=1,...,ℓ formula ℓ ∆(a )= a′ ⊗a′′. 0 0,i 0,i Xi=0 WenowgiveadefinitionofdualHochschildcohomologywithcoefficientsinthebase fieldk, ratherthaninthecoalgebraAitself. Naturally,onecouldwritedownadefinition ofdualHochschildcohomologywithcoefficientsinany“A-bicomodule,”inawaythatis basicallyobviousonceonehastakenaglanceatDefinitions2.2and2.4. Forthepurposes ofthispaper,however,wewillonlyeverneedcoefficientsinkandinA. Definition2.4. LetkbeafieldandAacopointedcoalgebraoverk,i.e.,acoalgebraover kequippedwithamorphismofk-coalgebrasη : k → A. Bythecycliccobarconstruction onAwithcoefficientsinkwemeanthecosimplicialk-vectorspace // // oo // oo // k oo //k⊗k A oo ////k⊗k A⊗k A oooo //... // THEHOCHSCHILDHOMOLOGYOFA(1). 7 withcofacemapsd0,d1,...,dn : A⊗kn →A⊗kn+1givenby η˜ (a )⊗a ⊗...⊗a R 0 1 n−1  ifi=0, di(a0⊗...⊗an−1)= iτaf0(0η˜⊗<(.a.i.<)⊗⊗na,ai−1⊗⊗.∆..(a⊗i)a⊗ai)+1⊗...⊗an−1  ifi=L n0, 1 n−1 ∼= whereη˜ :k→k⊗ AisηcomposedwiththeusualisomorphismA−→k⊗ Asendingato R k k ∼ = 1⊗a,whereη˜ :k →A⊗ kisηcomposedwiththeusualisomorphismA−→A⊗ ksending L k k a to a⊗1, and where τ is as in Definition2.2. The codegeneracymaps are constructed fromthecounit(augmentation)maponAintheusualway. BythecycliccobarcomplexofAwithcoefficientsink,denotedcoCH•(A,k),wemean thealternatingsigncochaincomplexofthecycliccobarconstructiononAwithcoefficients ink.WewritecoHH∗(A,k)foritscohomology,whichwecalldualHochschildcohomology withcoefficientsink. Proposition2.5. LetkbeafieldandletAbeak-algebrawhichisfinite-dimensionalasa k-vectorspace. LetA∗ denotethek-lineardualcoalgebraofA. Then,foreachn ∈ N,the nthHochschildhomologyk-vectorspaceofAandthenthdualHochschildcohomologyk- vectorspaceofA∗aremutuallyk-linearlydual.Thatis,wehaveisomorphismsofk-vector spaces: (2.0.3) hom (HH (A,A),k)∼=coHHn(A∗,A∗), k n (2.0.4) hom (coHHn(A∗,A∗),k)∼=HH (A,A), k n aswellasisomorphisms (2.0.5) hom (HH (A,k),k)∼=coHHn(A∗,k), k n (2.0.6) hom (coHHn(A∗,k),k)∼=HH (A,k). k n Proof. By construction, the cyclic cobar construction is simply the k-linear dual of the usualcyclicbarconstruction,sobytheuniversalcoefficienttheorem(initsformforchain complexes),wehaveisomorphisms coHHn(A∗,A∗)∼= Hn(coCH•(A,A)) ∼= Hn(hom (CH (A,A),k)) k • ∼ =hom (H (CH (A,A)),k) k n • ∼=hom (HH (A∗,A∗),k), k n givingusisomorphism2.0.3. Finite-dimensionalityof A asa k-vectorspaceimpliesthat CH (A,A) is finite-dimensionalas a k-vectorspace, hence the doubledualofCH (A,A) n n recoversCH (A,A) again, giving us isomorphism 2.0.4. Essentially the same argument n givesisomorphisms2.0.5and2.0.6. (cid:3) Proposition2.6. (MayspectralsequencefordualHochschildcohomology.)Letkbea fieldandletAbeak-coalgebra.Let (2.0.7) F A⊆ F A⊆ F A⊆... 0 1 2 8 ANDREWSALCH beafiltrationofAwhichiscomultiplicative,thatis,if x ∈ F A,then∆(x) ∈ m F A⊗ m n=0 n Fm−nA. Thenthereexistsaspectralsequence P Es,t ∼=coHHs,t(E0A,E0A)⇒coHHs(A,A) 1 ds,t :Es,t →Es+1,t−r. r r r ThebigradingsuperscriptscoHHs,t are asfollows: sisthe usualcohomologicaldegree, while t is the May degree, defined and computed as follows: given a cohomology class x ∈ coHHs(E0A,E0A),itsMaydegreeisthetotaldegree(inthegradingonE0Ainduced bythefiltrationonA)ofanyhomogeneouscocyclerepresentativeforxinthecycliccobar complex. Thisspectralsequenceenjoysthefollowingadditionalproperties: (1) If the filtration 2.0.7 is finite, i.e., F A = 0 for some n ∈ N, then the spectral n sequenceconvergesstrongly. (2) IfAisalsoagradedcocommutativek-coalgebraandthefiltrationlayersF Aare n generated (as two-sided coideals) by homogeneous elements, then this spectral sequence is a spectral sequence of graded k-vector spaces, i.e., the differential preservesthegrading. (3) IfAistheunderlyingcoalgebraofthek-lineardualHopfalgebraofacommutative HopfalgebraBoverk,andthefiltration2.0.7isafiltrationbyHopfideals,then E0A is is also a commutative Hopf algebra, and the E -term and the abutment 1 of the spectral sequence each have a natural ring structure. Furthermore, the spectralsequenceismultiplicative,i.e., the differentialsinthe spectralsequence obeythegradedLeibnizrule,andtheproductinthespectralsequenceconverges totheproductontheabutment,moduloexoticmultiplicativeextensions(thisisthe usualsituationinspectralsequencesofdifferentialgradedalgebras). (4) Thedifferentialinthespectralsequenceis(likeanyotherspectralsequenceofa filtered cochain complex) computed on a class x ∈ coHH∗,∗(E0A,E0A) by com- putingahomogeneouscocyclerepresentativeyfor xinthecycliccobarcomplex for E0A,liftingytoahomogeneouscochainy˜ inthecycliccobarcomplexfor A, applying the cyclic cobar differential d to y˜, then taking the image of dy˜ in the cycliccobarcomplexforE0A. Proof. Thisis, ofcourse, allformallydualto Proposition2.1. Theonlythingthatneeds some explanation is the ring structure. The underlyingfiltered DGA of this spectral se- quencehasaringstructuregivenbythecomposite ∼ = coCH•(A,A)⊗ coCH•(A,A)−→(CH (B,B))∗⊗ (CH (B,B))∗ k • k • ∼ = −→(CH (B,B)⊗ CH (B,B))∗ • k • −→≃ (CH (B⊗ B,B⊗ B))∗ • k k −∆→∗ CH (B,B)∗ • ∼ = −→coCH•(A,A), where the map marked ≃ is the k-linear dual of the Alexander-Whitney map, the map marked∆∗ isthek-lineardualofCH appliedtothecomultiplicationmapon B(whichis • well-defined,sinceCH isfunctorialonk-algebramapsandsinceBisassumedcocommu- • tative,sothatitscomultiplicationisak-algebramorphism).Therestisformal. (cid:3) THEHOCHSCHILDHOMOLOGYOFA(1). 9 Proposition2.7. (MayspectralsequencefordualHochschildcohomology,withcoeffi- cientsinthebasefield.)LetkbeafieldandletAbeak-coalgebra.SupposeAisequipped withacomultiplicativefiltrationasin2.0.7.Thenthereexistsaspectralsequence Es,t ∼=coHHs,t(E0A,k)⇒coHHs(A,k) 1 ds,t :Es,t →Es+1,t−r. r r r ThebigradingsuperscriptscoHHs,tareasinProposition2.6. Thisspectralsequenceenjoysproperties1,2,and4fromProposition2.6. Proof. EssentiallyidenticaltoProposition2.6. (cid:3) 3. TheMayandabelianizingfiltrations. We aim to compute HH (A(1),A(1)),the Hochschildhomologyof A(1). By Proposi- ∗ tion 2.5, this amounts to computing coHH∗(A(1)∗,A(1)∗), and then taking the F -linear 2 dual.Wenowgoaboutdoingthis. Definition 3.1. Recall that the May filtration on A(1) is the filtration by powers of the augmentationideal I. We write F˙n(A(1))forthe nthfiltrationlayerin thisfiltration,i.e., F˙nA(1) = In, andwewrite E˙ A(1)fortheassociatedgradedF -algebra. If x ∈ A(1),we 0 2 sometimeswrite x˙fortheassociatedelementinE˙ A(1). 0 Proposition3.2. TheF -algebraE˙ A(1)isthegradedF -algebrawithgeneratorsS˙q1and 2 0 2 S˙q2ingradingdegrees1and2,respectively,andrelations 0=S˙q1S˙q1 =S˙q2S˙q2 =S˙q1S˙q2S˙q1S˙q2+S˙q2S˙q1S˙q2S˙q1. TheF -lineardualsofA(1)andE˙ (A(1))are,asHopfalgebras, 2 0 A(1)∗ =F [ξ ,ξ ]/ξ4,ξ2, 2 1 2 1 2 2 ∆(ξ )=ξ ⊗1+ξ ⊗ξ +1⊗ξ , 2 2 1 1 2 E˙ A(1) ∗ =F [ξ ,ξ ,ξ ]/ξ2 ,ξ2 ,ξ2 , 0 2 1,0 1,1 2,0 1,0 1,1 2,0 (cid:16) (cid:17) ∆(ξ )=ξ ⊗1+ξ ⊗ξ +1⊗ξ , 2,0 2,0 1,0 1,1 2,0 withξ ,ξ ,ξ allprimitive.ThenotationxistraditionalfortheconjugateofxinaHopf 1 1,0 1,1 algebra,andinthiscase,ξ = ξ andξ = ξ +ξ3. Thenotationξ isusedtodenotethe 1 1 2 2 1 i,j imageofξ2j ∈A(1)inE˙ (A(1)). i 0 Proof. Well-knownconsequenceofthecomputationofthedualSteenrodalgebra,asin[9]. (cid:3) Proposition 3.3. The F -algebra E˙ A(1) is isomorphic to the group ring F [D ] of the 2 0 2 8 dihedralgroupD . 8 Proof. Weusethepresentation D =ha,b|a2,b2,abab=babai 8 forD . TheF -algebramap 8 2 f :F [D ]→E˙ A(1) 2 8 0 10 ANDREWSALCH givenby f(a)=1+Sq˙1 f(b)=1+Sq˙2 iswell-defined,since f(a)2 = 1= f(b)2and f(a)f(b)f(a)f(b)= f(b)f(a)f(b)f(a). (Here it is essential that we are using E˙ A(1) and not A(1), since S˙q2 2 = 0 in E˙ A(1) but 0 0 (Sq2)2 6=0inA(1).) Onecheckseasilythat(a+1)2 = 0=(b+(cid:16)1)2(cid:17)and(a+1)(b+1)(a+ 1)(b+1)=(b+1)(a+1)(b+1)(a+1),hencewehaveaF -algebramap 2 f−1 :E˙ A(1)→F [D ] 0 2 8 sendingS˙q1toa+1andsendingS˙q2tob+1. Clearly f−1 isinverseto f. (cid:3) We nowuse the well-knowncomputationof theHochschildhomologyofgrouprings (seee.g. Corollary9.7.5of[13]): Theorem3.4. SupposeG is a discrete group, k a field. Let hGi be the set of conjugacy classesofelementsinG,andgivenaconjugacyclassS,letC (S)denotethecentralizer G ofS inG. Thenthereexistsanisomorphismofgradedk-vectorspaces HH (k[G],k[G])∼=⊕ H (C (S);k). ∗ S∈hGi ∗ G (Theorem3.4seemstobewell-known,butIdonotknowwhotoattributetheresultto, ifanyone!) Corollary3.5. ThedimensionofHH (E˙ A(1),E˙ A(1))asak-vectorspaceis n 0 0 dim HH (E˙ A(1),E˙ A(1))=3n+5. k n 0 0 Proof. WeuseProposition3.3andTheorem3.4. Therearefiveconjugacyclassesofele- mentsinD =hx,y| x2,y4,xy=y3xi: 8 1,{x,y2x},{yx,y3x},{y,y3},{y2}, withcentralizers D ,hx,y2 | x2,(y2)2i,hy2 |(y2)2i,hy|y4i,D , 8 8 respectively.Thesecentralizersubgroupsareisomorphicto D ,C ×C ,C ,C ,D , 8 2 2 2 4 8 respectively.Thehomology,withF coefficients,ofthesegroupsiswell-known: 2 dimF2Hn(D8;F2)=n+1 dimF2 Hn(C2×C2;F2)=n+1 dimF2Hn(C2;F2)=1 dimF2Hn(C4;F2)=1, hence dimF2 HHn(E˙0A(1),E˙0A(1))=dimF2HHn(F2[D8],F2[D8]) =dimF2Hn(D8;F2)+dimF2Hn(C2×C2;F2) +dimF2Hn(C2;F2)+dimF2Hn(C4;F2) +dimF2Hn(D8;F2) =3n+5. (cid:3)