Examples of Involutions on Algebraic K-Theory of Bimonoidal Categories PDF
Preview Examples of Involutions on Algebraic K-Theory of Bimonoidal Categories
Examples of Involutions on Algebraic K-Theory of Bimonoidal Categories Marc Lange Diplomarbeit am Fachbereich Mathematik, Universita¨t Hamburg Betreuerin: Prof. Dr. Birgit Richter (Korrigierte Version) 16. April 2011 Introduction AlgebraicK-theoryofbimonoidalcategoriesarisesasastructureofinterestin“Two- vectorbundlesandformsofellipticcohomology”[3]byNilsBaas,BjørnIanDundas andJohnRognesanditsfollow-uppapers“Ringcompletionofrigcategories”[1]and “Stablebundlesoverrigcategories”[2]bythesethreeauthorsjointwithBirgitRichter. (These papers formerly were the one paper “Two-vector bundles define a form of el- lipticcohomology”.) Specificallytheyinvestigatethecategoryoftwo-vector-bundles. By analogy to the case of principal bundles over a topological space they study two- vector-bundlesbyexaminingarepresentedfunctor[X,K(V)], whereK(V)istheal- gebraicK-theoryspectrumofthebimonoidalcategoryoffinite-dimensionalcomplex vector spaces V. This furthermore embeds into the context of interpolating between the complexity of singular homology, which only captures few phenomena, and the complexity of complex bordism, which detects all levels of periodic phenomena at once. ThecohomologytheorydefinedbyK(V)isinaprecisemanneronelevelmore complexthantopologicalK-theoryis. Thetopicofinvolutionsisintroducedbythefactthatitwouldbenegligenttoignore that the category of complex vector spaces is equipped with an involution naturally inducedbyconjugationincomplexnumbers. ThisdiplomathesisstudiestheK-theoryandHochschildhomologyofringswithin- volution. Indetailthediplomathesisarosefromthemotivationtoexaminenon-trivial involutions on K-theory of bimonoidal categories by studying non-trivial involutions on rings. To that end after introducing the basic concepts I define the algebraic K- theoryofstrictbimonoidalcategoriesfollowingBirgitRichter’s“Aninvolutiononthe K-theoryofbimonoidalcategorieswithanti-involution”[21]. Furthermoreifanobject has an associated anti-involution, this gives an associated involution on its K-theory, whichwedefineaccordingto[21]aswell. As an example I explicitly present K-theory and involutions of group rings and more specifically Laurent polynomials. These examples illustrate the limitations that involutionscanbestudiedonK-groupsdirectlywiththesamedifficultythatK-groups canbecomputed. Thereisaneedforfurthertools. 3 The last chapter is the heart of this diploma thesis. I investigate the Dennis trace map Dtr: K (R) → HH (R) n n andthemapassociatedtoananti-involutiononHochschildhomology. Themainresult 5.4.3 of this diploma thesis is that the trace map Dtr does commute with the induced involutions. Thus the trace map provides an additional tool to study involutions on K-theory and can help to prove non-triviality. The result should be compared to the statement by Bjørn Ian Dundas in the introduction of [8] that his functorial definition ofatracemapinparticularimpliesthatitrespectsinvolutionsonK-theoryandtopo- logicalHochschildhomology. Thisdiplomathesisservesasanalgebraicanalogueof afactknownonthetopologicallevel,althoughitisnotdirectlyimpliedbythat. Finally I discuss the example of the integers with an adjoined prime root of unity Z[ζ ] and discuss the non-triviality in that case in contrast to Laurent polynomials, p whichgoestoshowtheusefulnessoftheDennistracemapaswellasitsdefects. Thistextisorganisedasfollows: Thefirsttwochaptersarededicatedtostudyalge- braicK-theoryinthestandardcontextofringsandspecificallystudyringswithinvolu- tionbyprovidinganinducedinvolutionontheK-groups. Thethirdchapteressentially is a summary of Birgit Richter’s “An involution on the K-theory of bimonoidal cate- gorieswithanti-involution”[21]. Furthermoretherelationsbetweenastrictbimonoidal category and its associated ring embed this paper into the non-triviality statements of the following chapters. I provide examples of rings with families of involutions in Chapter 4. All of these examples have non-trivial involutions on K . Finally, chap- 1 ter 5 introduces the Dennis trace map as an example of a useful tool which can be extended to the context of rings with involution and provide examples to evaluate its usefulnessintheringcontext. Acknowledgements Let me take this opportunity to thank several hands full of people, who were directly orindirectlyinvolvedinthisdiplomathesis: Ofcourse myprimarythanks areaddressedatBirgit Richterforsuggesting atopic that allowed to explore a lot of mathematics, lots of which did not make it into this thesis, and giving me the freedom to do so, which has taken a considerably bigger amountoftimethanitusuallywould:) Furthermoreinanorder,whereeachplaceisequivalenttofirstplace,Ithank: Stephanie Ziegenhagen, for loads of non-mathematical and mathematical discus- sions alike, the inspiration of your strict, thorough, clear head and warm, just and 4 honestly outspoken heart is of ever-growing and indescribable importance to me. I agree: Maytheforceofuniversalpropertiessaveusfromcoordinates! Hannah Ko¨nig for giving me the quickest introduction known to mankind to my courseofstudiesaboutfiveyearsago,thussavinggenerationsofkidsfrommyschool teaching:) ThomasNikolausforallowingmetopickyourheadeveryonceinawhiletoprovide anotherviewonmathematics,universityorotherthingsasawhole. The founders of the internet for giving me access to invaluable things such as Google, Google scholar, dict.leo.org, Ubuntu, YouTube, Allen Hatcher’s “Algebraic Topology”,thearXivandlotsofotherdistractingorhelpfulthings. DonaldKnuthandnumerousotherpersonsforLATEX. ChristophSchweigertforseveralhelpfuldiscussions,actions,coursesandinpartic- ularamotivatingstartingpointin“LinearAlgebra”. Quote: “LinearAlgebraorganises the brain enormously.” Today I know your influence on that course strengthens that effecttremendously. AstridDo¨rho¨ferandEvaKuhlmannforwarmingupourfloorbypurepresenceand fornon-bureaucratichelpwheneverneeded. OfcoursetherearepeopleremainingoutsidetheGeomatikumaswell: Mark Wroblewski and Raluca Oancea for putting up with my peculiarities for all thoseyears-stress-inducedandnotstress-inducedalike. Additionalthanksforgiving me an occasional kick, whenever I might need it, for sports, spare time, games, dis- cussionsorjustgeneralfun:) andofcoursejustbeingthewonderfulfactorofmylife youeachonyourownandtogetherare. Ute Zwicker for providing yet another view on life, which helped to keep me sane lotsandlotsoftimes,forjustgeneralkillingoftimeonthetrainbydiscussions,which couldhavehappilywentonformorehoursthananytraincouldeverbedelayed. LastbutnotleastIthankmyparentsforprovidingforme-financiallyandotherwise - for all that time, although I am aware that I can be an expensive beast regarding money,nervesandotherthingseveryonceinawhileorpermanently:) Allofyoushouldknowthatthisdiplomathesiswouldnotexistwithoutyou. 5 Contents 1 Introduction to K-Theory of Rings 9 1.1 K-TheoryofRings . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.1 ClassifyingSpaceofaGroup . . . . . . . . . . . . . . . . . 10 1.1.2 BarConstructionontheGroupRing . . . . . . . . . . . . . . 11 1.1.3 Plus-ConstructiononClassifyingSpaces . . . . . . . . . . . 12 2 K-Theory of Rings with (Anti-)Involution 17 2.1 TheInducedInvolutionontheK-TheorySpace . . . . . . . . . . . . 17 2.2 InvolutionandDeterminant . . . . . . . . . . . . . . . . . . . . . . . 22 3 Involutions on Bimonoidal Categories 25 3.1 BarConstructionforMonoidalCategories . . . . . . . . . . . . . . . 29 3.2 K-TheoryofaStrictBimonoidalCategory . . . . . . . . . . . . . . 31 3.3 BimonoidalCategorieswithInvolution . . . . . . . . . . . . . . . . . 33 4 Non-trivial Involutions 39 4.1 InvolutiononGroupRings . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 InvolutionsonLaurentPolynomials . . . . . . . . . . . . . . . . . . 40 4.2.1 UnitsinLaurentPolynomials . . . . . . . . . . . . . . . . . 40 4.2.2 InvolutionsonR[t±,...,t±] . . . . . . . . . . . . . . . . . . 42 1 k 5 Hochschild Homology of Rings 47 5.1 DefinitionofHochschildHomology . . . . . . . . . . . . . . . . . . 47 5.2 ConstructionoftheTraceMap . . . . . . . . . . . . . . . . . . . . . 50 5.3 InvolutiononHochschildHomology . . . . . . . . . . . . . . . . . . 54 5.3.1 OpposingtheSimplicialStructureontheHochschildComplex 54 5.4 TheDennisTraceMapCommuteswithInvolutions . . . . . . . . . . 57 5.5 DetectingInvolutionswiththeTraceMap . . . . . . . . . . . . . . . 62 7 1 Introduction to K-Theory of Rings In order to investigate K-theory of rings with involution I first want to give a solid foundation of K-theory of general rings. My approach will diverge drastically from the historical development in an effort to homogenise the appearance of K-theory throughoutthisdiplomathesis. But in order to do the actual history justice, let me give some remarks on the his- tory of (algebraic) K-theory. The K in the designation K-theory is reminiscent of Grothendieck’sapproachtotheRiemann-RochTheoreminwhichhestudiedcoherent sheavesoveranalgebraicvarietyandlookedatclasses-whichis“Klassen”inGerman -ofthosesheavesmoduloexactsequencesin1957. Thisapproachhasbeenpublished byBorelandSerrein“Lethe´ore`medeRiemann-Roch”[6](1958). Thisfirstgroupis whatisnowadayscalledK . 0 Aboutsevenyearslater,BassprovidedadefinitionofthegrouphecalledK (A)for 1 Aaringandprovidedanexactsequence K (A,q) → K (A) → K (A/q) → K (A,q) → K (A) → K (q) 1 1 1 0 0 0 in the paper “K-Theory and Stable Algebras” [4] (1964). Since this exact sequence wasbuiltupontodefinehigherK-theory,thenameK forthisgroupisstillthecom- 1 monone. The next group K was defined by Milnor in 1968 according to Bass’ “K and 2 2 symbols”[5]andagainhadanexactsequenceconnectingittoK ,alongwithapairing 1 K (A)⊗K (A) → K (A) 1 1 2 whichisbilinearandantisymmetric,thuslookedlikeasegmentofagradedring. Since theexactsequence,thispairingandfurtherrelationslookfartoonaturaltobeacoin- cidence,onewaslookingforageneralsequenceofK-groupsK (R),whichcouldbe n associatedtoaringandwhichwouldyieldamorestructuralexplanationoftheknown results. ThiswasgivenbyMilnorin“AlgebraicK-TheoryandQuadraticForms”[19], but Milnor himself described his definition as “purely ad hoc” (in [19] as well). Further- more Milnor’s extension is restricted to be K-theory of fields, otherwise it would not agreewiththefirstthreeknownK-groups. 9 1.1. K-THEORYOFRINGS ThenextattemptatdefiningK-theoryforeachnaturalnumberwasgivenbyQuillen in 1973 in his paper “Higher Algebraic K-Theory: I” [20]. Quillen defines a topo- logical space, proves that its homotopy groups agree with the known definitions and derivessomestructuralresults,whichtranslatealgebraicrelationsintotopologicalre- lations between these newly defined spaces. These are the K-groups that are studied inthisdiplomathesis. 1.1 K-Theory of Rings Iwillgenerallyassumeringstobeunitalbutnotnecessarilycommutativerings. As said before, I will deviate from the historical viewpoint and just define the K- theoryspaceinordertogiveamorelinearconcisesummaryforK-theoryofrings. In preparationforthattherearesomenecessaryprerequisitesingrouphomology. 1.1.1 Classifying Space of a Group Thefirstinvestigationfocusesonaspace|BG|,whicharisesinthecontextofclassify- ingprincipalG-bundlesforagroupG. Ityieldstheresultthatisomorphismclassesof G-bundlesoveraspaceX areclassifiedbyhomotopyclassesofmapsfromX to|BG|. Itisrelevantinthecontextofthisthesishoweverbecauseofitshomotopygroups(cf. Lemma 1.1.2), so I will not elaborate on bundles any further. For simplicial meth- odsconsultLoday’sAppendixBin“CyclicHomology”[15],the“basicdefinitions”in ChapterIofGoerss-Jardine“Simplicialhomotopytheory”[12]andMay’s“Simplicial ObjectsinAlgebraicTopology”[17]. Definition1.1.1. ForGagroupdefineasimplicialsetasfollows: • Thesetofn-simplicesisgivenas BG := Gn = G×...×G, n i.e. n-simplicesaren-tuplesofelementsofG, • Thefacemapsd : BG → BG aregivenbytheequations i n n−1 (g ,...,g ) for i = 0 2 n di(g1,...,gn) := (g1,...,gigi+1,...,gn) for i = 1,...,n−1 (g ,...,g ) for i = n. 1 n−1 • Thedegeneraciess : BG → BG aregivenbytheequations i n n+1 s (g ,...,g ) := (g ,...,g ,1,g ,...,g ) for i = 0,...,n i 1 n 1 i−1 i n 10
The list of books you might like
