Z 2 -graded Čech Cohomology in Noncommutative Geometry

Transcription

1 arxiv:math/ v2 [mathkt] 24 Sep 2004 Z 2 -graded Čech Cohomology in Noncommutative Geometry Do Ngoc Diep July 19, 2018 Astract TheZ 2 -graded Čech cohomology theory is considered intheframework of noncommutative geometry over complex numer field and in particular the homotopy invariance and Morita invariance are proven In some special case we deduce an isomorphism etween this noncommutative theory and the classical Z 2 -graded Čech cohomology theory Keywords: Čech cohomology of a sheaf, cyclic theory Mathematics Suject Classification 2000: 14F20, 19K35, 46L80, 46M20 1 Introduction Let us fix in this paper the field of complex numers as the ground field for algeras, modules, etc The general Čech cohomology presheaf Ȟq for an aritrary presheaf X on a category C, Ȟ q (X,G) = Ȟq (G)(X) was introduced in [SGA42] TheideaofČechcohomologyfornoncommutativegeometrywasappeared in [KR], [R] In this paper we use this idea to define the corresponding (periodic) Z 2 -graded Čech theory We prove the homotopy invariance and Morita invariance of Čech cohomology in the framework of noncommutative geometry Our main result is The work was supported in part y Vietnam National Project of Research in Fundamental Sciences and The Adus Salam ICTP, NESCO 1

2 Do Ngoc Diep 2 ased on a detailed analysis of the structure of C*-algeras A crucial oservation is the fact that for C*-algeras the category of *-representations defines exactly the C*-algera itself, y the well-known Gelfand-Naimark- Segal Theorem From this we can deduce the Morita invariance and homotopy invariance of the Čech cohomology, the same properties of periodic cyclic homology of the C*-algera Since our result is valid not only for C*- algeras, we work in the general context of a noncommutative algera over complex numers The paper is organized as follows Taking the Čech cohomology in place of the de Rham theory in the periodic cyclic theory of A Connes, in 2 we define the Z 2 -graded Čech cohomology theory Then in section 3 we prove two important properties of the theory as the homotopy invariance and Morita invariance In the last section 4 we deduce also some kind of Connes- Hochschild-Kostant-Rosenerg theorem This lets us see a clear relation with the classical case of commutative algeras and ordinary Čech cohomology theory Notations: We follow the notations from [SGA42] and [O] 2 Z 2 -graded Čech cohomology 21 Grothendieck topos The main purpose of this section is to formulate and define the functor of (periodiccyclic) Z 2 -gradedčechcohomology Thewell-known periodiccyclic homology is ased on the cyclic homology theory of A Connes, which is an algeraic framework of the Z 2 -graded de Rham cohomology theories In the algeraic context, it was defined y J Cuntz and D Quillen in terms of X-complexes and it has ecome a new chapter of noncommutative algeraic geometry The most general Gronthendieck algeraic geometry is purely asedintermsofcategories Inthegenericcasethisturnsouttothealgeraic version of the Čech cohomology in place of de Rham cohomology theories We follows the work of Orlov [O] in particular to formulate the theory The main references are [SGA42] and [O] Many of our results could e otained in the fields of other characteristics, ut we restrict ourselves to the complex case Let us denote y C a fixed category and Set the category of sets Any contravariant functor X from C to Set is called a presheaf of sets and the

3 Do Ngoc Diep 3 category of all presheaves of sets on C is denoted y Ĉ The category C can e considered as a sucategory of Ĉ, consisting of representale functors h = h R : C Set If R is an oject of C, then there is a natural isomorphism HomĈ(h R,X) = X(R) For any oject X Ĉ the category over X is the category of pairs (R,Φ), where R is an oject of C and Φ X(R), and is denoted y C/X Recall that a sieve in the category C is a full sucategory D C such that any oject of C for which there exists a morphism from it to some oject in D is contained in Oj(D) A sieve on R is nothing more than a supresheaf of R in the category Ĉ A Grothendieck topology T on a category C is defined y giving for each oject R in C a set J(R) of the so called covering sieves satisfying the following axioms: (T1) For any oject R the maximal sieve C/R is in J(R) (T2) If T J(R) and f : S R a morphism in C, then the induced sieve f (T) := { α S fα T} is in J(S) (T3) If T J(R) is a covering sieve and is a sieve on R such that f () J(S) for all f : S R in T, then J(R) AGrothendiecksiteΦ = (C,T )isacategoryc andequippedwithagrothendieck topology T It is reasonale to remind the Jacoson topology on the set of all representations of a group or Zariski topology on algeraic varieties For the categories with fier product, a Grothendieck topology can e given y a Grothendieck pretopology which is defined y giving for each oject R in C a family Cov(R) of morphism to R such that (P1) For any family {R α R} α I in Cov(R) and S R a morphism of C, the fier product family R α R S S is also in Cov(R) (P2) If {R α R} α R is in Cov(R) and {R βα R α } βα J α is in Cov(R α ) for each α I, then the total family {R γ R} γ α I Jα is in Cov(R) (P3) The trivial family {id R : R R} is in Cov(R)

4 Do Ngoc Diep 4 Any Grothendieck pretopology P on C generates a Grothendieck topology T such that a sieve is covering in T if and only if it contains some covering family in P The topos on the category of functors from a category C to another one D is defined y the usual rule 22 The standard cosimplicial complex of a continuous functor We define in this susection the Z 2 -graded Čech cohomology theory Let us recall the definition of the Čech cohomology with coefficients in a sheaf M Let = ( i X) i I e a covering sieve of X in the category C Suppose that the cover has the property that all fier product and pushout diagrams exist, see ([SGA42], Exp 4) Denote A the associated standard simplicial complex: A : (i,j,k) I 3 A( i j k ) (i,j) I 2 A( i j ) i I A( i) Let us consider again a ringed cite (category) (C,A), Ĉ the topos of presheaves on C, ε : C Ĉ the canonical functor associating to each oject X C the functor h X, presented y X Define H q (h X,M) = H q (X,M), see ([SGA42], Exp IV, 231), as derived functor of the projective limit functor : H q (S,M) := R q lim M S C/S For any A-module M, denote C (,M) := Hom A (A,M) : C (,M) : i I M( i) (i,j) I 2 M( i j ) One defines H q (,M) := H q (C (,M)) If R is a covering sieve generalized y the family then H q (,M) = H q (R,M) and the functor H q (,) commutes with restriction of scalars, see ([SGA42], Exp IV, Proposition 234)

5 Do Ngoc Diep 5 One defines, ([SGA42], Exp IV, 24) H 0 (M)(X) := H 0 (X,M) = M(X), H q (M)(X) := H q (M,M) For an aritrary presheaf G of A-module G, the groups H q (R,G) are called the Čech cohomology with respect to the covering sieve R, with coefficients in G For a sheaf M of A-modules over C, the group H q (,M) := H q (,H 0 (M)) is defined as the Čech cohomology group of the sheaf M with respect to he the cover One has also Ȟ 0 (G)(X) = lim G(R) R X and therefore H q (G)(X) = lim H q (R,G) R X which is called the presheaf of Čech cohomology Define Ȟ q (X,G) := Ȟq (G)(X), one has also Ȟ q (X,G) = lim H q (,G) For a sheaf M of A-modules, one has Ȟ q (X,M) = Ȟq (X,H 0 (M)), Ȟ q (M) = Ȟq (Ȟ0 (M)) The groups Ȟq (X,M) are called the Čech cohomology groups of the sheaf M 23 The periodic cyclic icomplex Lemma 21 (The action of Z k+1 ) There is a natural action of the cyclic group Z k+1 on the Čech cohomology cochain complex C (,M) associated with a covering

6 Do Ngoc Diep 6 Proof The action of the cyclic group Z k+1 is defined a cyclic permutation of indices of s, ie (λm)( i0 ik ) := M( ik i0 ik 1 ) It is not hard also to see that for a covering sieve = { α X} there is a natural isomorphism M( i0 ik ) = M( i0 ) M( ik ) Therefore the Čech cohomology complex ecomes the cyclic complex for lim M() Corollary 22 (Hochschild differentials and Cyclic operations) The wellknown Hochschild differentials and and Connes cyclic operators λ, N = 1+λ++λ k, s are well-defined on Z 2 -graded Čech cocycles Definition 23 (Periodic icomplex) Let(C,A) earinged-cite, Ĉ the topos of sheaves, M a sheaf of A-modules Then the icomplex (i 1,i 2,i 3 ) I 3 N M( i1 i2 i3 ) (i 1,i 2,i 3 ) I 3 M( i1 i2 i3 ) (i,j) I 2 M( i j ) i I M( i) N N (i,j) I 2 M( i j ) i I M( i) is well-defined and is called the (periodic) Čech icomplex Definition 24 (The total complex and Z 2 -graded The associated total complex of which is defined as TotC(,M) ± := C i,j, i+j=±(mod 2) Čech cohomology)

7 Do Ngoc Diep 7 where C i,j := (i 1,,i k ) I M( k i1 ik ) and ± = ev (even) or od (odd) The cohomology of this total complex is called the Z 2 -graded Čech cohomologyof M anddenotedyz 2 Ȟ (,M)andZ 2 Ȟ(X,M) := lim Z2Ȟ(,M) It can e also realized as the cohomology of the total complex related with the process of passing through direct limits, ie the direct limit i-complex (i 1,i 2,i 3 ) I 3 N lim M( i1 i2 i3 ) (i 1,i 2,i 3 ) I 3 lim M( i1 i2 i3 ) lim (i,j) I 2 M( i j ) lim i I M( i) N N lim (i,j) I 2 M( i j ) lim i I M( i) For a C*-algera A, we define it Z 2 -graded Čech cohomology Z2Ȟ(A) = Z 2 Ȟ(A)astheZ 2 -gradedčechcohomologyofthecategoryof*-representations of A Remark 25 In the first periodic i-complex without direct limits, all the horizontal lines are acylic, ut it is in general not the case for the second periodic i-complex with direct limits 3 Homotopy invariance and Morita invariance We prove in this section two main properties of the (periodic cyclic) Z 2 - graded Čech cohomology theory: homotopy invariance and Morita invariance, which make the theory easier to compute and eing a generalized homology theory Definition 31 (Chain Homotopy of functors) Let us consider two functors F,G : C D Denote the corresponding chain functors etween complexes y {F n },{G n }, where F n,g n : C n D n for complexes 0 C 0 C 1 C 2 F 0,G 0 F 1,G 1 F 2,G 2 0 D 0 D 1 D 2

8 Do Ngoc Diep 8 We say that F and G are chain homotopic if there exist augmentation functors s n : C n D n 1 such that for all n F n G n = s n n 1 + n s n+1 Lemma 32 Two functors F,G : A B are homotopic if and only if for any covering sieve = ( i X) i I, there exists a chain homotopy of chain complexes and i I i I A( i ) B( i ) i,j I I i,j I I A( i X j ) B( i X j ) Remark 33 In the case of smooth manifolds the chain complex homotopy is realized y integration of the so called Cartan homotopy formula for the Lie derivative L ξ = ı(ξ) d+d ı(ξ) etween de Rham complexes Lemma 34 Let A and B e C*-algeras, and let A (resp B) e the category of -modules Then the categories A and B are homotopic one-to-another if and only if the two algeras A and B are homotopic Proof Because of the Gelfand-Naimark-Segal theorem, the C*-algeras are exactly defined y the category of *-representations, the category of *- representations of B C[0, 1] is isomorphic to the category of*-representations of B Lemma 35 Let A e a C*-algera and A the category of *-representations (ie A-modules) of A, then Z 2 Ȟ (A) = HP (A) Proof Let us consider affine covering sieve i X = dual oject of A Â = SpecA, the

9 Do Ngoc Diep 9 Theorem 36 (Homotopy Invariance) Let ϕ t : A B,t I = [0,1] e a homotopy of algeras, then Z 2 Ȟ (A) = Z 2 Ȟ (B) Proof Let A(resp B) e the category of A-modules(resp, B-modules) Step 1 Change the homotopy y a piecewise-linear homotopy in the space of functors from the category A to the category B Step 2 A piecewise-linear homotopy gives rise to a chain complex homotopy Step 3 Two chain complex homotopical functors induces the same isomorphism of Čech cohomology groups It is an easy consequence from the results of homological algera: If F n and G n are chain complex homotopic than the induced morphisms satisfies F n G n = n 1 s n +s n+1 n The second summand is a zero morphism on cohomology and the first summand is a oundary The sum on the right is therefore a zero morphism Lemma 37 Let us denote y Mat n (C) the algera of all square n n- matrices with complex entries Then we have a natural isomorphism Z 2 Ȟ (Mat n (C)) = Z 2 Ȟ (C) Proof Every complex matrix can e homotopic to a unitary one Then, every unitary matrix can e y conjugation reduced to a diagonal matrix of complex numers of module 1 Every elementary lock(in this case, diagonal element)[e iθ ] is homotopic to identity [1] y the classical homotopy [e iθt ] 0 t 1, ie [ cosθ sinθ sinθ cosθ ] I = [ Lemma 38 (Adjoint functors) There is a natural equivalence of functors Hom and : ] Hom(R Mn(C),M) = Hom(R,Hom(Mn(C),M))

10 Do Ngoc Diep 10 Proof This isomorphism of functors is a particular case of the general adjointness etween Hom and in homological algera Lemma 39 There is a natural isomorphism of derived functors R q Hom(Mn(C),M) = R q Hom(C,M) Proof It is an easy exercise from homological algera Theorem 310 (Morita Invariance) Z 2 Ȟ (A Mat n (C)) = Z 2 Ȟ (A) Proof Let us remark that A A Mat n (C) is a fier undle Now apply the Grothendieck s Leray-Serre spectral sequence for this firation Following the previous lemmas 38 and 39, there is a natural isomorphism of functors R p Hom(R Mn(C),R q (M) = R p Hom(R,R q Hom(Mn(C),M)), which are the E 2 term of a Leray-Serre spectral sequence converging to the Čech cohomology Corollary 311 The Z 2 -graded Čech cohomology theory is a generalized cohomology theory Čech coho- 4 Comparison with the classical mology theory In this section we show that a generalization of the Connes-Hochschild- Kostant-Rosenerg Theorem can e otained easily Theorem 41 Let A e a stale continuous C*-algera with spectrum a smooth compact manifold X, in fact A = C(X,K(P)) is the algera of continuous sections of a smooth, locally trivial undle K(P) := P P K on X with fire the algera K of compact operators on a separale Hilert space associated to a principal P undle P on X via the adjoint action of P on K Let δ(p) H 3 (X;Z) e the Dixmier-Douady invariant, that classifies such algeras A and c(p) some closed 3-form on X, that presents the class 2πiδ(P) in the real cohomology Let A e the category of all *-representations of the C*-algera AThen the Z 2 -graded Čech cohomology Z 2Ȟ (A) is isomorphic to the de Rham cohomology H (X;c(P)) which is isomorphic to the classical Z 2 -graded Čech cohomology Z 2Ȟ (X;c(P))

11 Do Ngoc Diep 11 Proof In this situation, the Z 2 -graded Čech cohomology of the category A is isomorphic to the Connes periodic cyclic homology HP (C (X,L 1 (P))), where C (X,L 1 (P)) is consisting of all smooth section of the su-undle L 1 (P) = P P L 1 of K(P) with fire the algera L 1 of trace class operators on the Hilert space with the same structure groups P, see [MS] for a more detailed proof in the language of periodic cyclic homology Acknowledgments The main part of this work was done while the author was visiting The Adus Salam ICTP in Trieste, Italy The author is grateful to ICTP and in particular would like to express his sincere thanks to Professor Dr Le Dung Trang for invitation and support References [KR] A Rosenerg and M Kontsevich Noncommutative smooth spaces, The Gelfand mathematical Seminar, , [MS] V Mathai and D Stevenson, On a generalized Connes- Hochschild-Kostant-Rosenerg theorem, arxiv: mathkt/ v1, 19 April 2004 [O] D Orlov, Quasicoherent sheaves in commutative and noncommutative geometry, MPI Preprint, 1999 [R] A L Rosenerg, Noncommutative schemes, Compositio Math 112(1998), [SGA42] M Artin, A Grothendieck, and J L Verdier, Théorie des topos et cohomologie des schémas, SGA 4, tome 2, Springer Lecture notes in Mathematics, No 270, 1972 Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, Cau Giay District, Hanoi, Vietnam dndiep@mathacvn