Homological methods in Non-commutative Geometry Tokyo, 2007/2008 1

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1 Homologcal methos n Non-commutatve Geometry Tokyo, 2007/ Lecture 2. Secon bcomplex for cyclc homology. Connes fferental. Cyclc homology an the e Rham cohomology n the HKR case. Homology of small categores. Smplcal vector spaces an homology of the category opp. 2.1 Secon bcomplex for cyclc homology. Recall that n the en of the last lecture, we have efne cyclc homology HC () of an assocatve untal algebra over a fel k as the homology of the total complex of a certan explct bcomplex (1.8) constructe from an ts tensor powers 1. Ths efnton s very a hoc. Hstorcally, t was arrve at as a result of a certan computaton of the homology of Le algebras of matrces over ; t s not clear at all what s the nvarant meanng of ths explct bcomplex. Next several lectures wll be evote mostly to varous alternatve efntons of cyclc homology. Unfortunately, all of them are a hoc to some egree, an none s completely satsfactory an shoul be regare as fnal. No really goo explanaton of what s gong on exsts to ths ay. ut we can at least o computatons. The frst thng to o s to notce that not only we know that the even-numbere columns C () of the cyclc bcomplex (1.8) are acyclc, but we actually have a contractng homotopy h for them gven by h(a 0 a n ) = 1 a 0 a n. Ths can be use to remove these acyclc columns entrely. The result s the secon bcomplex for cyclc homology whch has the form b (2.1) b b 2 b 2 b 3 b 2 b b 3 b 4, b wth the horzontal fferental : n (n+1) gven by = ( τ) h ( +τ + + τ n 1 ). Ths fferental s known as the Connes fferental, or the Connes-Tsygan fferental, or the Rnehart fferental. In the commutatve case, t was scovere by G. Rnehart back n the 1960es; then t was forgotten, an rescovere nepenently by. Connes an. Tsygan n about 1982 (n the general assocatve case). Lemma 2.1. The agram (2.1) s a bcomplex whose total complex s quassomorphc to the total complex of (1.8). 1 y the way, a goo reference for everythng relate to cyclc homology s J.-L. Loay s book Cyclc homology, Sprnger, Personally, I fn also very useful an ol overvew artcle. Fegn,. Tsygan, tve K-theory, n Lecture Notes n Math, vol

2 Homologcal methos n Non-commutatve Geometry Tokyo, 2007/ Proof. Ths s a general fact from lnear algebra whch has nothng to o wth the specfcs of the stuaton. ssume gven a bcomplex K, wth fferentals 1,0, 0,1, an assume gven a contractng homotopy h for the complex K,, 0,1 for every o 1. Defne the agram K,, 1,0, 0,1 by K,j = K 2,j, 0,1 = 0,1, 1,0 = 1,0 h 1,0. Then 1,0 1,0 = 1,0 h 2 1,0 h 1,0 = 0, an 1,0 0,1 + 0,1 1,0 = 1,0 h 1,0 0,1 + 0,1 1,0 h 1,0 = 1,0 h 0,1 1,0 1,0 0,1 h 1,0 = 1,0 (h 0,1 + 0,1 h) 1,0 = 1,0 1,0 = 0, so that K, s nee a bcomplex, an one checks easly that the map ( 1) ( 1) +1 (h 1,0 ) : K, = K 2, 2 K, s a chan homotopy equvalence between the total complexes of K, an K,. Exercse 2.1. Check ths. 2.2 Comparson wth e Rham cohomology. The man avantage of the complex (2.1) wth respect to (1.8) s that t allows the comparson wth the usual e Rham cohomology n the commutatve case. Proposton 2.2. In the assumptons of the Hochschl-Kostant-Rosenberg Theorem, enote n = m Spec, an assume that n! s nvertble n the base fel k (thus ether char k = 0, or char k > n). Then the HKR somorphsm HH () = Ω extens to a quassomorphsm between the bcomplex (2.1) an the bcomplex Ω 2 0 Ω 2 Ω Ω 2 Ω 3 Ω 4, 0 0 where the vertcal fferental s 0, an the horzontal fferental s the e Rham fferental. Proof. Frst we show that uner the atonal assumpton of the Proposton, the HKR somorphsm extens to a canoncal quassomorphsm P between the Hochschl complex an the complex Ω, 0. Ths quassomorphsm P s gven by 0 0 P (a 0 a 1 a ) = 1! a 0a 1 a.

3 Homologcal methos n Non-commutatve Geometry Tokyo, 2007/ Ths s obvously a map of complexes: nee, snce (a 1 a 2 ) = a 1 a 2 + a 2 a 1 by the Lebntz rule, the expresson for P (b(a 0 a )) conssts of terms of the form a 0 a j a 1 a j 1 a j+1 a, every such term appears exactly twce, an wth opposte sgns. Thus P nuces a map p : HH () Ω. y HKR, both ses are somorphc flat fntely generate -moules; by Nakayama Lemma, to prove that p an somorphsm, t suffces to prove that t s surjectve. Ths s clear snce s commutatve, the alternatng sum sgn(σ)a 0 σ(a 1 a ) σ over all the permutatons σ of the nces 1,..., s a Hochschl cycle for any a 0,..., a, an we have ( ) P sgn(σ)a 0 σ(a 1 a ) = a 0 a 1 a. σ So, p s an somorphsm, an P s nee a quassomorphsm. It remans to prove that t sens the Connes-Tsygan fferental to the e Rham fferental that s, we have P = P. Ths s also very easy to see. Inee, every term n (a 0... a ) contans 1 as one of the factors. Snce 1 s annhlate by the e Rham fferental, the only non-trval contrbuton to P ((a 0... a )) comes from the terms whch contan 1 as the frst factor, so that we have 1 P ((a 0... a )) = P (h(τ j (a 0 a ))) = 1 1 τ j (a 0 a )! = j=0 1 ( 1)! a 0 a, j=0 whch s exactly (P (a 0 a )). Corollary 2.3. In the assumptons of Proposton 2.2, we have a natural somorphsm HP () = H DR(Spec )((u)). Proof. Clear. Remark 2.4. For example, the Connes-Tsygan fferental n the lowest egree, : 2, s gven by (a) = 1 a + a 1, whch s very close to the formula a 1 1 a whch gves the unversal fferental Ω 1 () nto the moule of Kähler fferentals Ω 1 () for a commutatve algebra. The fference n the sgn s rrelevant because of the HKR entfcaton of HH 1 () an Ω 1 () f one works out explctly the entfcaton gven n Lecture 1, one checks that 1 a goes to 0, so that t oes not matter wth whch sgn we take t. The comparson map P n the lowest egree just sens a b to ab, so that P ((a)) = a.

4 Homologcal methos n Non-commutatve Geometry Tokyo, 2007/ Generaltes on small categores. Our next goal s to gve a slghtly less a hoc efnton of cyclc homology also ntrouce by. Connes. Ths s base on the technques of the so-calle homology of small categores. Let us escrbe t. For any small category Γ an any base fel k, the category Fun(Γ, k) of functors from Γ to k-vector spaces s an abelan category, an the rect lmt functor lm Γ s rght-exact. Its erve functors are calle homology functors of the category Γ an enote by H (Γ, E) for any E Fun(Γ, k). For nstance, f Γ s a groupo wth one object wth automorphsm group G, then Fun(Γ, k) s the category of k-representatons of the group G; the homology H (Γ, ) s then tautologcally the same as the group homology H (G, ). nalogously, the nverse lmt functor lm Γ s left-exact, an ts erve functors H (Γ, ) are the cohomology functors of the category Γ. In the group case, ths correspons to the usual cohomology of the group. y efnton of the nverse lmt, we have H (Γ, E) = Ext (k Γ, E), where k Γ enotes the constant functor from Γ to k -Vect. In partcular, H (Γ, k Γ ) = Ext (k Γ, k Γ ) s an algebra, an the homology H (Γ, k Γ ) wth constant coeffcents s a moule over ths algebra. In general, t s not easy to compute the homology of a small category Γ wth arbtrary coeffcents E Fun(Γ, k). One way to o t s to use resolutons by the representable functors k [a], [a] Γ these are by efnton gven by k [a] ([b]) = k[γ([a], [b])] for any [b] Γ, where Γ([a], [b]) s the set of maps from [a] to [b] n Γ, an k[ ] enotes the k-lnear span. y Yonea Lemma, we have Hom(k [a], E ) = E ([a]) for any E Fun(Γ, k); therefore k [a] s a projectve object n Fun(Γ, k), hgher homology groups H (Γ, k [a] ), 1 vansh, an agan by Yonea Lemma, we have (2.2) Hom(lm Γ k [a], k) = Hom(k [a], k Γ ) = k Γ ([a]) = k, so that H 0 (Γ, k [a] ) = k. Every functor E Fun(Γ, k) amts a resoluton by sums of representable functors for example, we have a natural ajuncton map E([a]) k [a] E, [a] Γ an ths map s obvously surjectve. nalogously, for cohomology, we can use co-representable functors k [a] gven by k [a] ([b]) = k[γ([b], [a])] ; they are njectve, H 0 (Γ, k [a] ) = k, an every E Fun(Γ, k) has a resoluton by proucts of functors of ths type. One can also thnk of functors n Fun(Γ, k) as presheaves of k-vector spaces on Γ opp. Ths s of course a very complcate name for a very smple thng, but t s useful because t brngs to mn famlar facts about sheaves on topologcal spaces or étale sheaves on schemes. Most of these facts hol for functor categores as well, an the proofs are actually much easer. Specfcally, t s convenent to use a verson of Grotheneck s formalsm of sx functors. Namely, f we are gven two small categores Γ, Γ, an a functor γ : Γ Γ, then we have an obvous restrcton functor γ : Fun(Γ, k) Fun(Γ, k). Ths functor has a left-ajont γ! an a rght-ajont f, calle the left an rght Kan extensons. (If you cannot remember whch s left an whch s rght, but are famlar wth sheaves, then the notaton γ!, γ wll be helpful.)

5 Homologcal methos n Non-commutatve Geometry Tokyo, 2007/ The rect an nverse lmt over a small category Γ are specal cases of ths constructon they are Kan extensons wth respect to the projecton Γ pt onto the pont category pt. The representable an co-representable functors k [a], k [a] are obtane by Kan extensons wth respect to the embeng pt Γ of the object [a] Γ. Gven three categores Γ, Γ, Γ, an two functors γ : Γ Γ, γ : Γ Γ, we obvously have (γ γ) = γ γ, whch mples by ajuncton γ! γ! = (γ γ)! an γ γ = (γ γ). In general, the Kan extensons γ!, γ have erve functors L f!, R f ; just as n the case of homology an cohomology, one can compute them by usng resolutons by representable resp. corepresentable functors. 2.4 Homology of the category opp. Probably the frst useful fact about homology of small categores s a escrpton of the homology of the category opp, the opposte to the category of fnte non-empty totally orere sets. We enote by [n] opp the set of carnalty n. Objects E Fun( opp, k) are known as smplcal k- vector spaces. Explctly, such an object s gven by k-vector spaces E([n]), n 1, an varous maps between them, among whch one tratonally stngushes the face maps n : E([n + 1]) E [n], 0 n the face map n correspons to the njectve map [n] [n + 1] whose mage oes not contan the ( + 1)-st element n [n + 1]. Lemma 2.5. For any smplcal vector space E Fun( opp, k), the homology H ( opp, E) can be compute by the stanar complex E gven by E n = E([n + 1]), n 0, wth fferental : E n E n 1, n 1, equal to = ( 1) n. 0 n Proof. y efnton, we have a map E 0 = E([1]) H 0 ( opp, E), whch obvously factors through the cokernel of the fferental, an ths s functoral n E. Denote by H ( opp, E) the homology groups of the stanar complex E. Then every short exact sequence of smplcal vector spaces nuces a long exact sequence of H ( opp, ), so that H ( opp, ) form a δ-functor. Moreover, H 0( opp, E) s by efnton the cokernel of the map = : E([2]) E([1]). Ths s the same as the rect lmt of the agram E([2]) E([1]) of two k-vector spaces E([2]), E([1]) an two maps 0 1, 1 1 between them (a rect lmt of ths type s calle a coequalzer). Snce ths agram has an obvous map to opp, we have a natural map H 0( opp, E) lm E = H 0 ( opp, E), an by the unversal property of erve functors, t extens to a canoncal map opp (2.3) H ( opp, E) H ( opp, E) of δ-functors. We have to prove that t s an somorphsm. Snce every E Fun( opp, k) amts a resoluton by sums of representable functors k [n], [n] opp, t suffces to prove that the map (2.3) s an somorphsm for all E = k [n] (ths s known as the metho of acyclc moels). Ths s clear: H ( opp, k [n] ) s k for = 0 an 0 otherwse, an the left-han se of (2.3) s the homology of the stanar complex of an n-smplex, whch s also k n egree 0 an 0 n hgher egrees. Exercse 2.2. Compute the cohomology H ( opp, E). Hnt: compute k [1].