COMPUTATIONSINTOPOLOGICALCOHOCHSCHILDHOMOLOGY By SarahKlanderman ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Mathematics|DoctorofPhilosophy 2020 ABSTRACT COMPUTATIONSINTOPOLOGICALCOHOCHSCHILDHOMOLOGY By SarahKlanderman Inrecentwork,HessandShipleyaninvariantofcoalgebraspectracalledtopo- logicalcoHochschildhomology(coTHH).In2018,Bohmann-Gerhardt-H˝genhaven-Shipley- ZiegenhagendevelopedaokstedtspectralsequencetocomputethehomologyofcoTHH forcoalgebrasoverthespherespectrum.However,examplesofcoalgebrasoverthesphere spectrumarelimited,andonewouldliketohavecomputationaltoolstostudycoalgebras overotherringspectra.Inthisthesis,weconstructarelativeokstedtspectralsequence tostudythetopologicalcoHochschildhomologyofmoregeneralcoalgebraspectra.Wecon- sider H F p , H Z H F p and H F p , BP @ n A H F p forcertainvaluesof n as H F p -coalgebrasand computethe E 2 -termofthespectralsequenceinthesecases.Further,weshowthatthis spectralsequencehasadditionalalgebraicstructure,andexploitthisstructuretocomplete coTHHcalculations.Finally,weshowthatcoHochschildhomologyisabicategoricalshadow, inthesenseofPonto. Tomygrandparents,whoinstilledinmealoveoflearningandwhohaveintentionally investedinmyeducationateverystepalongtheway. iii ACKNOWLEDGMENTS Tomyadvisor,TeenaGerhardt,thankyouforyourgenerositywithyourtime,your helpfulfeedbackthroughoutmygraduatejourney,andforbelievinginme.Youhavebeen anincrediblementorandinspiration.Thankyoualsototheothermembersofmycommittee, MatthewHedden,JosePerea,andparticularlyAnnaMarieBohmannforbeingwillingto serve.Fromthealgebraictopologycommunitysp,thankyoutoCaryMalkiewich forgivingmetheideatotakemyworktotheshadowrealmandJonathanCampbellfor walkingmethroughTHHshadowasItraversedhisnightmareofduals. Ihavebeengratefulforthemanyfriendswhohavegonethroughthegradschooljourney withme,especiallyHiteshGakhar,HarrisonandJessicaLeFrois,ReshmaMenon,Joshua Ruiter,RaniSatyam,CharlotteUre,LydiaWassink,andAllisonYoung.Further,mylove ofteachinghasbeeninstrumentallyshapedbytheincredibleteamwehaveatMSU;ahuge thankyoutoAndyKrauseandTsvetaSendovasplly. Iowemanythankstofantasticconferencebuddieswhohavelistened,askedgreatques- tions,andpatientlyexplainedideastomeovertheyears,includingKatharineAdamyk, DuncanClark,ShaneClark,CloverMay,EmilyRudman,andNikoSchonsheck.Inparticu- lar,thankyoutoMaxPerouxformanycoalgebradiscussionsandGabeAngelini-Knollfor havingpatiencewithmymanyill-formedquestions. Iamdeeplygratefultomyimmediatefamily:Dave,Barb,William,andRebecca,and thefamilythatIhavebeenluckyenoughtomarryinto:Mai,Helen,Jason,andSara.A hugethankyoutoBob,Marj,Ahna,andDanwhowentoutoftheirwaytomakeMichigan ahomeawayfromhome.ToJosha,youbrightenandblessmylifeincountlessways;thank youforinspiringandchallengingmewhileconsistentlyprovidingloveandencouragement. iv TABLEOFCONTENTS Chapter1Introduction ................................... 1 Chapter2Background ................................... 8 2.1HochschildHomology..................................8 2.2TopologicalHochschildHomology...........................11 2.3Coalgebras.........................................13 2.4CoHochschildHomology................................17 2.5CoalgebrasinSpectra..................................20 2.6TopologicalCoHochschildHomology.........................26 2.7TheClassicalokstedtSpectralSequence......................28 Chapter3Constructionofarelativeokstedtspectralsequence ..... 33 Chapter4Algebraicstructuresinthe(relative)okstedtspectralse- quence ....................................... 40 4.1Hopfalgebrastructureintheokstedtspectralsequence.............41 4.2 j -Hopfalgebrastructureintheokstedtspectralsequence..........50 4.3 j -Hopfalgebrastructureintherelativecookstedtspectralsequence.....52 Chapter5Explicitcalculations ............................. 56 5.1 E 2 -pageExamples....................................57 5.2ComputationalTools..................................63 5.3ExteriorInputs......................................65 5.4DividedPowerInput...................................73 Chapter6Shadows ...................................... 76 6.1(Co)BarConstructions.................................77 6.2ShadowBackground...................................80 6.3CoHochschildHomologyisaShadow.........................88 6.4MoritaInvariance....................................97 APPENDIX ............................................ 104 BIBLIOGRAPHY ........................................ 106 v Chapter1 Introduction Hochschildhomology,whichwewilldenoteasHH,isaclassicalalgebraicinvariantofrings thatcanbeextendedtopologicallytogiveaninvariantofringspectracalledtopologi- calHochschildhomology(THH).Inthe1970's,Doi[14]studiedaconstructiondualto Hochschildhomologyforcoalgebras,calledcoHochschildhomology(coHH).Recentworkof HessandShipley[20]dethetopologicalanalogofDoi'swork,topologicalcoHochschild homology(coTHH),tostudy co algebraspectra. WorkofMalkiewich[25]andHess-Shipley[20]showscoTHHofsuspensionspectrais relatedtoTHHforsimplyconnectedspaces X via coTHH ‹ ª X “ THH ‹ ª ‹ X ““ ª L X; wherethelastequivalencecomesfromworkofokstedtandWaldhausen.ThuscoTHHis relevantforstudyingthehomologyoffreeloopspaces, L X ,themaintopicoftheof stringtopology[12,13].Further,becauseTHHisdirectlyrelatedtoalgebraic K -theoryvia tracemethods,coTHHalsohasapplicationsforalgebraic K -theoryofspaces. InthispaperwewilldevelopcomputationaltoolsforstudyingtopologicalcoHochschild homology.Theprimarytoolsusedtocomputetopological(co)Hochschildhomologyarespec- tralsequences.Inthelate1980's,okstedtidenthe E 2 -termofthespectralsequence 1 inducedfromtheskeletalofthesimplicialobjectTHH ‹ A “ Y tobethefamiliar algebraictheoryofHochschildhomology: E 2 ⁄ ; ⁄ HH ⁄ ‹ H ⁄ ‹ A ; k ““ Ô H ⁄ ‹ THH ‹ A “ ; k “ : Thisspectralsequenceisreferredtoasthe okstedtspectralsequence foraringspectrum A . In2018,Bohmann-Gerhardt-H˝genhaven-Shipley-Ziegenhagenshowedthatinthedual situationthereisa cokstedtspectralsequence .Thisisthespectralsequence forthecosimplicialspectrumcoTHH ‹ C “ Y ,for C acoalgebraspectrumoverthespherespec- trum[4].Aswewouldhope,thisspectralsequencehasclassicalcoHochschildhomologyas its E 2 -term,andincaseswhenitdoesindeedconvergewehave: E ⁄ ; ⁄ 2 coHH ⁄ ‹ H ⁄ ‹ C ; k ““ Ô H ⁄ ‹ coTHH ⁄ ‹ C “ ; k “ : Intheirworkhowever,thesetoolsareonlysetuptostudycoalgebraspectraoverthe spherespectrum S ,i.e : for C withcomultiplicationmap C C , S C .Examplesofthissort arecloselyrelatedtosuspensionspectraofspacesandarefairlylimitedasshownbyrecent workofPeroux-Shipley[30].Inthispaper,webroadenthesetoolstoapplytocoTHHfor coalgebrasoveranycommutativeringspectrum. Inordertomotivatetheneedforarelativeokstedtspectralsequence,weexamine avarietyofexamplesofcoalgebrasoverspectraotherthan S .Forexample,thefollowing propositiongivesawayofgeneratingcoalgebraspectraoveracommutativeringspectrum B . 2 Proposition1.0.1 Amapofcommutativeringspectra A B inducesa B -coalgebrastructureonthespectrum B , A B . WecallthespectralsequencethatallowsustostudythetopologicalcoHochschildho- mologyofcoalgebrasoveranarbitrarycommutativeringspectrum R the relativecokstedt spectralsequence Theorem1.0.2 Let E and R becommutativeringspectra, C an R -coalgebraspectrumthatistas an R -module,and N a ‹ C;C “ -bicomodulespectrum.If E ⁄ ‹ C “ isover E ⁄ ‹ R “ ,thenthere existsaspectralsequenceforthecosimplicialspectrumcoTHH R ‹ N;C “ Y that abutsto E t s ‹ coTHH R ‹ N;C ““ with E 2 -page E s;t 2 coHH E ⁄ ‹ R “ s;t ‹ E ⁄ ‹ N “ ;E ⁄ ‹ C ““ givenbytheclassicalcoHochschildhomologyof E ⁄ ‹ C “ withcotsin E ⁄ ‹ N “ . Noteinparticularthatthisholdsforanygeneralizedhomologytheory E inadditionto beingoverthemoregeneralringspectrum R .Further,weidentifyconditionsforconvergence ofthisspectralsequence.Inparticular,forthecasewhen E S ,ifforevery s thereexists some r sothat E s;s i r E s;s i ª thentherelativeokstedtspectralsequenceconverges completelyto ˇ ⁄ ‹ coTHH R ‹ N;C ““ . Aquestioniswhetherthe E 2 -termofthisspectralsequenceiscomputable.By theaboveproposition, H F p , H Z H F p and H F p , BP @ n A H F p for n 0 ; 1andfor n 2at theprimes p 2 ; 3are H F p -coalgebras.Inthesecasesforexample,the E 2 -termisindeed computable: 3 Proposition1.0.3 Forthe H F p -coalgebra H F p , H Z H F p ,the E 2 -pageofthespectralsequencecalculating ˇ t s ‹ coTHH H F p ‹ H F p , H Z H F p ““ is E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , H Z H F p ““ F p ‹ ˝ “ a F p ! for SS ˝ SS ‹ 0 ; 1 “ ; SS ! SS ‹ 1 ; 1 “ . Proposition1.0.4 Forthe H F p -coalgebra H F p , BP @ n A H F p for n 0 ; 1andfor n 2attheprimes p 2 ; 3,the E 2 -pageofthespectralsequencecalculating ˇ t s ‹ coTHH H F p ‹ H F p , BP @ n A H F p ““ is E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , BP @ n A H F p ““ F p ‹ ˝ 0 ;:::˝ n “ a F p ! 0 ;:::! n for SS ˝ i SS ‹ 0 ; 2 p i 1 “ ; SS ! i SS ‹ 1 ; 2 p i 1 “ . Becausecomputationswiththisrelativeokstedtspectralsequencearequitecompli- cated,anyadditionalstructureonthespectralsequencecanhelpinthesecalculations.By workofAngeltveit-Rognes,theclassicalokstedtspectralsequenceforacommutativering spectrumhasthestructureofaspectralsequenceofHopfalgebrasundersomecondi- tions[1].Bohmann-Gerhardt-Shipleyshowthatunderappropriateconditions,the okstedtspectralsequenceforacocommutativecoalgebraspectrumhaswhatiscalleda j -Hopfalgebrastructure ,ananalogofaHopfalgebrastructureforworkingoveracoalgebra [5].ThefollowingpropositionfollowsfromBohmann-Gerhardt-Shipley'swork: Theorem1.0.5 For C acocommutativecoalgebraspectrum,iffor r C 2each E ⁄ ; ⁄ r ‹ C “ iscoover ˇ ⁄ ‹ C “ , 4 thentherelativeokstedtspectralsequenceisaspectralsequenceof j ˇ ⁄ ‹ C “ -Hopfalgebras. Thisadditional j -Hopfalgebrastructureisverycomputationallyuseful.Forinstance, wecanextendtheworkofBohmann-Gerhardt-Shipley[5]toshowthefollowing: Theorem1.0.6 Fora k ,let C beacocommutative Hk -coalgebraspectrumsuchthatcoHH ⁄ ‹ ˇ ⁄ ‹ C ““ is over ˇ ⁄ ‹ C “ andthegradedcoalgebra ˇ ⁄ ‹ C “ isconnected.Thenthe E 2 -termofthe relativeokstedtspectralsequencecalculating ˇ ⁄ ‹ coTHH Hk ‹ C ““ , E ⁄ ; ⁄ 2 ‹ C “ coHH k ⁄ ‹ ˇ ⁄ ‹ C ““ ; isa j ˇ ⁄ ‹ C “ -bialgebra,andtheshortestnon-zerotial d s;t r inlowesttotaldegree s t mapsfroma j ˇ ⁄ ‹ C “ -algebraindecomposabletoa j ˇ ⁄ ‹ C “ -coalgebraprimitive. Thisalgebraicstructureprovesveryusefulforexplicitcomputationswiththeokstedt spectralsequence.Weusetherelativeokstedtspectralsequencetoshow: Theorem1.0.7 Fora k ,let C beacocommutative Hk -coalgebraspectrumthatisrantasan Hk - modulewith ˇ ⁄ ‹ C “ k ‹ y “ for S y S oddandgreaterthan1.Thentherelativecookstedt spectralsequencecollapsesand ˇ ⁄ ‹ coTHH Hk ‹ C ““ k ‹ y “ a k w asgraded k -modulesfor S w S S y S 1. Theorem1.0.8 Let k beadandlet p char ‹ k “ ,including0.For C acocommutative Hk -coalgebra 5 spectrumthatistasan Hk -modulewith ˇ ⁄ ‹ C “ k ‹ y 1 ;y 2 “ for S y 1 S ; S y 2 S bothodd andgreaterthan1,if p m isnotequalto S y 2 S 1 S y 1 S 1 or S y 2 S 1 S y 1 S 1 1forall m C 0,thentherelative okstedtspectralsequencecollapsesand ˇ ⁄ ‹ coTHH Hk ‹ C ““ k ‹ y 1 ;y 2 “ a k w 1 ;w 2 ; asgraded k -modulesfor S w i S S y i S 1. Further,inaresultanalogoustotheworkofBohmann-Gerhardt-H˝genhaven-Shipley- Ziegenhagen[4]wehave Theorem1.0.9 Let k beaandlet C beacocommutativecoassociative Hk -coalgebraspectrumthatis tasan Hk -modulespectrum,andwhosehomotopycoalgebrais ˇ ⁄ ‹ C “ k x 1 ;x 2 ;::: ; wherethe x i areinnon-negativeevendegreesandthereareonlymanyofthem ineachdegree.Thentherelativeokstedtspectralsequencecalculatingthehomotopy groupsofthetopologicalcoHochschildhomologyof C collapsesat E 2 ,and ˇ ⁄ ‹ coTHH Hk ‹ C ““ k x 1 ;x 2 ;::: a k ‹ z 1 ;z 2 ;::: “ as k -modules,with z i indegree S x i S 1. Finally,wewillshowthatcoHochschildhomology(coHH)isabicategoricalshadow. Ponto[31]andPonto-Shulman[32]developedtheoriginalframeworkforshadowsandtraces 6 inbicategories.Morerecently,workofCampbell-Ponto[11]usedthisfundamentalframework toshowthatTHHisanexampleofashadow.Inparticular,theshadowstructureformally providesmanyofthedesirablepropertiesofTHH,includingMoritainvariance.Therefore inthesamevein,wewillshowthatcoHochschildhomology(coHH)isalsoashadowforthe appropriatelybicategory. Thisthesisisorganizedasfollows.Chapter2introducescoalgebrasinspectraandtopo- logicalcoHochschildhomology.InChapter3weconstructtherelativeokstedtspectral sequence.InChapter4westudythealgebraicstructuresofthisspectralsequence,and Chapter5discussessomeexplicittopologicalcoHochschildhomologycalculations.Finally inChapter6wedelveintothetheoreticalframeworkofcoHochschildhomologyandshow thatitisashadow. 7 Chapter2 Background 2.1HochschildHomology Tobegin,werecalltheofHochschildhomologyforassociative k -algebras. 2.1.1 Let k beacommutativering, A anassociativealgebraover k ,and M an ‹ A;A “ -bimodule. Recallthatthisalgebraicstructuregivesusamultiplicationmap A a k A A and aunitmap k A .LetHH ‹ A;M “ Y denotethesimplicial k -modulewith r -simplices HH ‹ A;M “ r M a A a r .Thefacemapsaregivenby d i ‹ m a a 1 a ::: a a r “ ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ ma 1 a ::: a a r i 0 m a a 1 a ::: a a i a i 1 a ::: a a r 1 B i @ r a r m a a 1 a ::: a a r 1 i r; andthedegeneracymapsinserttheunitmap.Inparticular, s i ‹ m a a 1 aa a r “ ¢ ¨ ¨ ¦ ¨ ¨ ¤ m a 1 a a 1 aa a r i 0 m a a 1 aa a i a 1 a a i 1 aa a r 1 B i B r Thissimplicialobjecthastheform: 8 M a k A a k A M a k A M Let C ⁄ ‹ A;M “ denotetheassociatedchaincomplexwithboundarymap d i ‹ 1 “ i d i .Then the q th Hochschildhomology of A withcotsin M is: HH q ‹ A;M “ H q ‹ C ⁄ ‹ A;M ““ : TheDold-Kancorrespondencegivesustheequivalent HH q ‹ A;M “ ˇ q ‹S HH ‹ A;M “ Y S“ forthegeometricrealizationofthesimplicial k -moduleHH ‹ A;M “ Y .Thislatterformulation makesitclearerhowwewouldextendthistocreateananalogoustopological theory. Remark2.1.2 Hochschildhomologyisherewithcotsinan ‹ A;A “ -bimodule M .When M A ,consideredasabimoduleoveritself,thisisdenotedbyHH q ‹ A “ . Remark2.1.3 If A M ,thentogetherwiththeextrastructureofacyclicoperator t n ‹ a 0 a a 1 a ::: a a n “ a n a a 0 a a 1 a ::: a a n 1 ; 9 thesefaceanddegeneracymapsdetermineacyclicobject,HH ‹ A “ Y .Thiscomplexiscalled the cyclicbarconstruction . OnemayalsotheHochschildhomologyofagradedalgebra: 2.1.4 Foraentialgradedalgebra ‹ A; “ withtrivialderivation ,let C ⁄ ‹ A; “ bethecyclic chaincomplexgivenby n ( ‹ A; “ a ‹ n 1 “ withfacemaps d i ‹ a 0 a ::: a a n “ ¢ ¨ ¨ ¦ ¨ ¨ ¤ a 0 a ::: a a i a i 1 a ::: a a n 0 B i @ n ‹ 1 “ S a n S‹S a 0 S S a 1 S ::: S a n 1 S“ a n a 0 a a 1 a ::: a a n 1 i n; degeneracymapsthatinserttheunit s i ‹ a 0 a ::: a a n “ a 0 a ::: a a i a 1 a a i 1 a ::: a a n ; andcyclicoperator t n ‹ a 0 a ::: a a n “ ‹ 1 “ S a n S‹S a 0 S S a 1 S ::: S a n 1 S“ n ‹ a n a a 0 a ::: a a n 1 “ : ThentheHochschildcomplexassociatedto C ⁄ ‹ A; “ isacomplexofcomplexeswithbound- arymap d givenbyanalternatingsumofthefacemaps d n Q i 0 ‹ 1 “ i d i C ⁄ ‹ A; “ a ‹ n 1 “ ÐÐ C ⁄ ‹ A; “ a n : 10 Thisbicomplexbelowhastrivialhorizontalmaps: ‹ A a A a A “ 0 ‹ A a A a A “ 1 ‹ A a A a A “ 2 ‹ A a A “ 0 ‹ A a A “ 1 ‹ A a A “ 2 A 0 A 1 A 2 d d d d d d andtheweightpartsofthetensorproductaregivenby: ‹ A a ‹ n 1 “ “ k ? i 0 ::: i n k A i 0 a ::: a A i n : Thentakingthehomologyofthetotalcomplexofthisbicomplexwithboundary d 0(since istrivial)givesthe Hochschildhomology ofthistialgradedalgebra,denotedby HH ⁄ ‹ A; “ . 2.2TopologicalHochschildHomology Motivatedbyapplicationstoalgebraic K -theory,inthelate1980'sokstedtconstructeda topologicalversionofHochschildhomology[6,7].TopologicalHochschildhomologyisan invariantofringspectraandcanbeasfollows: 2.2.1 Foracommutativeringspectrum R ,an R -algebra A ,andan ‹ A;A “ -bimodulespectrum M , wehaveamultiplicationmap A , R A A andaunitmap R A alongwithleftand 11 rightactionsof A on M A , R M M M , R A M: LetTHH R ‹ A;M “ Y bethesimplicial R -modulespectrumwith r -simplicesTHH R ‹ A;M “ r M , R A , R r andfacemaps d i ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ , Id , ‹ r 1 “ i 0 Id , i , , Id , ‹ r i 1 “ 1 B i @ r ‹ , Id , ‹ r 1 “ “ X ti r where t isthecyclicpermutationbringingthelastfactoraroundtothefront.Thedegeneracy mapswillagaininserttheunitmapappropriately.Thenthe topologicalHochschild homology relativeto R of A withcotsin M isthegeometricrealization THH R ‹ A;M “ S THH R ‹ A;M “ Y S Remark2.2.2 Asbefore,when M A wetheneliminateitfromthenotationandwrite THH R ‹ A “ S THH R ‹ A;A “ Y S Insummary,wehavenowintroducedthealgebraicnofHochschildhomology anditstopologicalanalog,topologicalHochschildhomology.Inthefollowingsectionswewill introduceanalogoustheoriesforcoalgebras,coHochschildhomology(coHH)andtopological coHochschildhomology(coTHH). 12 Algebra Topology Algebras: HH ‹ A “ THH ‹ A “ Coalgebras: coHH ‹ C “ coTHH ‹ C “ 2.3Coalgebras Firstwewillrecalltheofcoalgebrasandcomodulesinordertointroducethe classicalcoHochschildhomologyofDoi[14]andcoHHofagradedcoalgebraoveragraded ring,sincewewillneedsuchstructureforthespectralsequence.Thenwewillintroduce coalgebrasinspectraandlookatexamples. 2.3.1 Let R beacommutativering.Thena(coassociative,counital) coalgebra C over R isan R - modulewith R linearmapsthatarethecomultiplication C C a C thatiscoassociative andcounit C R thatiscounital,i.e.thefollowingcoassociativityandcounitality diagramscommute: C C a C C a C C a C a C Id a a Id C C a C C a C R a C C C a R Id Id a a Id Example2.3.2 Fora k ,thepolynomialcoalgebra k w 1 ;w 2 ;::: for w i inevendegreeisthevectorspace withbasisgivenby Ÿ w j i š for j C 0and i C 1.Ithascoproduct 13 ‹ w j i “ Q k ‰ j k ’ w k i a w j k i andcounit ‹ w j i “ ¢ ¨ ¨ ¦ ¨ ¨ ¤ 1if j 0 0if j x 0 : Example2.3.3 Fora k ,theexteriorcoalgebra k ‹ y 1 ;y 2 ;::: “ for y i inodddegreesisthevectorspace withbasisgivenby Ÿ 1 ;y i š for i C 1,whichhascoproduct ‹ y i “ 1 a y i y i a 1 ‹ 1 “ 1 a 1 andcounit ‹ y i “ 0 ‹ 1 “ 1 Example2.3.4 Fora k ,thedividedpowercoalgebra k x 1 ;x 2 ;::: with x i inevendegreesisthevector spacewithbasisgivenby Ÿ j ‹ x i “š for j C 0and i C 1.Ithascoproduct ‹ j ‹ x i ““ Q a b j a ‹ x i “ a b ‹ x i “ where 0 ‹ x i “ 1 ; 1 ‹ x i “ x i ,andcounit 14 ‹ j ‹ x i ““ ¢ ¨ ¨ ¦ ¨ ¨ ¤ 1if j 0 0if j x 0 : UnderstandingHopfalgebraswillalsobeessentialforthiswork,sowerecallthe here.Firstweintroducebialgebras. 2.3.5 A bialgebra A overacommutativering R isaunitalassociative R -algebrawithmultipli- cation A a R A A andunit R A alongwithcomultiplication A A a R A andcounit A R suchthat A isalsoacounitalcoassociative R -coalgebrasatisfyingthe followingcommutativediagrams,whereall a belowareover R : 1. A a A / / a A / / A a A A a A a A a A Id a t a Id / / A a A a A a A a O O where t swapsthetwocomponentsof A a A . 2. A a A a & & / / A y y R a R R 3. R a R R a x x % % A a AA o o 15 4. R & & Id A x x R 2.3.6 A Hopfalgebra A isabialgebraoveracommutativering R togetherwithamapof R - modules ˜ A A calledthe antipode suchthatthefollowingdiagramcommutes: A a A ˜ a Id / / A a A # # A ; ; # # / / R / / A A a A Id a ˜ / / A a A ; ; where isthemultiplication,isthecomultiplication, istheunit,and isthecounit. Itwillalsobeusefultohavethe\dual"notionofmodulesonwhich C coacts : 2.3.7 Let R beacommutativeringand C an R -coalgebra.Then N isa right C -comodule ifit isan R -moduletogetherwithan R -linearmap N N a R C thatiscoassociativeand counital,i.e.thatmakesthefollowingdiagramscommute: N N a R C N a R C N a R C a R C Id a a Id N N a R C N Id Id a isreferredtoasa right C -coaction . Similarly,a left C -comodule isan R -moduletogetherwithan R -linearmap N C a R N 16 thatiscoassociativeandcounital,i.e.thatmakesthefollowingdiagramscommute: N C a R N C a R N C a R C a R N a Id Id a N C a R N N Id a Id isreferredtoasa left C -coaction . Similartothewayinwhichrightandleftmodulestructurestogethermaycreatea bimodulestructure,theanalogousholdsforcomodules: 2.3.8 For R -coalgebras C;D ,a ‹ C;D “ - bicomodule N isaleft C -comodulewithleftcoaction N C a R N andright D -comodulewithrightcoaction N N a R D that thefollowingcommutativediagram: N N a R D C a R N C a R N a R D a Id Id a 2.4CoHochschildHomology In[14],DoicoHochschildhomologyasaninvariantofcoalgebras: 2.4.1 Let k beacommutativering, C a(coassociative,counital) k -coalgebra,and N a ‹ C;C “ - bicomodule.Then C comesequippedwithacoassociativecomultiplication C C a C 17 andcounit C k .Webuildthecochaincomplex C ⁄ ‹ N;C “ : ÐÐ N a C a C ÐÐ N a C ÐÐ N ÐÐ 0 withcoboundarymap i ‹ 1 “ i i for i givenby i ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ a Id a r i 0 Id a i a a Id a ‹ r i “ 1 B i B r ~ t X ‹ a Id a r “ i r 1 where denotestherightcoaction, denotestheleftcoaction,and ~ t isthemapthattwists thefactortothelast.Thentheq th coHochschildhomology of C withcotsin N is coHH q ‹ N;C “ H q ‹ C ⁄ ‹ N;C ““ : Remark2.4.2 WedenotecoHochschildhomologywithcotsin C bycoHH q ‹ C “ when C isviewedas a ‹ C;C “ -bicomoduleoveritself. Remark2.4.3 WorkofHess-Parent-Scott[19]showsthatthecoHochschildhomologyofatialgraded coalgebraoveragradedringfollowsasabovewiththeadditionofsignsthatfollowfromthe Koszulrule.Inthespectralsequenceofthenextchapterwewillusethefollowing ofcoHHofagradedcoalgebraoveragradedringbasedontheirwithtrivial tial: 2.4.4 Foratialgradedcoalgebra ‹ D;@ “ withtrivialcoderivation @ ,let C ⁄ ‹ D;@ “ bethe 18 cycliccochaincomplexgivenby n ( ‹ D;@ “ a ‹ n 1 “ withcofacemaps i ¢ ¨ ¨ ¦ ¨ ¨ ¤ Id a i a a Id a ‹ r i “ 0 B i B r ‹ 1 “ S d 0 S‹S d 1 S S d 2 S ::: S d n S“ ~ t X ‹ a Id a r “ i r 1 ; where ~ t isthemapthattwiststhefactortothelast,codegeneracymapsthatinsertthe counit ˙ i Id a ‹ i 1 “ a a Id r i ; andcocyclicoperator t n ‹ d 0 a ::: a d n “ ‹ 1 “ S d 0 S‹S d 1 S S d 2 S ::: S d n S“ n ‹ d 1 a ::: a d n a d 0 “ : ThenthecoHochschildcomplexassociatedto C ⁄ ‹ D;@ “ hascoboundarymapgivenbyan alternatingsumofthecofacemaps n Q i 0 ‹ 1 “ i i C ⁄ ‹ D;@ “ a n ÐÐ C ⁄ ‹ D;@ “ a ‹ n 1 “ ; 19 andthebicomplexbelowhastrivialhorizontalmaps: ‹ D a D a D “ 0 ‹ D a D a D “ 1 ‹ D a D a D “ 2 ‹ D a D “ 0 ‹ D a D “ 1 ‹ D a D “ 2 D 0 D 1 D 2 @ @ @ @ @ @ wheretheweightpartsofthetensorproductaregivenby: ‹ D a ‹ n 1 “ “ k ? i 0 ::: i n k D i 0 a ::: a D i n : Takingthecohomologyofthetotalcomplexofthisbicomplexwithcoboundary 0givesthe coHochschildhomology ofthistialgradedcoalgebra,denotedbycoHH ⁄ ‹ D;@ “ . 2.5CoalgebrasinSpectra Nowwewillintroducecoalgebrasinspectra.Firstwestatetheofacoalgebrafor thegeneralsettingofasymmetricmonoidalcategory. 2.5.1 Let ‹ D ; a ; 1 “ beasymmetricmonoidalcategory.Thena(coassociative,counital) coalgebra C > D hasacomultiplication C C a C thatiscoassociativeandacounitmorphism C 1thatiscounital,i.e.thefollowingcoassociativityandcounitalitydiagramscommute: 20 C C a C C a C C a C a C Id a a Id C C a C C a C 1 a C C C a 1 Id Id a a Id 2.5.2 A coalgebraspectrum isacoalgebrainoneofthesymmetricmonoidalcategoriesof spectra. 2.5.3 Foracommutativeringspectrum R ,an R -coalgebraspectrum C isacoalgebrainthe symmetricmonoidalcategoryof R -modules.Ithascomultiplication C C , R C and counit C R ,satisfyingthecoassociativityandcounitalityconditions. Example2.5.4 Foraspace X ,thediagonalmap X X , X ontopologicalspacesinducesacomultiplication maponthesuspensionspectrum ª ‹ X “ : ª ‹ X “ ª ‹ X , X “ ª ‹ X “ , ª ‹ X “ ; making ª ‹ X “ intoacoalgebraspectrum. However,itshouldbenotedthatmostspectradonothavediagonalmapsandthusex- amplesofthisformarequitelimited,asshowninworkofPeroux-Shipley[30].Inparticular, theyprovethatallcoalgebraspectraover S arecocommutativeinstrictmonoidalcategories ofspectra.Aswesawintheaboveexample,somecoalgebrasoverthespherespectrum comefromsuspensionspectra,butPeroux-Shipleyfurthershowthatinmodelcategoriesall 21 S -coalgebrasarecloselyrelatedtosuspensionspectra.Thisrigidstructureof S -coalgebras thereforeprovidesmotivationforstudyingotherkindsofcoalgebraspectra. Becauseexamplesofcoalgebrasinspectraover S arelimited,aprimarygoalofthiswork istodeveloptoolstostudycoalgebrasoverothercommutativeringspectra.Onesourceof suchcoalgebraspectraisthefollowinggeneralconstruction: Proposition2.5.5 Amapofcommutativeringspectra ˚ A B inducesa B -coalgebrastructureonthe spectrum B , A B . Proof. Tohaveacoalgebrastructure,wewantacomultiplicationmap B , A B ‹ B , A B “ , B ‹ B , A B “ B , A B , A B Butwehavetheequivalence i A , Id B , A B B , A A , A B .Themap i A insertsanextra copyof A : i A B / / B , A A B , S Id , A / / B , A O O forunitmap A .Applying ˚ yields B , A B B , A A , A B Id , ˚ , Id ÐÐÐÐÐ B , A B , A B whichinducesourdesiredcomultiplicationmap. Wefurtherneedacounitmap B , A B B .By B , A B isthecoequalizer 22 of B , A , B Id , ÐÐÐ ÐÐÐ , Id B , B B , A B formoduleactions A , B B B , B ˚ , Id m B , A B B , B Id , ˚ m wheretheringmap m B , B B isthemultiplicationinthecommutativeringspectrum B .Considermappingto B inthisdiagram: B , A , B Id , / / , Id / / B , B / / m B , A B B Diagram2.1 IfweconsiderthetwocompositesinDiagram2.1, B , A , B Id , / / Id , ˚ , Id B , B / / m / / B B , B , B Id , m 8 8 B , A , B , Id / / Id , ˚ , Id B , B / / m / / B B , B , B m , Id 8 8 weseethattheyagreesince B isinparticularanassociativeringspectrum,sowehavea mapfrom B , A , B B ,makingthediagram2.1commute.Bytheuniversalpropertyof coequalizers,thereexistsauniquemap B , A B B inthisdiagram 23 B , A , B Id , / / , Id / / * * B , B / / m B , A B y y B andthismap givesusthedesiredcounitmap.Nowwemustcheckthatthisisindeeda coalgebrabythatitthenecessarycoassociativityandcounitalitydia- grams. First,weneedtocheckthatwesatisfycoassociativityofthecomultiplication,thatis ‹ Id , “ ‹ , Id “ Thispropertycanbeshownbyprovingthefollowingdiagram commutes: B , A B ‹ B , A B “ , B ‹ B , A B “ ‹ B , A B “ , B ‹ B , A B “ ‹ B , A B “ , B ‹ B , A B “ , B ‹ B , A B “ Id , , Id Thisresultfollowsfromadiagramchasethroughtheofthecomultiplication. Recallthat i A inserts , A A andsimilarly i B willinsert , B B ,sothatthemapinthis settingisthecomposition: B , A B i A , Id / / B , A A , A B Id , ˚ , Id / / B , A B , A B Id , i B , Id / / ‹ B , A B “ , B ‹ B , A B “ : Sonowexpandingtheabovediagramwiththisdecompositionofthecomultiplicationyields Diagram1givenintheAppendix.Commutativityofeachofthesquaresinthatdiagram followsbecauseeachofthevertical(andhorizontal)mapsareequivalentateachlevelwith additionalcopiesoftheidentityinsertedasappropriate.Thusthecoassociativityaxiomis Counitalityfollowsfromshowing ‹ , Id “ Id ‹ Id , “ Because B iscommutative, theleftandrightmoduleactionsof B on B , A B coincide: 24 ‹ B , A B “ , B ‹ B , A B “ Id , / / ‹ B , A B “ , B B B , A B B , B ‹ B , A B “ ‹ B , A B “ , B ‹ B , A B “ , Id o o B , A B l l 2 2 soittoshowwhyoneofthesetrianglescommutes.Theright-handcompositioncan bebrokendownas B , A B i A , Id ÐÐÐ B , A A , A B Id , ˚ , Id ÐÐÐÐ B , A B , A B Id , i B , Id ÐÐÐÐÐ B , A B , B B , A B , Id ÐÐ B , B ‹ B , A B “ B , A B: Observethatthelastfactorof B isunchangedbythismap,andsothiscompositionmap comesfrom B B , S B , A B , B B; Id , A Id , B Id , ˚ m wherethepartofthiscompositemustbeId , B bytheofmapofringspectra. However,unitalityof B impliesthat m ‹ Id , B “ Id B ,andthereforetherighttriangleinour diagramisequivalenttotheidentityon B , A B .Asimilarcanbeusedtoshow thelefttrianglecommutesaswell,sowehaveshownthat B , A B comingfromthemapof commutativeringspectra ˚ A B isacoassociative,counital B -coalgebraspectrum. Examplesofthesekindsofcoalgebraspectrainclude H F p -coalgebrassuchas H F p , H F p , H F p , H Z H F p ,and H F p , BP @ n A H F p for n 0 ; 1andfor n 2attheprimes p 2 ; 3,some ofwhichwewillexaminelateroninfurtherdetail. 25 2.6TopologicalCoHochschildHomology Wewill(topological)coHochschildhomologyforanygeneralsymmetricmonoidal categoryasinBohmann-Gerhardt-H˝genhaven-Shipley-Ziegenhagen[4].Thereafterwewill beprimarilyusingtheasitappliestospectra,althoughclassicalcoHochschild homologyasbyDoi[14]canberecoveredfromthefollowingmoregeneral byconsideringthecategoryofcoalgebrasovera 2.6.1 Let ‹ D ; a ; 1 “ beasymmetricmonoidalmodelcategoryandlet C > D beacoalgebrawith coassociativecomultiplication C C a C andcounit C 1.Further,let N bea ‹ C;C “ -bicomodulewithleftandrightcoactions N C a N and N N a C .Thenlet coTHH ‹ N;C “ Y bethecosimplicialobjectwithr-simplicescoTHH ‹ N;C “ r N a C a r ,with cofacemaps i ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ a Id a r i 0 Id a i a a Id a ‹ r i “ 0 @ i B r ~ t X ‹ a Id a r “ i r 1 where ~ t isthemapthattwiststherstfactortothelast,andwithcodegeneracymaps ˙ i N a C a ‹ r 1 “ N a C a r for0 B i B r ˙ i Id a ‹ i 1 “ a a Id a r i : Thisgivesacosimplicialobjectoftheform 26 N a C a C N a C N Thenthe topologicalcoHochschildhomology ofthecoalgebra C withcoientsin N isgivenby coTHH ‹ N;C “ Tot ‹ R coTHH ‹ N;C “ Y “ where R istheReedytreplacementandTotrepresentsthetotalization. Remark2.6.2 WetopologicalcoHochschildhomologywithcotsina ‹ C;C “ -bicomodule N , butwhenweconsider C asabicomoduleoveritself,wewritecoTHH ‹ C “ forcots in C .AswithtopologicalHochschildhomology,wewillfurtherdecoratethenotationwith coTHH R ‹ C “ whenweconsiderthetopologicalcoHochschildhomologyof C relativeto R , i.e.coTHHofan R -coalgebra C for R acommutativeringspectrum. Remark2.6.3 Recallthatforringsandringspectra A ,HH ‹ A “ Y andTHH ‹ A “ Y areexamplesof cyclicbar constructions .Similarlyforacoalgebra C ,thecoHochschildcomplexfor C togetherwitha cyclicoperator ~ t n ‹ a 0 a a 1 aa a n “ a 1 aa a n 1 a a n a a 0 : iscalledthe cycliccobarconstruction ,andbothcoHH ‹ C “ Y andcoTHH ‹ C “ Y areexam- ples. 27 Asmentionedabove,workinginthecategoryofcoalgebrasoverarecoverstheclas- sicalcoHHofDoi[14].Ourgeneralconventionwillbetosprefertothisconstruc- tionas topological coHochschildhomologywhenweareconsideringasinputsomecoalgebra spectrum .Forinstance,wewillrefertotheworkofHess-Parent-Scott[19]asstudying coHochschildhomology oftialgradedcoalgebras(dg-coalgebras)overa 2.7TheClassicalokstedtSpectralSequence BeforeconstructingtheokstedtspectralsequenceforcoTHH,werecalltheclassical okstedtspectralsequenceforTHH,duetookstedtin[6].Recallthatforaringspectrum A (i.e.an S -algebrawithmultiplication A , A A andunit A S ),wecanbuildthe simplicialspectrumTHH ‹ A “ Y via A , A , A A , A A wherethefacemaps d i A , ‹ r 1 “ A , r aregivenby d i ¢ ¨ ¨ ¦ ¨ ¨ ¤ Id , i , , Id , ‹ r i 1 “ 0 B i @ r ‹ , Id , i “ X ti r for t thatcyclicallypermutesthelastelementtothefront,andwherethedegeneracymaps 28 s i A , ‹ r 1 “ A , ‹ r 2 “ inserttheunitmapfor0 B i B r : s i Id , ‹ i 1 “ , , Id , ‹ r i “ ThenforthissimplicialspectrumTHH ‹ A “ Y ,onecanconsiderthespectralsequencethat comesfromitsskeletalandconvergesto H ⁄ ‹ THH ‹ A “ ; k “ ,where k isaFor thisspectralsequence, E 1 ⁄ ;q isisomorphictothenormalizedchaincomplexof H q ‹ THH ‹ A “ Y “ . SoweconsiderthehomologyappliedtoeachsimpliciallevelofTHH ‹ A “ Y : H ⁄ ‹ A , A , A ; k “ H ⁄ ‹ A , A ; k “ H ⁄ ‹ A ; k “ : Notethenthat H ⁄ ‹ A , A ,, A ; k “ ˇ ⁄ ‹ A , A ,, A , Hk “ ˇ ⁄ ‹‹ A , Hk “ , Hk ‹ A , Hk “ , Hk , Hk ‹ A , Hk ““ ˇ ⁄ ‹ A , Hk “ a ˇ ⁄ Hk ˇ ⁄ ‹ A , Hk “ a ˇ ⁄ Hk a ˇ ⁄ Hk ˇ ⁄ ‹ A , Hk “ wherethelastisomorphismfollowsfromtheKunnethspectralsequencebecause ˇ ⁄ ‹ A , Hk “ isover ˇ ⁄ Hk k .Thuswecanrewriteeachleveltoget: H ⁄ ‹ A , A ,, A ; k “ H ⁄ ‹ A ; k “ a k H ⁄ ‹ A ; k “ a k a k H ⁄ ‹ A ; k “ Further,the d 1 tialofthespectralsequenceunderthisidenagreeswith 29 thetialofthecomplexcomputingHH ⁄ ‹ H ⁄ ‹ A ; k ““ .Thereforethe E 2 -termofthis spectralsequenceisHH ⁄ ‹ H ⁄ ‹ A ; k ““ ,theclassicalHochschildhomologyof H ⁄ ‹ A ; k “ . Sincethisstructurefollowsfromthegeneralspectralsequencethatarisesfromtheskeletal ofthesimplicialspectrumTHH ‹ A “ Y ,thiscanbeextendedtoanyhomologytheory, whichisformallystatedinthecontextof S -modulesinthefollowingtheoremsofElmendorf- Kriz-Mandell-May[15]: Theorem ([15]ThmIX.2.9) Let E beacommutativeringspectrum, A aringspectrum,and M acell ‹ A;A “ - bimodule.If E ⁄ ‹ A “ is E ⁄ thenthereisaspectralsequenceoftheform E 2 p;q HH E ⁄ p;q ‹ E ⁄ ‹ A “ ;E ⁄ ‹ M ““ Ô E p q ‹ THH S ‹ A;M ““ Here E ⁄ isthehomologytheoryassociatedtothecommutativeringspectrum E ,i.e. E ⁄ ‹ A “ ˇ ⁄ ‹ E , A “ .Thustheaboveresultcomesfromthespectralsequencederivedfrom thesimplicialationofTHH R ‹ A;M “ (forthecase R S )asgivenin: Theorem ([15]ThmX.2.9) Let E and R beringspectraand K ⁄ beapropersimplicial R -modulespectrum. Thenthereisanaturalhomologicalspectralsequence Ÿ E r p;q K ⁄ š suchthat E 2 p;q K ⁄ H p ‹ E q ‹ K ⁄ ““ and Ÿ E r p;q K ⁄ š convergesstronglyto E ⁄ ‹S K ⁄ S“ . 30 NotethatthisTheoremX.2.9from[15]givesamoregeneralstatementoftheokstedt spectralsequencefortopologicalHochschildhomologyofan R -algebra,whichwestatehere forclarity. Theorem2.7.1 Suppose E and R arecommutativeringspectra, A isan R -algebra,and M isacell ‹ A;A “ - bimodule.Thenif E ⁄ ‹ A “ isover E ⁄ ‹ R “ ,thenthereexistsaspectralsequence E 2 p;q HH E ⁄ ‹ R “ p;q ‹ E ⁄ ‹ A “ ;E ⁄ ‹ M ““ Ô E p q ‹ THH R ‹ A;M ““ Wequicklyverifythatthisspectralsequencehastheindicated E 2 -term.Forthisspectral sequence, E 1 ⁄ ;q isisomorphictothenormalizedchaincomplexof E q ‹ THH R ‹ A;M “ Y “ : Sowe considerthe E -homologyappliedtoeachsimpliciallevelofTHH R ‹ A;M “ Y : E ⁄ ‹ M , R A , R , R A “ ˇ ⁄ ‹ E , M , R A , R , R A “ ˇ ⁄ ‹ E , M , E , R E , R , R A , E , R E , R , R , E , R E , R , R A “ ˇ ⁄ ‹ E , M , E , R E , A , E , R E , A , E , R E , A “ ˇ ⁄ ‹ E , M “ a ˇ ⁄ ‹ E , R “ ˇ ⁄ ‹ E , A “ a ˇ ⁄ ‹ E , R “ a ˇ ⁄ ‹ E , R “ ˇ ⁄ ‹ E , A “ (since ˇ ⁄ ‹ E , A “ isover ˇ ⁄ ‹ E , R “ byhypothesis) E ⁄ ‹ M “ a E ⁄ ‹ R “ E ⁄ ‹ A “ a E ⁄ ‹ R “ a E ⁄ ‹ R “ E ⁄ ‹ A “ The d 1 tialofthespectralsequenceunderthisidenagreeswiththedif- ferentialofthecomplexcomputingHH E ⁄ ‹ R “ ‹ E ⁄ ‹ A “ ;E ⁄ ‹ M ““ ,soweidentifythe E 2 -term: E 2 p;q HH E ⁄ ‹ R “ p;q ‹ E ⁄ ‹ A “ ;E ⁄ ‹ M ““ Ô E p q ‹ THH R ‹ A;M ““ Nowthatwe'reequippedwiththerelativestatementofthetheoremforthetheoriesof HHandTHH,wewanttoinvestigatewhatiscurrentlyknownaboutthedualsituation 31 forcoTHH,andthenseehowwecanextendthatworkinordertohaveatoolforgeneral R -coalgebraspectra. 32 Chapter3 Constructionofarelativeokstedt spectralsequence Associatedtoacosimplicialspectrumisaspectralsequence[8].Applied tothecosimplicialspectrumcoTHH ‹ C “ Y thisyieldsaspectralsequencewhose E 2 -term wasideninBohmann-Gerhardt-H˝genhaven-Shipley-Ziegenhagen[4]astheclassical coHochschildhomologyofcoalgebrasinthesenseofDoi[14].Sp,theyshow: Theorem ([4]Thm4.1) Let k beaand C acoalgebraspectrumthatistasaspectrum.Then thespectralsequenceforthecosimplicialspectrumcoTHH ‹ C “ Y givesaokstedtspectralsequenceforcalculating H t s ‹ coTHH ‹ C “ ; k “ with E 2 -page E s;t 2 coHH k s;t ‹ H ⁄ ‹ C ; k ““ givenbytheclassicalcoHochschildhomologyof H ⁄ ‹ C ; k “ asagraded k -module. Bearinmindthatthespectralsequence doesnotalwaysconverge , althoughtheauthorsdospecifyconditionsunderwhichtheokstedtspectralsequence 33 willconvergecompletely. 1 Notethatthisspectralsequenceappliestotheordinaryhomology ofcoTHH ‹ C “ where C isan S -coalgebra.Nowwewanttocreatearelativeversionof thistheoremfor R -coalgebraspectrathatwouldgivethespectralsequence computingthehomologyofcoTHH R ‹ C “ .AswesawintheTHHsetting,wewouldexpect thatsomeconditionsmustbe Forcommutativeringspectra E and R; an R -coalgebraspectrum C ,anda ‹ C;C “ - bicomodule N ,wewillseethatif E ⁄ ‹ C “ isover E ⁄ ‹ R “ ,thentheKanspec- tralsequenceforthecosimplicialspectrumcoTHH R ‹ N;C “ Y canbeusedincalculating E t s ‹ coTHH R ‹ N;C ““ with E 2 -page E s;t 2 coHH E ⁄ ‹ R “ s;t ‹ E ⁄ ‹ N “ ;E ⁄ ‹ C ““ : Wewillrefertothisspectralsequenceasthe relativecokstedtspectralsequence .Notein particularthatthisholdsforanygeneralizedhomologytheoryinadditiontobeingoverthe moregeneralringspectrum R . Wewillformallystateandprovethatthisrelativeokstedtspectralsequence exists,andthenidentifyacorollarythatwillbeusefulforlatercomputations.Further,we willdescribeconditionsforconvergenceofthisspectralsequence. Theorem3.0.1 Let E and R becommutativeringspectra, C an R -coalgebraspectrumthatistas an R -module,and N a ‹ C;C “ -bicomodulespectrum.If E ⁄ ‹ C “ isover E ⁄ ‹ R “ ,thenthere existsaspectralsequenceforthecosimplicial R -modulecoTHH R ‹ N;C “ Y that 1 Forinstance,theokstedtspectralsequenceconvergeswhen C isasuspensionspectrum ª X for simplyconnected X [4]. 34 abutsto E t s ‹ coTHH R ‹ N;C ““ with E 2 -page E s;t 2 coHH E ⁄ ‹ R “ s;t ‹ E ⁄ ‹ N “ ;E ⁄ ‹ C ““ givenbytheclassicalcoHochschildhomologyof E ⁄ ‹ C “ withcotsin E ⁄ ‹ N “ . Proof. Tobegin,wewillrecallthegeneralconstructionofthespectralse- quence[8]forageneralReedytcosimplicial R -module X Y . Letbethecosimplicialspacewiththestandard n -simplex n asits n th level.The categoryof R -modulesiscotensoredoverpointedspaces(seee.g.[2]),andthenotation D K willbeusedforthecotensorofan R -module D withasimplicialspace K .SoforourReedy tcosimplicial R -module X Y thetotalizationof X Y isgivenby: Tot ‹ X Y “ eq › M n C 0 ‹ X n “ n ’ M > a ; b ‹ X b “ a ” : Let sk s ` bethecosimplicialsubspacewith n th level sk s n thatisthe s -skeletonof the n -simplex.Thenonecan Tot s ‹ X Y “ eq › M n C 0 ‹ X n “ sk s n ’ M > a ; b ‹ X b “ sk s a ” : Theinclusions sk s 0 sk s 1 induceatowerof Tot s ‹ X Y “ p s ÐÐ Tot s 1 ‹ X Y “ p s 1 ÐÐÐ Tot s 2 ‹ X Y “ p 1 ÐÐ Tot 0 ‹ X Y “ X 0 : i s i s 1 i s 2 i 0 F s F s 1 F s 2 F 0 35 Wethenhaveanassociatedexactcouple ˇ ⁄ ‹ Tot ⁄ ‹ X Y ““ ˇ ⁄ ‹ Tot ⁄ ‹ X Y ““ ˇ ⁄ ‹ F ⁄ “ p ⁄ @ i ⁄ thatyieldsahalfplanecohomologicalspectralsequence Ÿ E r ;d r š withtials d r E s;t r E s r;t r 1 r : Wenowwanttoidentifytheer F s .Recallthatthenormalizedcochaincomplex N s X Y istobe: N s X Y s 1 ˛ i 0 ker ‹ ˙ i X s X s 1 “ forcodegeneracymaps ˙ i asgivenbythecosimplicialstructure. Theneacher F s canbeidenedwith F s s ‹ N s X Y “ : The E 1 -termofthespectralsequencecanthusbeiden E s;t 1 ˇ t s ‹ F s “ ˇ t s ‹ s ‹ N s X Y ““ ˇ t ‹ N s X Y “ N s ˇ t ‹ X Y “ withtial d 1 N s ˇ t ‹ X Y “ N s 1 ˇ t ‹ X Y “ .Thismapcanthenbeidenwith 36 ‹ 1 “ i ˇ t ‹ i “ ,andwehave H ⁄ ‹ N s ˇ t ‹ X Y ““ H s ‹ ˇ t ‹ X Y ““ Ô E s;t 2 H s ‹ ˇ t ‹ X Y “ ; ‹ 1 “ i ˇ t ‹ i ““ Herewecareaboutthespcasewhen X Y R ‹ E , coTHH R ‹ N;C “ Y “ ,where R indicatestheReedytreplacement,andsoweget ˇ ⁄ ‹ X Y “ ˇ ⁄ ‹ R ‹ E , coTHH R ‹ N;C “ Y “ ˇ ⁄ ‹ E , coTHH R ‹ N;C “ Y “ : RecallthatcoTHH R ‹ N;C “ hascosimplicialstructure: N , R C , R C N , R C N sowhenwetake ˇ ⁄ ‹ E , “ ,atthe n th levelweseethat: ˇ ⁄ ‹ E , N , R C , R , R C “ ˇ ⁄ ‹ E , N , E , R E , R , R C , E , R E , R , R , E , R E , R , R C “ ˇ ⁄ ‹ E , N , E , R E , C , E , R E ,, E , R E , C “ ˇ ⁄ ‹ E , N “ a ˇ ⁄ ‹ E , R “ ˇ ⁄ ‹ E , C “ a ˇ ⁄ ‹ E , R “ a ˇ ⁄ ‹ E , R “ ˇ ⁄ ‹ E , C “ (since ˇ ⁄ ‹ E , C “ isover ˇ ⁄ ‹ E , R “ byhypothesis) E ⁄ ‹ N “ a E ⁄ ‹ R “ E ⁄ ‹ C “ a E ⁄ ‹ R “ a E ⁄ ‹ R “ E ⁄ ‹ C “ andsoweget 37 ˇ ⁄ R ‹ E , coTHH R ‹ N;C “ n “ ˇ ⁄ ‹ E , coTHH R ‹ N;C “ n “ E ⁄ ‹ N “ a E ⁄ ‹ R “ E ⁄ ‹ C “ a E ⁄ ‹ R “ n : Then ‹ 1 “ i ˇ ⁄ ‹ i “ givesthecoHochschildtialunderthisidenandthuswe getthecoHochschildcomplex: E s;t 2 H s ‹ ˇ t ‹ X Y “ ; ‹ 1 “ i ˇ t ‹ i ““ H s ‹ E t ‹ N “ a E ⁄ ‹ R “ E t ‹ C “ a E ⁄ ‹ R “ n ; ‹ 1 “ i ˇ t ‹ i ““ coHH E ⁄ ‹ R “ s;t ‹ E ⁄ ‹ N “ ;E ⁄ ‹ C ““ Thereforetheresultistheld-Kanspectralsequence E s;t 2 coHH E ⁄ ‹ R “ s;t ‹ E ⁄ ‹ N “ ;E ⁄ ‹ C ““ ?? Ô E t s ‹ coTHH R ‹ N;C ““ whereweuse??asareminderthatthissequencedoesnotconvergeingeneral. Becauseitwillbeparticularlyusefulinfutureexamples,westatethefollowingspecial casewhen E S asacorollary: Corollary3.0.2 Let R beacommutativeringspectrumand C an R -coalgebraspectrum.If ˇ ⁄ ‹ C “ isover ˇ ⁄ ‹ R “ ,thenthereexistsaspectralsequencethatabutsto ˇ t s ‹ coTHH R ‹ C ““ with E 2 -page E s;t 2 coHH ˇ ⁄ ‹ R “ s;t ‹ ˇ ⁄ ‹ C ““ givenbytheclassicalcoHochschildhomologyof ˇ ⁄ ‹ C “ . Nowwewanttoseetheconditionswerequireforconvergence.Basedonworkof 38 Kan[8]andBohmann-Gerhardt-H˝genhaven-Shipley-Ziegenhagen[4],wehavethefollowing convergenceresult. Proposition3.0.3 Ifforevery s thereexistssome r sothat E s;s i r E s;s i ª ,thentherelativeokstedtspectral sequenceforcoTHH R ‹ C “ convergescompletelyto ˇ ⁄ Tot R ‹ E , coTHH R ‹ C “ Y “ ConditionsforcompleteconvergencecanbefoundinGoerss-Jardine[18].Further,from thenaturalconstructionofamapHom ‹ X;Y “ , Z Hom ‹ X;Y , Z “ wegetanaturalmap P E , Tot ‹ R coTHH R ‹ C “ Y “ Tot R ‹ E , coTHH R ‹ C “ Y “ Applyinghomotopytotheabovegivesusthefollowingcorollary. Corollary3.0.4 Ifforevery s thereexistssome r sothat E s;s i r E s;s i ª and P E , Tot ‹ R coTHH R ‹ C “ Y “ Tot R ‹ E , coTHH R ‹ C “ Y “ inducesanisomorphisminhomotopy,thentherelativeokstedt spectralsequenceforcoTHH R ‹ C “ convergescompletelyto E ⁄ ‹ coTHH R ‹ C ““ . FortheexamplesinwhichwewillcomputetopologicalcoHochschildhomologyinthis thesis,wearetaking E S andsotheconditiononthemap P is.Weformallystate thisspccasehereforeasyreference: Corollary3.0.5 Whenconsidering E S ,ifforevery s thereexistssome r sothat E s;s i r E s;s i ª thenthe relativeokstedtspectralsequenceconvergescompletelyto ˇ ⁄ ‹ coTHH R ‹ C ““ . 39 Chapter4 Algebraicstructuresinthe(relative) okstedtspectralsequence Understandingadditionalalgebraicstructureinaspectralsequencecanhelpfacilitatecalcu- lations.Inthissectionwestudythestructureoftherelativeokstedtspectralsequence. ByworkofAngeltveit-Rognes,theclassicalokstedtspectralsequenceforacommutative ringspectrumhasthestructureofaspectralsequenceofHopfalgebrasundersome conditions[1].Bohmann-Gerhardt-Shipleyshowthatunderappropriatecondi- tionstheokstedtspectralsequenceforacocommutativecoalgebraspectrumhaswhatis calleda j -Hopfalgebrastructure ,ananalogofaHopfalgebrastructureworkingoveracoal- gebra[5].ItfollowsfromBohmann-Gerhardt-Shipley'sworkthattherelativeokstedt spectralsequencealsohasthistypeof j -Hopfalgebrastructure,andthisadditionalalgebraic structureiscomputationallyuseful.Forinstance,withthisstructuretheshortestnonzero tialmapsfromanalgebraindecomposabletoacoalgebraprimitive. Inthissection,wewillbeginbyexaminingthestructureoftheokstedtspectralsequence andtheimplicationsofthatalgebraicstructureasinAngeltveit-Rognes[1].Wewillthen introducethenecessaryandtheoremsforthe j -Hopfalgebrastructureandstate theresultofBohmann-Gerhardt-Shipley[5]thattheokstedtspectralsequencehasthis structure.Finallywewillseehowtherelativeokstedtspectralsequencebyextension 40 alsohasthis j -Hopfalgebrastructure. 4.1Hopfalgebrastructureintheokstedtspectral sequence InordertoshowthattheokstedtspectralsequenceisaspectralsequenceofHopfalgebras, Angeltveit-Rognes[1]showasinElmendorf-Kriz-Mandell-May[15]thatforacommutative ringspectrum A ,THH ‹ A “ isaHopfalgebraover A itselfinthehomotopycategory.One canidentifyTHH ‹ A “ with A a S 1 ascommutative S -algebras[26],whichinducesmapson THHthatgivetheHopfstructureinthehomotopycategory.Inparticular,THH ‹ A “ isa commutativeHopfalgebra,butitisnotcocommutativeingeneralbecausethecoproductis inducedfromthepinchmap,whichisnothomotopycocommutative.Angeltveit-Rognesalso studyananalogousHopfstructureontheentireokstedtspectralsequence,under conditions[1]. 4.1.1 Thestandard simplicialcircle S 1 Y isgivenby 1 ~ @ 1 fortheonesimplex 1 .Then 1 r haselements Ÿ x 0 ;:::;x r 1 š for x j r 1 thatsends j termsto0.Identifying x 0 x r 1 createsthequotient S 1 r withfacemaps d i ‹ x r “ ¢ ¨ ¨ ¦ ¨ ¨ ¤ x r r B i x r 1 r A i anddegeneracymaps 41 s i ‹ x r “ ¢ ¨ ¨ ¦ ¨ ¨ ¤ x r r B i x r 1 r A i: InthecaseofTHH ‹ A “ Y A a S 1 Y ,theHopfalgebrastructureover A onTHH ‹ A “ is inducedbysimplicialmapson S 1 Y : ‹ Theinclusionofthebasepoint ⁄ S 1 Y inducestheunitmap A THH ‹ A “ Y . ‹ Theretraction S 1 Y ⁄ inducesthecounitmap THH ‹ A “ Y A . ‹ Thefoldmap ˚ S 1 Y - S 1 Y S 1 Y inducesthemultiplicationmap ˚ THH ‹ A “ Y , A THH ‹ A “ Y THH ‹ A “ Y : Thereisnosimplicialpinchmap S 1 Y S 1 Y - S 1 Y forthissimplicialmodelofof S 1 ,so[1]needed toalsoemployadoublecirclemodel.Thedoublecircle dS 1 Y isthequotientofthedouble 1-simplexgivenby dS 1 Y ‹ 1 O 1 “ O @ 1 N @ 1 @ 1 : Theythenusethedoublecirclemodel dS 1 Y tothecoproduct.Thereisasimplicial pinchmap dS 1 Y S 1 Y - S 1 Y takingthedoublecircletothewedgeoftwocircles,anda simplicialmap ˜ dS 1 Y dS 1 Y ,whichinterchangesthetwocopiesof 1 .These mapsinducemapsonacorresponding\doublemodel"ofTHH,which[1]showsisweakly equivalenttoordinaryTHH.Therefore,onegetsthefollowingcoproductandantipodemaps inthehomotopycategorythatmakeTHHintoan A -Hopfalgebrainthehomotopycategory: 42 THH ‹ A “ THH ‹ A “ , A THH ‹ A “ ˜ THH ‹ A “ THH ‹ A “ : Angeltveit-Rognesfurtherprovethatthesesimplicialmapsonthecircleyieldstructure intheokstedtspectralsequenceaswell. Theorem4.1.2 ([1]4.5) If A isacommutativeringspectrum,then: 1. If H ⁄ ‹ THH ‹ A “ ; F p “ isover H ⁄ ‹ A ; F p “ ,thenthereisacoproduct H ⁄ ‹ THH ‹ A “ ; F p “ H ⁄ ‹ THH ‹ A “ ; F p “ a H ⁄ ‹ A ; F p “ H ⁄ ‹ THH ‹ A “ ; F p “ and H ⁄ ‹ THH ‹ A “ ; F p “ isan A ⁄ -comodule H ⁄ ‹ A ; F p “ -Hopfalgebra,where A ⁄ isthe dualSteenrodalgebra. 2. Ifeachterm E r ⁄ ; ⁄ ‹ A “ for r C 2isover H ⁄ ‹ A ; F p “ ,thenthereisacoproduct E r ⁄ ; ⁄ ‹ A “ E r ⁄ ; ⁄ ‹ A “ a H ⁄ ‹ A ; F p “ E r ⁄ ; ⁄ ‹ A “ and E r ⁄ ; ⁄ ‹ A “ isan A ⁄ -comodule H ⁄ ‹ A ; F p “ -Hopfalgebraspectralsequence.Inpartic- ular,thetials d r respectthecoproduct . Asmentionedabove,weareusingthenotation A ⁄ forthedualSteenrodalgebra,which 43 isthe F p -Hopfalgebra: A ⁄ ¢ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¤ F p ˘ 1 ;˘ 2 ;::: a F p ‹ ˝ 0 ;˝ 1 ;::: “ p odd F p ˘ 1 ;˘ 2 ;::: p 2 for S ˘ i S 2 ‹ p i 1 “ (or2 i 1if p 2), S ˝ i S 2 p i 1[28]. InordertounderstandthisspectralsequencestructureinthesettingofAngeltveit-Rognes [1],werecallafew 4.1.3 Foranaugmentedalgebra A overacommutativering R withaugmentation A R ,the indecomposableelements of A ,denotedbythe R -module QA ,aregivenbytheshort exactsequence IA a IA ÐÐ IA ÐÐ QA ÐÐ 0 formultiplicationmap and IA ker ‹ “ . Example4.1.4 Indecomposableelementsinthepolynomialalgebra k w 1 ;w 2 ;::: areclassesoftheform w i . Theaugmentationinthiscaseis k w 1 ;w 2 ;::: k w i ( 0 so IA ker ‹ “ ‹ w 1 ;w 2 ;::: “ .Sothentheimageoftheproducton IA willbetermsofthe form w m i i :::w m j j for m k A 1,whichmeans QA isgivenbyelementsoftheform w i . 44 Example4.1.5 Similarly,intheexterioralgebra k ‹ y 1 ;y 2 ;::: “ ,indecomposableelementsareclassesofthe form y i .Theaugmentationisgivenby k ‹ y 1 ;y 2 ;::: “ k y i ( 0 so IA ker ‹ “ ‹ y 1 ;y 2 ;::: “ .Theimageoftheproducton IA willbetermsoftheform y i 1 ;y i 2 :::y i n for n A 1,whichmeans QA isgivenbyelementsoftheform y i . 4.1.6 Foracoaugmentedcoalgebra C overacommutativering R withcoaugmentation R C andcounit C R ,the primitiveelements of C ,denotedbythe R -module PC ,are givenbytheshortexactsequence 0 ÐÐ PC ÐÐ JC ÐÐ JC a JC forcomultiplicationmapand JC coker ‹ “ .Anelement x > ker ‹ “ isprimitiveifits imageunderthequotientby Im ‹ “ in JC isin PC . Remark4.1.7 Inacoaugmentedcoalgebra C , x isprimitiveif ‹ x “ 1 a x x a 1.Notethatthisformulation isequivalenttotheabovebecausethecoproducton x > IC ker ‹ “ isgivenby ‹ x “ 1 a x x a 1 i x œ i a x œœ i : Since C iscoaugmented,itsplitsas R ` IC ,whichmeansthat 45 C a C ‹ R a R “ ` ‹ IC a R “ ` ‹ R a IC “ ` ‹ IC a IC “ : Because C iscounital, Id ‹ a Id “ X ‹ Id a “ X ; so i x œ i a x œœ i > IC a IC .But R ÐÐ C R ` IC r ( ‹ r; 0 “ hascokernel JC IC ,soforprimitive x > IC JC , 0 ÐÐ PC ÐÐ JC ÐÐ JC a JC x ( x ( 0 meansthat i x œ i a x œœ i > JC a JC mustbezero,andso ‹ x “ 1 a x x a 1asdesired. Example4.1.8 Primitiveelementsinthepolynomialcoalgebra k w 1 ;w 2 ;::: areclassesoftheform w p m i for p char ‹ k “ .Thecoaugmentation k ÐÐ k w 1 ;w 2 ;::: 1 ( 1 hascokernel JC withbasis Ÿ w j 1 ;w j 2 ;::: š forall j C 1.Recallthecomultiplicationisgivenby ‹ w j i “ Q k ‰ j k ’ w k i a w j k i 46 Since p isthecharacteristicof k , ‹ w p m i “ 1 a w p m i w p m i a 1 so w p m i isprimitive.Theother w n i arenotprimitivebecause ‹ w n i “ x 1 a w n i w n i a 1since thosebinomialcotsdonotvanish. Example4.1.9 Intheexteriorcoalgebra k ‹ y 1 ;y 2 ;::: “ ,primitiveelementsareclassesoftheform y i .Recall thatthecoproducton k ‹ y 1 ;y 2 ;::: “ isgivenby ‹ y i “ 1 a y i y i a 1andthereforethethose termsareprimitive. Example4.1.10 Primitiveelementsinthedividedpowercoalgebra k x 1 ;x 2 ;::: areclassesoftheform x i . Recallthatthedividedpowercoalgebrahascomultiplication ‹ j ‹ x i ““ Q a b j a ‹ x i “ a b ‹ x i “ Sosince 0 ‹ x i “ 1and 1 ‹ x i “ x i ,wehave ‹ x i “ 1 a x i x i a 1 : Theother j ‹ x i “ for j A 1arenotprimitivebecausetheirimageunderwillhaveadditional a ‹ x i “ a b ‹ x i “ terms. Studyingprimitiveandindecomposableelementscanbeparticularlyusefulbecauseof resultslikethefollowingfromAngeltveitandRognes: 47 Theorem (Prop4.8[1]) Let A beacommutative S -algebrawith H ⁄ ‹ A ; k “ connectedandsuchthatHH ⁄ ‹ H ⁄ ‹ A ; k ““ isover H ⁄ ‹ A ; k “ .Thenthe E 2 -termoftheokstedtspectralsequence E 2 ⁄ ; ⁄ ‹ A “ HH ⁄ ‹ H ⁄ ‹ A ; k ““ isan H ⁄ ‹ A ; k “ -Hopfalgebra,andashortestnon-zerotial d r s;t inlowesttotaldegree s t ,ifoneexists,mustmapfromanalgebraindecomposabletoacoalgebraprimitivein HH ⁄ ‹ H ⁄ ‹ A ; k ““ . Proof. WewillgothroughtheproofaspresentedbyAngeltveit-Rognes[1]sincetheproof oftheanalogousresultfortheokstedtspectralsequencewillbesimilar. Firstwejustifywhy E 2 ⁄ ; ⁄ ‹ A “ isan H ⁄ ‹ A ; k “ -Hopfalgebra.RecallthattheHopfalgebra structureincludesacomultiplicationmap,multiplicationmap,counitmap,unitmap,and antipode.Aswesawabove,THHhasan A -Hopfalgebrastructureinthehomotopycate- gorythatcomesfromidentifyingtopologicalHochschildhomologywiththesimplicialtensor THH ‹ A “ A a S 1 .Thenthissimplicialstructuregivesathatinducesaspectral sequence.The E 2 -terminthiscaseisHochschildhomologyof H ⁄ ‹ A ; k “ ,andtheproduct andcoproductdescendto E 2 . Nowsuppose d 2 ;:::;d r 1 areallzero.Since E 2 isan H ⁄ ‹ A ; k “ -Hopfalgebra,thismeans E 2 ⁄ ; ⁄ ‹ A “ E r ⁄ ; ⁄ ‹ A “ isstillan H ⁄ ‹ A ; k “ -Hopfalgebra.Iftheclass xy isdecomposablefor classes x;y withpositivedegreesuchthat d r ‹ xy “ x 0,thenapplyingtheLeibnizRuleyields: d r ‹ xy “ d r ‹ x “ y xd r ‹ y “ ; 48 whichimpliesthat d r ‹ x “ x 0or d r ‹ y “ x 0.Therefore xy cannotbeinthelowestpossibletotal degreeforthesourceofthetial,andthusthelowesttotaldegreenonzerotial mustmapfromanalgebraindecomposableinstead. Ontheotherhand,ifweassume d r ‹ z “ isnotacoalgebraprimitive,thentheco-Leibniz Rulesays X d r ‹ z “ ‹ d r a 1 1 a d r “ ‹ z “ ‹ d r a 1 1 a d r “‹ z a 1 1 a z i z œ i a z œœ i “ ‹ d r ‹ z “ a 1 d r ‹ 1 “ a z i d r ‹ z œ i “ a z œœ i “ ‹ z a d r ‹ 1 “ 1 a d r ‹ z “ i z œ i a d r ‹ z œœ i ““ ‹ d r ‹ z “ a 1 i d r ‹ z œ i “ a z œœ i “ ‹ 1 a d r ‹ z “ i z œ i a d r ‹ z œœ i ““ (since d r ‹ 1 “ 0) wherethetensorproductsareover H ⁄ ‹ A ; k “ .Sothisimpliesthat d r ‹ z œ i “ x 0or d r ‹ z œœ i “ x 0 forsome i ,becauseifthey'reallzerothen ‹ d r ‹ z ““ d r ‹ z “ a 1 1 a d r ‹ z “ ; whichbythatsaysthat d r ‹ z “ isprimitive,contradictingourinitialassumption. Sosinceeither d r ‹ z œ i “ x 0or d r ‹ z œœ i “ x 0andthecoproductpreservesdegree(i.e. deg ‹ z œ i “ deg ‹ z œœ i “ deg ‹ z “ for z œ i and z œœ i inpositivedegree), deg ‹ z œ i “ @ deg ‹ z “ and deg ‹ z œœ i “ @ deg ‹ z “ . Butthen z œ i and z œœ i areinlowertotaldegreethan z ,sotheshortestnon-zerotialin lowesttotaldegreehastohitacoalgebraprimitive. Remark4.1.11 TheresultinAngeltveit-Rognes[1]furthershowsthatwhen k F p thereisan A ⁄ -comodule structure,butsincewedon'tuseananalogousstructureinthecoalgebrasettingwechose nottoincludeitintheabovediscussion. 49 4.2 j -Hopfalgebrastructureintheokstedtspec- tralsequence Wehavenowexaminedcertainpropertiesthatwereknownabouttheokstedtspectral sequenceandcomputationalimplicationsofthatstructure,andsowenowwanttoaddress theanalogousokstedtspectralsequencesetting. Aswesawintheprevioussection,underappropriateconditionstheokstedt spectralsequenceforacommutativeringspectrum A isaspectralsequenceofHopfalgebras overthecommutativering H ⁄ ‹ A ; k “ .However,inthisdualsettingwewouldthenwantto showthatwehaveaHopfalgebraoverthecoalgebra H ⁄ ‹ C ; k “ .However,thisrequiresa notionofaHopfalgebraoveracoalgebra,whichBohmann-Gerhardt-Shipley[5]calla j -Hopf algebra.Tostart,wewillthusrecallbackgroundinformationaboutthecotensorproduct j beforestatingthattheokstedtspectralsequencehasa j -Hopfalgebrastructure. Foran R -coalgebra C ,aright C -comodule M with M M a C ,andaleft C -comodule N with N C a N ,thecotensorof M and N over C istobetheequalizerin R -modules: M j C N eq „‹ M a R N “ a Id N / / Id M a / / M a R C a R N ‚ : Notethatthecotensordoesnotalwaysyielda C -comodule,butundersomeconditions itdoes.Inparticular,if C isacoalgebraoveraand M and N are C -bicomodules,then M j C N isa C -bicomodule. Inordertogivetheofa j C -Hopfalgebraforacoalgebra C overa k ,we needtheofa j C -coalgebraanda j C -bialgebra. 4.2.1 ([5]) Let C beacoalgebraoveraA j C -coalgebra D isa C -bicomodulealongwitha 50 comultiplicationmap D D j C D andacounitmap D C thatarecoassociative andcounitalmapsof C -comodules. 4.2.2 ([5]) Let C beacoalgebraoveraA j C -bialgebra D isa j C -coalgebrathatisalsoequipped withamultiplicationmap D j C D D andaunitmap D C thatsatisfyassociativity andunitality.Themultiplicationmustalsobecompatiblewiththe j C -coalgebrastructure. A j C -Hopfalgebra D isa j C -bialgebraalongwithanantipode ˜ D D thatisa C -comodulemapsatisfyingthecorrespondinghexagonalantipodediagram. See[5]formoredetailsonthediagramsforcoassociativityandcounitalityandthose specifyingtheinteractionsbetweenthealgebraandcoalgebrastructures. Aswesawabove,inthecommutativecasewecanidentifyTHH ‹ A “ Y asthetensorof A with S 1 Y .Similarly,coTHH ‹ C “ Y canbeviewedasa co tensorwith S 1 Y .Wealsodiscussedhow [1]showsthatthesimplicialpinchandfoldmapsinducemapsonTHHandtheokstedt spectralsequence.Inthesameway,workofBohmann-Gerhardt-Shipleyshowsthatsim- plicialpinchandfoldmapsinthedualsettinginducemapsontheokstedtspectral sequence.In[5],theydescribetopologicalcoHochschildhomologyasthefollowingcotensor with S 1 coTHH Y ‹ C “ C S 1 Y ; sothatthesimplicialmapson S 1 ultimatelyinducea j -Hopfstructureontheokstedt spectralsequence.Inordertostatethisresult,werequirethefollowing 4.2.3 Foracoalgebra C overa k ,arightcomodule M over C iscalled if M j C is exactasafunctorfromleft C -comodulesto k -modules. 51 Wecannowstatetheanalogof[1,Theorem4.5]forcoTHH. Theorem4.2.4 ([5]) For C acocommutativecoalgebraspectrum,iffor r C 2each E ⁄ ; ⁄ r ‹ C “ isover H ⁄ ‹ C ; k “ , thentheokstedtspectralsequenceisaspectralsequenceof j H ⁄ ‹ C ; k “ -bialgebras. Theyfurthershowthatthisbialgebrastructuremakestheokstedtspectralsequence intoaspectralsequenceof j H ⁄ ‹ C ; k “ -Hopfalgebrasbytheappropriateantipode map. 4.3 j -Hopfalgebrastructureintherelativeokstedt spectralsequence Wenowwanttoseehowthe j -Hopfalgebrastructureextendstotherelativeokstedt spectralsequence.Wewillconsidercoalgebrasover Hk ,for k aAdaptingthenotation fromBohmann-Gerhardt-Shipley'swork[5],wethecotensorinthequasicategory ofcocommutative Hk -coalgebras,denotedby CoCAlg Hk .Notethat[5]usesthenotation CoCAlg ‹ Mod Hk “ basedonthesymmetricmonoidalquasicategoryof Hk -modules, Mod Hk , sothisisminorcondensingofnotation. 4.3.1 Given C > CoCAlg Hk andasimplicialset X Y ,wewrite C X Y forthe cotensor of C withthe simplicialset X Y .Onthe n th cosimpliciallevelthisis: ‹ C X Y “ n M x > X n C: 52 Sousingthisnotionofcotensorin CoCAlg Hk andthesimplicialcircle S 1 Y fromthelast section,coTHH Hk ‹ C “ Y C S 1 Y . Remark4.3.2 Notethatsincetherelativeokstedtspectralsequencewasstatedingeneralityforany homologytheory E ,wenowneedtorestrictourselvestotheappropriateconditionsforthe j -coalgebrastructure.Inthisthesiswearespeinterestedinsuchexampleswhere E S thatare Hk -coalgebras. Theorem4.3.3 ([5]) For C acocommutativecoalgebraspectrum,iffor r C 2each E ⁄ ; ⁄ r ‹ C “ iscoover ˇ ⁄ ‹ C “ , thentherelativeokstedtspectralsequenceisaspectralsequenceof j ˇ ⁄ ‹ C “ -Hopfalgebras. Theprooffollowsasin[5],inwhichcotensoringwiththesimplicialfoldmap S 1 Y - S 1 Y S 1 Y inducesthecomultiplication,andthemultiplicationfurthercomesfromthesimplicialpinch maponthedoublecirclesimplicialmodel dS 1 Y S 1 Y - S 1 Y .Aswesawintheokstedtspectral sequence,wenowwanttousetheadditionalalgebraicstructuretounderstandtials inthespectralsequence.However,inordertomakesenseoftheseideasinthedualsetting, wealsoneedthefollowingandresultsregardingindecomposableandprimitive elements. 4.3.4 Aunital j C -algebra A withmultiplication A j C A A andunit C A is augmented ifthereexistsanaugmentationmap A C suchthat j and Id. 4.3.5 Acounital j C -coalgebra D withcomultiplication D D j C D andcounit D C is 53 coaugmented ifthereexistsacoaugmentationmap C D suchthat j and Id. 4.3.6 ([5]) Givenacoaugmented j C -coalgebra D ,let PD benedbytheshortexactsequence 0 ÐÐ PD ÐÐ JD ÐÐ JD j C JD; where JD coker ‹ “ .Anelementinker ‹ “ is primitive ifitsimagein JD isin PD . 4.3.7 ([5]) Foranaugmented j C -algebra A ,the indecomposables of A ,denotedby QA ,are bytheshortexactsequence IA j C IA ÐÐ IA ÐÐ QA ÐÐ 0 ; where IA ker ‹ “ . SincethetheoremregardingtheHopfstructureintherelativeokstedtspectralse- quencewillrequire ˇ ⁄ ‹ C “ tobeconnected,wethattermhere: 4.3.8 ([5]) Agraded k -coalgebra D ⁄ isconnectedif D ⁄ 0when ⁄ @ 0,andthecounitmap D ⁄ k isanisomorphismindegreezero. Theorem4.3.9 Fora k ,let C beacocommutative Hk -coalgebraspectrumsuchthatcoHH ⁄ ‹ ˇ ⁄ ‹ C ““ is over ˇ ⁄ ‹ C “ andthegradedcoalgebra ˇ ⁄ ‹ C “ isconnected.Thenthe E 2 -termofthe 54 relativeokstedtspectralsequencecalculating ˇ ⁄ ‹ coTHH Hk ‹ C ““ , E ⁄ ; ⁄ 2 ‹ C “ coHH k ⁄ ‹ ˇ ⁄ ‹ C ““ ; isa j ˇ ⁄ ‹ C “ -bialgebra,andtheshortestnon-zerotial d s;t r inlowesttotaldegree s t mapsfroma j ˇ ⁄ ‹ C “ -algebraindecomposabletoa j ˇ ⁄ ‹ C “ -coalgebraprimitive. Proof. Theprooffollowsasinthenon-relativeversionin[5].Notethattherequirement thatcoHH ‹ ˇ ⁄ ‹ C ““ isatover ˇ ⁄ ‹ C “ isreallyaconditionon E 2 .Howeversincewecan dothisargumentpagebypage,notialsonthe E 2 -pageimpliesthat E 2 E 3 and sothesameconditionwillholdforthatpage.Thus,sincewe'rejustconcerned abouttheofthenon-zerotials,theonlyconditionwehavetosatisfyistheone weneedforthe E 2 -page. 55 Chapter5 Explicitcalculations AnaturalquestionthatcomesupwhenstudyingcoTHHistoaskwhatkindsofcoalgebra spectraexist,andforthosethatexist,isthe E 2 -pageoftherelativeokstedtspectral sequencecomputable?AlthoughBohmann-Gerhardt-H˝genhaven-Shipley-Ziegenhagen[4] demonstratethattheokstedtspectralsequencecaninputexamplesoftheform ª X forsimplyconnected X ,Peroux-Shipleyshowthatexamplesof S -coalgebrasarestillquite limited[30].Sonowthatwehaveawayofgeneratingnew R -coalgebraspectraoftheform B , A B andstudyingthemviatherelativeokstedtspectralsequence,wewillgothrough afewspexamples.Inparticular,wewillexaminethecoalgebras H F p , H Z H F p and H F p , BP @ n A H F p (forthose n and p suchthat BP @ n A iscommutative).Inthischapter,we willstartbycomputingthe E 2 -termoftherelativeokstedtspectralsequencecomputing ˇ ⁄ ‹ coTHH ‹ C ““ fortheseexamples. Theseresultsthatwecanindeedcomputethe E 2 -pagesoftherelativeokstedt spectralsequencecalculatingrelativetopologicalcoHochschildhomologyofsomeexamples of,inthiscase, H F p -coalgebras.Asdiscussedinthelastchapter,inordertousethe j - Hopfalgebrastructuretothe E ª -pageandcompletethecomputationofthehomotopy groupsofcoTHH,weneedourcoalgebraspectratobecocommutative.Asaresult,these examplescannotbeusingthe j -Hopfalgebratechniquesbecausetheyarenot cocommutative.However,wewillconsidercocommutativecoalgebraswithhomotopythat 56 issimilartotheabove E 2 -pageexampleslaterinthischapter. 5.1 E 2 -pageExamples Proposition5.1.1 Forthe H F p -coalgebra H F p , H Z H F p ,the E 2 -pageofthespectralsequencecalculating ˇ t s ‹ coTHH H F p ‹ H F p , H Z H F p ““ is E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , H Z H F p ““ F p ‹ ˝ “ a F p F p ! for SS ˝ SS ‹ 0 ; 1 “ ; SS ! SS ‹ 1 ; 1 “ . Proof. ByProposition2.5.5, H F p , H Z H F p isan H F p -coalgebracomingfromthemap ˚ H Z H F p ,whichisinducedby Z mod p ÐÐÐÐ F p . Wewantto E ⁄ ‹ coTHH R ‹ C ““ for R H F p , E S ,and C H F p , H Z H F p asin theCorollary3.0.2.Notethatwesatisfytheconditionthat ˇ ⁄ ‹ H F p , H Z H F p “ is over ˇ ⁄ ‹ H F p “ F p ,becausemodulesareoverCorollary3.0.2statesthatthe relativeokstedtspectralsequencehastheform: E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , H Z H F p ““ ?? Ô ˇ t s ‹ coTHH H F p ‹ H F p , H Z H F p ““ wherethe??serveasareminderthatconvergenceforthisspectralsequence cannotbeautomaticallyassumed.WecanusetheKunnethspectralsequencetocalculate ˇ ⁄ ‹ H F p , H Z H F p “ : Tor E ⁄ ‹ R “ p;q ‹ E ⁄ ‹ M “ ;E ⁄ ‹ N ““ E p q ‹ M , R N “ 57 whichexistsif E ⁄ ‹ R “ isaright R ⁄ -module[15,TheoremIV.6.2].Herewehave E S and R H Z ,sosince E ⁄ ‹ R “ ˇ ⁄ ‹ H Z “ Z isindeedtover R ⁄ ˇ ⁄ ‹ H Z “ Z ,wemay applytheKunnethspectralsequencetoget: Tor ˇ ⁄ ‹ H Z “ p;q ‹ ˇ ⁄ ‹ H F p “ ;ˇ ⁄ ‹ H F p ““ Tor Z p;q ‹ F p ; F p “ ˇ p q ‹ H F p , H Z H F p “ Tocomputethis E 2 -term,wewillneedtocreateaprojectiveresolutionof F p asa Z - module: Z p ÐÐ Z mod p ÐÐÐÐ F p Ð 0 : Thenwecantruncateand a Z F p toget Z a Z F p p ÐÐ Z a Z F p Ð 0 whichto F p p ÐÐ 0 F p Ð 0 Thuswehave F p indegree0and1.Asacoalgebrathisistheexteriorcoalgebrawith asinglegeneratorindegree1.Nowthatweknowthat ˇ ⁄ ‹ H F p , H Z H F p “ isanexterior coalgebraover F p ,the E 2 -pagelookslike: 58 E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , H Z H F p ““ coHH F p s;t ‹ F p ‹ ˝ ““ ( S ˝ S 1) F p ‹ ˝ “ a F p ! (byProposition5.1in[4]) withbidegrees SS ˝ SS ‹ 0 ; 1 “ and SS ! SS ‹ 1 ; 1 “ .Thusthis E 2 -pagelookslike: 0 1 2 3 4 0 1 2 3 4 1 ˝ ! ˝! ! 2 ˝! 2 ! 3 ˝! 3 ! 4 Thereforewehaveshownthedesiredresult. Nowthatwe'veseenonecalculationofan E 2 -termfortherelativeokstedtspec- tralsequence,let'sconsiderasimilar H F p -coalgebraexample, H F p , BP @ n A H F p .Firstwe introduce BP basedonthecomplexcobordismspectrum MU asin[33]. 5.1.2 ([9]) Thespectrum BP ,calledthe Brown-Petersonspectrum ,isnamedbecauseBrownand Petersonshowedthat MU localizedataprimecanbesplitintoawedgeproductofsuspen- sionsof BP .Inparticularitischaracterizedby ˇ ⁄ ‹ BP “ Z ‹ p “ v 1 ;v 2 ;::: ; 59 for S v i S 2 ‹ p i 1 “ .TherearealsotruncatedBrown-Petersonspectra BP @ n A ,with ˇ ⁄ ‹ BP @ n A “ Z ‹ p “ v 1 ;:::;v n asshownby[22]. Remark5.1.3 Notethatinordertoconsider B , A B asa B -coalgebra,weneed A tobeacommutativering spectrum.Because BP @ 0 H Z ‹ p “ isanEilenberg-MacLanespectrumofacommutative ring,itwillalsobecommutative.Similarly,[27]showthat BP @ 1 ` iscommutative sinceitisequivalenttothealgebraic K -theoryofacommutativeringspectrum.However for n 2, BP @ 2 A isonlyknowntobecommutativefor p 2[23]and p 3[21],andsowe limitourexamplestothesecases. Proposition5.1.4 Forthe H F p -coalgebra H F p , BP @ n A H F p for n 0 ; 1andfor n 2attheprimes p 2 ; 3,the E 2 -pageofthespectralsequencecalculating ˇ t s ‹ coTHH H F p ‹ H F p , BP @ n A H F p ““ is E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , BP @ n A H F p ““ F p ‹ ˝ 0 ;:::˝ n “ a F p ! 0 ;:::! n for SS ˝ i SS ‹ 0 ; 2 p i 1 “ ; SS ! i SS ‹ 1 ; 2 p i 1 “ . Proof. H F p , BP @ n A H F p isan H F p -coalgebrabecausebasedonthefor BP @ n A wehave ˇ ⁄ BP @ n Z ‹ p “ v 1 ;v 2 ;:::;v n 60 for S v i S 2 p i 2 ; sothereisamapofringspectra BP @ n A H Z ‹ p “ H F p ; givenbymappingtotheEilenberg-MacLanespectrumon ˇ 0 .Thecompositiongivesa coalgebrastructurebyProposition2.5.5for n 0 ; 1andfor n 2attheprimes p 2 ; 3. WeapplytherelativeokstedtspectralsequenceofCorollary3.0.2with R H F p , E S ,and C H F p , BP @ n A H F p ,givingus: E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , BP @ n A H F p ““ ?? Ô ˇ t s ‹ coTHH H F p ‹ H F p , BP @ n A H F p ““ Nowtocompute ˇ ⁄ ‹ H F p , BP @ n A H F p “ ,wemayagainusetheKunnethspectralsequence: Tor E ⁄ ‹ R “ p;q ‹ E ⁄ ‹ M “ ;E ⁄ ‹ N ““ E p q ‹ M , R N “ whichexistsif E ⁄ ‹ R “ isaright R ⁄ -module.Soherewehave Tor ˇ ⁄ BP @ n A p;q ‹ ˇ ⁄ H F p ;ˇ ⁄ H F p “ Tor Z ‹ p “ v 1 ;v 2 ;:::;v n p;q ‹ F p ; F p “ ˇ p q ‹ H F p , BP @ n A H F p “ Tocomputethis,wewillneedtocreateaprojectiveresolutionof F p asa Z ‹ p “ v 1 ;v 2 ;:::;v n - module.WemayusetheKoszulcomplex,whichwewillincludefor n 2sincethatisthe largestcasewewillconsider: 61 S v 1 v 2 S Z ‹ p “ v 1 ;v 2 ÐÐ S v 1 S Z ‹ p “ v 1 ;v 2 ` S v 2 S Z ‹ p “ v 1 ;v 2 ` S v 1 v 2 S Z ‹ p “ v 1 ;v 2 1 ( ‹ v 2 ; v 1 ;p “ ÐÐ Z ‹ p “ v 1 ;v 2 ` S v 1 S Z ‹ p “ v 1 ;v 2 ` S v 2 S Z ‹ p “ v 1 ;v 2 ÐÐ Z ‹ p “ v 1 ;v 2 ÐÐ F p ÐÐ 0 : ‹ 1 ; 0 ; 0 “ ( ‹ v 1 ; p; 0 “‹ 1 ; 0 ; 0 “ ( p ‹ 0 ; 1 ; 0 “ ( ‹ v 2 ; 0 ; p “‹ 0 ; 1 ; 0 “ ( v 1 ‹ 0 ; 0 ; 1 “ ( ‹ 0 ;v 2 ; v 1 “‹ 0 ; 0 ; 1 “ ( v 2 where isthemodulemap Z ‹ p “ v 1 ;v 2 mod p ÐÐÐÐ F p thatsends v 1 and v 2 to0.Wethen truncateand a Z ‹ p “ v 1 ;v 2 F p togettheresultingcomplex: S v 1 v 2 S F p 0 ÐÐ S v 1 S F p ` S v 2 S F p ` S v 1 v 2 S F p 0 ÐÐ F p ` S v 1 S F p ` S v 2 S F p 0 ÐÐ F p ÐÐ 0 Sincecoassociativecomultiplicationpreservestotaldegree,theresulting F p ` F p ` S v 1 S 1 F p ` S v 2 S 1 F p isanexteriorcoalgebraover F p generatedbywhatwewillcall ˝ 0 ;˝ 1 ;˝ 2 indegrees S ˝ 0 S 1, S ˝ 1 S S v 1 S 1,and S ˝ 2 S S v 2 S 1. 1 AsinTilson[35],thespectralsequencecomputing ˇ ⁄ ‹ H F p , BP @ n A H F p “ thencollapsesatthis E 2 -page,andwerecoverthecalculation: ˇ ⁄ ‹ H F p , BP @ n A H F p “ F p ‹ ˝ 0 ;˝ 1 ;:::;˝ n “ for S ˝ i S S v i S 1 2 p i 1,andourcomputationamountstothefamiliar: E s;t 2 coHH F p s;t ‹ ˇ ⁄ ‹ H F p , BP @ n A H F p ““ coHH F p s;t ‹ F p ‹ ˝ 0 ;˝ 1 ;:::;˝ n ““ ( S ˝ S 2 p i 1) F p ‹ ˝ 0 ;:::;˝ n “ a F p F p ! 0 ;:::;! n (byLemma5.1in[4]) 1 In[35,Prop5.6]thatspexaminesthecasewhere n 2and p 2,thesegeneratorsarecalled 2 ; v 1 ; and v 2 . 62 withbidegrees SS ˝ i SS ‹ 0 ; 2 p i 1 “ and SS ! i SS ‹ 1 ; 2 p i 1 “ . Remark5.1.5 ThankstoaconversationwithMikeHill,wealsohavethefollowingquotientsofthedual Steenrodalgebrafor p 2thatemergeasthehomotopygroupsof H F 2 -coalgebrasofthe form B , A B for S ˘ i S 2 i 1: ˇ ⁄ H F 2 , H F 2 H F 2 F 2 H F 2 , H Z H F 2 ‹ ˘ 1 “ H F 2 , ku H F 2 ‹ ˘ 1 ;˘ 2 “ H F 2 , ko H F 2 F 2 ˘ 1 ;˘ 2 ˘ 4 1 ;˘ 2 2 H F 2 , tmf 1 ‹ 3 “ H F 2 ‹ ˘ 1 ;˘ 2 ;˘ 3 “ H F 2 , tmf H F 2 F 2 ˘ 1 ;˘ 2 ;˘ 3 ˘ 8 1 ;˘ 4 2 ;˘ 2 3 SobyLemma5.1in[4]asintheexamplesthatwesawabove,wecouldsimilarlythe E 2 -pages: E s;t 2 coHH F 2 s;t ‹ ˇ ⁄ ‹ H F 2 , ku H F 2 ““ F 2 ‹ ˘ 1 ;˘ 2 “ a F 2 ! 1 ;! 2 E s;t 2 coHH F 2 s;t ‹ ˇ ⁄ ‹ H F 2 , tmf 1 ‹ 3 “ H F 2 ““ F 2 ‹ ˘ 1 ;˘ 2 ;˘ 3 “ a F 2 ! 1 ;! 2 ;! 3 5.2ComputationalTools Wearegoingtousethe j -Hopfstructureofthelastchaptertogivefurthercomputational tools.Recallfromthepreviouschapterthattheshortestnonzerotialmustgofrom a j -Hopfalgebraindecomposabletoa j -coalgebraprimitive.Westudytheprimitives of j C -coalgebrasoftheform C a D . 63 Proposition5.2.1 ([5]) Forcoaugmented k -coalgebras C and D , C a D isa j C -coalgebraandanelement c a d > C a D isprimitiveasanelementofthe j C -coalgebra C a D ifandonlyif d isprimitiveinthe k - coalgebra D . Bohmann-Gerhardt-ShipleyprovethatifcoHH ‹ D “ isover D thencoHH ‹ D “ isa j D -algebra[5].Wewillfurtherneedtoidentifytheindecomposableelements,butinthat casewewillrestricttothespcomputationalsettingwewillneed. Thelasttoolweintroducehereisthatfor Hk -coalgebrastherelativeokstedtspectral sequenceisitselfaspectralsequenceof k -coalgebras,whichwillthenallowustorestrict tialsevenfurthertotargetsthatare k -coalgebraprimitives.Bohmann-Gerhardt- H˝genhaven-Shipley-Ziegenhagen[4]showedthefollowingresultforthecookstedtspectral sequence,andtherelativecasefollowsfromtheirwork. Theorem5.2.2 If C isaconnectedcocommutative Hk -coalgebrathatistasan Hk -module,then therelativeokstedtspectralsequencefor E S isaspectralsequenceof k -coalgebras. Inparticular,forevery r A 1thereisacoproduct E ⁄ ; ⁄ r ÐÐ E ⁄ ; ⁄ r a k E ⁄ ; ⁄ r ; andthetials d r respectthecoproduct. Proof. Thisprooffollowsasin[4]sincefor E S wearealreadyinthesettingofcosimplicial Hk -modules. 64 5.3ExteriorInputs ThegoalofthissectionistocomputethehomotopygroupsofthetopologicalcoHochschild homologyofcoalgebraspectrawithanexteriorhomotopycoalgebra.Nowbecauseweproved inthepreviouschapterthatourspectralsequencehasa j -Hopfalgebrastructure,wewill usethattheshortestnonzerotialgoesfromanalgebraindecomposabletoacoalgebra primitive. Theorem5.3.1 Fora k ,let C beacocommutative Hk -coalgebraspectrumthatisrantasan Hk - modulewith ˇ ⁄ ‹ C “ k ‹ y “ for S y S oddandgreaterthan1.Thentherelativecookstedt spectralsequencecollapsesand ˇ ⁄ ‹ coTHH Hk ‹ C ““ k ‹ y “ a k w asgraded k -modulesfor S w S S y S 1. Proof. Recallthattheconditionoftherelativeokstedtspectralsequenceis becausewe'retaking E S and R Hk ,sothe E 2 -pageis E s;t 2 coHH k s;t ‹ k ‹ y ““ k ‹ y “ a k w byProposition5.1in[4].Butnowbecausethedegreeof y isbothoddandgreaterthan1, wewillshowthatthespectralsequenceissparseenoughthatallditialswillbezero. ByProposition4.3.9weknowthattheshortestnontrivialtialinlowesttotal degreemustmapfroma j k ‹ y “ -algebraindecomposabletoa j k ‹ y “ -coalgebraprimitive. Sincethe E 2 -pageisgivenby k ‹ y “ a k w ,Proposition5.2.1impliesthatelementsinthis 65 j k ‹ y “ -coalgebrawillbeprimitiveifandonlyifthecomponentfrom k w isprimitiveinthe k -coalgebra k w .Recallthatprimitivesinthe k -coalgebra k w 1 ;w 2 ;::: moregenerallyare oftheform w p m i for p char ‹ k “ ,sohere k ‹ y “ a › primitivesin k w ” k ‹ y “ a w p m Thereforetheonlytermsinthespectralsequencethatarepossibletargetsoferentials are w p m and yw p m for m C 0andprime p . Bohmann-Gerhardt-Shipleyalsoidentifytheindecomposableelementsforthe j k ‹ y “ - algebra k ‹ y “ a k w asthoseoftheform k ‹ y “ a w sincetheindecomposableelements of k w 1 ;w 2 ;::: moregenerallyare w i [5].Thustheonlytermsinthespectralsequence thatarepossiblesourcesoftialsare y and yw ,since yw j isdecomposablefor j A 1. Becausethepossibletargetsareoftheform w p m and yw p m ,andthe yw p m appearinthe samediagonalasboth y and yw ,thoseelementscannotbehitbyany ‹ r;r 1 “ -bidegree tial.Thusweneedonlyjustifywhytialsfrom y and yw cannothittermsof theform w p m . Notethattheelementsweareconsideringliveinthefollowingbidegrees( SS SS )and ‹ t s “ totaldegrees( S S )for m;n C 1: SS y SS ‹ 0 ; 2 n 1 “ (since S y S 2 n 1isoddand A 1) SS w p m SS ‹ p m ;p m ‹ 2 n 1 ““ ‹ p m ; 2 np m p m “ (since SS w SS ‹ 1 ; 2 n 1 “ ) SS yw SS ‹ 1 ; 4 n 2 “ SS d r ‹ y “SS ‹ r; S y S r 1 “ ‹ r; 2 n 1 r 1 “ ‹ r; 2 n r “ SS d r ‹ yw “SS ‹ 1 r; 4 n 2 r 1 “ ‹ 1 r; 4 n r 1 “ First,wewilljustifythat SS d r ‹ y “SS x SS w p m SS forany m C 1.Supposebycontradictionthat 66 thesetermswereinthesamebidegrees.Thenthecoordinatetellsusthat r p m ,sowe havefromthesecondcoordinate: 2 n p m 2 np m p m 2 n 2 np m 1 p m (since n C 1) but p isprimeand m C 1,sowehaveacontradiction. Secondwejustifythat SS d r ‹ yw “SS x SS w p m SS forany m C 1.Supposebycontradictionthat thesetermswereinthesamebidegrees.Thenthecoordinatetellsusthat r 1 p m ,so wehavefromthesecondcoordinate: 4 n p m 2 np m p m 4 n 2 np m 2 p m ; whichistrueonlywhen m 1and p 2.However,if m 1then r 1 p m 2 1 impliesthat r 1andwearealreadyconsideringthe E 2 -page,sonosuchtialexists. NowwewanttomakesuretheconvergenceconditionsofCorollary3.0.5hold;thatis,if forevery s thereexistssome r sothat E s;s i r E s;s i ª thentherelativeokstedtspectral sequenceconvergescompletelyto ˇ ⁄ ‹ coTHH Hk ‹ C ““ .However,becausethetials startingatthe E 2 -pagemustbetrivial,wesatisfythisconditionforconvergence,which yields: E 2 E ª k ‹ y “ a k w andsowehaveanisomorphismwith ˇ ⁄ ‹ coTHH Hk ‹ C ““ asgraded k -modules. 67 Nextweconsiderthecomputationwhenweincreasethenumberofcogenerators.Bohmann- Gerhardt-Shipleyidentheindecomposableelementsforthissettingmoregenerally: Proposition5.3.2 ([5]) Theindecomposableelementsinthe j k ‹ y 1 ;y 2 ;:::;y n “ -algebra coHH ‹ k ‹ y 1 ;y 2 ;:::;y n ““ k ‹ y 1 ;y 2 ;:::;y n “ a k w 1 ;w 2 ;:::;w n aregivenby k ‹ y 1 ;y 2 ;:::;y n “ a w i . Theorem5.3.3 Let k bealdandlet p char ‹ k “ ,including0.For C acocommutative Hk -coalgebra spectrumthatistasan Hk -modulewith ˇ ⁄ ‹ C “ k ‹ y 1 ;y 2 “ for S y 1 S ; S y 2 S bothodd andgreaterthan1,if p m isnotequalto S y 2 S 1 S y 1 S 1 or S y 2 S 1 S y 1 S 1 1forall m C 0,thentherelative okstedtspectralsequencecollapsesand ˇ ⁄ ‹ coTHH Hk ‹ C ““ k ‹ y 1 ;y 2 “ a k w 1 ;w 2 ; asgraded k -modulesfor S w i S S y i S 1. Proof. Suppose S y 1 S a and S y 2 S b sothatonthe E 2 -pageofthespectralsequence y 1 appears inbidegree ‹ 0 ;a “ ; and y 2 appearsinbidegree ‹ 0 ;b “ ,whichimplies SS w 1 SS ‹ 1 ;a “ ; SS w 2 SS ‹ 1 ;b “ . ThenweassumeWLOGthat b C a andwewilldetermineifthereisthepossibilityfor tialsbyexaminingthedegreesofthetermsinthespectralsequence.Wewillreferto ourassumptionsthat p m isnotequalto S y 2 S 1 S y 1 S 1 as condition1 andnotequalto S y 2 S 1 S y 1 S 1 1as condition2 . Notethatbecauseofthe j -coalgebrastructurefromProposition4.3.9theshortestnon- 68 trivialtialhastohitacoalgebraprimitive.If char ‹ k “ p aprime,thenbyProposi- tion5.2.1coalgebraprimitiveswillbeoftheform k ‹ y 1 ;y 2 “ a w p m i sincetheprimitivesin k w 1 ;w 2 areoftheform w p m 1 or w p n 2 .However,byTheorem5.2.2 therelativeokstedtspectralsequenceinthissettingalsohasacoalgebrastructureover k .Thereforethenontrivialtialhastohita k -coalgebraprimitive,thatisonly classesoftheform y i or w p m i (andnotanyoftheirtensoredcombinations).Sincethe y i s appearinthezerocolumn,theycannotbehitbyanytials,soouronlypossibletargets areclasses w p m 1 or w p n 2 .Similarly,if char ‹ k “ 0thentheonlyprimitivesin k w 1 ;w 2 are w 1 and w 2 . Further,thesourceoftheshortestnontrivialntialmustbea j -algebraindecompos- able,whichbyProposition5.3.2willbeoftheform k ‹ y 1 ;y 2 “ a w i .Thusweonlyconsider tialsfromthefollowingsourcesthatlandinbidegrees: SS d r ‹ y 1 “SS ‹ r;a r 1 “ SS d r ‹ y 2 “SS ‹ r;b r 1 “ SS d r ‹ y 1 y 2 “SS ‹ r;a b r 1 “ SS d r ‹ w 1 “SS ‹ 1 r;a r 1 “ SS d r ‹ w 2 “SS ‹ 1 r;b r 1 “ SS d r ‹ y 1 w 1 “SS ‹ 1 r; 2 a r 1 “ SS d r ‹ y 2 w 1 “SS ‹ 1 r;a b r 1 “ SS d r ‹ y 1 w 2 “SS ‹ 1 r;a b r 1 “ SS d r ‹ y 2 w 2 “SS ‹ 1 r; 2 b r 1 “ SS d r ‹ y 1 y 2 w 1 “SS ‹ 1 r; 2 a b r 1 “ SS d r ‹ y 1 y 2 w 2 “SS ‹ 1 r;a 2 b r 1 “ 69 Theprimitiveelementsthatcouldserveaspossibletargetsliveinbidegrees: SS w p m 1 SS ‹ p m ;ap m “ SS w p m 2 SS ‹ p m ;bp m “ Notethatifthereisanonzerotialhittingoneoftheseclasses,comparingthe degreeofthecoordinateimpliesinformationabouteither r or1 r .Inthe char ‹ k “ 0 case, S w 1 S ‹ 1 ;a “ and S w 2 S ‹ 1 ;b “ implythatnonontrivialtialsexistsinceweare alreadyonthe E 2 -page.Thusweassume char ‹ k “ p isprimesothatthecoordinate implies r p m or1 r p m ,whichwewillusetosimplifythesecondcoordinateofthe bidegree. Suppose d r ‹ y 1 “ hitsaclass w p m 1 .Thenbycomparingdegrees: a p m 1 ap m : Thisgiveseitherthat a 1(exceptwe'reassuming S y i S A 1)orthat m 0(then a couldbe anything),butinthatcase r p m 1,andwe'realreadyonthe E 2 -page.Thusthereisno suchpossibletial.Asimilarargumentcanbeusedtojustifywhy d r ‹ y 2 “ x w p m 2 . Suppose d r ‹ y 1 “ hitsaclass w p m 2 .Thenbycomparingdegrees: a p m 1 bp m : So a 1 b 1 p m ,butweassumedthat b C a ,sothisequalityonlyholdsif p m r 1.Butwe areconsideringthe E 2 -page,sonosuchtialexists.Asimilarregarding 70 r determinesthat d r ‹ y 1 w 1 “ x w p m 1 and d r ‹ y 2 w 2 “ x w p m 2 . Suppose d r ‹ y 2 “ hitsaclass w p m 1 .Thenbycomparingdegrees: b p m 1 ap m so p m b 1 a 1 .Nowweassumedin condition1 that S y 2 S 1 S y 1 S 1 x p m ,sonosuchtialexists. Suppose d r ‹ y 1 y 2 “ hitsaclass w p m 1 .Then a b p m 1 ap m ; so b ‹ p m 1 “‹ a 1 “ ,but b isoddand a 1isevenandwecan'thaveequalitydue totheparityissue,sotherearenosuchpossibletials.Similarparityissuesarise toshow d r ‹ y 1 y 2 “ x w p m 2 ,aswellasfor d r ‹ w 1 “ x w p m 1 or w p m 2 , d r ‹ w 2 “ x w p m 1 or w p m 2 , d r ‹ y 1 y 2 w 1 “ x w p m 1 or w p m 2 ,and d r ‹ y 1 y 2 w 2 “ x w p m 1 or w p m 2 . Nowsuppose d r ‹ y 1 w 1 “ hitsaclass w p m 2 .Thenthecoordinateimpliesthat r 1 p m , sothesecondcoordinategives: 2 a r 1 b ‹ 1 r “ ; so2 a 1 b 1 r 1.But a B b so2 a 1 b 1 B 2 ‹ 1 “ @ 3 B r 1since r C 2andsonosuchtial exists.Similarbasedontheassumptionthat a B b allowustoconcludethat d r ‹ y 2 w 1 “ x w p m 2 and d r ‹ y 1 w 2 “ x w p m 2 . Suppose d r ‹ y 2 w 1 “ hitsaclass w p m 1 .Then a b p m 2 ap m ; 71 so b 1 a 1 p m 1.Howeverweassumedin condition2 that p m cannotbeequalto S y 2 S 1 S y 1 S 1 1, sonosuchtialexists.Thisconditionalsoarisesinthecase d r ‹ y 1 w 2 “ x w p m 1 . Finallysuppose d r ‹ y 2 w 2 “ hitsaclass w p m 1 .Then 2 b p m 2 ap m ; so2 b 1 a 1 p m .However,weclaimthattheassumption p m x 2 S y 2 S 1 S y 1 S 1 isalreadyeliminated bytheexistingconditions.First,if m 0then p m 1andthisassumptiondoesnotapply sinceweassumedabovethat S y 2 S C S y 1 S .Therefore,weneedonlyjustifythat p m x 2 S y 2 S 1 S y 1 S 1 for m C 1.If p isodd,anoddprime p toanypowerwillstillbeoddandso p m x 2 S y 2 S 1 S y 1 S 1 dueto parity. If p 2,considerthecasewhere S y 2 S 1 S y 1 S 1 isodd.Then2 S y 2 S 1 S y 1 S 1 willonlybeequaltoapower of p 2ifthepoweris1.But m 1wouldimplyherethat r 1andwearealreadyonthe E 2 -page.If S y 2 S 1 S y 1 S 1 iseven,thenverifyingthat p m x 2 S y 2 S 1 S y 1 S 1 for m C 1isequivalenttochecking that p n x S y 2 S 1 S y 1 S 1 for n C 0,i.e. condition1 .So,nosuchrentialfrom y 2 w 2 to w p m 1 exists if condition1 is Wehavenowviacombinatoricswhyallpossibletialscanbeeliminated, whetherthatisforparityreasons,becausewe'realreadyonthe E 2 -page,orbecausewe restrictedvaluesof p m basedontheconditionslistedinthehypotheses.Thusthespectral sequencecollapses,andtheconvergenceconditionsofCorollary3.0.5holdsowehavethe desiredresult. Remark5.3.4 Notethattheconditionson p m allowustoavoidcaseslike S y 1 S 3 ; S y 2 S 5,whichhasa possible d 2 tialfrom y 2 to w 2 1 fortheprime p 2(whichisinthiscaseiseliminated 72 by condition1 ). 5.4DividedPowerInput AlongwithprovingtheexistenceoftheokstedtspectralsequenceinBohmann-Gerhardt- H˝genhaven-Shipley-Ziegenhagen[4],theyshowthefollowingtcomputationalre- sult: Theorem5.4.1 ([4]5.4) Let C beacocommutativecoassociativecoalgebraspectrumthatistasaspectrum, andwhosehomologycoalgebrais H ⁄ ‹ C ; k “ k x 1 ;x 2 ;::: ; wherethe x i arecogeneratorsinnon-negativeevendegreesandthereareonlymany cogeneratorsineachdegree.Thentheokstedtspectralsequencefor C collapsesat E 2 , and E 2 E ª k x 1 ;x 2 :::: a k ‹ z 1 ;z 2 ;::: “ with x i indegree ‹ 0 ; S x i S“ and z i indegree ‹ 1 ; S x i S“ . Nowwewouldliketohaveananalogousresultfortherelativeokstedtspectralse- quenceforthecasewhen E S and R Hk fora k .Recallthattheserestrictionsallow ustousethe j -Hopfalgebrastructureofthespectralsequencetoeliminatecertainpossible tials. Theorem5.4.2 Let C beacocommutativecoassociative Hk -coalgebraspectrumthatistasan 73 Hk -modulespectrum,andwhosehomotopycoalgebrais ˇ ⁄ ‹ C “ k x 1 ;x 2 ;::: ; wherethe x i areinnon-negativeevendegreesandthereareonlymanyofthem ineachdegree.Thentherelativeokstedtspectralsequencecalculatingthehomotopy groupsofthetopologicalcoHochschildhomologyof C collapsesat E 2 ,and ˇ ⁄ ‹ coTHH Hk ‹ C ““ k x 1 ;x 2 ;::: a k ‹ z 1 ;z 2 ;::: “ as k -modules,with z i indegree S x i S 1. Proof. Since E ⁄ ‹ C “ ˇ ⁄ ‹ C “ k x 1 ;x 2 ;::: istover E ⁄ ‹ R “ ˇ ⁄ ‹ Hk “ k ,therelative okstedtspectralsequencethatabutsto ˇ t s ‹ coTHH Hk ‹ C ““ has E 2 -page E s;t 2 coHH k s;t ‹ k x 1 ;x 2 ;::: ByProposition5.1in[4], coHH k ⁄ ; ⁄ ‹ k x 1 ;x 2 ;::: k x 1 ;x 2 ;::: a k ‹ z 1 ;z 2 ;::: “ ; where SS z i SS ‹ 1 ; S x i S“ .Nowwewanttoexaminethetialsonthis E 2 -pageofour spectralsequence.Inparticular,Theorem4.3.9saysthatthecoalgebrastructureimpliesthat theshortestnonzerotialhastohita j -coalgebraprimitive.SincecoHH ⁄ ‹ ˇ ⁄ ‹ C ““ is a j -coalgebraover ˇ ⁄ ‹ C “ k x 1 ;x 2 ;::: ,weknowbyProposition5.2.1thattheprimitive 74 elementswillbeoftheform k x 1 ;x 2 ;::: a › primitivesin k ‹ z 1 ;z 2 ;::: “ ” ; wheretheprimitivesin k ‹ z 1 ;z 2 ;::: “ viewedasa k -coalgebraareoftheform z i . Notethatsinceallofthe x i 'sappearindegree ‹ 0 ; S x i S“ ,all x i 'sandallthedividedpowers willstayinthezerocolumn.Similarly,theexteriorcogenerator z i isindegree ‹ 1 ; S x i S“ ,and soallpossibletargets,i.e.combinationsof x i 'swithasingle z i ,willbeinthecolumn. Becauseweareonthe E 2 -page,thedtialsofbidegree ‹ 2 ; 1 “ willbemappingoutside ofthesetwocolumns,aswillallpossible d r tialsonlater E r -pages.Thusbeyond thezeroandcolumns,theonlyelementsthatmaybehitbytialsarethosethat includeatleast z i z j .However,aswesaidabove,suchelementsarenotprimitive,andthe shortestnon-zerotial d s;t r inlowesttotaldegree s t hastohita j ˇ ⁄ ‹ C “ -coalgebra primitive.Therefore,ourspectralsequencecollapsesat E 2 . NowwewanttomakesuretheconvergenceconditionsofCorollary3.0.5hold;thatis,if forevery s thereexistssome r sothat E s;s i r E s;s i ª thentherelativeokstedtspectral sequenceconvergescompletelyto ˇ ⁄ ‹ coTHH Hk ‹ C ““ .However,becausethetials startingatthe E 2 -pagemustbetrivial,wesatisfythisconditionforconvergence,andsowe havethefollowingisomorphismof k -modules: ˇ ⁄ ‹ coTHH Hk ‹ C ““ k x 1 ;x 2 ;::: a k ‹ z 1 ;z 2 ;::: “ : . 75 Chapter6 Shadows Inasymmetricmonoidalcategory ‹ C ; a ; 1 “ ,anobject C iscalled dualizable withdual D > C ifthereisacoevaluationmap 1 C a D andevaluationmap D a C 1thatsatisfy thetriangleidentities: ‹ Id C a “ X ‹ a Id C “ C C a D a C C Id C ‹ a Id D “ X ‹ Id D a “ D D a C a D D Id D Usingthisstructure,onecanethetraceofamap f C C as 1 Ð C a D f a Id ÐÐÐ C a D D a C ÐÐ 1 : Observethatthesymmetricmonoidalsettingcriticallyprovidesthesymmetryisomorphism C a D D a C .Onemightwanttoextendthenotionoftracetobicategories.Fortwo objects C and D inabicategory,thereisahorizontalcomposition C b D .However,one wouldnotexpecttohaveasymmetryisomorphismrelating C b D and D b C .Indeed, C b D and D b C maynotevenliveinthesamecategory. WorkofPonto[31]andPonto-Shulman[32]developsanotionofabicategoricalshadow toaddressthisissue.Morerecently,workofCampbell-Ponto[11]usedthisframeworkto showthatTHHisashadow.InthischapterwewillshowthatcoHochschildhomology 76 (coHH)isalsoashadow.Noteinparticularthatoncewehavethestructureofashadow, otherpropertiessuchasMoritainvariancefollowasaconsequence.Someoftheseproperties werealreadyshownviaothermethods,buttheframeworkofshadowsgivesusanother perspective. 6.1(Co)BarConstructions Becausewewillneedbarandcobarconstructionstogiveexamplesofshadowsinthischapter, westatethosehere. 6.1.1 Let k beacommutativering, A a k -algebra, M aright A -module,and N aleft A -module. asimplicial k -module M a k A a k A a k N M a k A a k N M a k N withfacemapsgivenby d i ‹ m a a 1 a ::: a a r a n “ ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ ma 1 a ::: a a r a ni 0 m a a 1 a ::: a a i a i 1 a ::: a a r a n 1 B i @ r m a a 1 a ::: a a r 1 a a r ni r; anddegeneracymapsthatinserttheunitmap: 77 s i ‹ m a a 1 aa a r a n “ ¢ ¨ ¨ ¦ ¨ ¨ ¤ m a 1 a a 1 aa a r a ni 0 m a a 1 aa a i a 1 a a i 1 aa a r a n 1 B i B r: Thenthe two-sidedbarcomplex Bar Y ‹ M;A;N “ isgivenbythesimplicial k -module above.Furtheronecanformachaincomplexof k -modulesviatheboundarymap d i ‹ 1 “ i d i tocreatethe two-sidedbarconstruction Bar ‹ M;A;N “ . 6.1.2 Let R beacommutativeringspectrum, A an R -algebra, M aright A -modulewithstructure map M , A M ,and N aleft A -modulewith A , N N ,whereall , inthis areover R .asimplicial R -module M , A , A , N M , A , N M , N withfacemapsgivenby d i ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ , Id , r i 0 Id , i , , Id , r i 1 B i @ r Id , r , i r; anddegeneracymapsthatinserttheunitmap: s i Id i 1 , , Id r i 1 ; for0 B i B r .Thenthe two-sidedbarcomplex Bar Y ‹ M;A;N “ isgivenbythesimplicial 78 R -moduleabove,whichwecanthengeometricallyrealizetogetthe two-sidedbar construction Bar ‹ M;A;N “ . 6.1.3 Let k beacommutativering, C a k -coalgebra, M aright C -comodulewithrightcoaction M M a k C ,and N aleft C -comodulewithleftcoaction N C a k N .a cosimplicial k -comodule M a k C a k C a k N M a k C a k N M a k N withcofacemapsgivenby i ¢ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¤ a Id a r 1 i 0 Id a i a a Id a ‹ r i 1 “ 1 B i B r Id a r 1 a i r 1 andcodegeneracymapsthatinsertthecounitmapfor0 B i B r 1: ˙ i Id a ‹ i 1 “ a a Id a r i : Wedenotethecosimplicial two-sidedcobarcomplex by coBar Y ‹ M;C;N “ .Onecanform acochaincomplexof k -comodulesviatheboundarymap i ‹ 1 “ i i tocreatethe two- sidedcobarconstruction coBar ‹ M;C;N “ . 79 6.2ShadowBackground Werecallsomebasic 6.2.1 ([32,11]) A bicategory B consistsof ‹ acollectionobjects, ob ‹ B “ ,called0-cells ‹ categories B ‹ R;T “ foreachpair R;T > ob ‹ B “ .Theobjectsinthesecategoriesare referredtoas1-cellsandthemorphismsas2-cells. ‹ unitfunctors U R > ob ‹ B ‹ R;R ““ forall R > ob ‹ B “ ‹ horizontalcompositionfunctorsfor R;T;V > ob ‹ B “ b B ‹ R;T “ B ‹ T;V “ B ‹ R;V “ whicharenotrequiredtobestrictlyassociativeorunital. ‹ naturalisomorphismsfor M > ob ‹ B ‹ R;T ““ , N > ob ‹ B ‹ T;V ““ ,and P > ob ‹ B ‹ V;W ““ , Q > ob ‹ B ‹ W;X ““ for R;T;V;W;X > ob ‹ B “ a ‹ M b N “ b P ÐÐ M b ‹ N b P “ l U R b M ÐÐ M r M b U T ÐÐ M thatsatisfythemonoidalcategorycoherenceaxioms(triangleidentityandpentagon identity): 80 ‹ M b U T “ b N M b ‹ U T b N “ M b N r b Id a Id b l ‹ M b N “ b ‹ P b Q “ ‹‹ M b N “ b P “ b Q M b ‹ N b ‹ P b Q ““ ‹ M b ‹ N b P ““ b Q M b ‹‹ N b P “ b Q “ a a a b Id a Id b a Example6.2.2 Thebicategory Mod ~ Ring whose0-cellsarerings,and Mod ~ Ring ‹ R;T “ R Mod T isthe categoryof ‹ R;T “ -bimodulesforrings R;T .Theunit U R isthe ‹ R;R “ -bimodule R ,and horizontalcompositionisgivenbythetensorproductofbimodules b B ‹ R;T “ B ‹ T;V “ B ‹ R;V “ ‹ M;N “ ( M b N M a T N Example6.2.3 Thebicategory D ‹ Ch ~ Ring “ has0-cellsthatareringsand D ‹ Ch ~ Ring “‹ R;T “ D ‹ R Mod T “ isthederivedcategoryof ‹ R;T “ -bimodules.Theunit U R isthe ‹ R;R “ -bimodule R viewed asachaincomplex,andhorizontalcompositionisgivenbythederivedtensorproduct a L : b B ‹ R;T “ B ‹ T;V “ B ‹ R;V “ ‹ M;N “ ( M b N M a L T N Notethat Bar Y ‹ M;T;N “ M a L T N viewedasatrivialsimplicialobjectviatheisomorphism 81 thatindegree j multipliestogetherallthefactorsof T .Thuswemayalsoconsiderthis horizontalcompositionasthetwo-sidedbarconstruction. Example6.2.4 Let D ‹ Mod ~ RingSpectra “ denotethebicategorywhose0-cellsareringspectra,andforring spectra R;T D ‹ Mod ~ RingSpectra “‹ R;T “ isthehomotopycategoryof ‹ R;T “ -bimodules. Theunit U R isthe ‹ R;R “ -bimodulespectrum R ,andhorizontalcompositionisgivenbythe derivedsmashproduct , L ofspectra b B ‹ R;T “ B ‹ T;V “ B ‹ R;V “ ‹ M;N “ ( M b N M , L T N Notethatasinthepreviousexample M , L T N Bar ‹ M;T;N “ [15,PropIV.7.5],andsowe mayalsoconsiderthishorizontalcompositionasthetwo-sidedbarconstruction. Nowthatwehavetheunderlyingbicategoricalstructure,wewillashadowonthat bicategory: 6.2.5 ([31,32]) A shadowfunctor forabicategory B consistsoffunctors `` ee C B ‹ C;C “ T forevery C > ob ‹ B “ andsomecategory T equippedwithanaturalisomorphismfor M > B ‹ C;D “ , N > B ‹ D;C “ `` M b N ee C ÐÐ `` N b M ee D : 82 For P > B ‹ C;C “ ,thesefunctorsmustsatisfythefollowingcommutativediagrams(when theymakesense): ``‹ M b N “ b P ee C / / `` a ee `` P b ‹ M b N “ee C `` a ee / / ``‹ P b M “ b N ee C `` M b ‹ N b P “ee C / / ``‹ N b P “ b M ee D `` a ee / / `` N b ‹ P b M “ee D O O `` P b U C ee C / / `` r ee ( ( `` U C b P ee C `` l ee / / `` P b U C ee C `` r ee v v `` P ee C Wecannowconsidershadowsforthebicategoriesthatweintroducedearlier. Example6.2.6 The\underivedversion"ofHochschildhomology(orHH 0 ‹ R ; M “ )isashadowonthebicat- egory Mod ~ Ring [32].Recallthat Mod ~ Ring ‹ R;R “ R Mod R ,solet R bearingand M be an ‹ R;R “ -bimoduleto `` ee R R Mod R A b M ( R a R a R op M HH 0 ‹ R ; M “ where A b isthecategoryofabeliangroupsandtheisomorphismabovefollowssince R a R a R op M H 0 ‹ R a L R a R op M “ Tor R a R op 0 ‹ R;M “ HH 0 ‹ R ; M “ Equivalentlywecouldthisshadowofthe ‹ R;R “ -bimodule M tobethecoequalizerof R a M / / / / M / / `` M ee ; where and aretherightandleftmoduleactionsrespectively. Themainpropertyofshadowsthatwewanttojustifyisthatforan ‹ R;T “ -bimodule M 83 andan ‹ T;R “ -bimodule N ,thereisanisomorphism `` M b N ee R ÐÐ `` N b M ee T Unpackingthiswesee `` M b N ee R `` M a T N ee R R a R a R op ‹ M a T N “ HH 0 ‹ R ; M a T N “ `` N b M ee T `` N a R M ee T T a T a T op ‹ N a R M “ HH 0 ‹ T ; N a R M “ Sojustifyingthatthereissuchanisomorphism comesdowntocomparing M a T N quo- tientedbytheactionof R a R op and N a R M quotientedbytheactionof T a T op . Recallthe0 th Hochschildhomologyofa k -algebra A withcotsinan ‹ A;A “ - bimodule B isgivenby: HH 0 ‹ A ; B “ B • @ ab ba A : Soto above,weneedamap M a T N • @ rm a n m a nr A ÐÐ N a R M • @ tn a m n a mt A : Wewouldliketo asthemapthatswapsthetensorfactors m a n ( n a m .However, inorderforthismaptobeawmap,weneedtoverifythat rm a n m a nr maps to0.Butbecauseoftheuniversalpropertyof a R ,wecanbring r throughthetensorsothat takes 84 rm a T n m a T nr z n a R rm nr a R m n a R rm n a R rm 0 : Thereforewehave inonedirection.Asimilarargumentjustithatitsinverse sends n a m ( m a n ,andtogetherthesegivethedesiredisomorphism: HH 0 ‹ R ; M a T N “ ÐÐ HH 0 ‹ T ; N a R M “ : WehaveseenabovethatHH 0 ‹ R ; “ isashadow.NowwewillseethatHochschild homologyisaswell. Example6.2.7 Hochschildhomologyisashadowonthecategory D ‹ Ch ~ Ring “ .Solet R bearingand M beachaincomplexof ‹ R;R “ -bimodulesto `` ee R D ‹ Ch ~ Ring “‹ R;R “ D ‹ Ch Z “ M ( R a L R a R op M HH ‹ R;M “ where Ch Z ischaincomplexesofabeliangroupsandHH ‹ R;M “ denotesthecomplexwhose homologygivesHochschildhomology.Theisomorphismabovefollowssince H i ‹ R a L R a R op M “ Tor R a R op i ‹ R;M “ HH i ‹ R ; M “ Again,theargumentamountstojustifyingthatfor M achaincomplexof ‹ R;T “ - bimodulesand N achaincomplexof ‹ T;R “ -bimodules,thereisanisomorphism 85 `` M b N ee R ÐÐ `` N b M ee T Unpackingthisandusingthefactthat M a L T N Bar ‹ M;T;N “ wesee `` M b N ee R `` M a L T N ee R R a L R a R op ‹ M a L T N “ HH ‹ R;M a L T N “ HH ‹ R;Bar ‹ M;T;N ““ `` N b M ee T `` N a L R M ee T T a L T a T op ‹ N a L R M “ HH ‹ T;N a L R M “ HH ‹ T;Bar ‹ N;R;M ““ Sojustifyingthatthereissuchanisomorphism amountstoconstructinganisomorphism: HH ‹ R;Bar ‹ M;T;N ““ ÐÐ HH ‹ T;Bar ‹ N;R;M ““ : RecallthatHochschildhomologyiscalculatedusingacyclicbarconstruction,andapplying theDold-Kancorrespondencebetweenchaincomplexesandsimplicial k -modulesallowsus toidentifybothoftheabovebisimplicialchaincomplexes,HH Y ‹ R ; Bar Y ‹ M;T;N ““ and HH Y ‹ T ; Bar Y ‹ N;R;M ““ withthebisimplicialobject H YY thatatthe ‹ i;j “ -spotisgivenby: R a R a ::: a R ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ i aa NM aa j ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ T a T a ::: a T wherethefacemapsaregivenbymultiplicationofadjacentterms.Thenthemap isgiven degree-wiseby: M a T aa T a N a R aa R ÐÐ N a R aa R a M a T aa T m a t 1 a t j a n a r 1 aa r i ( n a r 1 aa r i a m a t 1 aa t j ; 86 wherethe isdeterminedbytheKoszulsignconvention.Thismapisbecause ofthe ‹ R;T “ -bimodulestructureon M andthe ‹ T;R “ -bimodulestructureon N .This bisimplicialidenthengivesanequivalenceofcomplexesandthusthedesired isomorphism .ThisapproachisoftenreferredtoasaDennis-WaldhausenMoritaArgument [36,3]. Example6.2.8 ([11]) TopologicalHochschildhomologyisashadowonthebicategory D ‹ Mod ~ RingSpectra “ .Let R bearingspectrumand M an ‹ R;R “ -bimodulespectrum,and `` ee R Ho ‹ Mod ‹ R;R “ “ Ho ‹ Sp “ M ( THH ‹ R;M “ whereHo ‹ Sp “ isthehomotopycategoryofspectra. Again,theargumentamountstojustifyingthatforan ‹ R;T “ -bimodule M anda ‹ T;R “ - bimodule N ,thereisanisomorphism `` M b N ee R ÐÐ `` N b M ee T Unpackingthisasbeforeshowsthatjustifyingthat isanisomorphismisequivalentto showingthatthereisanisomorphism THH ‹ R;Bar ‹ M;T;N ““ ÐÐ THH ‹ T;Bar ‹ N;R;M ““ : AsabovewemayapplytheDennis-WaldhausenMoritaArgumenttoidentifyboth THH Y ‹ R;Bar Y ‹ M;T;N ““ andTHH Y ‹ T;Bar Y ‹ N;R;M ““ withthebisimplicialspectrumthat atthe ‹ i;j “ -spotlookslike 87 R , R , ::: , R ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ i ,, NM: ,, j ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ T , T , ::: , T Thenthegeometricrealizationyieldsthedesiredequivalence[3]. Nowwewillseehowtoextendthisworktothedualsituationwithcoalgebrainputs. 6.3CoHochschildHomologyisaShadow WewanttoshowthatcoHochschildhomologyisalsoashadow.Wedescribethe followingbicategory. 6.3.1 Fora k ,thebicategory CoAlg k has0-cellsthatarecoalgebrasover k ,say C;D ,and CoAlg k ‹ C;D “ isthecategoryof ‹ C;D “ -bicomodules.Theunit U C isthe ‹ C;C “ -bicomodule C ,andhorizontalcompositionisgivenbythecotensorproduct j givenby b B ‹ C;D “ B ‹ D;E “ B ‹ C;E “ ‹ M;N “ ( M b N M j D N For ‹ C;D “ -bicomodule M , ‹ D;E “ -bicomodule N ,and ‹ E;F “ -bicomodule P thenatural isomorphisms a ‹ M b N “ b P ÐÐ M b ‹ N b P “ l U C b M ÐÐ M r M b U D ÐÐ M followasin[14]fromthenaturalisomorphisms 88 ‹ M j D N “ j E P M j D ‹ N j E P “ C j C M M M j D D M: Themapsabovearegivenby: a ‹ M j D N “ j E P M j D ‹ N j E P “ ‹ m a n “ a p ( m a ‹ n a p “ l C j C M M i c i a m i ( i ‹ c i “ m i r M j D D M i m i a d i ( i m i ‹ d i “ Wereferto[10,11.6]fortheproofthatthecotensorisassociativesincethe k ensures Theorem6.3.2 The0 th Hochschildhomology,coHH 0 ,isashadowonthebicategory CoAlg k .Thatis,it givesafamilyoffunctors coHH 0 ‹ ;C “ CoMod ‹ C;C “ A b N ( N j C a C op C coHH 0 ‹ N;C “ thatsatisfytherequiredshadowproperties. Proof. Weusethebicategorywith1-and2-cellsfrom CoMod ‹ C;C “ ,whichisthecategory of ‹ C;C “ -bicomodulesasabove.Recallthehorizontalcomposition: 89 b B ‹ C;D “ B ‹ D;E “ B ‹ C;E “ ‹ M;N “ ( M b N M j D N Notethatsinceweareworkingwithcoalgebrasovera k ,thecotensor M j D N isa bicomodule. Wewanttotherequiredfunctor: `` ee C CoMod ‹ C;C “ A b N ( N j C a C op C coHH 0 ‹ N;C “ where A b isagainthecategoryofabeliangroups.Theisomorphismabovefollowsfrom N j C a C op C H 0 ‹ N j C a C op C “ CoTor 0 C a C op ‹ N;C “ coHH 0 ‹ N;C “ NowthebruntofwhatweneedtojustifytoshowthatcoHH 0 isashadowisthatfor M a ‹ C;D “ -bicomoduleand N a ‹ D;C “ -bicomodule,wehaveanisomorphism `` M b N ee C ÐÐ `` N b M ee D Butunpackingournotationgives `` M b N ee C `` M j D N ee C ‹ M j D N “ j C a C op C coHH 0 ‹ M j D N;C “ `` N b M ee D `` N j M ee D ‹ N j C M “ j D a D op D coHH 0 ‹ N j C M;D “ : 90 Recallthatthe0 th coHochschildhomologyofa k -coalgebra A withcocientsinan ‹ A;A “ -bicomodule B withcoactions B B a A and B A a B isgivenby: coHH 0 ‹ B;A “ Ÿ b > B S ~ t ‹ b “ ‹ b “š where ~ t isthetwistmap[14].Wewanttoamap Ÿ m a n > M j D N S ~ t 1 ‹ m a n “ 1 ‹ m a n “š ÐÐ Ÿ n a m > N j C M S ~ t 2 ‹ n a m “ 2 ‹ n a m “š forthefollowingcomoduleandstructuremaps 1 M j D N M j D N a C m a n ( m a N ‹ n “ 1 M j D N C a M j D N m a n ( M ‹ m “ a n 2 N j C M N j C M a D n a m ( n a M ‹ m “ 2 N j C M D a N j C M n a m ( N ‹ n “ a m M M M a D M M C a M N N N a C N N D a N: Wewouldliketo asthemapthatswapsthefactors m a n ( n a m .Inorder forthismaptobewweneedtoverifythat ~ t 2 ‹ n a m “ 2 ‹ n a m “ .Notethe ofcotensor M j D N ÐÐ ‹ M a N “ M a Id N / / Id M a N / / M a D a N impliesthatfor m a n > M j D N , M ‹ m “ a n m a N ‹ n “ ,verifying ~ t 2 ‹ n a m “ ~ t ‹ N ‹ n “ a m “ n a M ‹ m “ 2 ‹ n a m “ : Asimilarargumentthemapintheotherdirectionaswell,sowecanthe 91 isomorphism by coHH 0 ‹ M j D N;C “ ÐÐ coHH 0 ‹ N j C M;D “ : m a n ( n a m Weneedtoshowthatforachaincomplexof ‹ C;C “ -bicomodules P thefollowingdiagrams arecommutativewhentheymakesense: ``‹ M b N “ b P ee / / `` a ee `` P b ‹ M b N “ee `` a ee / / ``‹ P b M “ b N ee `` M b ‹ N b P “ee / / ``‹ N b P “ b M ee `` a ee / / `` N b ‹ P b M “ee O O `` P b U C ee / / `` r ee ' ' `` U C b P ee `` l ee / / `` P b U C ee `` r ee w w `` P ee Butnoticethatthediagramaboveisequivalentto coHH 0 ‹‹ M j D N “ j C P;C “ / / `` a ee coHH 0 ‹ P j C ‹ M j D N “ ;C “ `` a ee / / coHH 0 ‹‹ P j C M “ j D N;C “ coHH 0 ‹ M j D ‹ N j C P “ ;C “ / / coHH 0 ‹‹ N j C P “ j C M;D “ `` a ee / / coHH 0 ‹ N j C ‹ P j C M “ ;D “ O O Soifweapplythesof a and fromabove,atediouscheckbasedonthe ofcoHH 0 showsthatthisdiagramcommutes. Furtherbytheoftheshadow,theseconddiagramisequivalentto: coHH 0 ‹ P j C C;C “ / / `` r ee * * coHH 0 ‹ C j C P;C “ `` l ee / / coHH 0 ‹ P j C C;C “ `` r ee t t coHH 0 ‹ P;C “ Thisdiagramcommutesbecause `` r ee and `` l ee justapplythecounit tothecopiesof C inthecotswhile coHH 0 ‹ P j C C;C “ coHH 0 ‹ C j C P;C “ byjust shthe P componenttotheappropriatespot.Thereforethe0 th coHochschildhomology isashadowinthisbicategoricalsetting. 92 6.3.3 Fora k ,let D ‹ CoAlg k “ denotethebicategorywhose0-cellsarecoalgebrasover k ,and D ‹ CoAlg k “‹ C;D “ D ‹ CoMod ‹ C;D “ “ isthederivedcategoryof ‹ C;D “ -bicomodules.The unit U C isthe ‹ C;C “ -bicomodule C viewedasacochaincomplex,andhorizontalcomposition isgivenbythederivedcotensorproduct,whichwedenoteby  j : b B ‹ C;D “ B ‹ D;E “ B ‹ C;E “ ‹ M;N “ ( M b N M  j D N Notethatthereisaquasi-isomorphismofcochaincomplexes M  j D N coBar ‹ M;D;N “ ,and sowemayalsoconsiderthishorizontalcompositionasthetwo-sidedcobarconstruction. Remark6.3.4 Usingthisequivalence,wethenaturalisomorphismsforthissettingas: a coBar ‹ coBar ‹ M;D;N “ ;E;P “ ÐÐ coBar ‹ M;D;coBar ‹ N;E;P ““ ‹ m a d 1 aa d i a n “ a e 1 aa e j a p ( m a d 1 aa d i a ‹ n a e 1 aa e j a p “ l coBar ‹ C;C;M “ ÐÐ M c a c 1 aa c i a m ( ‹ c “ ‹ c 1 “ ‹ c i “ m r coBar ‹ M;D;D “ ÐÐ M m a d 1 aa d j a d ( ‹ d 1 “ ‹ d j “ ‹ d “ : where i;j denotethenumberoftensoredcopiesinthecobarconstruction. Theorem6.3.5 CoHochschildhomology,coHH,isashadowonthebicategory D ‹ CoAlg k “ .Thatis,itgives afamilyoffunctors 93 coHH ‹ ;C “ D ‹ CoMod ‹ C;C “ “ D ‹ CoCh k “ N ( C  j C a C op N coHH ‹ N;C “ asacomplexthatsatisfytherequiredproperties,where CoCh k isthecategoryofcochain complexes. Proof. Weusethebicategory D ‹ CoMod ‹ C;C “ “ ,whichisthederivedcategoryof ‹ C;C “ - bicomodulesasabove.Recallthehorizontalcomposition: b B ‹ C;D “ B ‹ D;E “ B ‹ C;E “ ‹ M;N “ ( M b N M  j D N coBar ‹ M;D;N “ Notethatthisrequires M tobeover k ,whichinthiscaseisbecause k isa Wewanttotherequiredfunctor: `` ee C D ‹ CoMod ‹ C;C “ “ D ‹ CoCh k “ N ( C  j C a C op N coHH ‹ N;C “ where CoCh k iscochaincomplexesof k -modulesandcoHH ‹ N;C “ denotesthecomplex whosehomologygivescoHochschildhomology.Theisomorphismabovefollowsfrom H i ‹ C  j C a C op N “ CoTor i C a C op ‹ C;N “ coHH i ‹ N;C “ NowthebruntofwhatweneedtojustifytoshowthatcoHHisashadowisthatfor M acochaincomplexof ‹ C;D “ -bicomodulesand N acochaincomplexof ‹ D;C “ -bicomodules, 94 wehaveanisomorphism `` M b N ee C ÐÐ `` N b M ee D Butunpackingournotationandusingthefactthat M  j D N coBar ‹ M;D;N “ gives `` M b N ee C `` M  j D N ee C C  j C a C op ‹ M  j D N “ coHH ‹ M  j D N;C “ coHH ‹ coBar ‹ M;D;N “ ;C “ `` N b M ee D `` N  j M ee D D  j D a D op ‹ N  j C M “ coHH ‹ N  j C M;D “ coHH ‹ coBar ‹ N;C;M “ ;D “ andsoshowingthatthereexistsanisomorphism amountsto coHH ‹ coBar ‹ M;D;N “ ;C “ ÐÐ coHH ‹ coBar ‹ N;C;M “ ;D “ : RecallthatcoHochschildhomologyiscalculatedusingacycliccobarconstruction,and bothcoHH Y ‹ coBar Y ‹ M;D;N “ ;C “ andcoHH Y ‹ coBar Y ‹ N;C;M “ ;D “ canbeidenwith thebicosimiplicialobject H YY thatatthe ‹ i;j “ -spotisgivenby: C a C a ::: a C ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ i aa NM aa j ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ D a D a ::: a D wherethecofacemapsaregivenbycomultiplicationattheappropriateindex.Thenthe map isgivendegree-wiseby: M a D aa D a N a C aa C ÐÐ N a C aa C a M a D aa D m a d 1 a d j a n a c 1 aa c i ( n a c 1 aa c i a m a d 1 aa d j ; wherethe isdeterminedbytheKoszulsign.Thismapbehaveswellwithrespectto 95 thecofacemapsbecauseofthe ‹ C;D “ -comodulestructureon M andthe ‹ D;C “ -comodule structureon N .Thisshthusgivesanequivalenceofcochaincomplexesand thedesiredisomorphism . Weneedtoshowthatforacochaincomplexof ‹ C;C “ -bicomodules P thefollowing diagramsarecommutativewhentheymakesense: ``‹ M b N “ b P ee / / `` a ee `` P b ‹ M b N “ee `` a ee / / ``‹ P b M “ b N ee `` M b ‹ N b P “ee / / ``‹ N b P “ b M ee `` a ee / / `` N b ‹ P b M “ee O O `` P b U C ee / / `` r ee ' ' `` U C b P ee `` l ee / / `` P b U C ee `` r ee w w `` P ee Butnoticethatthediagramaboveisequivalentto coHH ‹‹ M  j D N “  j C P;C “ / / `` a ee coHH ‹ P  j C ‹ M  j D N “ ;C “ `` a ee / / coHH ‹‹ P  j C M “  j D N;C “ coHH ‹ M  j D ‹ N  j C P “ ;C “ / / coHH ‹‹ N  j C P “  j C M;D “ `` a ee / / coHH ‹ N  j C ‹ P  j C M “ ;D “ : O O Soifweapplytheof a and fromabove,atediouscheckbasedontheDennis- WaldhausenMoritaArgumentshowsthatthisdiagram,whichisexpandedinDiagram2of theAppendix,commutes. Furtherthankstothenaturalisomorphismsofthebicategoricalstructure,thesecond diagramisequivalentto: coHH ‹ P  j C C;C “ / / `` r ee ) ) coHH ‹ C  j C P;C “ `` l ee / / coHH ‹ P  j C C;C “ `` r ee u u coHH ‹ P;C “ Thisdiagramcommutesbecause `` r ee and `` l ee justapplythecounit tothecopiesof C in thecotswhile coHH ‹ coBar ‹ P;C;C “ ;C “ coHH ‹ coBar ‹ C;C;P “ ;C “ by justshthe P componenttotheappropriatespot.ThereforecoHHisashadow. 96 6.4MoritaInvariance Classically,wecanMoritaequivalenceinthecontextofringsandbimodules: 6.4.1 ([29]) Tworings R and T are Moritaequivalent ifthereexistbimodules R M T and T N R sothat M a T N RN a R M T as R -and T -bimodulesrespectively. HochschildhomologyisknowntobeMoritainvariant: Theorem6.4.2 ([24]) If R and T areMoritaequivalentrings,thenthereisanaturalisomorphism HH ⁄ ‹ R “ HH ⁄ ‹ T “ : Moritaequivalenceinthedualsettingofcoalgebrasisoftenreferredtoas Morita-Takeuchi invariance thankstoworkofTakeuchi[34].ThisworkwasfurtherdevelopedbyFarinati andSolotar[17],BrezenzinskiandWisbauer[10]andHess-Shipley[20]. 6.4.3 ([10]) Twocoalgebras C and D are Morita-Takeuchiequivalent ifthereexistsa ‹ C;D “ -bicomodule M anda ‹ D;C “ -bicomodule N suchthattherearebicomoduleisomorphisms M j D N C and N j C M D . Thisgivesanequivalenceofcategories: N j C CoMod C CoMod D 97 CoHochschildhomologyissimilarlyknowntobeMorita-Takeuchiinvariant,byworkof Farinati-Solotar: Theorem6.4.4 ([16]) If C and D areMorita-Takeuchiequivalentcoalgebras,thenthereisanaturalisomorphism coHH ⁄ ‹ C “ coHH ⁄ ‹ D “ : Moregenerally,Moritaequivalenceisanaturalnotionofequivalenceinbicategories.We recalltheofMoritaequivalenceaspresented,forinstance,inCampbell-Ponto[11]. Inordertoitinthebicategoricalsetting,weneedthefollowing 6.4.5 Forabicategory B ,a1-cell M > B ‹ C;D “ is rightdualizable ifthereexistsa1-cell N > B ‹ D;C “ ,calledthe rightdual ,alongwithacoevaluation2-cell ‹ M;N “ U C M b N and anevaluation2-cell ‹ M;N “ N b M U D suchthattheysatisfythetriangleidentities: Id M ‹ Id M b ‹ M;N “ “ X ‹ ‹ M;N “ b Id M “ M U C b M M b N b M M b U D M Id N ‹ ‹ M;N “ b Id N “ X ‹ Id N b ‹ M;N “ “ N N b U C N b M b N U D b N N Thepair ‹ M;N “ iscalleda dualpair ,and N is leftdualizable with leftdual M . 6.4.6 ([11]) Let B beabicategory.Then C;D > ob ‹ B “ are Moritaequivalent ifthereexist1-cells M > B ‹ C;D “ and N > B ‹ D;C “ suchthat ‹ M;N “ and ‹ N;M “ aredualpairsandthe coevaluationmaps ‹ M;N “ U C M b N , ‹ N;M “ U D N b M andtheevaluationmaps ‹ M;N “ N b M U D , ‹ N;M “ M b N U C areinverses.Thatis, 98 ‹ N;M “ X ‹ M;N “ Id N b M ‹ N;M “ X ‹ M;N “ Id U C ‹ M;N “ X ‹ N;M “ Id M b N ‹ M;N “ X ‹ N;M “ Id U D Example6.4.7 Usingthisstructure,weseethatinthebicategory Mod ~ Ring , R and T areMoritaequivalent ifthereexistsan ‹ R;T “ -bimodule M anda ‹ T;R “ -bimodule N suchthat ‹ M;N “ and ‹ N;M “ aredualpairs,giving M a T N R N a R M T: NotethatsinceHH 0 isashadow: `` M b N ee R `` N b M ee T HH 0 ‹ R ; M a T N “ HH 0 ‹ T ; N a R M “ : Moritainvariancesaysthat M a T N R and N a R M T .Sothenweget HH 0 ‹ R ; R “ HH 0 ‹ R ; M a T N “ (Moritaequivalence) HH 0 ‹ T ; N a R M “ (equivalence) HH 0 ‹ T ; T “ (Moritainvariance) soourMoritaequivalentrings R and T haveequivalent\underived"Hochschildhomology. Example6.4.8 Inthebicategory D ‹ Ch ~ Ring “ , R and T areMoritaequivalentifthereexistchaincomplexes M of ‹ R;T “ -bimodulesand N of ‹ T;R “ -bimodulessuchthat ‹ M;N “ and ‹ N;M “ aredual pairs,giving 99 M a L T N R N a L R M T: Again,sinceHHisashadow: `` M b N ee R `` N b M ee T HH ‹ R;M a L T N “ HH ‹ T;N a L R M “ : Moritainvariancesaysthat M a L T N R and N a L R M T .Sothenweget HH ‹ R;R “ HH ‹ R;M a L T N “ (Moritaequivalence) HH ‹ T;N a L R M “ (shadows) HH ‹ T;T “ (Moritaequivalence) soourMoritainvariantrings R and T haveequivalentHochschildhomologyaswell. Example6.4.9 Since[11]furthershowthatTHHisashadow,thesamenotionofMoritainvarianceholdsfor thebicategoricalsetting D ‹ Mod ~ RingSpectra “ .Thatis,ringspectra R and T areMorita equivalentifthereexistsan ‹ R;T “ -bimodulespectrum M anda ‹ T;R “ -bimodulespectrum N suchthat ‹ M;N “ and ‹ N;M “ aredualpairs,yielding M , L T N R N , L R M T: ThensinceTHHisashadow, 100 `` M b N ee R `` N b M ee T THH ‹ R;M , L T N “ THH ‹ T;N , L R M “ : SinceMoritainvariancesaysthat M , L T N R and N , L R M T , THH ‹ R;R “ THH ‹ R;M , L T N “ (Moritaequivalence) THH ‹ T;N , L R M “ (shadows) THH ‹ T;T “ (Moritaequivalence) soourMoritaequivalentringspectra R and T haveequivalenttopologicalHochschildho- mologyaswell. NowweconsiderwhichobjectsareMoritaequivalentinthebicategory CoAlg k . Example6.4.10 Inthebicategory CoAlg k ,coalgebras C and D areMoritaequivalentifthereexistsa ‹ C;D “ - bicomodule M anda ‹ D;C “ -bicomodule N suchthat ‹ M;N “ and ‹ N;M “ aredualpairs(i.e. thereexistcoevaluationandevaluationmapsfor M and N satisfyingtheconditionsofthe yielding M j D N C N j C M D: ThisrecoverstheclassicalnotionofMorita-Takeuchiequivalenceasin[34].According tothebicategoricalshadowstructurewehavethefollowingMoritainvarianceresultsfor coHochschildhomology. 101 Proposition6.4.11 If C and D areMoritaequivalentcoalgebrasinthebicategory CoAlg k then coHH 0 ‹ C “ coHH 0 ‹ D “ : Proof. BecausecoHH 0 isashadow, `` M b N ee C `` N b M ee D coHH 0 ‹ M j D N;C “ coHH 0 ‹ N j C M;D “ : Suppose C and D areMoritaequivalent.By M j D N C and N j C M D and therefore coHH 0 ‹ C;C “ coHH 0 ‹ M j D N;C “ (Moritaequivalence) coHH 0 ‹ N j R M;D “ (shadows) coHH 0 ‹ D;D “ (Moritaequivalence) Thus C and D havethesame0 th coHochschildhomology. NowweconsiderMoritaequivalentobjectsinthebicategory D ‹ CoAlg k “ . Example6.4.12 Inthebicategory D ‹ CoAlg k “ ,coalgebras C and D areMoritaequivalentifthereexistsa chaincomplexof ‹ C;D “ -bicomodules M andachaincomplexof ‹ D;C “ -bicomodules N such that ‹ M;N “ and ‹ N;M “ aredualpairs(i.e.thereexistcoevaluationandevaluationmaps for M and N satisfyingtheconditionsoftheyielding 102 M  j D N C N  j C M D: Moritaequivalentobjectsinthebicategory D ‹ CoAlg k “ areclassicallyMorita-Takeuchi equivalentaswell. Proposition6.4.13 If C and D areMoritaequivalentcoalgebrasinthebicategory D ‹ CoAlg k “ then coHH ⁄ ‹ C “ coHH ⁄ ‹ D “ : Proof. BecausecoHHisashadow, coHH ‹ M  j D N;C “ coHH ‹ N  j C M;D “ : Suppose C and D areMoritaequivalent.By M  j D N C and N  j C M D and therefore coHH ‹ C;C “ coHH ‹ M  j D N;C “ (Moritaequivalence) coHH ‹ N  j C M;D “ (shadows) coHH ‹ D;D “ (Moritaequivalence) ThusMoritaequivalentcoalgebras C and D haveequivalentcoHochschildhomology. ThisrecoversaresultofFarinati-Solotar[16],usingtheperspectiveofshadows. 103 APPENDIX 104 105 Appendix Thefollowingdiagramsarereferencedinthisthesis. Diagram1:CoassociativityfromproofofProposition2.5.5 B , A B i A , Id / / i A , Id B , A A , A B Id , ˚ , Id / / Id , i A , Id B , A B , A B Id , i B , Id / / Id , i A , Id ‹ B , A B “ , B ‹ B , A B “ Id , Id , i A , Id B , A A , A B Id , ˚ , Id i A , Id , Id / / B , A A , A A , A B Id , ˚ , Id , Id / / Id , Id , ˚ , Id B , A B , A A , A B Id , i B , Id , Id / / Id , Id , ˚ , Id ‹ B , A B “ , B ‹ B , A A , A B “ Id , Id , Id , ˚ , Id B , A B , A B Id , i B , Id i A , Id , Id / / B , A A , A B , A B Id , ˚ , Id , Id / / Id , Id , i B , Id ‹ B , A B “ , A ‹ B , A B “ Id , i B , Id , Id / / Id , Id , i B , Id ‹ B , A B “ , B ‹ B , A B , A B “ Id , Id , Id , i B , Id ‹ B , A B “ , B ‹ B , A B “ i A , Id , Id , Id / / ‹ B , A A , A B “ , B ‹ B , A B “ Id , ˚ , Id , Id , Id / / ‹ B , A B , A B “ , B ‹ B , A B “ Id , i B , Id , Id , Id / / ‹ B , A B “ , B ‹ B , A B “ , B ‹ B , A B “ Diagram2:ShadowpropertiesofcoHHfromproofofTheorem6.3.5 coHH ‹ coBar ‹ coBar ‹ M;D;N “ ;C;P “ ;C “ / / `` a ee coHH ‹ coBar ‹ P;C;coBar ‹ M;D;N ““ ;C “ `` a 1 ee / / coHH ‹ coBar ‹ coBar ‹ P;C;M “ ;D;N “ ;C “ coHH ‹ coBar ‹ M;D;coBar ‹ N;C;P ““ ;C “ / / coHH ‹ coBar ‹ coBar ‹ N;C;P “ ;C;M “ ;D “ `` a ee / / coHH ‹ coBar ‹ N;C;coBar ‹ P;C;M ““ ;D “ O O 105 BIBLIOGRAPHY 106 BIBLIOGRAPHY [1] VigleikAngeltveitandJohnRognes.HopfalgebrastructureontopologicalHochschild homology. Algebr.Geom.Topol. ,5:1223{1290,2005. [2] DavidBarnesandConstanzeRoitzheim. IntroductiontoStableHomotopyTheory , volume185.CambridgeUniversityPress,2020. [3] AndrewJ.BlumbergandMichaelA.Mandell.Localizationtheoremsintopological Hochschildhomologyandtopologicalcyclichomology. Geom.Topol. ,16(2):1053{1120, 2012. [4] AnnaMarieBohmann,TeenaGerhardt,AmalieH˝genhaven,BrookeShipley,and StephanieZiegenhagen.ComputationaltoolsfortopologicalcoHochschildhomology. TopologyAppl. ,235:185{213,2018. [5] AnnaMarieBohmann,TeenaGerhardt,andBrookeShipley.TopologicalcoHochschild homologyandthehomologyoffreeloopspaces. inpreparation ,2020. [6] Marcelokstedt.TopologicalHochschildhomology. preprint,Bielefeld ,3,1985. [7] Marcelokstedt. ThetopologicalHochschildhomologyofZandZ/p .UnivatBiele- feld,FatfurMathematik,1985. [8] A.K.andD.M.Kan. Homotopylimits,completionsandlocalizations .Lecture NotesinMathematics,Vol.304.Springer-Verlag,Berlin-NewYork,1972. [9] EdgarH.Brown,Jr.andFranklinP.Peterson.Aspectrumwhose Z p cohomologyis thealgebraofreduced p th powers. Topology ,5:149{154,1966. [10] TomaszBrzezinskiandRobertWisbauer. Coringsandcomodules ,volume309of London MathematicalSocietyLectureNoteSeries .CambridgeUniversityPress,Cambridge, 2003. [11] JonathanA.CampbellandKatePonto.TopologicalHochschildhomologyandhigher characteristics. Algebr.Geom.Topol. ,19(2):965{1017,2019. [12] MoiraChasandDennisSullivan.Stringtopology. arXivpreprintmath/9911159 ,1999. [13] RalphL.Cohen,JohnD.S.Jones,andJunYan.Theloophomologyalgebraofspheres andprojectivespaces.In Categoricaldecompositiontechniquesinalgebraictopology (IsleofSkye,2001) ,volume215of Progr.Math. ,pages77{92.auser,Basel,2004. 107 [14] YukioDoi.Homologicalcoalgebra. J.Math.Soc.Japan ,33(1):31{50,1981. [15] A.D.Elmendorf,I.Kriz,M.A.Mandell,andJ.P.May. Rings,modules,andalgebrasin stablehomotopytheory ,volume47of MathematicalSurveysandMonographs .American MathematicalSociety,Providence,RI,1997.WithanappendixbyM.Cole. [16] MarcoA.FarinatiandAndreaSolotar.Morita-Takeuchiequivalence,cohomology ofcoalgebrasandAzumayacoalgebras.In Rings,Hopfalgebras,andBrauergroups (Antwerp/Brussels,1996) ,volume197of LectureNotesinPureandAppl.Math. ,pages 119{146.Dekker,NewYork,1998. [17] MarcoA.FarinatiandAndreaSolotar.Cycliccohomologyofcoalgebras,coderivations anddeRhamcohomology.In Hopfalgebrasandquantumgroups(Brussels,1998) , volume209of LectureNotesinPureandAppl.Math. ,pages105{129.Dekker,New York,2000. [18] PaulG.GoerssandJohnF.Jardine. Simplicialhomotopytheory ,volume174of Progress inMathematics .auserVerlag,Basel,1999. [19] KathrynHess,Paul-EugeneParent,andJonathanScott.CoHochschildhomologyof chaincoalgebras. J.PureAppl.Algebra ,213(4):536{556,2009. [20] KathrynHessandBrookeShipley.InvariancepropertiesofcoHochschildhomology. arXivpreprintarXiv:1811.06508 ,2018. [21] MichaelHillandTylerLawson.Automorphicformsandcohomologytheorieson Shimuracurvesofsmalldiscriminant. Adv.Math. ,225(2):1013{1045,2010. [22] DavidCopelandJohnsonandW.StephenWilson.ProjectivedimensionandBrown- Petersonhomology. Topology ,12:327{353,1973. [23] TylerLawsonandNikoNaumann.CommutativityconditionsfortruncatedBrown- Petersonspectraofheight2. J.Topol. ,5(1):137{168,2012. [24] Jean-LouisLoday. Cyclichomology ,volume301.SpringerScience&BusinessMedia, 2013. [25] CaryMalkiewich.CyclotomicstructureinthetopologicalHochschildhomologyof DX . Algebr.Geom.Topol. ,17(4):2307{2356,2017. [26] J.McClure,R.Schanzl,andR.Vogt. THH ‹ R “ R a S 1 for E ª ringspectra. J.Pure Appl.Algebra ,121(2):137{159,1997. [27] JamesEMcClureandRESOnthetopologicalHochschildhomologyofbu,I. AmericanJournalofMathematics ,115(1):1{45,1993. 108 [28] JohnMilnor.TheSteenrodalgebraanditsdual. Ann.ofMath.(2) ,67:150{171,1958. [29] KiitiMorita.Dualityformodulesanditsapplicationstothetheoryofringswith minimumcondition. Sci.Rep.TokyoKyoikuDaigakuSect.A ,6:83{142,1958. [30] MaximilienPerouxandBrookeShipley.Coalgebrasinsymmetricmonoidalcategories ofspectra. HomologyHomotopyAppl. ,21(1):1{18,2019. [31] KatePonto.Fixedpointtheoryandtraceforbicategories. Asterisque ,pagesxii+102, 2010. [32] KatePontoandMichaelShulman.Shadowsandtracesinbicategories. J.Homotopy Relat.Struct. ,8(2):151{200,2013. [33] DouglasC.Ravenel. Complexcobordismandstablehomotopygroupsofspheres ,volume 121of PureandAppliedMathematics .AcademicPress,Inc.,Orlando,FL,1986. [34] MitsuhiroTakeuchi.Moritatheoremsforcategoriesofcomodules. J.Fac.Sci.Univ. TokyoSect.IAMath. ,24(3):629{644,1977. [35] SeanTilson.PoweroperationsintheKunnethspectralsequenceandcommutative H F p -algebras. arXiv:1602.06736AlgebraicTopology ,2016. [36] FriedhelmWaldhausen.Algebraic K -theoryoftopologicalspaces.II.In Algebraic topology,Aarhus1978(Proc.Sympos.,Univ.Aarhus,Aarhus,1978) ,volume763of LectureNotesinMath. ,pages356{394.Springer,Berlin,1979. 109