DOCSLIB.ORG

CYCLIC HOMOLOGY Contents 1. Introduction [01/08/18] 1 2. Derived

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CYCLIC HOMOLOGY Contents 1. Introduction [01/08/18] 1 2. Derived CYCLIC HOMOLOGY BORIS TSYGAN Contents 1. Introduction [01/08/18]1 2. Derived tensor product, traces [01/10/18]5 3. Where does B come from? [01/12/18]7 3.1. Noncommutative forms8 4. January 17, 20179 5. January 19, 2017 11 6. January 22, 2018 13 1. Introduction [01/08/18] Let A be an associative algebra over a ring k. Most of the well-established theory is for k a field of characteristic zero, but we will try to also focus on the case of Z and characteristic p. Later on in the course we will try to get to the case of dg-categories or ring spectra. In particular our viewpoint throughout the course will be to try to understand A as \functions" on some noncommutative space. We will especially focus on trying to understand differential forms (or their analogs). Let us establish some inuition. Let A be a k-algebra for k some (unital) commu- tative algebra. To think of elements of A as functions on some space X you might think of, for example, matrix-valued functions. Then what would differential forms on X be? Well it would be something written as n ! = a0da1da2 ··· dan 2 Ω (X); where the symbols dai will not (super)commute as they do in the classical case. Thus we think of, crudely, our space of n-forms as A ⊗ A¯⊗n where A¯ = A=k · 1. Here we are quotienting by the unit because we expect d(1) = 0. Classically recall we have a de Rham differential d :Ωn ! Ωn+1, which is given d : a0da1 ··· dan 7! 1da0da1 ··· dan: We can do this in our situation as well by taking A ⊗ A¯⊗n −!d A ⊗ A¯⊗n+1 to send a0 ⊗ a1 ⊗ · · · 7! 1 ⊗ a0 ⊗ a1 ⊗ · · · . But this is quite dull|it does not even see the product in A (there is no Leibniz rule). We have that d2 = 0 of course, but there is no cohomology except H0 =∼ k. In other words, in this crude formulation the Poincar´elemma holds for all of our noncommutative spaces X. Let's keep this in mind and try something else. Define now ⊗n Cn(A) = Cn(A; A) = A ⊗ A¯ : Date: Winter 2018. 1 2 BORIS TSYGAN We define differential b : Cn(A) ! Cn−1(A) n−1 X j a0 ⊗ a1 ⊗ · · · ⊗ an 7! (−1) a0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ an j=0 n + (−1) ana0 ⊗ a1 ⊗ · · · ⊗ an−1: One checks directly that this is well-defined and that b2 = 0. Let's look at some low degrees explicitly: ··· A ⊗ A¯⊗2 A ⊗ A¯ A The last arrow sends a0 ⊗ a1 7! a0a1 − a1a0 and the next arrow sends a0 ⊗ a1 ⊗ a2 7! a0a1 ⊗ a2 − a0 ⊗ a1a2 + a2a0 ⊗ a1: How do we think about this complex? Well let's say X is a manifold and A is some algebra of (matrix-valued) functions on X. A matrix-valued function can be reduced to a scalar function: a0 7! tr(a0). Moreover given two such functions we can produce a one-form a0 ⊗ a1 7! tr(a0da1). Notice that b(a0 ⊗ a1 ⊗ a2) is sent to zero under this reduction. In general, we construct HKR n Cn(A; A) −−−! Ω (X) 1 a ⊗ · · · ⊗ a 7! tr(a da ··· da ) 0 n n! 0 1 n called the Hochschild-Kostant-Rosenberg map. This is a map of complexes if we equip the de Rham complex with the zero differential (and treat it homologically instead of cohomologically). Of course this is all in the specific case where A is an algebra of matrix-valued functions. Can we think of the Cn(A) as forming a cohomological complex? Our naive differential d from before does not play nicely with b (it does not supercommute). There is however a construction of Rinehart in the '60s B : Cn(A) ! Cn+1(A): n X nj B(a0 ⊗ · · · ⊗ an) = (−1) 1 ⊗ aj ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ aj−1: j=0 This is a differential such that B2 = bB + Bb = b2 = 0: Exercise 1. Check that (over characteristic zero) HKR ◦ B = ddR ◦ HKR: For example we have a commutative diagram 1 ⊗ a0 ddR tr(a0) B ddR a0 tr a0 There seems to be a sign and so on. error for n odd. CYCLIC HOMOLOGY 3 We can construct a sort of double complex that starts like C0 C1 C2 C3 C0 C1 C2 C0 C1 C0 We define then per Y CPn(A) = CCn (A) = Cj(A) j≡n mod 2 with differential b + B. We call this the periodic cyclic complex. It is a periodic complex (since b + B squares to zero) of order 2 and its homology the periodic cyclic homology of A. To give you a flavor of the sort of things you get, consider the following result. Theorem 2 (Feigin-Tsygan '85). Let A be a finitely generated commutative unital algebra over C. Let X = specmA ⊂ CN . Then for n 2 Z ∼ Y j HPn(A) = H (X; C); j≡2 mod 2 where on the right we have singular cohomology. In particular periodic cyclic homology is very rigid: Theorem 3 (Goodwillie). If Q ⊂ k and I,! B A where I is a nilpotent ideal, then ∼ HP•(B) −! HP•(A): There is also a complex known as the negative cyclic complex − b+B − − ···! CC1 (A) −−−! CC0 (A) ! CC−1(A) !··· − whose homology is the negative cyclic homology HC• (A) for the index in Z. This complex is perhaps the least used of the ones we have introduced some for. In the notation of Ezra Getzler, let u be a formal variable of degree −2. Then we can write neatly all of the above as: per −1 CC• (A) = C•(A)[[u ; u]] with differential b + uB and − CC• (A) = C•(A)[[u]] with differential b + uB and C•(A) = C•(A) with differential b. Let's turn to some extensions. Let M be an A-bimodule. Then take ⊗n Cn(A; M) = M ⊗ A¯ 4 BORIS TSYGAN with differential X b(a0 ⊗ a1 ⊗ · · · ⊗ an) = ±a0 ⊗ · · · ajaj+1 ⊗ · · · + ∓ana0 ⊗ · · · where a0 2 M. In particular note that b(m ⊗ a1) = ma1 − a1m. In particular, HH0(A; M) = M=ham − ma j a 2 A; m 2 Mi = M ⊗A⊗Aop A: More generally we have the following. Proposition 4. Hochschild homology computes the derived tensor product ∼ A⊗Aop HHn(A; M) = Torn (A; M): This in turn motivates us to look for invariant definitions for the other homologies defined above. CYCLIC HOMOLOGY 5 2. Derived tensor product, traces [01/10/18] Recall from last time, given an associative unital algebra A over k we defined for n ≥ 0 ⊗n Cn(A; A) = Cn(A) = A ⊗ A¯ where A¯ = A=k · 1. We defined two differentials b : Cn(A) ! Cn−1(A) B : Cn(A) ! Cn+1(A) given by n−1 X j b(a0 ⊗ a1 ⊗ · · · ⊗ an) = (−1) a0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ an j=0 n + (−1) ana0 ⊗ a1 ⊗ · · · ⊗ an−1 n X nj B(a0 ⊗ · · · ⊗ an) = (−1) 1 ⊗ aj ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ aj−1: j=0 Recall that we had b2 = bB + Bb = B2 = 0. Remark 5. Notice that B2 = 0 for a stupid reason, as it places the unit in a position other than the first tensor slot. Of course, we could construct a variant ⊗n+1 where Cn(A) = A , in which case we modify B as X X B(a0 ⊗ · · · ⊗ an) = ±1 ⊗ − ∓ ± − ⊗1 where by the dash we mean what we have in the formula above. The sign in the second sum is the sign of the permutation. ⊗n Recall that we had similar definitions Cn(A; M) = M ⊗ A¯ for when M is an A-bimodule. In this case we saw that ∼ H0(A; M) = M=[A; M] = M ⊗A⊗Aop A: Theorem 6. We have, for when A is a flat k-module (e.g. k is a field), ∼ A⊗Aop Hn(A; M) = Torn (A; M): Proof. We have a bar resolution of A as an A ⊗ Aop-module: 0 0 0 ··· A ⊗ A¯⊗2 ⊗ A b A ⊗ A¯ ⊗ A b A ⊗ A b A ⊗n In particular we let Bn = A ⊗ A¯ ⊗ A for n ≥ 0 where n 0 X j b (a0 ⊗ · · · ⊗ an+1) = (−1) a0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ an+1: j=0 02 op One finds that b = 0 and moreover that each Bn is an A ⊗ A -bimodule via multiplication on the left and right. It is clear that b0 is a morphism of bimodules. Notice that this resolution is acyclic|it has no homology. In particular there are work this out maps s : Bn ! Bn+1 given s(a⊗ ··· an+1) = 1 ⊗ a0 ⊗ · · · ⊗ an+1 such that b0s + sb0 = id : 6 BORIS TSYGAN op The flatness condition in the hypothesis gives us that Bn is a flat A⊗A -bimodule. Hence the bar resolution is a flat A ⊗ Aop-bimodule resolution of A. Tensoring now with M on the left, we find that ∼ M ⊗A⊗Aop Bn(A) = Cn(A; M): Explicitly, we map Cn(A; M) ! M ⊗A⊗Aop Bn(A) via m ⊗ (a1 ⊗ · · · ⊗ an) 7! m ⊗ (1 ⊗ a1 ⊗ · · · ⊗ 1): Careful about M being a The differential b0 induces the differential b. right-bimodule? Before proceeding with systematic theory, let's look at how to motivate the differential B. First let's look at the trace property of HH0(A; M). Observe that algebras form a 2-category: for every pair A; B of algebras we have the category C(A; B) of (A; B)-bimodules. We have a composition ⊗B : C(A; B) × C(B; C) !C(A; C) which is reasonably associative (not quite strict, but coherently).
Load more

Recommended publications
  • Cyclic Homology and Equivariant Homology
    Invent.math. 87, 403-423 (1987) [nffelltiolle$ mathematicae Springer-Verlag1987 Cyclic homology and equivariant homology John D.S. Jones Mathematics Institute,University of Warwick, Coventry CV4 7AL, UK Introduction The purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of Connes [9-11], see also Loday and Quillen [20], and "IF equivariant homology and cohomology theories. Here II" is the circle group. The most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w3. In this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply. Let A be an associative algebra over a commutative ring K. Then one can form the cyclic homology HC.(A) and cohomology HC*(A) of A. These groups have periodicity operators HC,(A)-.~HC,,_z(A), HC"(A)-.HC"+Z(A). Connes [10] has defined products in cyclic cohomology and using this product structure HC*(K) becomes a polynomial ring K[u] where u has degree 2. The groups HC*(A) now become modules over this polynomial ring and the action of u corresponds to the periodicity operator. Thus it seems reasonable to regard K[u] as the natural coefficients for cyclic cohomology and then to make HC.(A) into a module over K[u] by using the periodicity operator. However it is clear that every element of HC.(A) is u-torsion so HC*(A) and HC.(A) cannot be dual over K[u].
    [Show full text]
  • Cyclic Homology of Affine Hypersurfaces with Isolated Singularities
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 120 (1997) 291-299 Cyclic homology of affine hypersurfaces with isolated singularities Ruth I. Michler * Department of Mathematics, University of North Texas, Denton, TX 76203-5116, USA Communicated by C.A. Weibel; received 20 September 1995; revised 17 January 1996 Abstract We consider reduced, affine hypersurfaces with only isolated singularities. We give an explicit computation of the Hodge-components of their cyclic homology in terms of de Rham cohomology and torsion modules of differentials for large n. It turns out that the vector spaces HC,(A) are finite dimensional for n 2 N - 1. @ 1997 Elsevier Science B.V. 1991 Math. Subj. Class.: 19, 14, 13 1. Introduction Let R = K[XI,&... ,X,I,] with K being an algebraically closed field of character- istic zero. Throughout this paper A denotes a reduced hypersurface with only isolated singularities given by A = R/(F), whith F E K[X~,&, . ,X,1. For a definition of the module of Kiihler differentials Qfi,K see for example [22, 8.8.1., p. 2941. The cohomol- ogy of the complex where d denotes the exterior differential, is called the de Rham cohomology of A and denoted by H&(A). In this paper we will compute the Hodge components H($)(A) of cyclic homology of a hypersurface with isolated singularities. We will use the iden- tification in [20] of the nth Hochschild homology groups with torsion submodules of differentials.
    [Show full text]
  • Open Uuyeotgonbayar-Dissertation
    The Pennsylvania State University The Graduate School LOCAL INDEX THEOREM IN NONCOMMUTATIVE GEOMETRY A Dissertation in Mathematics by Uuye Otgonbayar c 2009 Uuye Otgonbayar Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2009 The Dissertation of Uuye Otgonbayar was reviewed and approved∗ by the following: Nigel Higson Evan Pugh Professor of Mathematics Dissertation Advisor, Chair of Committee John Roe Professor of Mathematics Head of the Department of Mathematics Nathanial Brown Associate Professor of Mathematics John Collins Professor of Physics ∗Signatures are on file in the Graduate School. Abstract We consider the multiplicative property of index cocycles associated to a spectral triple. We construct a perturbation of the JLO-cocyle and prove that it is multiplicative on the cochain level with respect to the exterior product of theta-summable spectral triples and the product in entire cyclic theory. Furthermore, we show that this cocycle admits an asymptotic expansion if the spectral triple admits the analogue of an Euler vector field. Using Laurent series, we obtain a multiplicative local index cocycle. iii Contents List of Symbols vi Acknowledgments xi Chapter 1 Introduction 1 The Atiyah-Singer Index Theorem. 1 Vector Bundle Modification and K-homology . 4 Spectral Triples . 4 Description of Contents . 5 Chapter 2 Spectral Triples and Index Theory 7 2.1 Definition and Examples of Spectral Triples . 7 2.1.1 Spectral Triples . 7 Sum and Product of Spectral Triples . 10 Perturbation of a Spectral Triple . 11 2.1.2 Multigraded Spectral Triples . 12 Clifford Algebras . 12 Multigraded Spectral Triples . 15 Odd Spectral Triples .
    [Show full text]
  • Algebraic K-Theory, Periodic Cyclic Homology, and the Connes-Moscovici Index Theorem
    Algebraic K-theory, periodic cyclic homology, and the Connes-Moscovici Index Theorem Master's thesis by Bram Mesland under supervision of prof dr. N.P. Landsman University of Amsterdam, Faculty of Science Korteweg-de Vries Institute for Mathematics Plantage Muidergracht 24, 1018 TV Amsterdam June 20th, 2005 2 3 Does it matter that this waste of time is what makes a life for you? -Frank Zappa 4 5 Algebraic K-theory, cyclic homology, and the Connes-Moscovici Index Theorem Abstract. We develop algebraic K-theory and cyclic homology from scratch. The boundary map in periodic cyclic cohomology is shown to be well-behaved with respect to the external product. Then we prove that the Chern-Connes character induces a natural transformation from the exact sequence in lower algebraic K-theory to the exact sequence in periodic cyclic homology. Using this, the Gohberg-Krein index theorem is easily derived. Finally, we prove the Connes-Moscovici index theorem, closely following Nistor in [20]. Keywords: Algebraic K-theory, cyclic homology, Chern-Connes char- acter, index theorem, noncommutative geometry. 6 Contents Introduction 9 1 Lower algebraic K-theory 15 1.1 Projective modules . 15 1.2 Grothendieck's K0 . 18 1.3 Idempotents . 20 1.4 Whitehead's K1 . 23 1.5 Relative K-theory . 25 1.6 Excision . 30 1.7 Topological K-theory . 33 1.8 C¤-algebras and index theorems . 38 2 Cyclic homology 41 2.1 The simplicial and cyclic categories . 41 2.2 Cyclic modules . 47 2.3 Hochschild homology . 49 2.4 Cyclic homology . 54 2.5 Periodic and negative cyclic homology .
    [Show full text]
  • Noncommutative Geometry, the Spectral Standpoint Arxiv
    Noncommutative Geometry, the spectral standpoint Alain Connes October 24, 2019 In memory of John Roe, and in recognition of his pioneering achievements in coarse geometry and index theory. Abstract We update our Year 2000 account of Noncommutative Geometry in [68]. There, we de- scribed the following basic features of the subject: I The natural “time evolution" that makes noncommutative spaces dynamical from their measure theory. I The new calculus which is based on operators in Hilbert space, the Dixmier trace and the Wodzicki residue. I The spectral geometric paradigm which extends the Riemannian paradigm to the non- commutative world and gives a new tool to understand the forces of nature as pure gravity on a more involved geometric structure mixture of continuum with the discrete. I The key examples such as duals of discrete groups, leaf spaces of foliations and de- formations of ordinary spaces, which showed, very early on, that the relevance of this new paradigm was going far beyond the framework of Riemannian geometry. Here, we shall report1 on the following highlights from among the many discoveries made since 2000: 1. The interplay of the geometry with the modular theory for noncommutative tori. 2. Great advances on the Baum-Connes conjecture, on coarse geometry and on higher in- dex theory. 3. The geometrization of the pseudo-differential calculi using smooth groupoids. 4. The development of Hopf cyclic cohomology. 5. The increasing role of topological cyclic homology in number theory, and of the lambda arXiv:1910.10407v1 [math.QA] 23 Oct 2019 operations in archimedean cohomology. 6. The understanding of the renormalization group as a motivic Galois group.
    [Show full text]
  • Summary of the Thesis
    Summary of the thesis The index theorem of Atiyah and Singer, discovered in 1963, is a striking result which relates many different fields in mathematics going from the analysis of partial differential equations to differential topology and geometry. To be more precise, this theorem relates the dimension of the space of some elliptic partial differential equations and topological invariants coming from (co)homology theories, and has important applications. Many major results from different fields (algebraic topology, differential topology, functional analysis) may be seen as corollaries of this result, or obtained from techniques developed in the framework of index theory. On another side, zeta functions associated to pseudodifferential operators on a closed Riemannian manifold contain in their analytic properties many interesting informations. For instance, the Weyl theorem on the asymptotic number of eigenvalues of a Laplacian may be recovered within the residues of the zeta function. This gives in particular the volume of the manifold, which is a geometric data. Using the framework of noncommutative geometry developed by Connes, this idea may be pushed further, yielding index theorems in the spirit of the one of Atiyah-Singer. The interest in this viewpoint is to be suitable for more delicate geometrical situations. The present thesis establishes results in this direction. The first chapter of the thesis aims at obtaining a general local index formula for "abstract elliptic operators". These formulas are derived from a cyclic cocycle expressed in terms of zeta functions residues, constructed by combining zeta functions renormalization techniques together with the excision property in cyclic cohomology. The formula also applies when the zeta function has multiple poles.
    [Show full text]
  • Cyclic Homology, Cdh-Cohomology and Negative K-Theory
    Annals of Mathematics, 167 (2008), 549–573 Cyclic homology, cdh-cohomology and negative K-theory By G. Cortinas,˜ C. Haesemeyer, M. Schlichting, and C. Weibel* Abstract We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero. Introduction The negative algebraic K-theory of a singular variety is related to its ge- ometry. This observation goes back to the classic study by Bass and Murthy [1], which implicitly calculated the negative K-theory of a curve X. By def- inition, the group K−n(X) describes a subgroup of the Grothendieck group 1 n K0(Y ) of vector bundles on Y = X × (A −{0}) . The following conjecture was made in 1980, based upon the Bass-Murthy calculations, and appeared in [38, 2.9]. Recall that if F is any contravariant functor on schemes, a scheme X is called F -regular if F (X) → F (X × Ar)is an isomorphism for all r ≥ 0. K-dimension Conjecture 0.1. Let X be a Noetherian scheme of di- mension d. Then Km(X)=0for m<−d and X is K−d-regular. In this paper we give a proof of this conjecture for X essentially of finite type over a field F of characteristic 0; see Theorem 6.2.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Arbeitsgemeinschaft: Topological Cyclic Homology
    Mathematisches Forschungsinstitut Oberwolfach Report No. 15/2018 DOI: 10.4171/OWR/2018/15 Arbeitsgemeinschaft: Topological Cyclic Homology Organised by Lars Hesselholt, Copenhagen/Nagoya Peter Scholze, Bonn 1 April – 7 April 2018 Abstract. Introduced by B¨okstedt-Hsiang-Madsen in the nineties, topolog- ical cyclic homology is a manifestation of the dual visions of Connes-Tsygan and Waldhausen to extend de Rham cohomology to a noncommutative set- ting and to replace algebra by higher algebra. The cohomology theory that ensues receives a denominator-free Chern character from algebraic K-theory, used by Hesselholt-Madsen to evaluate the p-adic K-theory of p-adic fields. More recently, Bhatt-Morrow-Scholze have defined a “motivic” filtration of topological cyclic homology and its variants, the filtration quotients of which give rise to their denominator-free p-adic Hodge theory AΩ. Mathematics Subject Classification (2010): 19D55, 14F30, 13D03, 19D50. Introduction by the Organisers Cyclic homology was introduced by Connes and Tsygan in the early eighties to serve as an extension of de Rham cohomology to a noncommutative setting. The negative version of cyclic homology receives a trace map from algebraic K-theory, which extends the classical Chern character and roughly records traces of powers of matrices. This trace map is a powerful rational invariant of algebraic K-theory. Indeed, a theorem of Goodwillie from 1986 shows that, rationally, the discrepancy for K-theory to be invariant under nilpotent extentions of rings agrees with that for negative cyclic homology; and a theorem of Corti˜nas from 2006 shows similarly that, rationally, the discrepancy for K-theory to preserve cartesian squares of rings agrees with that for negative cyclic homology.
    [Show full text]
  • Additive Invariants of Orbifolds
    Additive invariants of orbifolds The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Tabuada, Gonçalo and Van den Bergh, Michel. "Additive invariants of orbifolds." Geometry & Topology 22, 5 (2018): 3003–3048 ©2018 Mathematical Science Publishers As Published http://dx.doi.org/10.2140/GT.2018.22.3003 Publisher Mathematical Sciences Publishers Version Final published version Citable link https://hdl.handle.net/1721.1/124685 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Geometry & Topology 22 (2018) 3003–3048 msp Additive invariants of orbifolds GONÇALO TABUADA MICHEL VAN DEN BERGH Using the recent theory of noncommutative motives, we compute the additive invari- ants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely “fixed- point data”. As a consequence, we recover, in a unified and conceptual way, the origi- nal results of Vistoli concerning algebraic K –theory, of Baranovsky concerning cyclic homology, of the second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Caldˇ araruˇ and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned computation is new in the literature. As an application, we verify Grothendieck’s standard conjectures of type C C and D, as well as Voevodsky’s smash-nilpotence conjecture, in the case of “low-dimensional” orbifolds. Finally, we establish a result of independent interest concerning nilpotency in the Grothendieck ring of an orbifold.
    [Show full text]
  • Topological Cyclic Homology
    1 Topological cyclic homology Lars Hesselholt and Thomas Nikolaus Abstract: This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B¨okstedtperiodicity that closely resembles B¨okstedt's original unpublished proof. We explain the extension of B¨okstedtperiodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients. Topological cyclic homology is a manifestation of Waldhausen's vision that the cyclic theory of Connes and Tsygan should be developed with the initial ring S of higher algebra as base. In his philosophy, such a theory should be meaningful integrally as opposed to rationally. B¨okstedtrealized this vision for Hochschild homology [9], and he made the fundamental calculation that THH∗(Fp) = HH∗(Fp=S) = Fp[x] is a polynomial algebra on a generator x in degree two [10]. By comparison, HH∗(Fp=Z) = Fphxi is the divided power algebra,1 so B¨okstedt'speriodicity theorem indeed shows that by replacing the base Z by the base S, denominators disappear. In fact, the base-change map HH∗(Fp=S) ! HH∗(Fp=Z) can be identified with the edge homomorphism of a spectral sequence 2 Ei;j = HHi(Fp/π∗(S))j ) HHi+j(Fp= S); arXiv:1905.08984v1 [math.KT] 22 May 2019 so apparently the stable homotopy groups of spheres have exactly the right size to eliminate the denominators in the divided power algebra.
    [Show full text]
  • Excision in Hochschild and Cyclic Homology Without Continuous Linear
    EXCISION IN HOCHSCHILD AND CYCLIC HOMOLOGY WITHOUT CONTINUOUS LINEAR SECTIONS RALF MEYER Abstract. We prove that continuous Hochschild and cyclic homology satisfy excision for extensions of nuclear H-unital Fr´echet algebras and use this to compute them for the algebra of Whitney functions on an arbitrary closed subset of a smooth manifold. Using a similar excision result for periodic cyclic homology, we also compute the periodic cyclic homology of algebras of smooth functions and Whitney functions on closed subsets of smooth manifolds. Contents 1. Introduction 1 2. Preparations: homological algebra and functional analysis 3 2.1. Some examples of symmetric monoidal categories 4 2.2. Exact category structures 6 2.3. Exact chain complexes, quasi-isomorphisms, and homology 8 2.4. Hochschild homology and cohomology 11 2.5. Pure conflations 12 3. Excision in Hochschild homology 15 4. Hochschild homology for algebras of smooth functions 19 4.1. Thealgebraofsmoothfunctionswithcompactsupport 21 5. Application to Whitney functions 23 6. Excision in periodic cyclic homology 28 7. Conclusion and outlook 30 References 31 arXiv:0912.3729v1 [math.KT] 18 Dec 2009 1. Introduction Hochschild and cyclic homology are invariants of non-commutative algebras that generalise differential forms and de Rham cohomology for smooth manifolds [4]. More precisely, let A := C∞(X) for a smooth manifold X be the Fr´echet algebra of smooth functions on a smooth manifold X (we impose no growth condition at infinity). Its continuous Hochschild homology HHn(A) for n ∈ N is naturally isomorphic to the space of differential n-forms Ωn(X) on X. Its continuous periodic cyclic homology HPn(A) for n ∈ Z/2 is naturally isomorphic to the de Rham 2000 Mathematics Subject Classification.
    [Show full text]
  • Lectures on Cyclic Cohomology
    Lectures on Cyclic Cohomology Masoud Khalkhali Mathematics Department, University of Western Ontario London ON, Canada Lectures at IMPAN, Warsaw, September 2016 Simons Semester on Noncommutative Geometry 2 Contents 0.1 Index theory in a noncommutative setting . .7 0.2 Elliptic operators . .7 0.3 The Gauss-Bonnet-Chern theorem as an index theorem . .8 0.4 The signature theorem . 11 0.5 Riemann-Roch-Hirzebruch theorem . 12 0.6 An odd index theorem . 12 0.7 Connes' noncommutative index theorem . 13 0.8 Hochschild cohomology . 15 0.9 Hochschild cohomology as a derived functor . 20 0.10 Deformation theory . 27 0.11 Topological algebras . 36 0.12 Hochschild (co)homology computations . 38 0.13 Cyclic Cohomology . 49 0.14 Connes' long exact sequence . 60 0.15 Connes' spectral sequence . 64 0.16 Cyclic cohomology computations . 66 0.17 What is the Chern character? . 79 0.18 Connes{Chern character in K-theory . 80 0.19 Connes{Chern character in K-homology . 90 0.20 Connes' index theorem . 101 Appendices 106 A Compact operators, Fredholm operators, and abstract index the- ory 107 B K-theory 115 Bibliography 119 3 4 Introduction Introduction In September 2016 I gave 5 introductory lectures on cyclic cohomology and some of its applications in IMPAN Warsaw, during the Simons Semester in Noncommu- tative Geometry. The audience consisted of graduate students and postdocs and my task was to introduce them to the subject. The following text is an expanded version of my lectures. In producing these lecture notes I have freely used mate- rial from my book, Basic Noncommutative Geometry.
    [Show full text]