Free products, cyclic homology, and the Gauss-Manin connection

FREE PR ODUCTS, CYC LIC HOMOLOGY, AND THE GA USS-MANIN CONNE CTION VICTOR GINZBUR G AND TRA VIS SCHEDLER with an App endix by Boris Tsygan Abstra ct. W e use the tec hniques of Cuntz and Quillen to present a new approac h to p eriod ic cyclic homolo gy . O ur co nstru ction is based on ((Ω q A )[ t ] , d + t · ı ∆ ), a nonc omm utative e quivariant de Rham c omplex of an asso ciativ e algebra A . Here d is the Karoubi-de Rham differential and ı ∆ is an operation analogous to contracti on with a ve ctor field. As a bypro duct, w e give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzl er, on the relative p eriodic cyclic homology of a flat family of asso ciative algebras ov er a central base ring. W e in tro duce and study fr e e-pr o duct deformations of an asso ciativ e algebra, a new typ e of defor- mation ov er a not necessari ly commutativ e base ring. Natural examples of free-pro duct deformations arise from prepro jectiv e algebras and group algebras for compact surface groups. Contents 1. In tro d uction 1 2. Noncomm utativ e calculus 3 3. Extended Karoubi-de Rham complex 5 4. Applications to Ho c hsc h ild and cyclic homology 9 5. Pro ofs 13 6. The Represen tation fun ctor 17 7. F ree p ro ducts and d eformations 20 App end ix A. On th e m orp hism from p erio d ic cyclic homology to equiv ariant cohomology , b y Boris Tsygan 26 References 30 1. In troduction Throughout, w e fix a field k of characte ristic 0 and write ⊗ = ⊗ k (although, up to p assing to reduced versions of homology theories, ev erythin g generalizes to the case that k is an arbitrary comm utativ e ring conta ining Q , pro vided th e algebra A has the prop ert y th at k → A i s a k -split cen tral injection). By an algebra we alwa ys mean an asso ciat ive un ital k -algebra, unless explicitly stated otherwise. Giv en an algebra A , w e view the space A ⊗ A as an A -bimo du le with resp ect to the outer bimo du le structure, which is defined by the formula b ( a ′ ⊗ a ′′ ) c := ( ba ′ ) ⊗ ( a ′′ c ), for any a ′ , a ′′ , b, c ∈ A . By an A -bimo du le, we will alw ays m ean an ( A ⊗ A op )-mo dule, i.e., an A -bimo dule on w hic h the left and righ t action of k coincide. 1.1. Double deriv ations. It is we ll-known that a regular v ector field on a smo oth affine algebraic v ariet y X is the same thing as a deriv ation k [ X ] → k [ X ] of the co ordinate ring of X . Th u s, deriv ations of a comm utativ e algebra pla y the role of v ector fields. 1 It has b een commonly accepted u n til recen tly that this p oint of view applies to noncommu- tativ e algebras as w ell. A first indicati on to wards a differen t p oint of view w as a disco very b y Cra wley-Bo ev ey [4] that, for a smo oth affine curve X with co ordinate ring A = k [ X ], the alge- bra of differentia l op erators on X can b e constru cted b y mea n s of double derivations A → A ⊗ A , rather than ordin ary deriv ations A → A . Since then, the significance of d ouble deriv ations in noncomm utativ e geometry w as explored further in [30] and [5 ]. T o explain the role of dou b le der iv ations in more detail we fir st recall some basic definitions. 1.2. (Double) deriv ations as infinitesimal automorphisms. Recall that a fr ee pro du ct of t w o algebras A and B , is an asso ciativ e algebra A ∗ B that contai n s A and B as subalgebras and whose ele ments are formal k -linear com b inations of w ords a 1 b 1 a 2 b 2 . . . a n b n , for an y n ≥ 1 and a 1 , . . . , a n ∈ A, b 1 , . . . , b n ∈ B . These words are tak en up to the equ iv alence imposed by the relation 1 A = 1 B ; for in stance, . . . b 1 A b ′ . . . = . . . b 1 B b ′ . . . = . . . ( b · b ′ ) . . . , for an y b, b ′ ∈ B . Let N b e an A - bimo dule . A k -linear map f : A → N is said to b e a deriv ation of A with co efficien ts in N if f ( a 1 a 2 ) = f ( a 1 ) a 2 + a 1 f ( a 2 ) , ∀ a 1 , a 2 ∈ A . Giv en a subalgebra R ⊂ A , w e let Der R ( A, N ) denote the space of R -linear deriv ations of A with resp ect to the su b algebra R , that is, of der iv ations A → N that annihilate the subalgebra R . Deriv ations of an algebra A ma y b e viewed as ‘infin itesimal automo r p hisms’. Sp ecifically , let A [ t ] = A ⊗ k [ t ] b e the p olynomial r ing in one v ariable with co efficient s in A . The natural algebra em b ed d ing A ֒ → A [ t ] mak es A [ t ] an A -bim o dule. A well-kno wn elemen tary calculation yields the follo wing. Lemma 1.2.1. The fol lowing pr op erties of a k -line ar map θ : A → A ar e e quivalent: • the map θ is a derivation of the algebr a A ; • the map A → t · A [ t ] , a 7→ t · θ ( a ) is a derivation of the algebr a A with c o e fficients in t · A [ t ] ; • the map a 7→ a + t · θ ( a ) gives an algebr a homomorph ism A → A [ t ] /t 2 · A [ t ] . All the ab o ve holds true, of course, no m atter whether the algebra A is comm u tativ e or not. Y et, th e elemen t t , the formal parameter, is by definition a c entr al elemen t of the algebra A [ t ]. In n oncomm utativ e geometry , th e assumption that the f orm al parameter t b e central is not quite natural, ho wev er . Put an other w a y , wh ile the tensor pro duct is a copro d uct in the category of comm utativ e asso ciativ e algebras, the free p ro duct is a copro duct in the category of not necessarily comm utativ e asso ciativ e algebras. W e see that, in noncommutat ive geometry , the algebra A t = A ∗ k [ t ], freely generated by A and an indeterminate t , should pla y the role of the p olynomial algebra A [ t ]. W e are going to argue that, once the p olynomial algebra A [ t ] is replaced by the algebra A t , it b ecomes more natural to replace deriv ations A → A by double derivations , i.e., by deriv ations A → A ⊗ A where A ⊗ A is viewed as an A -bimo dule with resp ect to the outer bimo d ule structure. T o see this, write A + t = A t · t · A t , a t w o-sided ideal generated by the element t . Then, there are n atural A -bimo dule isomorphisms A t / A + t ∼ → A, and A t / ( A + t ) 2 ∼ → A ⊕ ( A ⊗ A ) , a + a ′ t a ′′ 7− → a ⊕ ( a ′ ⊗ a ′′ ) . (1.2.2) Giv en a k -linear map Θ : A → A ⊗ A we will u se sy mb olic Swe edler notation to write this map as a 7→ Θ ′ ( a ) ⊗ Θ ′′ ( a ), w here we sys tematica lly sup press the summation sym b ol. No w, a free p ro duct analogue of Lemma 1.2.1 reads as follo ws . Lemma 1.2.3. The fol lowing pr op erties of a k -line ar map Θ : A → A ⊗ A ar e e quivalent: • the map Θ is a double derivation; • the map A → A + t , a 7→ Θ ′ ( a ) t Θ ′′ ( a ) , is a derivation of the algebr a A with c o efficients in the A -bimo dule A + t ; • the map a 7→ a + Θ ′ ( a ) t Θ ′′ ( a ) giv e s an algebr a homomorph ism A → A t / ( A + t ) 2 . 2 1.3. La yo ut of the pap er. In § 3, w e r ecall the definition of th e DG algebra of n oncomm utativ e differen tial forms [2, 3], follo wing [6], and that of the K arou b i-de Rham complex [18]. W e also in tro d u ce an extende d Kar oubi -de Rham c omplex , that will pla y a crucial role later. In § 2, we dev elop the basics of n oncomm utativ e calculus inv olving the action of d ou b le deriv ations on the extended Karoubi-de Rh am complex, via Lie deriv ativ e and con traction op erations. In § 4, we state three main results of the pap er. The first tw o, Theorem 4.1.1 and Theorem 4.3.2, pro vide a description, in terms of the Karoubi-de Rham complex, of the Hochsc h ild homolog y of an algebra A and of the p eriod ic cyclic h omology of A , resp ectiv ely . The third r esult, Th eorem 4.4.1, giv es a formula for the Gauss-Manin connection on p erio dic cyclic homology of a family of algebras, [13], in a wa y that av oids complicated formulas and resem bles equiv arian t cohomology . These results are pro ve d in § 5, using prop erties of the Karoubi op erator and the harmonic decomp osition of noncomm utativ e differen tial forms in tro d uced by Cuntz and Qu illen, [6, 7]. In § 6, w e establish a connection b et ween cyclic homology and equiv arian t cohomology via the represent ation fu nctor. More precisely , we giv e a homomorphism from our non commutativ e equi- v ariant d e Rham complex (wh ich extends the complex u s ed to compu te cyclic homology) to the equiv ariant de Rham complex computing equiv arian t cohomology of th e repr esen tation v ariet y . In § 7, w e in tro duce a n ew notion of free p r o duct deformation ov er a not necessarily commuta tive base. W e extend classic results of Gerstenhab er concerning deformations of an asso ciativ e algebra A to our n ew setting of f ree pr o duct deformations. T o th is end , we consider a doub le-graded Ho c hsc h ild complex  ⊕ p,k ≥ 2 C p ( A, A ⊗ k ) , b  . W e define a new asso ciativ e pro d uct f , g 7→ f ∨ g on that complex, and stud y Maur er-Cartan equations of the form b ( β ) + 1 2 β ∨ β = 0. 1.4. Ac knowledgemen ts. W e thank Y an Soib elman, Boris Tsyg an, and Michel V an den Bergh for useful comments. The first a uthor was partially supp or ted by the NSF g rant DMS-03 03465 and CRDF gra nt RM1-254 5 -MO-03 . The second author was partially supp orted b y a n NSF GRF. 2. No ncommut a tive cal culus 2.1. The comm utator quotient. Let B = ⊕ k ∈ Z B k b e a Z -graded algebra and M = ⊕ k ∈ Z M k a Z -graded B -bimo dule. Giv en a homogeneous elemen t u ∈ B k or u ∈ M k , we p ut | u | := k . A linear map f : B q → M q + n is said to b e a de gr e e n gr ade d deriv ation if, for any h omogeneous u, v ∈ B , f ( uv ) = f ( u ) · v + ( − 1) n | u | u · f ( v ). Given a graded subalgebra R ⊂ B , we let Der n R ( B , M ) denote the v ector space of d egree n graded R -linear d eriv ations. The direct s u m Der q R B := L n ∈ Z Der n R ( B , B ) has a n atural Lie sup e r -algebra structure giv en by the sup er -comm utator. Let [ B , B ] b e the k -linear span of the s et { b 1 b 2 − ( − 1) pq b 2 b 1 , b 1 ∈ B p , b 2 ∈ B q , p, q ∈ Z } . W e put B cyc := B / [ B , B ]. An y (ungraded) algebra ma y b e regarded as a graded algebra concentrat ed in degree zero. Thus, for an algebra B w ithout grading, w e h a v e the sub space [ B , B ] ⊂ B spann ed b y ordinary comm utators and the corresp onding comm utator qu otien t space B cyc = B / [ B , B ]. W e write T B M = ⊕ n ≥ 0 T n B M for th e tensor algebra of a B -bimo du le M . Th us, T q B M is a graded asso ciativ e algebra with T 0 B M = B . 2.2. Noncomm utativ e differen tial forms. Fix an algebra A and a subalgebra R ⊂ A . (After this section, w e will only tak e R to b e either k or k [ t ], but other in teresting examples include when R = k I and A is a qu otien t of the path algebra of a quiv er with vertex set I .) Let Ω 1 R A := Ker( m ) b e the kernel of the multiplic ation map m : A ⊗ R A → A , and write i ∆ : Ω 1 R A ֒ → A ⊗ R A for the tautological em b edding. Thus, Ω 1 R A is a A -bimo d ule, called the b imo dule of nonc ommutative one-forms on the algebra A r elativ e to the su balgebra R . One has a short exact s equ ence of A -bimo dules, see [6, § 2], 0 − → Ω 1 R A i ∆ − → A ⊗ R A m − → A − → 0 . (2.2.1) 3 The assignm ent a 7→ d a := 1 ⊗ a − a ⊗ 1 giv es a canonical deriv ation d : A → Ω 1 R A . Th is deriv ation is ‘u n iv ersal’ in the sense th at, for any A -b im o dule M , th ere is a canonical bijection Der R ( A, M ) ∼ → Hom A - bimod (Ω 1 R A, M ) , θ 7→ i θ , (2.2.2) where i θ : Ω 1 R A → M stands for a A -bimo d u le map defin ed by the form ula i θ ( u d v ) := u · θ ( v ). The map d extends un iquely to a degree 1 d eriv ation of Ω R A := T q A (Ω 1 R A ), the tensor algebra of the A -bimo dule Ω 1 R A . Thus, (Ω q R A, d ) is a DG algebra called the algebra of n oncomm utativ e differen tial f orm s on A relativ e to the subalgebra R (w e will interc h angeably use the notation Ω R A or Ω q R A dep en ding on w hether we w ant to emphasize the grading or not). F or eac h n ≥ 1, there is a standard isomorph ism of left A -mo du les, see [6], Ω n R A = A ⊗ R T n R ( A/R ); usually , one writes a 0 d a 1 d a 2 . . . d a n ∈ Ω n R A for the n -form corresp onding to an eleme nt a 0 ⊗ ( a 1 ⊗ . . . ⊗ a n ) ∈ A ⊗ R T n R ( A/R ) und er this isomorph ism . The de Rham differenti al d : Ω q R A → Ω q +1 R A is giv en by the form ula d : a 0 d a 1 d a 2 . . . d a n 7→ d a 0 d a 1 d a 2 . . . d a n . F ollo wing K aroubi [18], we defi ne the (relativ e) n oncomm utativ e de Rham complex of A as DR R A := (Ω R A ) cyc = Ω R A/ [Ω R A, Ω R A ] , the s up er-commuta tor quotien t of the gr ade d algebra Ω q R A . Th e space DR R A comes equipp ed w ith a natural gradin g and with the de Rh am differen tial d : DR q R A → DR q +1 R A , in d uced from the one on Ω q R A . In degrees 0 and 1, DR 0 R A = A/ [ A, A ] and DR 1 R A = Ω 1 R A/ [ A, Ω 1 R A ], resp ectiv ely . In the ‘absolute’ case R = k we will use unadorned notation Ω n A := Ω n k A, DR A := DR k A , etc. The DG algebra (Ω q R A, d ) ca n b e c haracterized by a univ ersal prop erty s aying that it is the unive rs al DG R -algebra generated by A (see [6, Corollary 2.2]). The univ ersal pr op ert y implies, in particular, for any algebras A and B , a canonical isomorphism Ω( A ∗ B ) ∼ = Ω A ∗ Ω B . (2.2.3) 2.3. Lie deriv ative and con traction for noncomm ut ativ e differen tial forms. In this section, w e introdu ce op erations of Lie d eriv ativ e and con traction on noncomm u tativ e different ial f orms. Fix an algebra A and a sub algebra R ⊂ A . Any deriv ation θ ∈ Der R A giv es rise, naturally , to a pair of graded deriv ations of the graded algebra Ω q R A , a con traction op eration i θ ∈ Der q − 1 A (Ω q R A ) and a Lie deriv ative op eration L θ ∈ Der q R (Ω q R A ), resp ectiv ely . T o define these op erations it is con v enient to use th e follo wing construction. Let K b e a graded Ω q R A -bimo dule and f : Ω 1 R A → K an A -bimo d ule map. One sho ws, by adapting the pr o of of [6, Prop ositio n 2.6] to a graded setting, that the assignment a 7→ 0 , d a 7→ f ( d a ) can b e extended uniquely to an A -linear deriv ation T q A (Ω 1 R A ) → K . Hence, one has the follo wing c h ain of canonical isomorph ism s Der q R ( A, K ) ∼ → (2.2.2) Hom q A - bimo d (Ω 1 R A, K ) ∼ → Der q − 1 A ( T q A (Ω 1 R A ) , K ) = Der q − 1 A (Ω R A, K ) . (2.3.1) Let θ ∈ Der q R ( A, K ) b e a graded deriv ation. Applying to θ the comp osite isomorph ism ab ov e one obtains a graded deriv ation i θ ∈ Der q − 1 A (Ω R A, K ), called c ontr action with θ . T o d efine con traction on differenti al forms, we put K = Ω q R A . F or any θ ∈ Der R A , the im age of θ in Der q − 1 A ( T q A (Ω 1 R A ) , K ) ab o ve p ro duces a graded deriv ation, wh ic h we also call i θ , on Ω q R A = T q A (Ω 1 R A ) (in fact, it is an A -deriv ation, not merely an R -deriv ation). F or a one-form u d v , w e ha ve i θ ( u d v ) = u · θ ( v ). Next, one d efines a Lie deriv ativ e map L θ : Ω q R A → Ω q R A by th e Cartan form ula L θ := i θ d + d i θ . Here, the expression on the righ t h and sid e is the sup er-comm utator of i θ , a graded deriv ation of degree − 1, with d , a graded deriv ation of d egree +1. It follo ws that L θ is a degree 0 deriv ation of the algebra Ω q R A . F or a one-form u d v w e hav e L θ ( u d v ) = ( θ ( u )) d v + u d ( θ ( v )). 4 Eac h of the maps L θ and i θ descends to a well-defined op eration on the de Rham complex DR q R A = (Ω R A q ) cyc . 2.4. Lie deriv ative a nd con traction for double deriv ations. W rite D er R A := Der R ( A, A ⊗ A ) for th e v ector sp ace of R -deriv ations A → A ⊗ A , whic h we refer to as “double deriv ations.” When R = k we write D er A . Double deriv ations do not give rise to natural op erations on the DG algebra Ω q R A itself. Instead, for any double deriv ation Θ : A → A ⊗ A , w e will define asso ciated con traction and Lie deriv ativ e op erations, which are double deriv ations Ω R A → Ω R A ⊗ Ω R A . T o do so, we apply the general construction based on (2.3.1) in the sp ecial case where K = Ω q R A ⊗ Ω q R A , an Ω q R A -bimo dule with resp ect to the outer action. F or Θ ∈ D er R A , w e consider the comp osition A → A ⊗ A = Ω 0 R A ⊗ Ω 0 R A ֒ → Ω q R A ⊗ Ω q R A = K . Th is is a degree zero deriv ation A → K so, u sing (2.3.1), w e obtain a co ntractio n map Ω R A → Ω R A ⊗ Ω R A . This map, to b e denoted i Θ , is an A -linear graded deriv ation of degree ( − 1). In the sp ecia l case n = 1, the con tr action i Θ reduces to the map i Θ : Ω 1 R A → A ⊗ A, α 7→ ( i ′ Θ α ) ⊗ ( i ′′ Θ α ) , that corresp onds to the d eriv ation Θ ∈ D er R A via the canonical b ijection (2.2.2) . More generally , for n ≥ 1, separating ind ividual h omogeneous comp onen ts, th e cont r action giv es maps i Θ : Ω n R A 7− → L 1 ≤ k ≤ n Ω k − 1 R A ⊗ Ω n − k R A. Explicitly , for any α 1 , . . . , α n ∈ Ω 1 R A , one fin ds i Θ ( α 1 α 2 . . . α n ) = X 1 ≤ k ≤ n ( − 1) k − 1 · α 1 . . . α k − 1 ( i ′ Θ α k ) ⊗ ( i ′′ Θ α k ) α k +1 . . . α n . (2.4.1) Next, we define the Lie deriv ativ e. T o this end, one fi rst extends the de Rham d ifferen tial on Ω R A to a degree one m ap d : Ω q R A ⊗ Ω q R A → Ω q R A ⊗ Ω q R A defin ed, f or any α ∈ Ω p R A and β ∈ Ω q R A , b y th e form ula d ( α ⊗ β ) := ( d α ) ⊗ β + ( − 1) p α ⊗ ( d β ) . No w, giv en Θ ∈ D er R A , we use Cartan’s form ula as a definition and p u t L Θ := i Θ d + d i Θ . Th is giv es a graded double der iv ation L Θ : Ω R A → Ω R A ⊗ Ω R A of degree 0. 3. Exte nded Karoubi-de Rham comp lex 3.1. Cyclic quotien ts for free pro ducts. Give n a graded algebra B = L q B q , we equip B t = B ∗ k [ t ], a fr ee pro du ct algebra, with a b i grading B t = L p,q B p,q t suc h that the homogeneous comp onen t B q , of the su balgebra B ⊂ B t is assigned bidegree (0 , q ) and the v ariable t is assigned bidegree (2 , 0). T h us, the p -grading counts twice the num b er of o ccur r ences of the v ariable t . The bigrading on B t descends to a bigrading ( B t ) cyc = L p,q ( B t ) p,q cyc on the su p er-comm u tator quotien t. In particular, ( B t ) 0 , q cyc = B q cyc . F urther, it is clear that the assignment u 1 t u 2 7→ ( − 1) | u 1 |·| u 2 | u 2 u 1 yields an isomorphism ( B t ) 2 , q cyc ∼ = B . More ge nerally , for an y n ≥ 1, th e space ( B t ) 2 n, q cyc is spanned b y cyclic words u 1 t u 2 t . . . t u n t , where cyclic means that, for instance, one has u 1 t u 2 t u 3 t = ( − 1) | u 1 | ( | u 2 | + | u 3 | ) t u 3 t u 1 t u 2 (mo dulo comm utators). Giv en a graded v ector space V , we let th e group Z /n Z act on V ⊗ n = T n k V by cyclic p ermutations tensored by the sign charact er and w rite V ⊗ n cyc := V ⊗ n / ( Z /n Z ) . Th u s, the assignments u 1 t u 2 t . . . t u n 7→ u 1 ⊗ u 2 ⊗ . . . ⊗ u n and u 1 t u 2 t . . . t u n t 7→ u 1 ⊗ u 2 ⊗ . . . ⊗ u n yield natural vecto r space isomorphisms B 2 n − 2 , q t ∼ → ( B q ) ⊗ n and ( B t ) 2 n, q cyc ∼ = ( B q ) ⊗ n cyc , ∀ n = 1 , 2 , . . . . (3.1.1) An y graded deriv ation θ : B → B has a n atural extension to a graded deriv ation θ t : B t → B t that restricts to θ on the subalgebra B ⊂ B t and send s t to 0. F u rther, an y double deriv ation 5 Θ : B → B ⊗ B giv es rise to a un ique deriv ation Θ t : B t → B t suc h that Θ t ( b ) = Θ ′ ( b ) t Θ ′′ ( b ), for an y b ∈ B , and Θ t ( t ) = 0. No w fix an un graded algebra B . W e view it as a graded algebra concen trated in degree zero. Then, the second grading on B t b ecomes trivial, so the bigrading effectiv ely reduces to a grading B t = L p B 2 p t , b y ev en integ ers. Then, we immediately d educe the follo win g. Lemma 3.1.2. Der p k [ t ] ( B t , B t ) = 0 for any p < 0 . The assignments θ 7→ θ t and Θ 7→ Θ t , yield isomorph isms Der( B , B ) ∼ → Der 0 k [ t ] ( B t , B t ) and D er( B , B ) ∼ → Der 2 k [ t ] ( B t , B t ) , r esp e ctively. 3.2. The extended de Rham complex. W e are going to in tro du ce an enlargemen t of the non- comm utativ e de Rham complex Ω A . This enlargemen t is a DG algebra that has three equiv alent definitions according to the follo wing lemma. Lemma 3.2.1. Ther e ar e natur al algebr a isomorphisms Ω k [ t ] ( A t ) ∼ = (Ω A ) ∗ k [ t ] ∼ = T A ( A ⊗ 2 ⊕ Ω 1 A ) . (3.2.2) The differ ential on T A ( A ⊗ 2 ⊕ Ω 1 A ) obtaine d by tr ansp orting the de R ham differ ential on Ω k [ t ] ( A t ) via the isomorphisms in (3.2.2) i s the derivation induc e d by the c omp osite i ∆ ◦ d : A → Ω 1 A ֒ → A ⊗ A (using (2.2 .1 ) ). Pr o of. Th e algebra Ω R ( A ∗ R ) is the quotien t of the algebra Ω( A ∗ R ) by the tw o-sided id eal generated b y the sp ace d R ⊂ Ω 1 R ⊂ Ω 1 ( A ∗ R ). S ince Ω( A ∗ R ) ∼ = (Ω A ) ∗ (Ω R ), by (2.2.3), w e d ed uce a DG algebra isomorp hism Ω R ( A ∗ R ) ∼ = (Ω A ) ∗ R. (3.2.3) Applying this in the sp ecial case R = k [ t ] yields the firs t isomorph ism in (3.2.2). T o p ro v e the second isomo r phism in (3.2.2), fix an A -b im o dule M . Vie w A ⊗ 2 ⊕ M as an A - bimo dule. The assignment ( a ′ ⊗ a ′′ ) ⊕ m 7→ a ′ ta ′′ + m clea r ly giv es an A -bimo dule map A ⊗ 2 ⊕ M → ( T A M ) t . This map can b e extended, by the univ ersal pr op ert y of the tensor algebra, to an algebra morphism T A ( A ⊗ 2 ⊕ M ) → ( T A M ) t . T o sh o w that this morp hism is an isomorphism , w e explicitly construct an inv erse map as follo w s . W e start with a natural alge b ra em b edd ing f : T A M ֒ → T A ( A ⊗ 2 ⊕ M ), ind u ced by the A - bimo dule em b edd in g M = 0 ⊕ M ֒ → A ⊗ 2 ⊕ M . Then, by the u niv ersal prop ert y of free pro du cts, w e can (uniquely) extend the map f to an algebra h omomorphism ( T A M ) t = ( T A M ) ∗ k [ t ] → T A ( A ⊗ 2 ⊕ M ) by sending t 7→ 1 A ⊗ 1 A ∈ A ⊗ 2 ⊂ T 1 A ( A ⊗ 2 ⊕ M ). It is str aightforw ard to chec k that the resulting h omomorphism is indeed an inv erse of the homomorp h ism constructed earlier. Applying the ab o ve in the sp ecial case M = Ω 1 A yields the second isomorphism of the lemma. The statemen t of the lemma concerning differen tials is left to th e reader.  It is con venien t to introdu ce a sp ecia l notation Ω t A := Ω k [ t ] ( A t ). T his algebra comes equipp ed with a natural bi -grading Ω t A = ⊕ p,q ≥ 0 Ω 2 p,q t A , w h ere the ev en p -grading is induced from the one on A t , and the q -comp onent corresp onds to the grading indu ced b y the n atural one on Ω q A . It is easy to see th at the p -grading corresp onds, und er the isomorphism (3 .2.2 ) to the grading on (Ω A ) ∗ k [ t ] that coun ts t wice the num b er of o ccurrences of the v ariable t . The extende d de Rham c omplex of A is defin ed as a sup er-comm u tator quotien t DR t A := DR k [ t ] ( A t ) = (Ω k [ t ] ( A t )) cyc ∼ = ((Ω A ) t ) cyc . The bigrading on Ω t A clearly descends to a bigrading on the extended de Rh am complex of A . The de Rh am d ifferen tial has bidegree (0 , 1): DR t A = ⊕ p,q DR 2 p,q t A, d : DR 2 p,q t A → DR 2 p,q +1 t A. 6 Next, we u s e the identificat ion (3.1.1) for B := Ω A , and equip (Ω q A ) ⊗ p with the tensor pr o duct grading that coun ts the total degree of differentia l form s in vo lved, e.g., given α i ∈ Ω k i A, i = 1 , . . . , p , f or α := α 1 ⊗ . . . ⊗ α p ∈ (Ω A ) ⊗ p cyc , we pu t deg α := k 1 + . . . + k p . Then, w e get DR 2 p,q t A = ( DR q A if p = 0; degree q comp onen t of (Ω q A ) ⊗ p cyc if p ≥ 1 . (3.2.4) 3.3. Op erations on the extended de Rham complex. Let θ ∈ Der A . The Lie deriv ativ e L θ is a deriv ation of the algebra Ω q A . Hence, there is an asso ciated deriv ation ( L θ ) t : (Ω A ) t → (Ω A ) t ; see Lemma 3.1.2(ii). On the other hand, we ma y fi rst extend θ to a deriv ation θ t : A t → A t , and then consider the L ie deriv ativ e L θ t , a deriv ation of the algebra Ω k [ t ] ( A t ) = Ω t A , of bidegree (0 , 0). V ery similarly , w e also ha ve graded deriv ations, ( i θ ) t and i θ t , of bid egree (0 , − 1). It is immediate to see that the t wo pr o cedures ab o ve agree with eac h other in the sense that, under the identificati on Ω k [ t ] ( A t ) ∼ = (Ω A ) t pro vided by (3.2.2 ), L θ t = ( L θ ) t , and i θ t = ( i θ ) t . (3.3.1) Next, we consider op erations ind uced by doub le deriv ations. Prop osition 3.3.2. Any double derivation Θ ∈ D er A gives a c anonic al Lie derivative op er ation, a gr ade d derivation L Θ ∈ Der k [ t ] (Ω t A ) of bide gr e e (2 , 0) , and a c ontr action op er ation, a g r ade d derivation i Θ ∈ Der k [ t ] (Ω t A ) of bide gr e e (2 , − 1) . Pr o of. Give n Θ ∈ D er A , w e fi rst extend it to a f r ee pro du ct d er iv ation Θ t : A t → A t , as in § 3.1. Hence, th er e are associated Lie deriv ativ e L Θ t , and con traction i Θ t , op eratio n s on the complex Ω k [ t ] ( A t ), of relativ e differen tial forms on the algebra A t . Th us, we ma y use (3.1.1) to inte rp ret L Θ t and i Θ t as op erations on Ω t A , to b e denoted by L Θ and i Θ , resp ectiv ely .  R emark 3.3.3 . On e can u se the ab ov e defin ed actions of doub le deriv ations on Ω t A to obtain an alternativ e construction of the op erations introd uced in § 2.4. Sp ecifically , we observe that the op erations L Θ t and i Θ t ma y b e v iewed, thank s to the fi rst isomorphism in (3.2.2), as deriv ations of the algebra (Ω A ) t of degree 2 w ith resp ect to the p -grading, that counts (t wice) the n u m b er of o ccurrences of t . Hence, app lying L emm a 3.1.2, we conclude that there exists a u nique d ouble deriv ation L Θ : Ω A → Ω A ⊗ Ω A su c h that, for the corresp ondin g map (Ω A ) t → (Ω A ) t , we hav e L Θ t = ( L Θ ) t . A similar argum ent yields a doub le deriv ation i Θ : Ω A → Ω A ⊗ Ω A su ch th at i Θ t = ( i Θ ) t . It is easy to c h eck that the deriv ations L Θ and i Θ th us defined are th e same as those in tro d u ced in § 2.4 in a differen t w ay . ♦ Both the Lie deriv ativ e and con traction op er ations on Ω t A d escend to the comm utator quotien t. This wa y , w e obtain the Lie deriv ativ e L Θ and the cont r action i Θ on DR t A , the extended de Rham complex. Explicitly , using isomorphisms (3.2.4), we can write the Lie der iv ativ e L Θ and con traction i Θ op erations as chains of maps of the f orm DR A − → Ω A − → (Ω A ) ⊗ 2 cyc − → (Ω A ) ⊗ 3 cyc − → . . . . (3.3.4) There are some standard ident ities inv olving the Lie d eriv ativ e and contrac tion op erations on Ω A asso ciated with ordinary deriv ations. S imilarly , the Lie d eriv ativ e and con traction op er ations on Ω t A resu lting from Prop ositio n 3.3.2 satisfy the follo wing id en tities: L Θ = d ◦ i Θ + i Θ ◦ d , i Θ ◦ i Φ + i Φ ◦ i Θ = 0 , i ξ ◦ i Θ + i Θ ◦ i ξ = 0 , ∀ Θ , Φ ∈ D er A, ξ ∈ Der A. (3.3.5) It follo ws, in particular, that the Lie deriv ativ e L Θ comm utes with the de Rham d ifferen tial d . T o p ro ve (3.3. 5 ), one first v erifies these iden tities on the generators of the algebra Ω t A = (Ω A ) t , that is, on differential forms of degrees 0 and 1, which is a simple computation. The general case 7 then follo ws by observing th at an y comm u tation relation b etw een (graded)-deriv ations th at holds on generators of the algebra holds true for all elemen ts of the algebra. It is imm ed iate that the indu ced op erations on DR t A also satisfy (3.3.5). 3.4. Reduced Lie deriv ativ e and con traction. The second comp onent (the q -comp onen t) of the bigrading on DR t A induces a grading on eac h of the sp aces (Ω A ) ⊗ k cyc , k = 1 , 2 , . . . , app earing in (3.3.4) . Eac h o f the maps in (3.3.4) corresp ond ing to the L ie d eriv ativ e induced by a double deriv ation Θ ∈ D er A preserves the q -grading. In the case of cont r action with Θ, all maps in th e corresp ondin g c hain (3.3.4) decrease the q -grading by one. The leftmost map in (3.3.4), to b e denoted ı Θ in th e con traction case and L Θ in th e Lie deriv ative case, will b e esp eciall y imp ortant for u s. These maps, whic h we will call the r e duc e d c ontr action and r e duc e d Lie derivative , resp ectiv ely , hav e the form ı Θ : DR q A − → Ω q − 1 A, and L Θ : DR q A − → Ω q A. (3.4.1) Explicitly , w e see from (2.4.1) that the op eration ı Θ , for instance, is giv en, for an y α 1 , α 2 , . . . , α n ∈ Ω 1 A , b y the follo wing formula: ı Θ ( α 1 α 2 . . . α n ) = n X k =1 ( − 1) ( k − 1)( n − 1) · ( i ′′ Θ α k ) · α k +1 . . . α n α 1 . . . α k − 1 · ( i ′ Θ α k ) . (3.4.2) An ad ho c defin ition of the maps in (3.4 .1 ) via explicit form ulas lik e (3.4.2) w as first giv en in [5]. Pro ving the commutatio n relations (3.3.5) u sing exp licit formulas is, ho wev er, ve r y painful; this w as carried out in [5] by rather long bru te-force compu tations. Our presen t approac h based on the free p ro duct construction yields th e comm utation relations for free. 3.5. The deriv ation ∆ . T here is a distinguished double deriv ation ∆ : A → A ⊗ A, a 7→ 1 ⊗ a − a ⊗ 1 . The corresp onding con traction map i ∆ : Ω 1 A → A ⊗ A is the ta utological em b edding (2.2.1). F urth ermore, the deriv ation ∆ t : A t → A t asso ciated with ∆, cf § 3.1, equals ad t : u 7→ t · u − u · t . Hence, th e Lie der iv ativ e map L ∆ : Ω t A → Ω t A reads ω 7→ ad t ( ω ) := t · ω − ω · t . Lemma 3.5.1. ( i ) F or any a 0 , a 1 , . . . , a n ∈ A , ı ∆ ( a 0 d a 1 . . . d a n ) = X 1 ≤ k ≤ n ( − 1) ( k − 1)( n − 1)+1 · [ a k , d a k +1 . . . d a n a 0 d a 1 . . . d a k − 1 ] . ( ii ) In Ω q A , we have ı ∆ ◦ d + d ◦ ı ∆ = 0 and d 2 = ( ı ∆ ) 2 = 0; similar e quations also hold in DR q t A , with i ∆ in plac e of ı ∆ . Pr o of. Part (i) is v erified by a straigh tforwa rd computation based on form ula (2.4.1). W e claim next that, in Ω t A , w e ha ve L ∆ = ad t . Indeed, it su ffices to c h eck this equalit y on the generators of the algebra Ω t A . It is clear that L ∆ ( t ) = 0 = ad t ( t ), and it is easy to see that b oth deriv ations agree on zero-forms and on one-forms. This p ro ve s th e claim. The equation ı ∆ ◦ d + d ◦ ı ∆ = 0, of part (ii) of the lemma, no w follo w s b y the Cartan form ula on the left of (3.3.5), since the equation L ∆ = ad t clearly implies that the map L ∆ : DR t A → DR t A , as w ell as the map L ∆ , v anishes. Finally , th e formula of part (i) sho ws that the image of the map ı ∆ is conta ined in [ A, Ω A ]. Hence, w e dedu ce ( ı ∆ ) 2 (Ω A ) ⊂ ı ∆ ([ A, Ω A ]) = 0, since the map ı ∆ v anishes on comm utators.  Let A τ := A ∗ k [ τ ] b e the graded algebra su c h th at A is placed in d egree zero and τ is an o dd v ariable placed in degree 1. Let d dτ b e the degree − 1 deriv ation of the algebra A τ that annihilates A and satisfies d dτ ( τ ) = 1. Similarly , let τ 2 d dτ b e the degree +1 graded d eriv ation of the algebra 8 A τ that annihilates A and satisfies τ 2 d dτ ( τ ) = τ 2 . F or an y homogeneous elemen t x ∈ A τ , put ad τ ( x ) := τ x − ( − 1) | x | τ ; in p articular, on e fi nds that ad τ ( τ ) = 2 τ 2 . It is easy to c heck th at eac h of the deriv ations d dτ , τ 2 d dτ , and ad τ − τ 2 d dτ squares to zero. Claim 3.5.2 . ( i ) The fol lowing assignment g ives a gr ade d algebr a emb e dding: j : Ω t A ֒ → A ∗ k [ τ ] , t 7→ τ 2 , a 0 d a 1 . . . d a n 7→ a 0 · [ τ , a 1 ] · . . . · [ τ , a n ] , Mor e over, the ab ove map intertwines the c ontr action op er ation i ∆ with the differ ential τ 2 d dτ , and the Kar oubi-de Rh am differ ential d with the differ e ntial ad τ − τ 2 d dτ . ( ii ) The image of the map j is annihilate d by the derivation d dτ . ( iii ) The c omplex  ( A τ ) cyc , d dτ  c omputes cyclic homolo gy of the algebr a A .  W e will neither use nor pro ve this result; cf. [6, Prop osition 1.4] an d [21, § 4.1 and Lemma 4.2.1]. 4. Applica tions to Hochs child and cycl ic homolo gy 4.1. Ho c hsc hild homology. Give n an algebra A and an A -b im o dule M , w e let H k ( A, M ) denote the k -th Ho c hschild h omology group of A with co efficien ts in M . Also, wr ite [ A, M ] ⊂ M for the k -linear span of the set { am − ma | a ∈ A, m ∈ M } . Thus, H 0 ( A, M ) = M / [ A, M ]. W e extend some ideas of C unt z and Quillen [7] to obtain our first imp ortant result. Theorem 4.1.1. F or any unital k -algebr a A , ther e is a natur al gr ade d sp ac e isomorphism H q ( A, A ) ∼ = Ker[ ı ∆ : DR q A → Ω q − 1 A ] . T o put Th eorem 4.1.1 in cont ext, recall that Cuntz and Quillen used noncommutat ive d ifferen tial forms to compute Ho c hschild homology . Sp ecifica lly , follo wing [6] and [7], consider a complex . . . b − → Ω 2 A b − → Ω 1 A b − → Ω 0 A − → 0. Here, b is the Ho chschild differ ential given by the formula b : α d a 7− → ( − 1) n · [ α, a ] , ∀ a ∈ A/ k , α ∈ Ω n A, n > 0 . (4.1.2) It wa s shown in [7] th at the complex (Ω q A, b ) can b e id entified with the stand ard Ho c hschild c hain complex. It follo ws that H q (Ω A, b ) = H q ( A, A ) are the Ho c hschild h omology groups of A . Theorem 4.1.1 will follo w d irectly from Prop ositio n 5.1.1 b elo w (see the discussion after this prop osition). Prop osition 5.1.1 itself will b e pr ov ed in § 5.2. R emark 4.1.3 . A somewhat more geomet ric in terpr etatio n of Theorem 4.1.1, from the p oint of view of represent ation fun ctors, is pro vided by the map (6.2.6): see Th eorem 6.2.5 of § 6 b elo w. 4.2. An a pplicat ion. T he algebra A is said to b e c onne cte d if th e follo w in g sequence is exact: 0 − → k − → DR 0 A d − → DR 1 A. (4.2.1) Prop osition 4.2.2. L et A b e a connected algebr a su c h that H 2 ( A, A ) = 0 . Then, • H 1 ( A, A ) = (DR 1 A ) closed and (DR 2 A ) exact = (DR 2 A ) closed . • Ther e is a natur al ve ctor sp ac e isomorphism (DR 2 A ) closed ∼ → [ A, A ] . Pr o of. W e will freely use th e notatio n of [5, § 4.1]. According to [5, Pr op osition 4.1.4] , for any connected algebra A , there is a map f µ nc , a lift of the nonc ommutative moment map , that fits in to the follo wing comm utativ e diagram: DR 1 A d / / ı ∆     (DR 2 A ) closed g µ nc u u ı ∆   [ A, A ] d / / [ A, Ω 1 A ] . (4.2.3) 9 Since H 2 ( A, A ) = 0, we deduce from the short exact sequence of Th eorem 4.1.1 for n = 2 that the righ t vertical map ı ∆ in diagram (4.2.3) is injectiv e. It follo ws, by comm utativit y of the low er righ t triangle in (4.2.3), that the map f µ nc m ust b e injectiv e. Thus, from Theorem 4.1.1 for n = 1, w e dedu ce H 1 ( A, A ) = k er( ı ∆ : DR 1 A → [ A, A ]) = (DR 1 A ) closed . Next, the left vertic al map ı ∆ in the diagram is give n by the formula a d b 7→ [ a, b ]. Hence, this map is s urjectiv e. Since the map f µ nc is injectiv e, it follo ws fr om th e commuta tivit y of th e upp er left triangle in (4.2.3) that f µ nc , as well as the map d in the top row of diagram (4.2.3), must b e surjectiv e. W e conclude that the map f µ nc yields an isomorphism (DR 2 A ) closed ∼ → [ A, A ] and also that (DR 2 A ) exact = (DR 2 A ) closed .  Prop osition 4.2.2 can b e easily extended to a relativ e setting wh ere the algebra A cont ains a subalgebra of the form R = k r , for some r ≥ 1. Then, the correct relat ive coun terpart of the comm utator space [ A, A ] turns out to b e the s u bspace [ A, A ] R ⊂ [ A, A ], formed by the elemen ts whic h commute w ith R . Th e corresp onding f orm alism has b een work ed out in [5]. Th e relativ e v ersion of Prop osition 4.2.2 reads as follo w s . Corollary 4.2.4. L e t R = k r . L e t A b e an algebr a c ontaining R and such that the se quenc e 0 → R → A → DR 1 R A is e xact and H 2 ( A, A ) = 0 . Then, ther e is a natur al ve ctor sp ac e isomorphism (DR 2 R A ) closed ∼ → [ A, A ] R .  An imp ortan t example where the ab o ve corollary applies is the case where A is the path algebra of a quiver with r ve rtices. R emark 4.2.5 . The isomorphism of Prop osition 4.2.2 and Corollary 4.2.4 plays a r ole in th e theory of Calabi-Y au algebras; see [14, Claim 3.9.11]. 4.3. Cyclic homology. W e r ecall some standard definitions, follo wing [22, Chapter 2 and p. 162]. F or any graded v ector sp ace M = ⊕ i ≥ 0 M i , we introdu ce a Z -graded k [ t, t − 1 ]-mo dule M ˆ ⊗ k [ t, t − 1 ] := L n ∈ Z  Y i ∈ Z t i M n − 2 i  , (4.3.1) where the grading is such that the space M ⊂ M ˆ ⊗ k [ t, t − 1 ] has th e natural grading, and | t | := 2. Belo w , we will u s e a complex of r e duc e d different ial forms, defined b y setting Ω 0 := Ω 0 A/ k = A/ k and Ω k := Ω k A for all k > 0. Let Ω • := L k ≥ 0 Ω k . The Ho c hs c hild differen tial induces a k [ t, t − 1 ]- linear differen tial b : Ω ˆ ⊗ k [ t, t − 1 ] → Ω ˆ ⊗ k [ t, t − 1 ] of degree − 1. W e also hav e the Connes d ifferen tial B : Ω • → Ω q +1 [2]. F ollo wing Lo day and Quillen [23], we extend it to a k [ t, t − 1 ]-linear differen tial on Ω ˆ ⊗ k [ t, t − 1 ] of degree +1. It is known that B 2 = b 2 = 0 and B ◦ b + b ◦ B = 0. T h us, the map B + t · b : Ω ˆ ⊗ k [ t, t − 1 ] → Ω ˆ ⊗ k [ t, t − 1 ] giv es a d egree +1 different ial on Ω ˆ ⊗ k [ t, t − 1 ]. W rite H P − q ( A ), where, ‘ − q ’ den otes inverting the degrees, f or the r e duc e d p erio dic cyclic ho- molo gy of A as d efined in [23] or [22, § 5.1], using a complex with differential of degree − 1. According to [7], the groups H P − q ( A ) turn out to b e isomorp hic to h omology groups of the complex ( Ω ˆ ⊗ k [ t, t − 1 ] , B + t · b ), with d ifferen tial of degree +1 (whic h is w hy we m us t inv ert the degrees). It is known th at the action of multiplicat ion by t yields p eriod icit y isomorph isms H P q ( A ) ∼ = H P q +2 ( A ) . Thus, up to isomorphism, there are only tw o groups, H P ev en ( A ) := H P 0 ( A ), and H P odd ( A ) := H P 1 ( A ). Next, we comp ose the map ı ∆ : DR q A → Ω q − 1 A with the natural pro jection Ω q A ։ DR q A to obtain a map Ω q A → Ω q − 1 A . The latter map descends to a map Ω • → Ω •− 1 . F urthermore, w e ma y extend this last map, as wel l as the de Rham differentia l d : Ω • → Ω q +1 , to k [ t, t − 1 ]-linear maps Ω ˆ ⊗ k [ t, t − 1 ] → Ω ˆ ⊗ k [ t, t − 1 ], of degrees − 1 and +1, resp ectiv ely . 10 The r esu lting maps d and ı ∆ satisfy d 2 = ( ı ∆ ) 2 = 0 and d ◦ ı ∆ + ı ∆ ◦ d = 0, by Lemma 3.5.1(ii). Th u s, the m ap d + t · ı ∆ giv es a d egree +1 differentia l on Ω ˆ ⊗ k [ t, t − 1 ]. This different ial ma y b e though t of as some sort of equiv arian t differentia l for th e ‘v ector field’ ∆. The follo wing theorem, to b e p ro ve d in § 5.4 b elo w, is one of the main results of the p ap er. It sho ws the imp ortance of the r educed con traction map ı ∆ for cyclic h omology . Theorem 4.3.2. The homolo gy of the c omplex ( Ω ˆ ⊗ k [ t, t − 1 ] , d + t · ı ∆ ) is isomorphic to H P − q ( A ) , the r e duc e d p erio dic cyclic homolo gy of A (with inverte d de g r e es). R emark 4.3.3 (Hodge filtration) . In [19, § 1.17], Kont sevich considers a ‘Ho d ge fi ltration’ on p erio dic cyclic homology . In terms of Th eorem 4.3.2, the Ho dge filtration F q Hod ge ma y b e defined as f ollo ws : • F n Hod ge H P ev en consists of those classes representable by sums P i ≥ n t − i γ 2 i , γ 2 i ∈ Ω 2 i ; • F n + 1 2 Hod ge H P odd consists of those classes rep resen table by sums P i ≥ n t − i γ 2 i +1 , γ 2 i +1 ∈ Ω 2 i +1 . 4.4. Gauss-Manin connection. It is w ell-kno wn that, giv en a smo oth family p : X → S of complex s c hemes ov er a smo oth b ase S , there is a canonical flat connection on the relativ e alg ebr aic de Rham cohomology groups H q D R ( X /S ), called the Gauss-M anin c onne ction . More alg ebr aically , let A b e a comm utativ e k -algebra whic h is smo oth ov er a regular subalgebra B ⊂ A . In suc h a case, the relativ e algebraic d e Rham cohomology m ay b e identi fi ed with H P B q ( A ), the relativ e p erio d ic cyclic homology; s ee, e.g., [12 ]. The Gauss-Manin connection therefore provides a flat conn ection on th e r elativ e p erio dic cyclic homology . In [13], Getzler extended th e definition of th e Gauss-Manin connection to a n oncomm utativ e setting. Sp ecifically , let A b e a (not necessarily comm utativ e) asso ciativ e alg ebr a equipp ed with a c entr al algebra em b edd ing B = k [ x 1 , . . . , x n ] ֒ → A . Ass u ming th at A is fr ee as a B -mo du le, Getzler has d efi ned a fl at connection on H P B q ( A ). Unfortunately , Getzler’s d efinition of the con- nection in v olve s qu ite complicate d calculatio ns in th e Ho c h sc hild complex that make it d ifficult to relate his definition with th e classical geometric construction of the Gauss-Manin connection on de Rham cohomology . Alternativ e approac hes to the d efinition of Getzler’s connection, also based on homologica l algebra, were suggested more recen tly by Kaledin [17] and b y Tsygan [29]. Belo w , we prop ose a new, geometrically m ore transparent (we b eliev e) appr oac h for the Gauss- Manin connection u s ing the construction of cyclic homology fr om th e previous su bsection. Unlik e earlier constructions, our formula for the connection on p eriod ic cyclic h omology is id en tical, es- sen tially , to the classic f orm ula for the Gauss-Manin connection in de Rham cohomology , though the ob jects inv olv ed h av e d ifferent meanings. Our v ersion of Getzle r ’s result reads as follo w s. Theorem 4.4.1. L et B b e a c ommutative algebr a. L et A b e an asso ciative algebr a e quipp e d with a c entr al algebr a emb e dding B ֒ → A such that the quotient A/B is a fr e e B -mo dule. Then, ther e is a c anonic al flat c onne ction ∇ GM on H P B q ( A ) . Notation 4.4 .2 . ( i ) Giv en an algebra R and a subs et J ⊂ R , let ( J ) d enote the t w o-sided ideal in R generated b y the set J . ( ii ) F or a comm utativ e algebra B , w e set Ω q comm B := Λ q B (Ω 1 comm B ), the sup er-comm utativ e DG algebra of d ifferen tial forms, generated b y the B -mo dule Ω 1 comm B of K¨ ahler d ifferen tials. Construction of the Gauss-Manin c onne ction. Giv en a c entr al algebra em b edd in g B ֒ → A , w e may realize the relativ e p erio d ic cyclic homology of A o ver B as follo ws. Firs t, we define the follo win g quotien t DG algebras of (Ω q A, d ): Ω B A := Ω q A/ ([Ω q A, Ω q B ]) , Ω( A ; B ) := Ω B A/ ( d B ) . 11 Th u s, w e ha v e a sup er-cen tr al DG algebra embed d ing Ω q comm B ֒ → Ω B A induced by the natural em b ed d ing Ω q B ֒ → Ω q A . W e in tro d uce the d escending filtration F q (Ω B A ) by p o w ers of the ideal ( d B ). F or the corresp onding associated grad ed algebra, there is a n atural surjection Ω q ( A ; B ) ⊗ B Ω i comm B ։ gr i F Ω B A, α ⊗ β 7→ αβ , ∀ α ∈ Ω B A, β ∈ Ω i comm B . (4.4.3) Belo w , w e will also mak e use of the ob jects Ω B A and Ω( A ; B ), obtained b y killing k ⊂ A = Ω 0 A . Th u s, Ω( A ; B ) ˆ ⊗ k [ t, t − 1 ] and Ω B A ˆ ⊗ k [ t, t − 1 ] are mo d ules ov er k [ t, t − 1 ]. Th er e is a n atural descending filtration F q on Ω B A ˆ ⊗ k [ t, t − 1 ] ind uced b y F q (Ω B A ) and suc h that k [ t, t − 1 ] is placed in fi ltration degree zero. T his filtration is ob viously stable und er the differen tial d . It is also stable un der the differen tial t · ı ∆ since the comm u tators that app ear in t · ı ∆ ( ω ) (see Lemma 3.5.1(i)) v anish , by definition of Ω B A . Therefore, the map (4.4.3) ind u ces a morp hism of d ouble complexes, equipp ed with the differen tials d ⊗ B Id and t · ı ∆ ⊗ B Id, Ω q ( A ; B ) ˆ ⊗ k [ t, t − 1 ] ⊗ B Ω i comm B ։ gr i F Ω B A ˆ ⊗ k [ t, t − 1 ] . (4.4.4) W e will show in § 5.6 b elo w that the assumptions of Theorem 4.4.1 ensure that the map (4.4.3 ) is an isomorp hism. Assume this for the momen t and co nsid er th e standard sp ectral sequence asso ciated with the filtration F q ( Ω B A ˆ ⊗ k [ t, t − 1 ]). The first page of th is sequence consists of terms gr F (Ω B A ˆ ⊗ k [ t, t − 1 ]). Under the ab o ve assumption, the LHS of (4.4.4), sum med o ver all i , comp oses the first page of the sp ectral sequence of  F q ( Ω B A ˆ ⊗ k [ t, t − 1 ]) , d + t · ı ∆  . T hen, for the second page of the sp ectral sequence w e get E 2 = H q  Ω( A ; B ) ˆ ⊗ k [ t, t − 1 ] , d + t · ı ∆  ⊗ B Ω q comm B . W e no w d escrib e the differential ∇ on the second page. Let ∇ GM : H q ( Ω( A ; B ) ˆ ⊗ k [ t, t − 1 ]) − → H q  Ω( A ; B ) ˆ ⊗ k [ t, t − 1 ]  ⊗ B Ω 1 comm B (4.4.5) b e the restriction of ∇ to degree zero. Then we immediately see that ∇ ( α ⊗ β ) = ∇ GM ( α ) ∧ β +( − 1) | α | α ⊗ ( d DR β ) , ∇ GM ( bα ) = b ∇ GM ( α )+( − 1) | α | α ⊗ ( d DR b ) , ∀ b ∈ B , where n o w d DR is the u sual de Rham d ifferen tial. F r om these equations, we deduce that the map ∇ GM , fr om (4.4.5) , giv es a flat connection on H i  Ω( A ; B ) ˆ ⊗ k [ t, t − 1 ]  for all i . Explicitly , we ma y describ e the connection ∇ GM as follo ws. Supp ose that ¯ α ∈ Ω( A ; B ) h as the prop erty that ( d + t · ı ∆ )( ¯ α ) = 0. Let α ∈ Ω B A b e an y lift, and consider ( d + t · ı ∆ )( α ). This must lie in ( d B ), and its image in Ω( A ; B ) ⊗ B Ω 1 comm B is th e desired elemen t. ♦ R emark 4.4.6 . In [13], Getzler tak es B = k [ [ x 1 , . . . , x n ] ], and tak es A to b e a formal deformation o v er B of an asso ciativ e algebra A 0 . Although suc h a setting is n ot formally cov ered by Theorem 4.4.1, our constru ction of the Gauss-Manin connection still applies. T o explain this, write m ⊂ B = k [ [ x 1 , . . . , x n ] ] for the augment ation ideal of the formal p o wer series with ou t constan t term. Let A 0 b e a k -v ector space w ith a fi xed nonzero elemen t 1 A , and let A = A 0 [ [ x 1 , . . . , x n ] ] b e the B -mo dule of formal p o wer series with co efficien ts in A 0 . W e equip B and A with the m -adic top ology , and view B as a B -submo dule in A via the em b edd ing b 7→ b · 1 A . Cor ol lary 4.4.7 . L et ⋆ : A × A → A b e a B -biline ar, c ontinuous asso ciative (not ne c essarily c om- mutative) pr o duct that makes 1 A the unit element. Then, the c onclusion of The or em 4.4.1 holds for H P B q ( A ) . 12 5. Proofs 5.1. The Karoubi op erator. Throughout this section, w e fix an algebra A and abbreviate Ω n := Ω n A and Ω := ⊕ n Ω n . Giv en an A -bimo du le M , p ut M ♮ := M / [ A, M ] = H 0 ( A, M ). In particular, an algebra h o- momorphism A → B mak es B an A -bimod ule. In suc h a case, one has a canonical pro jection B ♮ = B / [ A, B ] ։ B cyc = B / [ B , B ]. This app lies for B = Ω q . So, w e g et a natural p ro jection Ω q ♮ → DR q A , whic h is not an isomorph ism, in general. F ollo wing C unt z and Quillen [7], w e consider a diagram Ω 0 d / / Ω 1 d / / b o o Ω 2 d / / b o o . . . . b o o Here, the de Rham d ifferen tial d and the Ho c hsc hild different ial b , defined in (4.1.2), are related via an imp ortan t Kar oubi op er ator κ : Ω q → Ω q [18]. Th e latter is defined by the form ula κ : α d a 7→ ( − 1) deg α d a α if deg α > 0, and κ ( α ) = α if α ∈ Ω 0 . By [18],[6], one h as b ◦ d + d ◦ b = Id − κ. It follo ws that κ commutes with b oth d and b . F u rthermore, it is easy to v erify (see [6] and the pro of of Lemma 5.2.1 b elo w) that the Karoubi op erator descends to a w ell-defined map κ : Ω n ♮ → Ω n ♮ , whic h is essent ially a cyclic p erm utation; sp ecifically , in Ω n ♮ , we hav e κ ( α 1 α 2 . . . α n − 1 α n ) = ( − 1) n − 1 α n α 1 α 2 . . . α n − 1 , ∀ α 1 , . . . , α n ∈ Ω 1 . Let ( − ) κ denote taking κ -inv arian ts. I n particular, write (Ω q ) κ ♮ := [(Ω q ) ♮ ] κ ⊂ (Ω q ) ♮ . Prop osition 5.1.1. F or any n ≥ 1 , we have an e quality ı ∆ = (1 + κ + κ 2 + . . . + κ n − 1 ) ◦ b as maps Ω n → Ω n − 1 . F u rthermor e , the map ı ∆ fits into a c anonic al short exact se quenc e 0 − → H n (Ω A, b ) − → DR n A ı ∆ − → [ A, Ω n − 1 A ] κ − → 0 . W e recall that the cohomology group H n (Ω A, b ) that o ccurs in the ab o v e d isp la y ed sh ort exact sequence is isomorp hic, as has b een men tioned in § 4.1 , to the Ho chsc h ild homology H n ( A, A ). Th us, Theorem 4.1.1 is an immediate consequence of the short exact s equence of the prop osition. Special cas e: H 1 ( A, A ) . F or one-forms, the formula of Prop osition 5.1.1 giv es ı ∆ = b . Thus, using the identificati on H 1 ( A, A ) = H 1 (Ω q , b ), the sh ort exact sequence of Prop osition 5.1.1 reads 0 − → H 1 ( A, A ) − → DR 1 A b = ı ∆ − − − − − → [ A, A ] − → 0 . (5.1.2) The short exact sequen ce (5.1.2) may b e obtained in an alternate wa y as follo ws. W e apply the righ t exact fun ctor ( − ) ♮ to (2.2.1) . The corresp onding long exact sequence of T or-groups reads . . . → H 1 ( A, A ⊗ A ) → H 1 ( A, A ) → (Ω 1 ) ♮ → ( A ⊗ A ) ♮ c → A ♮ → 0 . No w, by the definition of T or, H k ( A, A ⊗ A ) = 0 for all k > 0 . Also, we hav e natural iden tifications (Ω 1 ) ♮ = DR 1 A and ( A ⊗ A ) ♮ ∼ = A . This w a y , the map c on the righ t of the displa yed form ula ab ov e ma y b e identified with th e natural p ro jection A ։ A/ [ A, A ]. Thus, Ker( c ) = [ A, A ], and the long exact sequence ab o ve reduces to the short exact sequen ce (5.1.2). It is immediate fr om defi n itions that the map b = ı ∆ in (5.1.2) is giv en b y Quillen’s formula u d v 7→ [ u, v ] [6 ]. In p articular, w e deduce that (DR 1 A ) exact ⊂ Ker( ı ∆ ) = H 1 ( A, A ) . 13 5.2. Pro of of Prop osition 5.1.1. W e fir st state and pro v e a lemma wh ic h w as implicit in [7] and [22, § 2.6] and wh ic h will p la y an imp ortant role in § 5 b elo w. Lemma 5.2.1. ( i ) The pr oje ction (Ω q ) ♮ → DR q A r estricts to a bije ction (Ω q ) κ ♮ ∼ → DR q A . ( ii ) The map b desc ends to a map b ♮ : (Ω q ) ♮ → Ω q − 1 . ( iii ) The kernel of the map b ♮ : (Ω q ) κ ♮ → Ω q − 1 , the r estriction of b ♮ to the sp ac e of κ -invariants, is isomorph ic to H n (Ω A, b ) . Pr o of of L emma. The argumen t b elo w follo ws th e pro of of [22, Lemma 2.6.8]. F rom defin itions, we get [ A, Ω] = b Ω and [ d A, Ω] = (Id − κ )Ω. Hence, we obtain, cf. [6]: [Ω , Ω ] = [ A, Ω ] + [ d A, Ω ] = b Ω + (Id − κ )Ω . W e deduce that Ω ♮ = Ω / b Ω and DR q A = Ω / [Ω , Ω] = Ω ♮ / (Id − κ )Ω ♮ . In p articular, since b 2 = 0, the map b descends to a well defined m ap b ♮ : Ω ♮ = Ω / b Ω → Ω. F urth er, one has the follo w ing standard identiti es [7, § 2], on Ω n for all n ≥ 1: κ n − Id = b ◦ κ n ◦ d , κ n +1 ◦ d = d . (5.2.2) The Karoub i op erator κ commutes with b , and hence induces a well-defined end omorphism of the vecto r sp ace Ω n / b Ω n , n = 1 , 2 , . . . . F urthermore, from the first iden tit y in (5.2.2) we see that κ n = Id on Ω n / b Ω n . Hence, w e ha ve a d irect sum decomp osition Ω ♮ = (Ω ♮ ) κ ⊕ (Id − κ )Ω ♮ . It follo ws that the natural pro j ection Ω ♮ = Ω / b Ω ։ DR q A = Ω ♮ / (Id − κ )Ω ♮ restricts to an isomorphism (Ω ♮ ) κ ∼ → DR q A . P arts (ii)–(iii) of Lemma 5.2.1 are clear from the pro of of [22], Lemma 2.6.8.  Pr o of of Pr op osition 5.1.1. Th e fir st statement of the p r op osition is immediate from the form ula of Lemma 3.5.1(i). T o prov e the second statemen t, we exploit the firs t identit y in (5.2.2 ). Usin g the form ula for ı ∆ and the fact that b comm utes with κ , w e compute that ( κ − 1) ◦ ı ∆ = b ◦ ( κ − 1) ◦ (1 + κ + κ 2 + . . . + κ n − 1 ) = b ◦ ( κ n − 1) = b 2 ◦ κ n ◦ d = 0 . (5.2.3) Hence, we deduce that the image of ı ∆ is conta ined in ( b Ω) κ . Con v ersely , giv en an y elemen t α = b ( β ) ∈ ( b Ω) κ , we fin d that ı ∆ ( β ) = (1 + κ + κ 2 + . . . + κ n − 1 ) ◦ b β = n · b β = n · α. Th u s, Im( ı ∆ ) = ( b Ω) κ = [ A, Ω ] κ , since b Ω = [ A, Ω]. F urthermore, since (1 + κ + κ 2 + . . . + κ n − 1 ) ◦ b = n b on (Ω q ) κ ♮ , the t wo m ap s hav e the same ke r n el. The exact sequen ce of the prop osition no w follo ws from Lemma 5.2.1.  5.3. Harmonic decomp osition. Our pro of of Theorem 4.3.2 is an adaptation of the strategy used in [7, § 2], based on the harmonic de c omp osition Ω = P Ω ⊕ P ⊥ Ω , where P Ω := Ker(Id − κ ) 2 , P ⊥ Ω := Im(Id − κ ) 2 . (5.3.1) The differential s B , b , and d comm ute with κ , hence preserv e the harmonic decomp osition. It will b e con venien t to introduce t wo d egree pr eserving linear maps N , N ! : Ω → Ω, such that, for any n ≥ 0, N | Ω n is m ultiplication by n, and N ! | Ω n is multiplicati on by n !. (5.3.2) Then, ( i ) B = N d P , and ( ii ) ı ∆ = b N P . (5.3.3) Here, equation (i) is p ro ve d b y Cu n tz and Q uillen [7, § 2, formula (11)] using the second ident ity in (5.2.2). 14 Pr o of of (5.3.3) (ii) . First, w e use that b commutes with κ . Therefore, app lying (5.2.3), w e find i ∆ ◦ (Id − κ ) 2 = ( κ − 1) ◦ i ∆ ◦ ( κ − 1) = 0. W e conclude that th e op eration ı ∆ annihilates P ⊥ Ω. It r emains to sho w that, on P Ω n , one has ı ∆ = ( n − 1) · b . T o this end, let α ∈ Ω n . F r om the first id en tit y in (5.2.2), α − κ n ( α ) ∈ b Ω. Hence, b α − κ n ( b α ) ∈ b 2 Ω = 0, since b 2 = 0. Thus, the op erator κ has finite order on b Ω, and hence on b ( P Ω). Bu t, for any op erator T of fin ite order, Ker(Id − T ) = Ker((Id − T ) 2 ). It follo ws that, if α ∈ P Ω n , then b α ∈ K er ((Id − κ ) 2 ) = Ker(Id − κ ). W e conclude that the elemen t b α is fixed by κ . Hence, (1 + κ + κ 2 + . . . + κ n − 1 ) ◦ b α = n · b α . Therefore, b y Prop ositio n 5.1.1, ı ∆ ( α ) = n · b α , and (5.3.3)(ii) is pro ved.  5.4. Pro of of Theorem 4.3.2 . Since the harm onic d ecomp osition is stable under all four differen- tials B , b , d , and ı ∆ , we may analyze the homology of eac h of the direct sum mands, P Ω and P ⊥ Ω, separately . First of all, we kn ow that B = 0 on P ⊥ Ω, by (5.3.3)(i), and moreov er it has b een sho wn by Cu n tz and Qu illen [7, Prop osition 4.1( 1)] th at ( P ⊥ Ω , b ) is acyclic . F u rthermore, since the complex ( Ω , d ) is acyclic (see [7, § 1] or [5] f ormula (2.5.1)), w e deduce the follo w ing. Each of the c omplexes ( P Ω , d ) and ( P ⊥ Ω , d ) is acyclic. (5.4.1) No w, the map ı ∆ v anishes on P ⊥ Ω by (5.3.3)(ii). Hence, on P ⊥ Ω ˆ ⊗ k [ t, t − 1 ], we hav e d + t · ı ∆ = d . Therefore, w e conclude using (5.4 .1 ) that ( P ⊥ Ω[ t ] , d ), and h ence also ( P ⊥ Ω[ t ] , d + t · ı ∆ ), are acyclic complexes. Th u s, to complete the pro of of the theorem, we m us t compare cohomol ogy of the complexes ( P Ω ˆ ⊗ k [ t, t − 1 ] , d + t · ı ∆  and ( P Ω ˆ ⊗ k [ t, t − 1 ] , B + t · b ). W e ha ve N · d + ( N + 1) − 1 · t · ı ∆ = B + t b . P ost-comp osing this by N ! (see (5.3.2)), w e obtain ( N !) · ( d + t · ı ∆ ) = ( B + t · b ) · ( N !) . W e deduce the follo wing isomorphism of complexes whic h completes the pro of of the theorem: N ! : ( P Ω ˆ ⊗ k [ t, t − 1 ] , d + t · ı ∆  ∼ → ( P Ω ˆ ⊗ k [ t, t − 1 ] , B + t · b ) . ✷ 5.5. Negativ e and ordinary cyclic homology. I t is p ossible to extend Theorem 4.3.2 to the case of (n onp erio dic) cyclic h omology and negativ e cyclic homology u sing harm onic d ecomp osition. T o explain this, put ( Ω ˆ ⊗ k [ t, t − 1 ]) + := L m ≥ 0  Y ii ( | a k | +1) P k ≤ i ( | a k | +1) . Using the f act that τ is a trace, re-indexing the N k , and re-orderin g the D ( a i ) (which cancels out the previous s ign), we see that the ab o ve is equal to ( n + 1)! N ! ( N + n + 1)! X P n 0 N k = N n X i =0 X N ′ i + N ′′ i = N i τ ( D ( a 0 ) x N 0 . . . D ( a n ) x N n ) = ( N + n + 1) ( n + 1)! N ! ( N + n + 1)! X P n 0 N k = N τ ( D ( a 0 ) x N 0 . . . D ( a n ) x N n ); b y the Leibniz iden tit y for D , w e rewrite this as ( n + 1)! N ! ( N + n )! ( X P n 0 N k = N D τ ( a 0 x N 0 . . . D ( a n ) x N n ) − 1 2 X P n 0 N k = N n X i =1 ± τ ( a 0 x N 0 . . . [ D , D ]( a i ) x N i . . . D ( a n ) x N n ) 28 − X P n 0 N k = N n X i =0 X N ′ i + N ′′ i = N i ± τ ( a 0 x N 0 . . . x N ′ i D ( x ) x N ′′ i . . . D ( a n ) x N n )); the signs app earing in the ab ov e terms from the Leibniz identit y , with an o v erall minus sign, are precisely the signs f rom th e defin ition of ι [ D ,D ] and ι D ( x ) . W e see that th e ab ov e is equ al to ( n + 1) Dχ ( D n x N ) − n ( n + 1) 2 χ ([ D , D ] D n − 1 x N ) − − ( n + 1) N χ ( D n D ( x ) x N − 1 ) . Therefore χ ( D n +1 x N )( B c ) = ( n + 1) D χ ( D n x N )( c ) − − n ( n + 1) 2 χ ([ D , D ] D n − 1 x N )( c ) − ( n + 1) N χ ( D n D ( x ) x N − 1 )( c ) . This is exactly the equalit y of the term s of the formula in the statement of the prop ositio n th at con tain u , in the sp ecia l cases X 1 = . . . = X n +1 = D and X n +2 = . . . = X N + n +1 = x (and with n replaced by N + n ). T o p ro ve the general case, te n s or our algebra by k [ t 1 , . . . , t n ] where | t i | = −| X i | , pu t X = t 1 X 1 + . . . t n X n , apply the sp ecial case to χ ( X , . . . , X ) , and look at the co efficien t at t 1 . . . t n .  A.3. Construction of the morphism. No w let A = Ω(Rep( A )) ⊗ End( V ), Rep( A ) b eing the sc heme of represen tations of an al gebra A in a giv en fin ite d im en sional space V . Consider the subalgebra End( V ) consisting of constant fu nctions. Let K = Ω(Rep( A )) and τ : A → K b e the matrix trace. Let g = End( V ) viewe d as a Lie algebra. Prop osition A.3.1. The map c, x → τ (exp( ι d + x ))( c ) , x ∈ g and c ∈ C −• ( A ) , c omp ose d with the morphism induc e d by ev : A → A , defines a morphism of c omplexes HKR( x ) : C −• ( A )(( u )) , b + uB → Ω(Rep( A ))[[ g ]] G (( u )) , ud + ι x Here ι x refers to the con traction of a form b y a ve ctor fi eld corresp onding to x . (Note that the v alue of this map at x = 0 is the HKR morphism). Pr o of. By Prop osition A.2.4, taking into accoun t that [ d, d ] = 0 and d ( x ) = 0 , HKR( x )(( b + uB )( c )) = ud HKR( x )( c ) + ∞ X K =0 1 K ! χ ( δ ( x )( d + x ) K )( c ) . If w e put c = α 0 ⊗ . . . ⊗ α n , α i ∈ A , then the second su m mand is equal to ∞ X N =0 X P n 0 N k = N N X i =1 tr( α 0 x N 0 dα 1 x N 1 . . . [ x, α i ] x N i . . . dα n x N n ) No w o b serv e that for x ∈ g , if L x ev( a ) = [ x, ev ( a )] w h ere L x denotes the Lie deriv ativ e b y the v ector field on Rep( A ) corresp ond ing to x . Therefore, for α i = ev ( a i ) , the ab o ve form u la turns into ∞ X N =0 X P n 0 N k = N N X i =1 tr( α 0 x N 0 dα 1 x N 1 . . . L x α i x N i . . . dα n x N n ) = ι x ∞ X N =0 X P n 0 N k = N tr( α 0 x N 0 dα 1 x N 1 . . . dα n x N n ) = ι x HKR( x )( c ) . 29  R emark A.3.2 . As for the Ho c hsc hild homology , the fact that the map HKR : C −• ( A ) → Ω(Rep( A )) (the v alue of HKR( x ) at x = 0) s en ds Ho chsc h ild cycles to basic forms follo ws from the form ula (used in th e p ro of ab ov e) ι x HKR( x )( c ) = HKR( xι ∆ ( c )), and the fact that ι ∆ kills the image of b . A.4. More on noncomm ut ativ e calculus. Let us fi nish b y saying a few words ab out noncom- m utativ e analogues of formulas (A.1.1 ) and other algebraic p rop erties of forms on manifolds. Note first that op erators ι X and L X can b e defined n ot just f or vect or fields and functions but for m ul- tiv ector fields; they sat isfy (A.1.1), the b rac k et b eing the Sc h outen-Nijenh uis brac k et. Since all the equations (A.1.1) are in term s of commutato rs , they can b e interpreted as an action of certain differen tial grad ed Lie algebras on the complex of forms. Th is latter formulatio n h as a noncom- m utativ e analogue as follo ws. F or an y (graded) asso ciativ e algebra A , let g ( A ) b e the graded Lie algebra of its Ho c hsc h ild co c hains, with the Gerstenhab er br ac k et and th e Ho c hschild differen tial δ . Then on ( C −• ( A )[[ u ]] , b + uB ) there is a k [[ u ]]-linear, ( u )-adically con tin uous s tr ucture of an L ∞ mo dule o v er ( g ( A )[ ǫ, u ] , δ + u ∂ ∂ ǫ ) , su c h that, for a co c hain D of degree ≤ 1 , D ǫ acts by − ι D as in (A.2.1), (A.2.2). F rom th at, one deduces a construction of a flat sup erconnection on th e p erio d ic cyclic complex of a family of alg ebras ([8], Prop osition 2). The formulas f or the L ∞ action are somewhat m ore complicated than the defin ition of χ in (A.2.3 ) . It w ould b e in teresting to b etter understand the relation b etw een the ab o ve and the construction of th e Gauss-Manin connection in the present pap er. Let us finish by men tioning one more feature of the classical calculus. The space of m ultivec tor fields is in fact a Gerstenhab er algebra, in particular a graded commutativ e algebra; on e h as ι X ι Y = ι X Y ; L X Y = ( − 1) | Y | L X ι Y + ι X L Y . (A.4.1) An algebraic system uniting (A.1.1) and (A.4.1) is called a calculus; it wa s sho wn in [20], [28], [9] that, for an asso ciat ive algebra A , the pair of complexes ( C ∗ ( A, A ) , C ∗ ( A, A )) is alw a ys a homotop y calculus. This structure quite inexplicit and not canonical; it dep ends on a c hoice of a Drinfeld asso ciator. References [1] D. J. An ic k, On the homolo gy of asso ciative algebr as. T rans. Amer. Math. So c. 296 (1986), 641–659. [2] A. Connes, Nonc ommutative di ffer ential ge ometry , Inst. Hautes ´ Etudes Sci. Publ. Math. 62 (1985), 257- 360. 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U niversi ty Ave, Chicago, IL 60637, USA ; ginzburg@m ath.uchicago.edu T.S. : Department of Mathematics, MIT, Cam bridge, MA 02139, USA . trasched@g mail.com 31