Equivariant Cartan Homotopy Formulae for DG -Algebra
aa r X i v : . [ m a t h . K T ] M a y EQUIVARIANT CARTAN HOMOTOPY FORMULAE FOR THECROSSED PRODUCT OF DG ALGEBRA
SAFDAR QUDDUS
Abstract.
We establish the equivariant Cartan Homotopy Formula for the crossed prod-uct of DG -algebra obtained by a finite group action. Introduction
The paracyclic modules were used in [GJ1] and [BGJ] to understand the cyclic homologyof the crossed product algebras. It is an analogue of the Eilenberg-Zilber theorem for bi-paracyclic modules. Let A be a unital DG algebra over a commutative ring k and let G be a finite discrete group which acts on A by automorphisms. The result is that the cyclichomology of the crossed product algebra A ⋊ G has the following decomposition. THEOREM 0.1 ([GJ1] [FT]) . If G is finite and | G | is invertible in k , then there is anatural isomorphism of cyclic homology and HC • ( A ⋊ G ; W ) = HC • ( H ( G, A ♮G ); W ) ,where H ( G, A ♮G ) is the cyclic module H ( G, A ♮G )( n ) = H ( G, k [ G ] ⊗ A ( n +1) ) . Where W is a finite dimensional graded module over the polynomial ring k [ u ], where deg ( u ) = −
2; the above result when considered for different coefficients W yield severaltheories, some are illustrated below:1) W = k [ u ] gives negative cyclic homology HC − ( A );2) W = k [ u, u − ] gives periodic cyclic homology HP • ( A );3) W = k [ u, u − ] /uk [ u ] gives cyclic homology HC • ( A );4) W = k [ u ] /uk [ u ] gives the Hochschild homology HH • ( A ).The above decomposition has been studied and used to understand the homological proper-ties of crossed product algebras [CGGV] [P1] [P2] [P3] [Q1] [Q2][Q3] [Q4] [ZH]. The primafact is that the (co)homology modules of the crossed product algebra decomposes intotwisted (co)homology modules relative to the conjugacy classes of G . Hence the studyingthe crossed product algebra is simplified. Date : June 2, 2020.
Key words and phrases.
Cartan Homotopy Formula, Hocshchild. .1. DG Algebra.
A unital DG algebra (
A, d ) is a differentially graded unital algebra A ,with k -bilinear maps A n × A m → A n + m , sending ( a, b ) ab such that d n + m ( ab ) = d n ( a ) b + ( − n ad m ( b )and such that ⊕ A n becomes an associative and unital k -algebra. Through out this article wedemand that the finite group action on ( A, d ) preserves the grading and commutes with thedifferential structure; i.e. for g ∈ G , dg = gd . The (co)homology theories on DG algebrashave been studied extensively [T] [K] [GJ2]. The above algebra can also be considered asa special A ∞ -algebra ( A, m i ) with m i = 0 for i >
2. Here m (2-co-chain for A ) definesproduct on A and m is the mixed differential.0.2. Cartan Homotopy Formulae.
The Cartan homotopy formulae for algebra were firstobserved by Rinehart [R] in the case where D is a derivation on commutative algebra, andlater in full generality by Getzler [G] for A ∞ -algebras. The formulae is stated below:Given D ∈ C k ( A, A ), we consider its Lie derivative L D and the Hochschild and cyclic chainmaps b and B . Let ι D denote the contraction associated to D and S D be the correspondingsuspension. THEOREM 0.2 (Cartan Homotopy Formulae) . [ b, L D ] + L δD = 0 , [ B, L D ] = 0 and [ L D , L E ] = L [ D,E ] [ b + B, ι D + S D ] = L D + ι δD + S δD where [ , ] is the graded Gerstenhaber bracket for C • ( A, A ).The above formulae was proved in generality for the A ∞ -algebras by Getzler [G]. The case k = 1 in his paper corresponds to the DG-algebra ( A, d ). Given a crossed product algebrawe can ask if the Cartan homotopy formulae hold for each components in the decompositiondescribed above. As to my knowledge, the question is unanswered even for an associativealgebra. We shall answer this question for DG -algebras when the group action preservesthe gradation. 1. Statement
For G a finite discrete group acting on a DG -algebra ( A, d ) over a ring k such that | G | isinvertible in k . For g ∈ G define a g -twisted A -left-module structure on A by the followingformula: a • ( u g m ) := u g g − ( a ) · m, where a, m ∈ A . We tag by u g the twisted element of A with this left bi-module structureand the left twisted bi-module is denoted by A g (refer [Q1], pp 332).Define C ( A ) g := A g and C n ( A ) g := A g ⊗ A ⊗ n . Let b g , B g denote the chain differentials of the complex C • ( A ) g and L gD be the twisted Liederivative associated to D ∈ C k ( A, A ). We shall produce the explicit expression of thesemaps in this literature. HEOREM 1.1.
The Lie derivative L gD , chain maps b g and B g satisfy the following: [ b g , L gD ]+ L gδD = 0 [ B g , L gD ] = 0 [ L gD , L gE ] = L g [ D,E ] and [ b g + B g , ι gD + S gD ] = L gD + ι gδD + S gδD . The [ , ] above is the Gerstenhaber bracket for C • ( A, A ) and δ is the Hochschild co-chainmap. 2. Paracyclic decomposition for cross product ( A, d ) ⋊ G Hochschild Chain Complex for ( A, d ) . Define the differentials d : C • ( A ) → C • ( A ), b : C • ( A ) → C • ( A )[ −
1] and B : C • ( A ) → C • ( A )[1] as follows. d ( a ⊗ · · · ⊗ a n ) = n X i =1 ( − P k | a i | −
1. The homology of ( C ( • ( A, A ) , b + d ) is the Hochschildhomology of ( A, d ) with coefficients in A endowed with the bi-module structure. Let W be a graded module over the polynomial ring k [ u ], where deg ( u ) = −
2; such that W hasfinite homological dimension then the module H • ( C • ⊠ W, b + d + uB ), where ( C • ⊠ W ) = C • [[ u ]] ⊗ k [ u ] W , is the cyclic homology of the mixed complex ( A, b + d, B ) with coefficientsin W . Some examples of cyclic (co)homology for various W are listed the statement ofTheorem 0.1. 3. Proof of the theorem
The paracyclic decomposition of the cyclic homology of (
A, d ) ⋊ G exists and the followingis the decomposition: HC • ( A ⋊ G ) = M [ g ] HC • ( H ( G g , A ♮g )) = M [ g ] HC • ( A g ) G with the maps; b g : C • ( A ) g → C • ( A ) g [ −
1] and B g : C • ( A ) g → C • ( A ) g [1] defined as: b g ( u g a ⊗ · · · ⊗ a n ) = n − X k =0 ( − P ki =0 ( | a i | +1)+1 ( u g a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n ) ( − | a n | +( | a n | +1) P n − i =0 ( | a i | +1)+1 ( u g g − ( a n ) a ⊗ · · · ⊗ a n − )and B g ( u g a ⊗· · ·⊗ a n ) = n X k =0 ( − P i ≤ k ( | a i | +1) P i ≥ k ( | a i | +1) ( u g e ⊗ g − ( a k +1 ) ⊗· · ·⊗ g − ( a n ) ⊗ a ⊗· · ·⊗ a k )To explicitly derive the above maps we consider the definition of these in terms of theelementary maps s i , t and d i [L] and decipher the twisted elementary maps [BG] and finallyformulate b g and B g . One important point here to be noted is that since the action of G on( A, d ) preserves grading, the crucial sign term ǫ k = P i ≤ k ( | a i | + 1) P i ≥ k ( | a i | + 1) remainsunchanged (in fact under this assumption the signs for A ∞ -algebras are also invariant underthe unified algebraic structure for chains and co-chains of A ∞ -algebras ) and is a necessarycondition to yield the Cartan homotopy formulae for twisted components.For D ∈ C k ( A, A ), the maps ι D and S gD can be computed using the signed cyclic permuta-tion map t [BGJ]. They are as follows: ι D ( u g a ⊗ · · · ⊗ a n ) = ( − | D || a | a D ( a , . . . , a n ); L gD ( u g a ⊗ · · · ⊗ a n ) = n − d X k =1 ( − ν k ( D,n ) u g a ⊗ · · · ⊗ D ( a k +1 , . . . , a k + d ) ⊗ . . . a n + n X k = n +1 − d ( − η k ( D,n ) u g D ( g − ( a k +1 ) , . . . , g − ( a n ) , a , . . . ) ⊗ · · · ⊗ a k . where the terms inside D in the second summand must contain a (without g action) andbe cyclically permuted. S D ( u g a ⊗· · ·⊗ a n ) = X j ≥ k ≥ j + d ( − ǫ jk ( D,n ) u g e ⊗ g − ( a k +1 ) ⊗ . . . g − ( a n ) ⊗ a ⊗· · ·⊗ D ( a j +1 , . . . , a j + d ) ⊗· · ·⊗ a k . where, | D | = (degree of the linear map D) + d , D ∈ C d ( A, A ) is being considered as alinear map D : A ⊗ d → A ; and the sign coefficients are ν k ( D, n ) = ( | D | +1)( | a | + P ki =1 ( | a i | +1)), η k ( D, n ) = | D | + P i ≤ k ( | a i | + 1) P i ≥ k ( | a i | + 1) and ǫ jk ( D, n ) = ( | D | + 1)( n X i = k +1 ( | a i | + 1) + | a | + j X i =1 ( | a i | + 1)) . We briefly describe the Gerstenhaber algebra structure on the co-homology H • ( A, A ), thecup product is defined as below, for D ∈ C d ( A, A ) and E ∈ C e ( A, A ).(
D ⌣ E )( a , . . . , a d + e ) = ( − | E | P i ≤ e ( | a i | +1) D ( a , . . . , a e ) E ( a d +1 , . . . , a d + e );and the product ◦ is defined as:( D ◦ E )( a , . . . , a d + e ) = X j ≥ ( − ( | E | +1) P ji =1 ( | a i | +1) D ( a , . . . , a j , E ( a j +1 , . . . , a j + e ) , . . . ) . he Lie bracket is hence [ D, E ] = D ◦ E − ( − | D | +1)( | E | +1) | E ◦ D . The co-chain map δ on C • ( A, A ) is ( δD )( a , . . . , a d +1 ) = ( − | a || D | + | D | +1 a D ( a , . . . , a d +1 )+ d X j =1 ( − | D | +1+ P ji =1 ( | a i | +1) D ( a , . . . , a j a j +1 , . . . , a d +1 )+( − | D | P di =1 ( | a i | +1) D ( a , . . . , a d ) a d +1 The operator δD can also be described as δD = [ m , D ]. Proof of Theorem 1.1. (3.1) [ b g , L gD ] + L gδD = b g L gD − L gD b g + L gδD . We evaluate the above expression on the element ( u g a ⊗ · · · ⊗ a n ). We define the barcomplex map on the equivariant Hochschild chain complex by b g ′ , it is as follows: b g ′ ( u g a ⊗ · · · ⊗ a n ) = n − X k =0 ( − P ki =0 ( | a i | +1)+1 ( u g a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n )Similarly we define the operator L g ′ D as follows: L g ′ D ( u g a ⊗ · · · ⊗ a n ) = n − d X k =1 ( − ν k ( D,n ) u g a ⊗ · · · ⊗ D ( a k +1 , . . . , a k + d ) ⊗ . . . a n . The untwisted expression in (3.1) is b g ′ L g ′ D − L g ′ D b g ′ + L g ′ δD . We collect the coefficients andthe terms cancel each other as signs mismatch, for example the expression u g a ⊗ · · · ⊗ a w D ( a w +1 , · · · , a w + d ) ⊗ . . . a n cancel only if ν w ( D, n ) + w X i =0 ( | a i | + 1) + 1 + ν w − ( δD, n ) + | a w || D | + | D | + 1 ≡
1( mod 2) . Which is true as | δD | = | D | + 1. The relation (3.1) is 0 in the untwisted case hencethe untwisted terms cancel each other once the signs mismatch. On the other hand thecancellation of the twisted terms in (3.1) is non-trivial. To see this we firstly observe that | D ( a i +1 , . . . , a i + d ) | = | D | i + d X j = i +1 ( | a j | + 1) and | g − ( a ) | = | a | Using the above relations we can see that the parity of signs remain the same and termscancel out as it did in the untwisted. For example, η k ( δD, n ) + | D | d X i =1 ( | b i | + 1) + η k ( D, n ) + | D ( b , . . . , b d ) | + 1 ≡
1( mod 2) , where ( b , . . . , b d ) = ( g − ( a k +1 ) , . . . , g − ( a n ) , a , . . . , a d − n + k ) and hence the terms of kind u g D ( g − ( a k +1 ) , . . . g − ( a n ) , a , . . . ) a d − n + k +1 ⊗ · · · ⊗ a k cancel each other. Similarly, it iseasy to check that all other types of twisted terms cancel each other and the appropriateparity of signs are ensured by the grade preserving group action.(3.2) [ L gD , L gE ] = L g [ D,E ]5 he proof of the above relation is straight forward, the bracket [ , ] is the Gerstenhaber Liealgebra commutator as described above.(3.3) [ B g , L gD ] = B g L gD − L gD B g .B g L gD ( u g a ⊗ · · · ⊗ a n ) = B g n n − d X l =1 ( − ν l ( D,n ) u g a ⊗ · · · ⊗ D ( a l +1 , . . . , a l + d ) ⊗ . . . a n + n X l = n +1 − d ( − η l ( D,n ) u g D ( g − ( a l +1 ) , . . . , g − ( a n ) , a , . . . ) ⊗ · · · ⊗ a l o = n − d X l =1 ( − ν l ( D,n ) n − d +1 X k =0 ( − P i ≤ k ( | a ′ i | +1) P i ≥ k ( | a ′ i | +1) u g e ⊗ g − ( a ′ k +1 ) ⊗· · ·⊗ g − ( a ′ n ) ⊗ a ′ · · ·⊗ a k Such that any of a ′ i could be D ( . . . ).+ n X l = n +1 − d ( − η l ( D,n ) n − d +1 X k =0 ( − P i ≤ k ( | a ′′ k | +1) P i ≥ k ( | a ′′ k | +1) u g e ⊗ g − ( a k +1 ) ⊗· · · D ( g − ( a l +1 ) , . . . , g − ( a n ) , a , . . . ) ⊗· · · ⊗ a k such that for each l one of the a ′′ k in the sign expression is D ( g − ( a l +1 ) , . . . , g − ( a n ) , a , . . . )and the rest are the remaining a i ’s that do not appear in the expression D ( g − ( a l +1 ) , . . . , g − ( a n ) , a , . . . ). L gD B g ( u g a ⊗· · ·⊗ a n ) = L gD n n X k =0 ( − P i ≤ k ( | a i | +1) P i ≥ k ( | a i | +1) ( u g e ⊗ g − ( a k +1 ) ⊗· · ·⊗ g − ( a n ) ⊗ a ⊗· · ·⊗ a k ) o = L g ′ D n n X k =0 ( − P i ≤ k ( | a i | +1) P i ≥ k ( | a i | +1) ( u g e ⊗ g − ( a k +1 ) ⊗ · · · ⊗ g − ( a n ) ⊗ a ⊗ · · · ⊗ a k ) o + 0= n X k =0 n +1 − d X l =1 ( − ν l ( D,n +1)+ P i ≤ k ( | a i | +1) P i ≥ k ( | a i | +1) u g e ⊗· · ·⊗ D ( g − ( a l +1 ) , . . . , g − ( a n ) , a , ... ) ⊗· · · a k . In the above expressions we have repeatedly used the fact that | g − ( a ) | = | a | . We alsoobserve that | D ( g − ( a l +1 ) , . . . , g − ( a n ) , a , ... ) | = | D | ( P d − n + lj =0 ( | a j | + 1) + P nj = l +1 ( | a j | + 1)).Hence the sign mismatch for cancellation to yield that L gD and B g commute.Finally we want to show that:(3.4) [ b g + B g , ι gD + S gD ] = L gD + ι gδD + S gδD The expression above can be written as below:[ b g , ι gD ] + [ B g , ι gD ] + [ b g , S gD ] + [ B g , S gD ] = P + Q + R + 0Observe that [ b g , ι gD ] = ι gδD because the twisted terms ( b g − b g ′ ) ι gD and ι gD ( b g − b g ′ ) canceleach other. Hence we are left to show that: Q + R = L gD + S gδD . The above relation can be seen be comparing the parity of sign indices for the terms, forexample the sign for the term u g D ( g − ( a k ) , . . . , g − ( a n ) , . . . , ) ⊗ · · · ⊗ a k ) for the RHS is k ( D, n ) while in the LHS it has sign coefficient as | D || e | + P i ≤ k ( | a i | + 1) P i ≥ k ( | a i | + 1);but indeed η k ( D, n ) ≡ | D || e | + X i ≤ k ( | a i | + 1) X i ≥ k ( | a i | + 1) mod 2since | e | = 1. Hence the terms of the given form cancel out. Similar relations involving η k ( D, n ), ν k ( D, n ), ǫ jk ( D, n )and ǫ jk ( δD, n ) yield the desired result. (cid:3) Acknowledgement
This result grew out of the discussion and collaboration with Xiang Tang and SayanChakraborty on equivariant Gauss-Manin connections, and the discussion has contributedto the outcome of the paper.