aa r X i v : . [ m a t h . QA ] S e p On some p -differential graded link homologies You Qi and Joshua SussanSeptember 15, 2020
Abstract
We show that the triply graded Khovanov-Rozansky homology of knots and links over a field ofpositive odd characteristic p descends to an invariant in the homotopy category finite-dimensional p -complexes.A p -extended differential on the triply graded homology discovered by Cautis is compatible with the p -DG structure. As a consequence we get a categorification of the Jones polynomial evaluated at an oddprime root of unity. Contents p -Hochschild homology and cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Relative homotopy categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Relative p -Hochschild homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 p -DG Soergel bimodules and braid relations 15 p -DG bimodules over the polynomial algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Elementary braiding complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Reidemeister II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Reidemeister III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 sl -homology theory 36 q is generic . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3 Example of the Jones invariant when q -prime root of unity . . . . . . . . . . . . . . . . . . . . 48 References 51
The Jones polynomial is a quantum invariant of oriented links which may be defined using the natural two-dimensional representation of quantum sl . Coloring the components of a link with other representations ofthis quantum group leads to a definition of the colored Jones polynomial. Witten constructed an invariantof -dimensional manifolds in a physical setting coming from Chern-Simons theory [Wit89] with a fixedlevel. If the -manifold is defined as surgery on a link L , Reshetikhin and Turaev [RT90] reconstructedWitten’s invariant by summing over colored Jones polynomials of L . In order for this summation to befinite, it is important that the colored Jones polynomials are evaluated at a root of unity, the order of whichis determined by the level of Witten’s theory. The Witten-Reshetikhin-Turaev (WRT) -manifold invariantfits into the framework of a (2 + 1) -dimensional TQFT.Crane and Frenkel [CF94] initiated the categorification program with the aim of lifting the (2+1) -d WRT-TQFT to a (3 + 1) -d TQFT. The first major success in this program was Khovanov’s categorification of theJones polynomial [Kho00]. Khovanov homology is a bigraded homology theory of links whose graded Eu-ler characteristic is the Jones polynomial. Since this discovery, there have been many other categorificationsof the Jones polynomial as well as their quantum sl n generalizations. One such construction was Kho-vanov and Rozansky’s categorification of the sl n and HOMFLYPT polynomials using matrix factorizations[KR08a, KR08b]. The HOMFLYPT homology theory is triply graded and the graded Euler characteristicrecovers the two-variable HOMFLYPT polynomial. Khovanov later recast this construction in the languageof Soergel bimodules [Kho07] building upon earlier work of Rouquier [Rou04] who gave a categorical con-struction of the braid group. This was later reproved by Rouquier in [Rou17]. In these constructions, onerepresents a link as the closure of a braid. To the braid, one associates a complex of Soergel bimodules.Taking Hochschild homology of each term yields a complex of bigraded vector spaces. Taking homologyof this complex results in a triply graded theory.The categorification of quantum groups and their associated link invariants at generic values of thequantum parameter has been the focus of a lot of research since Crane and Frenkel’s work. The first ap-proach towards categorically specializing the quantum parameter to a root of unity was due to Khovanov[Kho16] and later expanded upon in [Qi14]. In this setup, one should consider algebraic structures overa field of characteristic p and search for a derivation ∂ such that ∂ p = 0 . This enhances the algebraicstructure to a module category over the tensor category of graded modules over a particular Hopf algebra H = k [ ∂ ] / ( ∂ p ) . Taking an appropriate homotopy or derived category gives rise to an action of the sta-ble category H − mod . Khovanov [Kho16] showed that the Grothendieck group of this stable category isisomorphic to the cyclotomic ring for the prime p , thus categorifying a structure at a prime root of unity.The first successful implementation of this idea was the categorification of the upper half of the smallquantum group for sl in [KQ15] by endowing the nilHecke algebra with a p -DG structure. It is an inter-esting open question how to import p -DG theory into the construction of Khovanov homology. A clearerpath towards categorifying link invariants at roots unity was described in [KQ15] where a p -differentialwas defined on Webster’s algebras [Web17]. One step in this direction was a categorification of the Buraurepresentation of the braid group at a prime root of unity which used a very special p -DG Webster algebra[QS16]. We also refer the reader to [QS17] for a survey in this direction. In this paper we propose a construction of a p -DG version of HOMFLYPT homology and its categoricalspecializations. We closely mimic the work of Khovanov and Rozansky in [KR16] where an action of the utline 3 Witt algebra on HOMFLYPT homology is constructed, and adapt their framework in the p -DG setting. Inparticular, the action of one of their Witt algebra generators (denoted L in [KR16]) corresponds to the p -differential ∂ considered in this work.Let R = k [ x , . . . , x n ] be the polynomial algebra generated by elements of degree two. The categoryof regular Soergel bimodules for gl n is the idempotent completion of the subcategory of ( R, R ) -bimodulesgenerated by the so-called Bott-Samelson bimodules. The Hopf algebra algebra H acts on the polynomialalgebra R determined by ∂ ( x i ) = x i . By the Leibniz rule, H also acts on tensor products of Bott-Samelsonbimodules. We may then form the category ( R, R ) H of such p -DG bimodules. We show that in an appro-priate homotopy category that there is a categorical braid group action, extending the result of Rouquier[Rou04]. For the proof, we follow the exposition of the braid group action in [KR16] very closely. Usinga certain p -extension functor, we then obtain a braid group action on a relative p -homotopy category. Itfollows that, to any braid group element β , there is a p -chain complex of H -equivariant Soergel bimodules pT β associated to β that is well-defined up to p -homotopy equivalences.We next turn our attention to extracting link invariants by taking various versions of Hochschild ho-mology. A p -analogue of the usual Hochschild homology p HH • , which goes back to the work of Mayer[May42a, May42b], is utilized. In the p -extended setting, we need to collapse the Hochschild and topolog-ical gradings into a single grading because of the Markov II invariance constraint. Thus the constructionyields just a doubly graded categorification of the HOMFLYPT polynomial where the a variable is nowspecialized to a prime root of unity.Let ζ C := P ni =1 x i ∂∂x i ∈ HH ( R ) be a Hochschild cohomology element of degree two, regarded asa derivation on R . The cap product of ζ C with an element in Hochschild homology yields a differential d C : HH i ( M ) → HH i − ( M ) of q -degree and a -degree − . This differential gives rise to, via p -extension, a p -differential ∂ C action on the p -Hochschild homology groups of any H -equivariant Soergel bimodule. Fora braid β , one may form a total p -differential ∂ T := ∂ t + ∂ C + ∂ q combining the topological differential ∂ t coming from the Rouquier complex with the derivation actions arising from ∂ C and H . The total differentialacts on p HH • ( pT β ) and gives rise to an invariant upon taking homology. Theorem.
Let L be a link presented as the closure of a braid β . The slash homology of p HH • ( pT β ) withrespect to ∂ T is a finite-dimensional framed link invariant whose Euler characteristic is the Jones polynomialevaluated at a prime root of unity.The link invariant using the action of the usual differential d C on Hochschild homology (ignoring theaction of H ) was first constructed by Cautis [Cau17], and further considered in other contexts by Robert-Wagner [RW20] and Queffelec-Rose-Sartori [QRS18]. The latter authors showed that it categorifies the Jonespolynomial for a generic value of the quantum parameter and is distinct from Khovanov homology. Theseworks actually utilized a degree N differential and categorified the link invariant arising from quantum sl N . We restrict to the case N = 2 due to the fact that ∂ q and ∂ C do not commute for arbitrary values of N .One may view this work as a combination of the results of [KR16] with [Cau17, QRS18, RW20].It is a natural problem to extend our result to categorify the colored Jones and sl n polynomials evaluatedat a prime root of unity. The first technical obstacle to overcome in that setting, is the construction of Koszulresolutions of the algebra of symmetric functions in the presence of a p -differential. We plan to explorethese questions in follow-up works. We now summarize the contents of each section.In Section 2 we review some constructions known in p -DG theory and develop some new ones suchas the p -extension functor, the totalization functor, the relative p -homotopy category, and (relative) p -Hochschild homology.A review of Soergel bimodules is given in Section 3, where a p -categorical braid group action is con-structed. Many of the techniques in this section parallel methods used in [KR16].Section 4 contains the construction of the categorification of the HOMFLYPT polynomial at a root ofunity. The main technical result in this section is invariance under the second Markov move. The proofbuilds upon the techniques in [RW20] which in turn used ideas from [Rou17] adapted to the H -equivariantand hopfological setting. cknowledgements. 4 A categorification of the Jones polynomial at a prime root of unity is developed in Section 5. We re-visit the proof of the second Markov move given in the previous section but now accounting for the extradifferential ∂ C . Here again we build upon ideas from [Cau17, RW20, QRS18].We conclude in Section 5 with the calculation of the homology theories developed in this work for (2 , n ) torus links. In particular, we exhibit non-trivial p -complexes as p -homologies of these links. The authors would like to thank Sabin Cautis, Mikhail Khovanov, and Louis-Hadrien Robert for helpfulconversations.Y.Q. is partially supported by the NSF grant DMS-1947532. J.S. is partially supported by the NSF grantDMS-1807161 and PSC CUNY Award 63047-00 51.
In this section, we recall some basic hopfological algebraic facts introduced in [Kho16, Qi14]. We alsodevelop the necessary constructions of p -analogues of classical Hochschild homology in the hopfologicalsetting.There will be several (super) differentials utilized in this section. We reserved the normal d for the superdifferential ( d = 0 ), and the symbol ∂ to denote a p -differential ( ∂ p = 0 ) over a field of finite characteristic p > . Various differentials will also be labeled with different subscripts to indicate their different meanings. Let A be an algebra over the ground field k of characteristic p > . We equip A with the trivial ( p -)differentialgraded structure by declaring that d ≡ , ∂ ≡ and A sits in degree zero. In this subsection, we study afunctor relating the usual homotopy category C ( A, d ) of A with its p -DG homotopy category C ( A, ∂ ) .To do this, recall that a chain complex of A -modules consists of a collection of A -modules and homo-morphisms d M : M i −→ M i − called boundary maps · · · d M / / M i +1 d M / / M i d M / / M i − d M / / M i − d M / / · · · , satisfying d M = 0 for all i ∈ Z . A null-homotopic map is a sequence of A -module maps f i : M i −→ N i , i ∈ Z ,of A -modules, as depicted in the diagram below, · · · d M / / M i +1 h i +1 ②②②② | | ②②②② d M / / f i +1 (cid:15) (cid:15) M ih i ②②②② | | ②②②② d M / / f i (cid:15) (cid:15) M i − h i − ②②② | | ②②② d M / / f i − (cid:15) (cid:15) M i − d M / / h i − ✇✇✇ { { ✇✇✇ f i − (cid:15) (cid:15) · · · h i − ②②②② | | ②②②② · · · d N / / N i +1 d N / / N i d N / / N i − d N / / N i − d N / / · · · which satisfy f i = d N ◦ h i + h i − ◦ d M for all i ∈ Z . The homotopy category C ( A, d ) , by construction, is thequotient of the category of chain complexes over A by the ideal of null-homotopic morphisms.For ease of notation, we will use bullet points to stand for a general ( p )-chain complex index in what fol-lows. Similarly, a p -chain complex of A -modules consists of a collection of A -modules and homomorphisms ∂ M : M i −→ M i − called p -boundary maps · · · ∂ M / / M i +1 ∂ M / / M i ∂ M / / M i − ∂ M / / M i − ∂ M / / · · · , satisfying ∂ pM ≡ . A p -chain complex can also be regarded as a graded module over the tensor productalgebra A ⊗ H , where H = k [ ∂ ] / ( ∂ p ) is a graded Hopf algebra where deg( ∂ ) = − and deg( A ) = 0 .We introduce some special notation for some specific indecomposable p -chain complexes over k , bysetting U i := H / ( ∂ i +10 ) , ≤ i ≤ p − . (2.1) omeexactfunctors 5 In particular, U i has dimension i + 1 . We will also use these modules with degree shifted up by an integer n ∈ Z , which we denote by U i { a } . Then U i { a } is concentrated in degrees a, a − , . . . , a − i : a k a − k · · · a − i k A map of p -complexes f : M • −→ N • of A -modules is said to be null-homotopic if there exists h : M • −→ N • + p − such that f = p − X i =0 ∂ iN ◦ h ◦ ∂ p − − iM . (2.2)The p -homotopy category, C ( A, ∂ ) , is then the quotient of p -chain complexes of A -modules by the idealof null-homotopic morphisms. It is a triangulated category, whose homological shift functor [1] ∂ is definedby M [1] ∂ := M ⊗ U p − { p − } (2.3)for any p -complex of A -modules. The inverse functor [ − ∂ is given by M [ − ∂ := M ⊗ U p − {− } , (2.4)which is a consequence of the fact that U p − ⊗ U p − decomposes into a direct sum of k { − p } and copies offree H -modules. Slash homology.
As an analogue of the usual homology functor, we have the notion of slash homologygroups [KQ15] of a p -complex. To recall its definition, let us set A = k . For each ≤ k ≤ p − form thegraded vector space H /k ( M ) = Ker( ∂ k +1 M )Im( ∂ p − k − M ) + Ker( ∂ kM ) . The original Z -grading on M gives a decomposition H /k • ( M ) = M i ∈ Z H /ki ( M ) . The differential ∂ M induces a map, also denoted ∂ M , which takes H /ki ( U ) to H /k − i − ( U ) . Define the slashhomology of M as H / • ( M ) = p − M k =0 H /k • ( M ) . (2.5)Also let H /i ( M ) := p − M k =0 H /ki ( M ) . We have the decompositions H / • ( M ) = M i ∈ Z H /i ( M ) = p − M k =0 H /ki ( M ) = M i ∈ Z p − M k =0 H /ki ( M ) . (2.6) H / • ( M ) is a bigraded k -vector space, equipped with an operator ∂ M of bidegree ( − , − , ∂ M : H /ki −→ H /k − i − .Forgetting the k -grading gives us a graded vector space H / • ( M ) with differential ∂ M , which we can viewas a graded H -module. H / • ( M ) is isomorphic to M in the homotopy category of p -complexes C ( k , ∂ ) , andwe can decompose M ∼ = H / • ( M ) ⊕ P ( M ) (2.7) omeexactfunctors 6 in the abelian category of H -modules, where P ( M ) is a maximal projective direct summand of M . Inparticular, we have H / • ( U i ) = ( U i , i = 0 , . . . , p − , i = p − (2.8)The slash homology group H / • ( M ) , viewed as an H -module, does not contain any direct summand iso-morphic to a free H -module.The assignment M H / • ( M ) is functorial in M and can be viewed as a functor H - mod −→C ( k , ∂ ) oras a functor C ( k , ∂ ) −→C ( k , ∂ ) . The latter functor is then isomorphic to the identity functor.As in the usual homological algebra case, we say a morphism f : M −→ N of p -complexes of A -modulesis a quasi-isomorphism if, upon taking slash homology, f induces an isomorphism f / : H / • ( M ) ∼ = H / • ( N ) . Theclass of quasi-isomorphisms constitutes a localizing class in C ( A, ∂ ) ([Kho16, Proposition 4]). Definition 2.1.
The p -derived category D ( A, ∂ ) is the localization of C ( A, ∂ ) at the class of quasi-isomorphisms.Alternatively, D ( A, ∂ ) is the Verdier quotient of C ( A, ∂ ) by the class of acyclic p -complexes , i.e., those p -complexes of A -modules annihilated by the slash-homology functor. p -Extension. We now define the p -extension functor P : C ( A, d ) −→C ( A, ∂ ) (2.9)as follows. Given a chain complex of A -modules, we repeat every term sitting in odd homological degrees ( p − times. More explicitly, for a given complex · · · d k +2 / / M k +1 d k +1 / / M k d k / / M k − d k − / / M k − d k − / / · · · , the p -extended complex looks like · · · d k +2 / / M k +1 · · · M k +1 d k +1 / / / / M k EDBCGF d k @A / / ❵❵❵❵❵❵❵ M k − · · · M k − d k − / / M k − d k − / / · · · . Likewise, for a chain-map · · · d k +3 / / M k +2 d k +2 / / f k +2 (cid:15) (cid:15) M k +1 d k +1 / / f k +1 (cid:15) (cid:15) M k d k / / f k (cid:15) (cid:15) · · ·· · · d k +3 / / N k +2 d k +2 / / N k +1 d k +1 / / N k d k / / · · · the obtained morphism of p -complexes of A -modules is given by · · · d k +3 / / M k +2 d k +2 / / f k +2 (cid:15) (cid:15) M k +1 f k +1 (cid:15) (cid:15) · · · M k +1 d k +1 / / f k +1 (cid:15) (cid:15) M k d k / / f k (cid:15) (cid:15) · · ·· · · d k +3 / / N k +2 d k +2 / / N k +1 · · · N k +1 d k +1 / / N k d k / / · · · This is clearly a functor from the abelian category of chain complexes over A into the category of p -DGmodules over ( A, ∂ ) , ( p -complexes of A -modules). Denote this functor by b P . Lemma 2.2.
The functor b P preserves the ideal of null-homotopic morphisms. omeexactfunctors 7 Proof.
It suffices to show that b P sends null-homotopic morphisms in C ( A, d ) to null-homotopic morphismsin C ( A, ∂ ) . Suppose f = dh + hd is a null-homotopic morphism in C ( A, d ) . We first extend h : M • −→ N • +1 to a map ˆ P ( h ) : ˆ P ( M ) • −→ ˆ P ( N ) • + p − . On unrepeated terms, ˆ P ( h ) sends M k to the copy of N k +1 sitting as the leftmost term in the repeated N k +1 ’s, while on the repeated terms, it only sends the rightmost M k +1 to the unrepeated N k +2 and actsby zero on the other repeated M k +1 ’s. Schematically, this has the effect as in the diagram below: · · · M k +3 d k +3 / / M k +2 d k +2 / / M k +10 ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ s s ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ · · · M k +1 d k +1 / / h k +1 ❣❣❣❣❣❣❣❣❣❣❣ s s ❣❣❣❣❣❣❣❣❣❣❣ M k d k / / h k ❣❣❣❣❣❣❣❣❣❣❣❣ s s ❣❣❣❣❣❣❣❣❣❣❣❣ M k − ❣❣❣❣❣❣❣❣❣❣❣❣❣❣ s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣ · · ·· · · N k +3 d k +3 / / N k +2 d k +2 / / N k +1 · · · N k +1 d k +1 / / N k d k / / N k − · · · . Now it is an easy exercise to check that b P ( f ) = p − X i =0 ∂ p − − iM ◦ b P ( h ) ◦ ∂ iN , (2.10)where ∂ M denotes the extended p -differential on b P ( M ) , and similarly for ∂ N . For instance, between therightmost repeated M k +1 and N k +1 , the left hand side of equation (2.10) equals f k +1 , while there areonly two non-zero terms contributing to the right hand side of (2.10), which are equal to, respectively, ∂ p − N ◦ b P ( h ) = d k +2 ◦ h k +1 , ∂ p − N ◦ b P ( h ) ◦ ∂ M = h k ◦ d k +1 . The sum of these two nonzero terms is precisely f k +1 by the null-homotopy assumption on h . One simi-larly checks for the other repeated and unrepeated terms, and the lemma follows.This lemma implies that b P descends to a functor P : C ( A, d ) −→C ( A, ∂ ) . (2.11)which we call the p -extension functor . Proposition 2.3.
The p -extension functor P is exact. Proof.
It suffices to show that P commutes with homological shifts in both categories, and sends distin-guished triangles to distinguished triangles.On C ( A, d ) , the homological shift [1] d moves every term of a complex one step to the left, while the ho-mological shift [1] ∂ on C ( A, ∂ ) is given by tensoring a p -complex of A -modules with the ( p − -dimensionalcomplex U p − { p − } = (cid:16) k · · · k (cid:17) where the underlined k sits in degree p − . Note that the collection of repeated terms can be identified with (cid:16) M k − · · · M k − , (cid:17) ∼ = M k − ⊗ U p − { kp − } , where the underlined term sits in degree kp − . Using the fact that U p − { p − } ⊗ U p − { kp − } ∼ = U { kp } ⊕ F where F is a direct sum of graded free H -modules, we see that U p − { p − } ⊗ ( M k − ⊗ U p − { kp − } ) ∼ = M k − { kp } omeexactfunctors 8 in the homotopy category C ( A, ∂ ) . From this it follows that U p − { p − } ⊗ P ( M ) is homotopy equivalentto the p -complex P ( M [1] d ) . Thus P commutes with homological shifts.To show that P sends distinguished triangles in C ( A, d ) to those in C ( A, ∂ ) , we use the characterizationof distinguished triangles in C ( A, d ) . Recall that a distinguished triangle P → Q → R in C ( A, d ) is ashort exact sequence → P → Q → R → of complexes of A -modules which split when ignoring thedifferentials. After applying the p -extension functor P to → P → Q → R → , one gets a short exactsequences of p -complexes which splits when ignoring the p -differentials. This is precisely the condition that P ( P ) → P ( Q ) → P ( R ) is a distinguished triangle in C ( A, ∂ ) (see [Qi14, Lemma 4.3]). Totalization.
Another useful functor is the totalization functor T , which we introduce next. To do so, wewill need the following result. Lemma 2.4.
Let ( K • , ∂ K ) be a p -complex of modules over A . Then K • is null-homotopic if and only if thereexists an A -module map σ : K • −→ K • +1 such that ∂ K σ − σ∂ K = Id K . Proof.
By definition (see equation (2.2)), a p -complex is null-homotopic if and only if there is an A -linearmap h : K • −→ K • + p − such that Id K = p − X i =0 ∂ p − − iK ◦ h ◦ ∂ iK . For any linear map φ on K • , let ad ∂ ( φ ) := [ ∂ K , φ ] . Now if σ is as given satisfies [ ∂ K , σ ] = Id K , then the map h := − σ p − satisfies [ ∂ K , h ] = − ( p − σ p − , and, inductively, ad r∂ ( h ) = − ( p − . . . ( p − r ) σ p − r − . In particular, we have p − X i =0 ( − i (cid:18) p − i (cid:19) ∂ p − − iK ◦ h ◦ ∂ iK = ad p − ∂ ( h ) = − ( p − K = Id K . Since (cid:0) p − i (cid:1) = ( − i in characteristic p , the null-homotopy formula follows.Conversely, if h is an A -linear null-homotopy map, then σ := ad p − ∂ ( h ) satisfies [ ∂ K , σ ] = Id K . As theiterated commutator of A -linear maps, σ is also A -linear. The lemma follows.Let ( A, ∂ A ) be a p -DG algebra where ∂ A has degree two . We regard A as an algebra object in the gradedmodule category of the graded Hopf algebra H q = k [ ∂ q ] / ( ∂ pq ) in which deg( ∂ q ) = 2 . The smash productalgebra A H q is then the graded algebra A ⊗ H q containing the subalgebras A ⊗ and ⊗ H q and subjectto the commutation relations (1 ⊗ ∂ q )( a ⊗
1) = a ⊗ ∂ q + ∂ A ( a ) ⊗ for any a ∈ A . Graded modules over A H q are also called p -DG modules over A , the collection of which willbe denoted ( A, ∂ A ) - mod .In analogy with the Hopf algebra H , we introduce the indecomposable balanced H q -complexes V i := − i k − i +2 k · · · i − k i k ! (2.12)for i = 0 , . . . , p − .As a matter of notation, we will denote the q -grading shifted copy of V i by q a V i , where the lowest degreeterm sits in degree a − i . Furthermore, if M is any p -DG module over A , we will denote by q a M the p -DGmodule whose underlying module is the same as M , but grading shifted up by a ∈ Z . This degree convention is to match the usual representation theoretical convention. See, for instance, [KQ15]. rothendieckrings 9
The p -DG homotopy category C ( A, ∂ A ) can be defined, similarly as for C ( A, ∂ ) before, by taking the quo-tient of the abelian category of p -DG modules by the ideal of null-homotopic morphisms , which consists ofhomogeneous A H q -module maps f : ( M, ∂ M ) −→ ( N, ∂ N ) of the form f = p − X i =0 ∂ iN ◦ h ◦ ∂ p − − iM , (2.13)where h is an A -linear homomorphism from ( M, ∂ M ) to ( N, ∂ N ) of degree − p .A p -DG homomorphism f : ( M, ∂ M ) −→ ( N, ∂ N ) is called a quasi-isomorphism if, again, f induces an iso-morphism of slash homolgy with respect to the p -differentials on M and N . Inverting quasi-isomorphismsin C ( A, ∂ A ) results in the p -DG derived category D ( A, ∂ A ) .Equip A H q with the zero p -differential ∂ , and ∂ carries an additional Z -grading that is independentof the original grading on A and H q . A p -complex of graded A H q -modules thus has a Z × Z -grading,where ∂ q has degree (2 , and ∂ has degree (0 , − . Consider the functor b T : ( A H q , ∂ ) - mod −→ ( A, ∂ A ) - mod (2.14)defined as follows. To a p -complex M of A H q -modules · · · ∂ M / / M i +1 ∂ M / / M i ∂ M / / M i − ∂ M / / M i − ∂ M / / · · · , where each term M i is a graded A -module together with an internal p -differential ∂ i , compatible with ∂ A ,we assign to it the singly graded A -module ⊕ i ∈ Z q − i M i whose new p -DG structure is given by ∂ T ( m ) := ∂ M ( m ) + ∂ i ( m ) ∈ q − i +2 M i − ⊕ q − i M i if m ∈ q − i M i . Lemma 2.5.
The functor b T descends to a triangulated functor on the p -DG homotopy categories: T : C ( A H q , ∂ ) −→C ( A, ∂ A ) . Proof. If Q • is a null-homotopic p -complex of A H q -modules, then, by Lemma 2.4, there exist an A H q -linear σ : Q • −→ Q • +1 such that [ ∂ Q , σ ] = Id Q . Since h commutes with the H q -actions, we have [ ∂ Q + ∂ q , σ ] = Id Q . It follows that T ( Q • ) is null-homotopic, and the functor T is well-defined on the p -homotopy categories. Itis then an easy exercise to verify that T preserves the triangulated structures on both sides. We will be considering the Grothendieck rings of the homotopy categories C ( k , ∂ ) and C ( k , ∂ q ) . Lemma 2.6.
The Grothendieck rings of the tensor triangulated categories C ( k , ∂ ) and C ( k , ∂ q ) are respec-tively isomorphic to K ( C ( k , ∂ ) ∼ = Z [ a, a − ] / (1 + a + · · · + a p − ) .K ( C ( k , ∂ q ) ∼ = Z [ q, q − ] / (1 + q + · · · + q p − ) . Proof.
See [Kho16, KQ15] for the proof and motivation of introducing these rings, especially the secondone . Again, we emphasize that setting ∂ q to be of degree two is to respect the usual convention in previous literature on Soergelbimodules, where polynomial generators are evenly graded. Alternatively, one may adapt the polynomial generators for Soergelbimodules in this paper to be of degree one, and both Grothendieck rings above are equal to the usual cyclotomic ring at a primitive p th root of unity. -Hochschildhomologyandcohomology 10 We will often abbreviate the Grothendieck rings by O p := K ( C ( k , ∂ )) ∼ = Z [ a, a − ] / (1 + a + · · · + a p − ) , (2.15a) O p := K ( C ( k , ∂ q )) ∼ = Z [ q, q − ] / (1 + q + · · · + q p − ) . (2.15b) Remark 2.7 (Grading shift functors) . In what follows, we will freely use the notation a i ( - ) and q i ( - ) , i ∈ Z ,to indicate the grading shift functors on C ( k , ∂ ) and C ( k , ∂ q ) . The functors then descend to multiplicationby the corresponding monomials in the Grothendieck rings.In this paper, we will be working with (finite-dimensional) a and q bigraded complexes over k equippedwith commuting differentials ∂ and ∂ q . On this category, one may consider the composition of slash-homology functors, first in the a -direction and then in the q -direction: H q ⊗ H - mod H / • in a -direction −−−−−−−−−−→ C ( H q , ∂ ) H / • in q -direction −−−−−−−−−−→ C ( C ( k , ∂ q ) , ∂ ) . (2.16)Here the last category stands for the homotopy category with object in C ( k , ∂ q ) . As ususal with takinghomology of the usual bicomplexes, these functors do not commute, and their order matters in the con-struction. Corollary 2.8.
The categories C ( H q , ∂ ) and C ( C ( k , ∂ q ) , ∂ ) have Grothendieck rings isomorphic to K ( C ( H q , ∂ )) ∼ = O p [ q, q − ] , K ( C ( C ( k , ∂ q ) , ∂ )) ∼ = O p ⊗ Z O p . Proof.
The bigraded abelian category H q ⊗ H - mod has its Grothendieck ring isomorphic to Z [ a ± , q ± ] . Anobject lying in the first slash homology functor has Euler characteristic in the ideal (1 + a + · · · + a p − ) ⊂ Z [ a ± , q ± ] . Similarly, a module lying inside the kernel of the composition functor has Euler characteristic in the ideal (1 + a + · · · + a p − , q + · · · q p − ) ⊂ Z [ a ± , q ± ] . The result follows. p -Hochschild homology and cohomology Now we come to the construction of the p -DG simplicial bar complex of Mayer [May42a, May42b] (see also[KW98]). The usual simplicial bar complex of a unital, associative algebra A is the complex: · · · d n +1 −−−→ A ⊗ ( n +2) d n −→ A ⊗ ( n +1) d n − −−−→ · · · d −→ A ⊗ d −→ A ⊗ −→ , (2.17a)where d i ( a ⊗ a ⊗ · · · ⊗ a i +1 ) = i +1 X k =0 ( − i a ⊗ · · · a k − ⊗ a k a k +1 ⊗ a k +2 ⊗ · · · ⊗ a i +1 (2.17b)The bar complex is a free bimodule resolution of A , as the augmented complex · · · d n +1 −−−→ A ⊗ ( n +2) d n −→ A ⊗ ( n +1) d n − −−−→ · · · d −→ A ⊗ d −→ A ⊗ −→ A −→ , (2.18)is acyclic. This can be seen by constructing a left A -module map σ : A ⊗ n −→ A ⊗ ( n +1) , x x ⊗ (2.19)as the null-homotopy.Let A be a k -algebra. In analogy with the usual simplicial bar complex, Mayer introduced on the usualaugmented bar complex (2.17) the linear map ∂ H ( a ⊗ a ⊗ · · · ⊗ a i +1 ) := i +1 X k =0 a ⊗ · · · a k − ⊗ a k a k +1 ⊗ a k +2 ⊗ · · · ⊗ a i +1 . -Hochschildhomologyandcohomology 11 Then it is an easy exercise to show that ∂ p ≡ . Furthermore, the null-homotopy map σ in (2.19) clearlysatisfies ∂ H σ − σ∂ H = Id . It follows that the augmented p -complex ( p ′• ( A ) , ∂ H ) := (cid:16) · · · ∂ H −−→ A ⊗ ( n +2) ∂ H −→ A ⊗ ( n +1) ∂ H −−→ · · · ∂ H −→ A ⊗ ∂ H −→ A ⊗ −→ A −→ (cid:17) (2.20)is acyclic.Assume next that ( A, ∂ A ) is a p -DG algebra. Extend the p -differential on A to any A ⊗ ( n +1) by the Leibnizrule so that, ∂ A ( a ⊗ a ⊗ · · · ⊗ a n ) := n X i =0 a ⊗ · · · ⊗ ∂ A ( a i ) ⊗ · · · ⊗ a n . As the multiplication map m : A ⊗ A −→ A commutes with ∂ A , it follows that the boundary maps in ( p ′• ( A ) , ∂ M ) commute with the internal differentials ∂ A on each A ⊗ n . We may thus consider the totalcomplex ( p ′• ( A ) , ∂ H + ∂ A ) . This construction is equivalent to the totalization T ( p ′• ( A ) , ∂ H ) . To make adistinction, we will denote the total differential on p ′• ( A ) by ∂ T := ∂ H + ∂ A in order to avoid potentialconfusion with the other differentials ∂ H and ∂ A .There is a natural inclusion map ι A : A −→ p ′• ( A ) of p -DG bimodules over A , whose cokernel is the p -DG bimodule ( e p • ( A ) , ∂ T ) := (cid:16) · · · ∂ H −−→ A ⊗ ( n +2) ∂ H −→ A ⊗ ( n +1) ∂ H −−→ · · · ∂ H −→ A ⊗ ∂ H −→ A ⊗ −→ (cid:17) . Recall here that each A ⊗ n also carries its internal differential ∂ A . Proposition 2.9.
The total p -complex ( p ′• ( A ) , ∂ T ) is acyclic. Furthermore, if Q is any p -DG module over A , p ′• ( A ) ⊗ A Q is also acyclic. Proof.
In order to show that p ′• ( A ) is acyclic, it suffices to check, by Lemma 2.4, that ( ∂ H + ∂ A ) σ − σ ( ∂ H + ∂ A ) = Id p ′ , where Id p ′ is the identity map of p ′• ( A ) This is clear since we have the easily verified commutator relations [ ∂ H , σ ] = Id p ′ , [ ∂ A , σ ] = 0 , [ ∂ H , ∂ A ] = 0 . The last statement is similar, as one just needs to replace the last copy of A in A ⊗ n by Q . Definition 2.10.
Suppose ( M, ∂ M ) is a left p -DG module over A . Set M [ − to be the tensor product of M with the ( p − -complex q p V p − (see equation (2.12)). The simplicial bar resolution for M is the p -DG module p • ( M ) := e p • ( A ) ⊗ A M [ − It inherits the p -differential from that of ∂ T and ∂ M via the Leibniz rule.Likewise, one defines the simplicial bar resolution for right p -DG modules. Proposition 2.11.
For any left p -DG module M over A , p • ( M ) is a cofibrant replacement of M . Proof.
To see the cofibrance of p • ( M ) , first note that p • ( M ) = (cid:0) · · · −→ A ⊗ ⊗ M [ − −→ A ⊗ ⊗ M [ − −→ A ⊗ M [ − −→ (cid:1) , which carries p -differentials on the arrows and internal differentials within each term. It has a naturalfiltration by p -DG submodules, whose subquotients have the form A ⊗ n ⊗ M [ − , ( n ≥ . -Hochschildhomologyandcohomology 12 As left p -DG modules over A , such modules are clearly direct sums of free p -DG A -modules. Therefore p • ( M ) satisfies the “Property P” criterion of [Qi14, Definition 6.3] and is thus cofibrant.By construction, there is a short exact sequence of p -DG modules over A −→ M [ − −→ p ′• ( A ) ⊗ A M [ − −→ p • ( M ) −→ . Since p • ( M ) is projective as a left A -module, the sequence splits when forgetting about p -differentials. By[Qi14, Lemma 4.3], the short exact sequence above gives rise to a distinguished triangle in the homotopycategory M [ − −→ p ′• ( A ) ⊗ A M [ − −→ p • ( M ) [1] −→ M .
Therefore there is a morphism f : p • ( M ) −→ M representing the [1] map on the last arrow. By Proposition2.9, f is a quasi-isomorphism.Using the bar resolution, we recall the derived tensor product functor construction in the p -DG setting. Definition 2.12.
Let M be a left p -DG module and N be a right p -DG module over A . The p -DG derivedtensor product of N and M is the object in H q − mod N ⊗ L A M := N ⊗ A p • ( M ) . As in [Qi14, Corollary 8.9], the functor is well-defined. Furthermore, it is readily seen that it is indepen-dent of cofibrant replacements one chooses for M (or N ). Corollary 2.13.
For any p -DG module M over ( A, ∂ A ) , there is a cofibrant p -DG replacement p • ( M ) ∼ = M in D ( A ) . (cid:3) We define the analogue of Hochschild homology in the p -DG setting. Definition 2.14.
Let A be a p -DG algebra, and M be a bimodule over A . Then the p -DG Hochschild (co)homology is the p -complex p HH • ( M ) := H / • ( p • ( A ) ⊗ A ⊗ A op M ) ( resp. p HH • ( M ) := H / • (HOM A ⊗ A op ( p • ( A ) , M )) . The p -DG Hochschild homology is a functorial “categorical trace” on the category of p -DG bimodulesover a p -DG algebra, similar to the usual Hochschild homology functor. Here, the functoriality means that,a morphism of p -DG bimodules f : M −→ N over A induces a morphism of p -Hochschild homology groups,which is defined by p HH • ( f ) := H / • (Id p • ( A ) ⊗ f ) : H / • ( p • ( A ) ⊗ A ⊗ A op M ) −→ H / • ( p • ( A ) ⊗ A ⊗ A op N ) . (2.21) Theorem 2.15.
Given two p -DG bimodules M and N over A , there is an isomorphism of p -complexes p HH • ( M ⊗ L A N ) ∼ = p HH • ( N ⊗ L A M ) . Proof.
This is more or less parallel to the classical Hochschild homology case. For this, we notice that byDefinition 2.12, M ⊗ L A N = M ⊗ A p • ( A ) ⊗ A N .
Then by Definition 2.14, one takes another tensor product over A ⊗ A op with p • ( A ) with respect to the left A -action on M and right A -action on N . This is best visualized as putting everything on a circle: M p • ( A ) N p • ( A ) . elativehomotopycategories 13 Here the connecting lines joining the p -DG modules in the picture stand for the usual tensor product over A . Rotating the picture by ◦ , one obtains the p -complex computing the p -Hochschild homology for thebimodule N ⊗ L A M . The result follows. For any ungraded algebra B over k , denote by d the zero ordinary differential and by ∂ the zero p -differential on B , while letting B sit in homological degree zero. When B is graded, the homologicalgrading is indepednent of the internal grading of B .Suppose ( A, ∂ A ) is a p -DG algebra. There is an exact forgetful functor between the usual homotopycategories of chain complexes of graded A H -modules F d : C ( A H q , d ) −→C ( A, d ) . An object K • in C ( A H q , d ) lies inside the kernel of the functor if and only if, when forgetting the H q -module structure on each term of K • , the complex of graded A modules F d ( K • ) is null-homotopic. Thenull-homotopy map on F d ( K • ) , though, is not required to intertwine H q -actions.Likewise, there is an exact forgetful functor F ∂ : C ( A H q , ∂ ) −→C ( A, ∂ ) . Similarly, an object K • in C ( A H q , ∂ ) lies inside the kernel of the functor if and only if, when forgettingthe H q -module structure on each term of K • , the p -complex of A modules F ( K • ) is null-homotopic. Thenull-homotopy map on F ( K • ) , though, is not required to intertwine H q -actions. Definition 2.16.
Given a p -DG algebra ( A, ∂ A ) , the relative homotopy category is the Verdier quotient C ∂ q ( A, d ) := C ( A H q , d )Ker( F d ) . Likewise, the relative p -homotopy category is the Verdier quotient C ∂ q ( A, ∂ ) := C ( A H q , ∂ )Ker( F ∂ ) . The subscripts in the definitions are to remind the reader of the H q -module structures on the objects.The categories C ∂ q ( A, d ) and C ∂ q ( A, ∂ ) are triangulated. By construction, there is a factoraization of theforgetful functor C ( A H q , d ) F d / / ' ' ❖❖❖❖❖❖❖❖❖❖❖ C ( A, d ) C ∂ q ( A, d ) rrrrrrrrrr , C ( A H q , ∂ ) F ∂ / / ' ' ❖❖❖❖❖❖❖❖❖❖❖ C ( A, ∂ ) C ∂ q ( A, ∂ ) rrrrrrrrrr . Proposition 2.17.
The p -extension functor P : C ( A H q , d ) −→C ( A H q , ∂ ) descends to a functor, stilldenoted by P , between the relative homotopy categories: P : C ∂ q ( A, d ) −→C ∂ q ( A, ∂ ) . Proof.
It suffices to show that, if K • ∈ Ker( F d ) , then P ( K • ) ∈ Ker( F ∂ ) . This is clear since, if h provides anull-homotopy in C ( A, d ) for F d ( K • ) , then P ( h ) is the null-homotopy for F ∂ ( P ( K • )) . elative p -Hochschildhomology 14 p -Hochschild homology In this paper, instead of the absolute version of p -Hochschild homology, we will need a relative version of p -Hochshild homology for a p -DG algebra, which we define now. An important reason for introducing therelative homotopy category is that the relative p -Hochschild homology functor descends to this category.Let ( A, ∂ A ) be a p -DG algebra. Equip the zero differential d and p -differential ∂ on A , and denote theresulting trivial ( p )-DG algebras by ( A , d ) and ( A , ∂ ) . Likewise, for a ( p -)DG bimodule M over A , wetemporarily denote by M the A -bimodule equipped with zero ( p -) differentials.The usual Hochschild homology of M over ( A , d ) in this case carries a natural H q -action, since the H q -action commutes with all differentials in the simplicial bar complex (2.17) for A . Definition 2.18.
The relative Hochschild homology of a p -DG bimodule ( M, ∂ M ) over ( A, ∂ A ) is the usualHochschild homology of M over ( A , d ) equipped with the induced H q -action from ∂ M and ∂ A , anddenoted HH ∂ q • ( M ) := HH • ( A , M ) Replacing the usual simplicial bar complex by Mayer’s p -simplicial bar complex p • ( A ) for ( A , ∂ ) , wemake the following definition. Definition 2.19.
The relative p -Hochschild homology of M is the p -complex of p HH ∂ q • ( M ) := H / • ( A ⊗ L A ⊗ A op0 M ) = H / • ( p ( A ) ⊗ A ⊗ A op0 M ) . Similar to p -Hochschild homology, the relative case is also covariant functor: if f : M −→ N is a mor-phism of p -DG bimodules over A , it induces p HH ∂ q • ( f ) := H / • (Id A ⊗ f ) : H / • ( A ⊗ L A ⊗ A op0 M ) −→ H / • ( A ⊗ L A ⊗ A op0 N ) . Proposition 2.20.
The relative p -Hochschild homology descends to a functor defined on the relative homo-topy category C ∂ q ( A, ∂ ) of p -DG bimodules over A . Proof.
An object that lies in the kernel F for p -DG modules over A ⊗ A op consists of null-homotopic p -complexes of bimodules over ( A , ∂ ) . Thus the relative p -Hochschild homology functor annihilates suchobjects, and descends to the quotient category.We also have the trace-like property for relative p -Hochschild homology. Proposition 2.21.
Given two p -DG bimodules M and N over A , there is an isomorphism of p -complexes of H q -modules p HH ∂ q • ( M ⊗ L A N ) ∼ = p HH ∂ q • ( N ⊗ L A M ) . Proof.
This follows from Theorem 2.15 by replacing ( A, ∂ A ) with ( A , ∂ ) .Our next goal is to show that we may relax the requirement that we utilize the simplicial bar resolu-tion when computing the relative Hochschild homology. For the next theorem, we use the fact that inthe simplicial bar complex p • ( A ) , all the p -complex maps are H q -equivariant since they are just sums ofmultiplications maps of A tensored with identities maps on A . Theorem 2.22.
Let M be a p -DG bimodule over A . Suppose f : Q • −→ M is a p -complex resolution of M over ( A , ∂ ) which is H q -equivariant, and each term of Q • is projective as an A ⊗ A op0 -module. Then f induces an isomorphism of H q -modules H / • ( A ⊗ A ⊗ A op0 Q • ) ∼ = p HH ∂ q • ( M ) . Proof.
By definition, there is a short exact sequence over ( A , ∂ ) : −→ M −→ C • ( f ) −→ Q • [1] −→ . The cone C • ( f ) , by construction, is equal to C • ( f ) ∼ = Q • ⊗ k [ ∂ ] / ( ∂ p ) ⊕ M n ( x ⊗ ∂ p − , f ( x )) | x ∈ Q • o , which is then an acyclic p -complex of bimodules over ( A, ∂ ) . Since the H q -actions on the modules Q • and M commute with the ∂ action, the above short exact sequence is an exact sequence of H q -modules.Taking the tensor product of p • ( A ) with the sequence, we get a short exact sequence since p • ( A ) isprojective as a module over A ⊗ A op0 −→ p • ( A ) ⊗ A ⊗ A op0 M −→ p • ( A ) ⊗ A ⊗ A op0 C • ( f ) −→ p • ( A ) ⊗ A ⊗ A op0 Q • [1] −→ . The middle term p • ( A ) ⊗ A ⊗ A op0 C • ( f ) is a p -complex of A -bimodules equipped with a filtration, whosesubquotients are isomorophic to grading shifts of C • ( f ) . Therefore it is acyclic. It follows that we have anisomorphism in D (( A ⊗ A op0 ) H q , ∂ ) p • ( A ) ⊗ A ⊗ A op0 Q • ∼ = p • ( A ) ⊗ A ⊗ A op0 M .
Taking p -Hochschild homology, we obtain an isomorphism of H q -modules H / • ( p • ( A ) ⊗ A ⊗ A op0 Q • ) ∼ = H / • ( p • ( A ) ⊗ A ⊗ A op0 M ) . Similarly, since Q • is projective as a bimodule over A , tensoring Mayer’s short exact sequence of bimoduleswith Q • over A ⊗ A op0 remains exact: −→ A ⊗ A ⊗ A op0 Q • −→ p ′• ( A ) ⊗ A ⊗ A op0 Q • −→ p • ( A )[1] ⊗ A ⊗ A op0 Q • −→ The middle term is, as above, acyclic since p ′• ( A ) is. It follows that A ⊗ A ⊗ A op0 Q • ∼ = p • ( A ) ⊗ A ⊗ A op0 Q • as objects in D ( H q , ∂ ) . Taking slash homology on both sides gives the desired result. p -DG Soergel bimodules and braid relations p -DG bimodules over the polynomial algebra Let k be a field of characteristic p > . The graded polynomial algebra R n = k [ x , . . . , x n ] has a naturalmodule-algebra structure over the graded Hopf algebra H q = k [ ∂ q ] / ( ∂ pq ) , where the generator ∂ q ∈ H q actsas a derivation determined by ∂ q ( x i ) = x i for i = 1 , . . . , n . Here the degree of each x i and ∂ q are both two,and will be referred to as the q -degree .When n is clear from the context, we will abbreviate R n by just R . The differential is invariant under thepermutation action of the symmetric group S n on the indices of the variables. Therefore let the subalgebraof polynomials symmetric in variables x i and x i +1 with its inherited H q -module structure be denoted by R in = k [ x , . . . , x i − , x i + x i +1 , x i x i +1 , x i +2 , . . . , x n ] . More generally, given any subgroup G ⊂ S n , the invariant subalgebra R Gn inherits an H q -algebra structurefrom R n (and is thus a p -DG algebra). In particular, we will also use the H q -subalgebra R i,i +1 n := R S n ,where S is the subgroup generated by permuting the indices i , i + 1 and i + 2 .When the number of variables n is clear from the context or is irrelevant of the statements, we willabbreviate R n by just R in what follows. This characteristic assumption is standard in the literature on Soergel bimodules. See, for instance [EK10, Section 2]. Remark 3.6also provides an explanation for this assumption. lementarybraidingcomplexes 16
The ( R, R ) -bimodule B i = R ⊗ R i R has the structure of an H q -module (and is thus a p -DG bimodule)where the differential acts via the Leibniz rule: for any h ⊗ g ∈ R ⊗ R i R , ∂ q ( h ⊗ g ) = ∂ q ( h ) ⊗ g + h ⊗ ∂ q ( g ) . The tensor category of ( R, R ) -bimodules generated by the B i , has an H q -module structure where the actioncomes from the comultiplication in H q . We denote this category by ( R, R ) H q - mod .Let f = P ni =1 a i x i ∈ F p [ x , . . . , x n ] ⊂ R be a linear function. We twist the H q -action on the bimodule B i to obtain a bimodule B fi defined as follows. As an ( R n , R n ) -bimodule, it is the same as B i but the action of H q is twisted by defining ∂ q (1 ⊗
1) = (1 ⊗ f. (3.1a)Similarly we define f B i where now ∂ q (1 ⊗
1) = f (1 ⊗ . (3.1b)For R n as a bimodule over itself, it is clear that f R n ∼ = R fn as p -DG bimodules. It follows that there are p n ways to put an H q -module structure on a rank-one free module over R n . Each such H q -module is quasi-isomorphic to a finite-dimensional p -complex. Choose numbers b i ∈ { , . . . , p } such that b i ≡ a i (mod p ) , i = 1 , . . . , n , and define the H q -ideal of RI = ( x p +1 − b , · · · , x p +1 − b n n ) . (3.2)Then the natural quotient map π : R fn ։ R fn / ( I · R fn ) (3.3)is readily seen to be a quasi-isomorphism. Lemma 3.1.
For each f = P ni =1 a i x i , the rank-one p -DG module R fn has finite-dimensional slash homology: H / • ( R fn ) ∼ = n O i =1 V p − b i { p − b i } . In particular, if any a i of f = P i a i x i is equal to one, then H / • ( R fn ) = 0 . Proof.
The first statement follows from the discussion before the lemma. For the second statement, notethat V p − is acyclic, and the tensor product of an acyclic H q -module with any H q -module is acyclic. Corollary 3.2.
Let M be a p -DG module over R which is equipped with a finite filtration, whose subquo-tients are isomorphic to R f for various f . Then M has finite-dimensional slash homology. Proof.
Induct on the length of the filtration, and apply the previous lemma.By Definition 2.16, a morphism f : A −→ B in the homotopy category C (( R, R ) H q , d ) is a relative iso-morphism if F d ( f ) is an isomorphism in the homotopy category C (( R, R ) , d ) . Localizing C (( R, R ) H q , d ) at all relative isomorphisms produces the relative homotopy category of ( R, R ) -bimodules C ∂ q ( R, R, d ) .Similarly we have the p -version of the relative homotopy category C ∂ q ( R, R, ∂ ) . Lemma 3.3.
There are ( R, R ) H q -module homomorphisms(i) rb i : R −→ q − B − ( x i + x i +1 ) i , where ( x i +1 ⊗ − ⊗ x i ) ;(ii) br i : B i −→ R , where ⊗ . eidemeisterII 17 Proof.
The fact that these maps are ( R, R ) -bimodule homomorphisms is well known. See, for instance,[EK10] for more details.In order to check that the homomorphism is compatible with the H q -module structure, we must checkthat rb i ( ∂ q (1)) = ∂ q ( rb i (1)) . Clearly rb i ( ∂ q (1)) = 0 . On the other hand ∂ q ( rb i (1)) = ∂ q ( x i +1 ⊗ − ⊗ x i )= x i +1 ⊗ − ⊗ x i + ( x i +1 ⊗ − ⊗ x i )( − x i − x i +1 )= 0 where the third term in the second equation above comes from the twist on B i .The second homomorphism clearly respects the H q -structure since ∂ q (1 ⊗
1) = 0 = ∂ q (1) .We now have complexes of ( R, R ) H q -modules T i := (cid:16) tB i br i −−→ R (cid:17) , T ′ i := (cid:16) R rb i −−→ q − t − B − ( x i + x i +1 ) i (cid:17) . (3.4)In the coming sections we will, for presentation reasons, often omit the various shifts built into the defini-tions of T i and T ′ i .We associate respectively to the left and right crossings σ i and σ ′ i between the i th and ( i + 1) st strandsin (3.5) the chain complexes of ( R, R ) H q -bimodules T i and T ′ i : σ i := · · · · · · σ ′ i := · · · · · · (3.5)More generally, if β ∈ Br n is a braid group element written as a product σ ǫ i i · · · σ ǫ k i k in the elementarygenerators where ǫ i ∈ {∅ , ′} , we assign the chain complex of ( R, R ) H q -bimodules T β := T ǫ i ⊗ R · · · ⊗ R T ǫ k i k . (3.6) Theorem 3.4.
Given any braid group element β ∈ Br n , the chain complex of T β associated to it is a welldefined element of the relative homotopy category C ∂ q ( R, R, d ) .The proof of the theorem, which is like the analogous proof in [KR16], will take up the rest of this section. The following lemma is crucial to proving the Reidemeister II braid relation and should be compared with[KR16, Lemma 4.3].
Lemma 3.5.
There exists an isomorphism of ( R, R ) H q -modules B i ⊗ R B i ∼ = B i ⊕ q B x i + x i +1 i defined by φ = (cid:18) φ φ (cid:19) : B i ⊗ R B i −→ B i ⊕ q B x i + x i +1 i , φ (1 ⊗ ⊗
1) = (cid:18) ⊗ (cid:19) , φ (1 ⊗ ( x i +1 − x i ) ⊗
1) = (cid:18) ⊗ (cid:19) ,ψ = ( ψ , ψ ) : B i ⊕ q B x i + x i +1 i −→ B i ⊗ R B i , ψ (1 ⊗
1) = 1 ⊗ ⊗ , ψ (1 ⊗
1) = 1 ⊗ ( x i +1 − x i ) ⊗ . Proof.
As an ( R, R ) -bimodule, B i ⊗ R B i is generated by ⊗ ⊗ and ⊗ ( x i +1 − x i ) ⊗ . It is a classical factthat the maps φ and ψ provide inverse isomorphisms of bimodules.It is straightforward to check that φ and ψ are compatible with the H q -module structure. Note that thiscondition forces the twist x i + x i +1 on the second B i term. Remark 3.6.
This lemma is where the characteristic p > assumption is crucially needed. Indeed, incharacteristic , the element ⊗ ( x i +1 − x i ) ⊗ ⊗ ( x i +1 + x i ) ⊗ x i +1 + x i ) ⊗ ⊗ becomes aredundant element in the standard set of bimodule generators for B i ⊗ R B i . eidemeisterIII 18Proposition 3.7. There are isomorphisms in the homotopy category C (( R, R ) H q , d ) T i ⊗ R T ′ i ∼ = Id ∼ = T ′ i ⊗ R T i . Proof.
In order to prove the isomorphism T ′ i ⊗ R T i ∼ = Id , we tensor the complexes for T ′ i and T i to obtain R ⊗ R R rb i ⊗ Id * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ R ⊗ R B i Id ⊗ br i ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ − rb i ⊗ Id ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ q − B − ( x i + x i +1 ) i ⊗ R Rq − B − ( x i + x i +1 ) i ⊗ R B i Id ⊗ br i ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (3.7)By Lemma 3.5, the complex (3.7) is isomorphic to R ⊗ R R rb i ⊗ Id * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ R ⊗ R B i Id ⊗ br i ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ γ γ ( ( PPPPPPPPPPPPP q − B − ( x i + x i +1 ) i ⊗ R Rq − B − ( x i + x i +1 ) i ⊕ B i (cid:16) δ δ (cid:17) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (3.8)where γ (1 ⊗
1) = 12 ( x i − x i +1 ) ⊗ , γ (1 ⊗
1) = −
12 (1 ⊗ , δ (1 ⊗
1) = 1 ⊗ , δ (1 ⊗
1) = 1 ⊗ ( x i +1 − x i ) . Contracting out the terms B i , one gets that (3.8) is homotopic to R rb i ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ q − B − ( x i + x i +1 ) i q − B − ( x i + x i +1 ) i Id ♠♠♠♠♠♠♠♠♠♠♠♠ (3.9)Finally, contracting out the terms B − ( x i + x i +1 ) , one gets T ′ i ⊗ R T i ∼ = R . Note that each homotopy γ used inthe contractions has the property ∂ ( γ ) = 0 . Let B i,i +1 ,i = R ⊗ R i,i +1 R where R i,i +1 = k [ x , . . . , x i − , e ( x i , x i +1 , x i +2 ) , e ( x i , x i +1 , x i +2 ) , e ( x i , x i +1 , x i +2 ) , x i +3 , . . . , x n ] . The following lemma is a rephrasing of [KR16, Lemma 4.7]. We will use the map φ from the statementof Lemma 3.5. eidemeisterIII 19Lemma 3.8. There exists a short exact sequence of ( R, R ) H q -modules which splits upon restricting to thecategory of ( R, R ) -bimodules / / B i,i +1 ,i f / / B i ⊗ B i +1 ⊗ B i f / / B x i + x i +1 i / / where f (1 ⊗
1) = 1 ⊗ ⊗ , f = φ ◦ (Id ⊗ br i +1 ⊗ Id) . Proof.
Ignoring the H q -module structure, this is a classical statement. See for example [EK10, Definition 3.9,Section 4.5].Clearly ∂ q ( f ) = 0 . Since ∂ q ( φ ) = 0 and ∂ q ( br i +1 ) = 0 , it follows that ∂ q ( f ) = 0 . Proposition 3.9.
There exist isomorphisms in the relative homotopy category C ∂ q ( R, R, d ) (i) T i T i +1 T i ∼ = T i +1 T i T i +1 ,(ii) T ′ i T ′ i +1 T ′ i ∼ = T ′ i +1 T ′ i T ′ i +1 . Proof.
We will prove the first isomorphism. The second follows from the first part and Proposition 3.7.By definition T i T i +1 T i ∼ = (cid:0) B i br i / / R (cid:1)(cid:0) B i +1 br i +1 / / R (cid:1)(cid:0) B i br i / / R (cid:1) . We will use the notation f and f defined in Lemma 3.8 and φ , φ , ψ , and ψ defined in Lemma 3.5. Thereis a short exact sequence of ( R, R ) H q -modules which splits when forgetting the H q -actions involved: −→ E −→ T i T i +1 T i −→ E −→ which we write vertically in diagram (3.10) below. Most of the maps in the complexes below transform a B i or B i +1 into an R (via br i or br i +1 ) and act as the identity on the other tensor factors. We use the followingshorthand notation. A “ + ” indicates that the coefficient of such a map is and a “ − ” indicates that thecoefficient of such a map is − . For example, the negative sign below stands for − := − br i ⊗ Id ⊗ Id : B i ⊗ R ⊗ B i −→ R ⊗ R ⊗ B i . eidemeisterIII 20 R ⊗ B i +1 ⊗ B i + / / − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ R ⊗ B i + % % ❑❑❑❑❑❑❑❑❑❑❑ E := (cid:15) (cid:15) B i,i +1 ,i φ ◦ + ◦ f / / + ◦ f ❧❧❧❧❧❧❧❧❧❧❧❧❧ + ◦ f ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ f (cid:15) (cid:15) B i −◦ ψ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ + ◦ ψ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ B i +1 ⊗ R + / / R Id (cid:15) (cid:15) B i ⊗ B i +1 ⊗ R − / / + ♠♠♠♠♠♠♠♠♠♠♠♠♠ Id ψ Id (cid:15) (cid:15) B i ⊗ R ⊗ R + sssssssssss Id Id Id (cid:15) (cid:15) R ⊗ B i +1 ⊗ B i + / / − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ R ⊗ B i + % % ▲▲▲▲▲▲▲▲▲▲▲ T i T i +1 T i = (cid:15) (cid:15) B i ⊗ B i +1 ⊗ B i + / / + ❧❧❧❧❧❧❧❧❧❧❧❧❧ + ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ f (cid:15) (cid:15) B i ⊗ R ⊗ B i − ♠♠♠♠♠♠♠♠♠♠♠♠♠ + ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ B i +1 ⊗ R + / / RB i ⊗ B i +1 ⊗ R − / / + ♠♠♠♠♠♠♠♠♠♠♠♠♠ (cid:16) φ (cid:17) (cid:15) (cid:15) B i ⊗ R ⊗ R + sssssssssss E := B x i + x i +1 i Id / / B x i + x i +1 i (3.10)All of the chain maps are annhilated by ∂ q and clearly E is homotopically equivalent to . Thus T i T i +1 T i is isomorphic to E in the relative homotopy category.There is an isomorphism of complexes E ∼ = E , via a basis change respecting the H q -structures, givenby the diagram R ⊗ B i +1 ⊗ B i / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ R ⊗ B i % % ❑❑❑❑❑❑❑❑❑❑❑ E = (cid:15) (cid:15) B i,i +1 ,i / / ♦♦♦♦♦♦♦♦♦♦♦ ' ' ❖❖❖❖❖❖❖❖❖❖❖ Id (cid:15) (cid:15) B i ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ B i +1 ⊗ R / / R Id (cid:15) (cid:15) B i ⊗ B i +1 ⊗ R / / ♠♠♠♠♠♠♠♠♠♠♠♠♠ Id Id Id (cid:15) (cid:15) B i ⊗ R ⊗ R sssssssssss Id IdId Id (cid:15) (cid:15) R ⊗ B i +1 ⊗ B i + / / − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ R ⊗ B i + % % ❑❑❑❑❑❑❑❑❑❑❑ E := B i,i +1 ,iφ ◦ (1 ⊗ br ⊗ ◦ f / / ( br ⊗ ⊗ ◦ f ♦♦♦♦♦♦♦♦♦♦♦ (1 ⊗ ⊗ br ) ◦ f ' ' ❖❖❖❖❖❖❖❖❖❖❖ B i (1 ⊗ br ) ◦ ψ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ B i +1 ⊗ R + / / RB i ⊗ B i +1 ⊗ R − / / + ♠♠♠♠♠♠♠♠♠♠♠♠♠ − = = ③③③③③③③③③③③③③③③③③③③③ B i ⊗ R ⊗ R There is a short exact sequence of complexes of ( R, R ) -bimodules compatible with the H q -structure whichsplits when forgetting the H q -structure E := (cid:15) (cid:15) B i Id / / (cid:15) (cid:15) B i (cid:15) (cid:15) R ⊗ B i +1 ⊗ B i / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ R ⊗ B i % % ❑❑❑❑❑❑❑❑❑❑❑ E = (cid:15) (cid:15) B i,i +1 ,i / / ♦♦♦♦♦♦♦♦♦♦♦ ' ' ❖❖❖❖❖❖❖❖❖❖❖ Id (cid:15) (cid:15) B i ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ R ⊗ B i +1 ⊗ R / / R Id (cid:15) (cid:15) B i ⊗ B i +1 ⊗ R / / ♠♠♠♠♠♠♠♠♠♠♠♠♠ = = ③③③③③③③③③③③③③③③③③③③③ Id 0 00 0 Id (cid:15) (cid:15) B i ⊗ R ⊗ R Id 0 00 Id 0 (cid:15) (cid:15) B i +1 ⊗ B i br i +1 ⊗ Id / / − Id ⊗ br i ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ B i br i % % ❑❑❑❑❑❑❑❑❑❑❑❑ E := B i,i +1 ,i ( br i ⊗ Id ⊗ Id) ◦ f ♦♦♦♦♦♦♦♦♦♦♦ (Id ⊗ Id ⊗ br i ) ◦ f ' ' ❖❖❖❖❖❖❖❖❖❖❖ RB i ⊗ B i +1 br i ⊗ Id / / − Id ⊗ br i +1 = = ③③③③③③③③③③③③③③③③③③③③ B i +1 br i +1 sssssssssss Since E is contractible, T i T i +1 T i ∼ = E . In the same exact way, one deduces that T i +1 T i T i +1 ∼ = E , whichproves the proposition. In this section we categorify the HOMFLYPT polynomial of any link using analogous arguments from[Cau17], [RW20] and [Rou17] adapted to the p -DG setting.For the next definition, we will allow complexes of Soergel bimodules to sit in half-integer degrees inthe Hochschild ( a ) and topological ( t ) degrees when considering the usual complexes of vector spaces. Wethen modify the elementary braiding complexes of equation (3.4) to be T i := ( at ) − q − (cid:16) tB i br i −−→ R (cid:17) , T ′ i := ( at ) q (cid:16) R rb i −−→ q − t − B − ( x i + x i +1 ) i (cid:17) . (4.1)Let β ∈ Br n be a braid group element on n strands. By Theorem 3.4, there is a chain complex of ( R n , R n ) H q -bimodules T β , well defined up to homotopy, associated with β . Then we write T β = (cid:16) · · · d −→ T i +1 β d −→ T iβ d −→ T i − β d −→ . . . (cid:17) . (4.2) Definition 4.1.
The untwisted H q -HOMFLYPT homology of β is the object [ HHH ∂ q ( β ) := a − n t n H • (cid:16) . . . −→ HH ∂ q • ( T i +1 β ) d t −→ HH ∂ q • ( T iβ ) d t −→ HH ∂ q • ( T i − β ) −→ . . . (cid:17) oublygradedtheory 22 in the category of triply graded H q -modules, where d t := HH ∂ q • ( d ) is the induced map of d on Hochschildhomology.By construction, the space [ HHH ∂ q ( β ) is triply graded by the topological ( t ) degree, the Hochschild ( a )degree as well as the quantum ( q ) degree. When necessary to emphasize each graded homogeneous pieceof the space, we will write [ HHH ∂ q i,j,k ( β ) to denote the homogeneous component concentrated in t -degree i , a -degree j and q -degree k .The following theorem is a particular case of the main result of [KR16], where we have only kept trackof the degree two p -nilpotent differential in finite characteristic p . The detailed verification given below,however, uses the main ideas of [Rou17] and differs from that of [KR16]. This proof serves as the model forthe other link homology theories in this paper. Theorem 4.2.
The untwisted H q -HOMFLYPT homology of β depends only on the braid closure of β as aframed link in R .As a convention for the framing number of braid closure, if a strand for a component of link is alteredas in the left of (4.3), then we say that the framing of the component is increased by (with respect to theblackboard framing). If a strand for a component of link is altered as in the right of (4.3), then we say thatthe framing of the component is decreased by . (4.3)Denote by f i ( L ) the framing number of the i th strand of a link L . Then, under the two Reidemeister movesof (4.3), f i ( L ) adds or subtracts respectively when changing from the corresponding left local picture tothe right local picture.Our main goal in this section is to establish this result. Due to Theorem 3.4, the proof reduces to showingthe invariance under the two Markov moves. We next seek to define the analogue of [ HHH in the category of p -complexes. This construction will serve asa precursor to the finite-dimensional sl -homology theory defined in the next section.We first would like to define pT β to be the p -complex of Soergel bimodules associated with β by pT β := P ( T β ) . (4.4)In other words, pT β should be a p -complex of the form pT β = (cid:16) · · · −→ T kβ d −→ T k − β = · · · = T k − β d −→ T k − β −→ · · · (cid:17) , where every term in odd topological degree is repeated p − times. We will denote the boundary maps inthe p -extended complex pT β by ∂ , in contrast to the usual topological differential d . Remark 4.3 (Half grading shifts for p -complexes) . Here, we point out that, unlike in the ordinary homotopycategory of complexes, we do not need to formally introduce half grading shifts in C ( k , ∂ ) in odd primecharacteristic. For instance, one may set a [ ] := a p +12 [ − a∂ , t [ ] := t p +12 [ − t∂ . (4.5)Using [ − a∂ ◦ [ − a∂ = [ − a∂ = a − p , one sees that, as functors, a [ ] ◦ a [ ] = a . Likewise, t [ ] ◦ t [ ] = t . oublygradedtheory 23 The same half grading shift functors can also be interpreted as a [ ] := a − p [1] a∂ , t [ ] := t − p [1] t∂ . (4.6)The two seemingly different definitions actually agree, as both are given by taking tensor product with the p -complex U p − { p − } in the a or t direction.However, the p -extension functor P does not intertwine between the ordinary half graded complexesand p -complexes, since, if we were to set P ( a ) = a [ ] , then we would have P ([1] ad ) = P ( a ) = P ( a ◦ a ) = a [ ] ◦ a [ ] = a = [1] a∂ . Thus we do not naively p -extend the braiding complexes (4.1) via P .We emphasize that the proof of invariance under both Markov II moves of our theory, forces us tocollapse the a and t gradings in this construction. The resulting homology theory will be doubly graded.More specifically, let us first collapse the a and t gradings in HHH into a single grading satisfying a = q t ,then we will categorically specialize t = [1] td into [1] t∂ by p -extension. We then lose the a -grading below,which is determined by the t -grading and q -grading. The t -degree remains independent of the q -degree onSoergel bimodules.We then define pT i := q − (cid:16) B i br i −−→ R [ − t∂ (cid:17) , pT ′ i := q (cid:16) R [1] t∂ rb i −−→ q − B − ( x i + x i +1 ) i (cid:17) . (4.7)Since pT i and pT ′ i are, up to t -grading shifts, obtained by applying P to T i and T ′ i , it follows that they satisfythe same braid relations, and can be used to define pT β similarly as done in Theorem 3.4. Definition 4.4.
Let β ∈ Br n be a braid group element written as a product σ ǫ i i · · · σ ǫ k i k in the elementarygenerators, where ǫ i ∈ {∅ , ′} . We assign to β the p -chain complex of ( R n , R n ) H q -bimodules pT β := pT ǫ i ⊗ R · · · ⊗ R pT ǫ k i k . (4.8)The boundary maps in the t -direction will be denoted ∂ for any pT β . Definition 4.5.
The untwisted H q -HOMFLYPT p -homology of β is the object p [ HHH ∂ q ( β ) := q − n H / • (cid:16) . . . −→ p HH ∂ q • ( pT i +1 β ) ∂ t −→ p HH ∂ q • ( pT iβ ) ∂ t −→ p HH ∂ q • ( pT i − β ) −→ . . . (cid:17) in the homotopy category of bigraded H q -modules. Here ∂ t stands for the induced map of the topologicaldifferentials on p -Hochschild homology groups ∂ t := p HH ∂ q • ( ∂ ) .In the definition of the H q -HOMFLYPT p -homology, we have applied the p -extensions in both the topo-logical and the Hochschild directions so that they can be collapsed into a single degree. The reason willbecome clearer later when categorifying the Jones polynomial at roots of unity. Therefore, in contrast to [ HHH( β ) , p [ HHH( β ) is only doubly graded, and we will adopt the notation p [ HHH i,j ( β ) as above to stand forits homogeneous components in topological degree i and q degree j . Further, the overall grading shift inthe definition will be utilized in the invariance under the Markov II moves below. Theorem 4.6.
The untwisted H q -HOMFLYPT p -homology of β depends only on the braid closure of β as aframed link in R .The proof of Theorems 4.2 and 4.6 will occupy the next few subsections, after we introduce the H q -equivariant ( p -)Koszul resolutions. oszulcomplexes 24 We recall here the Koszul resolution C n of the polynomial algebra R n as well as its adaption in the p -DGsetting. In the one variable case, we have a short exact sequence of k [ x ] -bimodules: −→ q k [ x ] ⊗ k [ x ] x ⊗ − ⊗ x −−−−−−→ k [ x ] ⊗ k [ x ] m −→ k [ x ] −→ . (4.9)In order to make the maps H q -equivariant, we twist the H q -action on the leftmost bimodule: −→ q k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x −−−−−−→ k [ x ] ⊗ k [ x ] m −→ k [ x ] −→ . (4.10)In the usual homotopy category of ( k [ x ] , k [ x ]) H q -modules, we have then an H q -equivariant relativereplacement of k [ x ] 0 −→ q a k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x −−−−−−→ k [ x ] ⊗ k [ x ] −→ (4.11)where we have inserted a in the leftmost non-zero term to emphasize the homological degree that it sits in.Denote by ( C , d ) this H q -equivariant Koszul resolution in the one-variable case.In the p -homotopy category, a relative replacement is then obtained by applying the p -extension functor P to C : −→ q a p − k [ x ] x ⊗ k [ x ] x = · · · = q a k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x −−−−−−→ k [ x ] ⊗ k [ x ] −→ (4.12)We abbreviate this p -resolution by pC .For the ease of notation, it will be useful to denote both replacements by −→ q k [ x ] x ⊗ k [ x ] x [1] a ∗ x ⊗ − ⊗ x −−−−−−→ k [ x ] ⊗ k [ x ] −→ where ∗ ∈ { d, ∂ } will label [1] as either the usual homological or p -homological shift in a -degree.For the polynomial ring R n = k [ x , . . . , x n ] , one takes the n -fold tensor product over k of the bimoduleresolution C of k [ x ] to get ( C n = C ⊗ n , d ) . We also define ( pC n , ∂ ) := pC ⊗ k · · · ⊗ k pC (4.13)as the n -fold tensor product of the one-variable resolution. Note that pC n is homotopic to, but bigger than,the p -complex of bimodules P ( C n ) . Example 4.7.
In characteristic , the -complex C is the total complex of the cube k [ x, y ] x ⊗ k [ x, y ] x = / / k [ x, y ] x ⊗ k [ x, y ] x x ⊗ − ⊗ x / / k [ x, y ] ⊗ k [ x, y ] k [ x, y ] x + y ⊗ k [ x, y ] x + y = / / y ⊗ − ⊗ y O O k [ x, y ] x + y ⊗ k [ x, y ] x + y x ⊗ − ⊗ x / / y ⊗ − ⊗ y O O k [ x, y ] y ⊗ k [ x, y ] yy ⊗ − ⊗ y O O k [ x, y ] x + y ⊗ k [ x, y ] x + y = / / = O O k [ x, y ] x + y ⊗ k [ x, y ] x + y x ⊗ − ⊗ x / / = O O k [ x, y ] y ⊗ k [ x, y ] y = O O Under the total p -differential, the copy of k [ x, y ] x + y ⊗ k [ x, y ] x + y sitting in the southwest corner generates anacyclic -subcomplex, modulo which one obtains the total -complex of P ( C ) : k [ x, y ] x ⊗ k [ x, y ] x = / / k [ x, y ] x ⊗ k [ x, y ] x x ⊗ − ⊗ x ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ k [ x, y ] x + y ⊗ k [ x, y ] x + y − y ⊗ ⊗ y ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ x ⊗ − ⊗ x ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ k [ x, y ] ⊗ k [ x, y ] k [ x, y ] y ⊗ k [ x, y ] y = / / k [ x, y ] y ⊗ k [ x, y ] y y ⊗ − ⊗ y ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ arkovI 25 Using this Koszul resolution, one immediately obtains, via the following result, that the relative p -Hochschild homology is completely determined by the classical relative Hochschild homology. Therefore,it may appear that one does not gain any more information by introducing p HHH ∂ • . However, this con-struction is essential for the collapsed homology theory to be defined in the next section. Proposition 4.8.
Let M be an ( R n , R n ) H q -bimodule. Then p HH ∂ q • ( M ) is determined by HH ∂ q • ( M ) by p HH ∂ q i ( M ) = ( HH ∂ q k − ( M ) ( k − p + 1 ≤ i ≤ kp − ∂ q k ( M ) i = kp (4.14) Proof.
Using the Koszul resolution, we have p HH ∂ • ( M ) = H / • ( M ⊗ ( R n ,R n ) P ( C n )) = H / • ( P ( M ⊗ ( R n ,R n ) C n )) = P (H • ( M ⊗ ( R n ,R n ) C n )) . The result follows.
Remark 4.9.
The result is true in more generality. If A is an (ungraded) algebra equipped with the zero p -differential, then its p -Hochschild homology is entirely determined by its usual Hochschild homology.This result is essentially due to Spanier [Spa49], but is also proved in more generality by [KW98].Indeed, if P • −→ A is any projective resolution of A over A ⊗ A op , then P ( P • ) provides a p -resolution of A , and p HH • ( A ) = H / • ( P ( P • ) ⊗ A ⊗ A op A ) = H / • ( P ( P • ⊗ A ⊗ A op A )) . (4.15)When computing the last slash homology, one may safely forget about the module structures involved in P ( P • ⊗ A ⊗ A op A ) and think of it as a direct sum, possibly infinite copies, of chain-complexes of the form −→ k −→ , −→ k −→ k −→ , (4.16)where the underlined term sits in some homological degree i . Under the p -extension functor, the last com-plex extends to a contractible p -complex, while the first complex becomes −→ k −→ , if i is even, or the ( p − -dimensional −→ k = · · · = k −→ if i is odd. The slash homology computation then follows. The usual HOMFLYPT homologies of two braid compositions β β and β β are isomorphic due to thetrace-like property of the usual Hochschild homology functor. The relative Hochschild homology alsoremembers the H q -action. Proposition 4.10.
Let β and β be two braids on n strands. Then [ HHH ∂ q ( β β ) ∼ = [ HHH ∂ q ( β β ) .The same property also holds for the HOMFLYPT pH q -homology groups. Proposition 4.11.
Let β and β be two braids on n strands. Then p [ HHH ∂ q ( β β ) ∼ = p [ HHH ∂ q ( β β ) . Proof.
This follows from Proposition 2.21, since we have the functorial isomorphism p HH ∂ q • ( pT iβ ⊗ R n pT iβ ) ∼ = p HH ∂ q • ( pT iβ ⊗ R n pT iβ ) for all i ∈ Z . Alternatively, this follows from combining the previous result with Proposition 4.8. See the remark below for more explanation of the slash homology computation. arkovII 26
In order to prove the second Markov move, one needs to show that for a (complex of) Soergel bimodules M over the polynomial p -DG algebra R n , that HOMFLYPT ( p ) H q -homologies of the bimodules (4.17) areisomorphic (up to shifts and twists). M · · ·· · · M · · ·· · · (4.17)Let Λ h x n +1 i be the exterior algebra in the variable x n +1 . Recall that R n = k [ x , . . . , x n ] and let M ∈ ( R n , R n ) H q - mod . Set C ′ = k [ x n +1 ] ⊗ Λ h x n +1 i ⊗ k [ x n +1 ] ∼ = C . Letting C n denote the Koszul resolu-tion of R n , we have inductively that C n +1 = C n ⊗ C ′ .As in the proof of Theorem 2.15, the Hochschild homology of M is depicted by the closure diagram MC C · · ·· · · , where the single strands connecting the boxes indicate tensor products over the one-variable polynomialrings labelling those strands.The proof of second Markov move essentially reduces to a computation of the partial Hochschild ho-mology with respect to the last variable x n +1 . This operation is diagramatically represented in (4.18). · · ·· · · · · ·· · · (4.18)This leads to an analysis of C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T n in Proposition 4.12.The following technical result will be the heart of establishing the invariance under the Markov II moves.We start with the usual HOMFLYPT homology case, under the Hopf algebra H q -action. Proposition 4.12.
Let β be a braid with n strands which is assigned a usual complex of Soergel bimodules M . Then there is an H q -equivariant isomorphism of the HOMFLYPT homology groups(i) [ HHH ∂ q (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) ∼ = [ HHH ∂ q ( M ) x n ,(ii) [ HHH ∂ q (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T ′ n ) ∼ = [ HHH ∂ q ( M ) − x n ,where we [ HHH ∂ q ( M ) ± x n denotes a twisting in the H q -action. arkovII 27 Proof.
Both identities are proved in a similar way. For the first statement, we note that by definition HH ∂ q • (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) = H v • ( C n +1 ⊗ ( R n +1 ,R n +1 ) (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n )) . where the (vertical) homology H v • above is taken with respect to the differential coming from the Koszulcomplex C n +1 .Note that C n +1 ⊗ ( R n +1 ,R n +1 ) (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) = ( C n ⊗ C ′ ) ⊗ ( R n +1 ,R n +1 ) (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) ∼ = C n ⊗ ( R n ,R n ) ( M ⊗ R n ( C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T n )) . (4.19)These isomorphisms, in terms of diagrammatics, can be interpreted as taking closures of the followingdiagrammatic equalities: MC n +1 · · ·· · · = MC n C ′ · · ·· · · = MC n C ′ · · ·· · · . Observe that C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T n is a bicomplex of ( R n +1 , R n +1 ) -bimodules a t (cid:0) x n +1 B x n +1 n (cid:1) br / / x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) a t − R x n +1 n +10 (cid:15) (cid:15) a − t q − B n br / / a − t − q − R n +1 . Here the grading shift conventions follow from equation (3.4). For ease of notation, we will mostly ignorethem within this proof below.It follows that there is a short exact sequence of bicomplexes of ( R n +1 , R n +1 ) -bimodules −→ Y −→ C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T n −→ Y −→ where the terms of the sequence are defined by arkovII 28 (cid:15) (cid:15) R x n +3 x n +1 n +1( x n +1 − x n ) ⊗ ⊗ ( x n +1 − x n ) (cid:29) (cid:29) x n +1 − x n ) / / (cid:15) (cid:15) R x n +1 n +1 (cid:15) (cid:15) Id (cid:2) (cid:2) := Y (cid:15) (cid:15) / / x n +1 B x n +1 n br / / x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) ˜ br ! ! R x n +1 n +10 (cid:15) (cid:15) = C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T n (cid:15) (cid:15) B n br / / R n +1 2Id (cid:1) (cid:1) e R x n + x n +1 n +1 / / ( x n +1 − x n ) ⊗ − ⊗ ( x n +1 − x n ) (cid:15) (cid:15) (cid:15) (cid:15) := Y (cid:15) (cid:15) B n br / / R n +1 (4.20)where e R n +1 is equal to R n +1 as a left R n +1 -module but the right action of R n +1 is twisted by the permuta-tion σ n ∈ S n +1 and ˜ br ( a ⊗ b ) = br ( aσ n ( b )) . It is a straightforward exercise to check that all maps above areequivariant with respect to the H q -action. We show it for the map φ := ( x n +1 − x n ) ⊗ ⊗ ( x n +1 − x n ) : R x n +3 x n +1 n +1 −→ x n +1 B x n +1 n . One calculates φ ( ∂ q (1)) = φ ( x n + x n +1 + 2 x n +1 )= ( x n +1 − x n ) ⊗ ⊗ ( x n +1 − x n ) + 2 x n +1 ( x n +1 − x n ) ⊗ x n +1 ⊗ ( x n +1 − x n ) , and ∂ q ( φ (1)) = ∂ (( x n +1 − x n ) ⊗ ⊗ ( x n +1 − x n ))= ( x n +1 − x n ) ⊗ ⊗ ( x n +1 − x n ) + x n +1 ( x n +1 − x n ) ⊗ x n +1 − x n ) ⊗ x n +1 + x n +1 ⊗ ( x n +1 − x n ) + 1 ⊗ ( x n +1 − x n ) x n +1 . Comparing the terms of φ ( ∂ q (1)) and ∂ q ( φ (1)) it suffices to check the identity inside the bimodule B n x n +1 ⊗ − x n +1 ⊗ x n = 1 ⊗ x n +1 − x n ⊗ x n +1 . Rearranging the terms of the above equation, we must show x n +1 ⊗ x n ⊗ x n +1 = 1 ⊗ x n +1 + x n +1 ⊗ x n . (4.21)Adding x n x n +1 ⊗ to both sides of (4.21) and using the fact that symmetric functions in x n and x n +1 maybe brought through a tensor product finishes the proof that φ ( ∂ q (1)) = ∂ q ( φ (1)) . arkovII 29 There is a splitting of the short exact sequence (4.20) regarded as a short exact sequence of ( R n , R n ) -bimodules, given by x n +1 B x n +1 n / / (cid:15) (cid:15) R x n +1 n +1 (cid:15) (cid:15) B n / / R n +1 ˜ R x n + x n +1 n +1 / / (cid:15) (cid:15) θ = = (cid:15) (cid:15) B n / / Id = = R n +1 Id [ [ (4.22)where θ ( f ( x , . . . , x n − ) x in x jn +1 ) = f ( x , . . . , x n − ) x in ⊗ x jn . We briefly explain why θ is a well-defined bimodule homomorphism. By definition θ ( x in x jn +1 ) = x in ⊗ x jn .Note that θ ( x in x jn +1 ) = θ ( x in · x jn +1 ) = x in θ ( x jn +1 ) = x in ⊗ x jn where we viewed x in as acting on the left of x jn +1 . Similarly, θ ( x in x jn +1 ) = θ ( x in · x jn ) = θ ( x in ) x jn = x in ⊗ x jn where we viewed x jn as acting on the right of x in . The short exact sequence (4.20) plugged back into (4.19) gives us a short exact sequence −→ C n ⊗ ( R n ,R n ) ( M ⊗ R n Y ) −→ ( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n −→ C n ⊗ ( R n ,R n ) ( M ⊗ R n Y ) −→ , (4.23)which is split as a sequence of ( R n , R n ) -bimodules. Taking homology with respect to the vertical differen-tials gives rise to a long exact sequence · · · → H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) → HH ∂ q i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) → H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) → · · · . Due to the splitting exactness of (4.23), the long exact sequence breaks up into short exact sequences of theform → H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) → HH ∂ q i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) → H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) → , (4.24)one for each i ∈ Z .So far we have ignored the topological (horizontal) differential in diagram (4.20). Each term in the shortexact sequence (4.24) carries the topological differential d t . Taking homology with respect to d t gives usanother long exact sequence · · · / / H hj H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) / / [ HHH ∂ q j,i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) EDBCGF@A / / H hj H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) / / H hj − H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) / / · · · . (4.25)We claim that H hj (H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y ))) = 0 for all j , (which we will show shortly), which impliesthat H hj H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) ∼ = [ HHH ∂ q j,i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) . The definition (4.20) of Y shows that H hj H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) is the homology of the two-termcomplex / / H vi ( C n ⊗ ( R n ,R n ) M ⊗ R n R x n +3 x n +1 n +1 ) x n +1 − x n ) / / H vi ( C n ⊗ ( R n ,R n ) M ⊗ R n R x n +1 n +1 ) / / . arkovII 30 Taking grading shifts back into account, this complex has homology concentrated in the second non-zeroterm, which is isomorphic to H hj H vi ( C n ⊗ ( R n ,R n ) M ⊗ R n R x n n ) ∼ = [ HHH ∂ q j,i ( M ) x n where the latter space is twisted as an H q -module by x n . This is part ( i ) of the proposition.We now prove the claim that H hj H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) = 0 . Note that Y fits into a short exactsequence of ( R n +1 , R n +1 ) -bimodules −→ Y ′′ −→ Y ψ −→ Y ′ −→ , where the surjective map ψ is given by e R x n + x n +1 n +1 / / (cid:15) (cid:15) (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) ψ : Y = := Y ′ B n / / br R n +1 Id R n +1 Id / / R n +1 . (4.26)Since the kernel of the multiplication map br : B n → R n +1 is generated as an ( R n +1 , R n +1 ) -bimodule by v = x n +1 ⊗ − ⊗ x n +1 , it is easy to check that x n v = vx n +1 and v generates a copy of bimodule isomorphicto e R x n + x n +1 n +1 . Thus the kernel of ψ is given by e R x n + x n +1 n +1 Id (cid:15) (cid:15) Y ′′ := e R x n + x n +1 n +1 . (4.27)Clearly, H v • ( Y ′′ ) = 0 , and it follows that H v • ( Y ) ∼ = H v • ( Y ′ ) . Taking homology with respect to the topological(horizontal) differential d t then yields H hj H vi ( Y ′ ) = 0 and H hj H vi ( Y ) = 0 .The computation of HH ∂ q • (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T ′ n ) is very similar. We only outline the necessary changes.Again, first note that C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T ′ n is a bicomplex of ( R n +1 , R n +1 ) -bimodules a t q R x n +1 n +1 rb / / (cid:15) (cid:15) a t − x n +1 B − x n nx n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) a t q R n +1 rb / / a t − B − ( x n + x n +1 ) n . (4.28)Ignore the grading shifts for now for ease of notation. There is a short exact sequence of bicomplexes of ( R n +1 , R n +1 ) -bimodules −→ Z −→ C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T ′ n −→ Z −→ , arkovII 31 whose terms are defined by (cid:15) (cid:15) R x n +1 n +1Id (cid:28) (cid:28) Id / / (cid:15) (cid:15) R x n +1 n +1 (cid:15) (cid:15) rb (cid:1) (cid:1) := Z (cid:15) (cid:15) / / R x n +1 n +1 rb / / (cid:15) (cid:15) x n +1 B − x n nx n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) ˜ br } } = C ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) T ′ n (cid:15) (cid:15) R n +1 rb / / Id B − ( x n + x n +1 ) n Id (cid:3) (cid:3) / / (cid:15) (cid:15) ˜ R n +1 x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) := Z (cid:15) (cid:15) R n +1 rb / / B − ( x n + x n +1 ) n (4.29)As in the previous part, there is a splitting of bicomplexes of ( R n , R n ) -bimodules given by R x n +1 n +1 rb / / (cid:15) (cid:15) x n +1 B − x n nx n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) R n +1 rb / / B − ( x n + x n +1 ) n / / (cid:15) (cid:15) ˜ R n +1 x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) φ ` ` R n +1 rb / / Id > > B − ( x n + x n +1 ) n Id a a (4.30)where φ was defined earlier. Thus we get short exact sequences of the form → H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Z )) → HH ∂ q i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T ′ n ) → H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Z )) → (4.31)for each i ∈ Z . Taking horizontal homology for this short exact sequence gives us a long exact sequence.However, since the homology of H hj H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Z )) is clearly always zero, we get that [ HHH j,i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T ′ n ) ∼ = H hj (H vi ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Z ))) arkovII 32 for all j ∈ Z . We need to analyze the latter homology space.There is a morphism of bicomplexes / / (cid:15) (cid:15) ˜ R n +1 x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) / / (cid:15) (cid:15) (cid:15) (cid:15) Z := := Z ′ R n +1 rb / / Id B − ( x n + x n +1 ) n br R n +1 x n +1 − x n / / R − ( x n + x n +1 ) n +1 (4.32)whose kernel is a vertical complex connected by the identity map. Thus [ HHH j,i (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T ′ n ) ∼ = H hj H vi (( M ⊗ k [ x n +1 ]) ⊗ Z ′ ) ∼ = [ HHH j,i ( M ) − x n . The result follows.The above proof serves as a model for the Markov II invariance of p [ HHH . We only present the necessarychanges.
Corollary 4.13.
Let β be a braid and M be the associated p -chain complex of Soergel bimodules. Then thereare isomorphisms of chain complexes of relative pH q -Hochschild homology groups:(i) p [ HHH ∂ q (( M ⊗ k [ x n +1 ]) ⊗ R n +1 pT n ) ∼ = p [ HHH ∂ q ( M ) x n ,(ii) p [ HHH ∂ q (( M ⊗ k [ x n +1 ]) ⊗ R n +1 pT ′ n ) ∼ = p [ HHH ∂ q ( M ) − x n . Proof.
To begin with, one replaces the Koszul complex C n utilized in the proof of Proposition 4.12 with the p -extended Koszul complex pC n . Also one needs to replace the vertical homology taken there by verticalslash homology (see Remark 4.12).For part ( i ) , one adapts equation (4.19) into pC n +1 ⊗ ( R n +1 ,R n +1 ) (( M ⊗ k [ x n +1 ]) ⊗ R n +1 pT n ) = ( pC n ⊗ pC ′ ) ⊗ ( R n +1 ,R n +1 ) (( M ⊗ k [ x n +1 ]) ⊗ R n +1 pT n ) ∼ = pC n ⊗ ( R n ,R n ) ( M ⊗ R n ( pC ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) pT n )) , (4.33)where pC ′ ∼ = pC is the p -extended complex of bimodules −→ q k [ x n +1 ] x n +1 ⊗ k [ x n +1 ] x n +1 [1] t∂ x n +1 ⊗ − ⊗ x n +1 −−−−−−−−−−−→ k [ x n +1 ] ⊗ k [ x n +1 ] −→ . (4.34) The monoidality of pC n is more convenient here than the homotopy-equivalent P ( C n ) . arkovII 33 Then, diagram (4.20) becomes the following p -extended version in the Hochschild direction: (cid:15) (cid:15) q R x n +3 x n +1 n +1 [1] t∂φ & & x n +1 − x n ) / / (cid:15) (cid:15) qR x n +1 n +1 (cid:15) (cid:15) Id (cid:2) (cid:2) := pY (cid:15) (cid:15) / / q ( x n +1 B x n +1 n )[1] t∂ br / / x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) ˜ br % % qR x n +1 n +10 (cid:15) (cid:15) = pC ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) pT n (cid:15) (cid:15) q − B n br / / % % q − R n +1 [ − t∂ (cid:2) (cid:2) q ˜ R x n + x n +1 n +1 [1] t∂ / / ( x n +1 − x n ) ⊗ − ⊗ ( x n +1 − x n ) (cid:15) (cid:15) (cid:15) (cid:15) := pY (cid:15) (cid:15) q − B n br / / q − R n +1 [ − t∂ . (4.35)Here φ is the map that sends to ( x n +1 − x n ) ⊗ ⊗ ( x n +1 − x n ) Now, as in the previous proof, one needs to show that pC n ⊗ ( R n ,R n ) ( M ⊗ R n pY ) does not contribute to p [ HHH ∂ q . This is easier since now pY is an acyclic p -complex. Furthermore, pY is quasi-isomorphic to the p -complex qR x n n sitting in t -degree zero. Hence we obtain the isomorphism p [ HHH ∂ q (( M ⊗ k [ x n +1 ]) ⊗ R n +1 pT n ) ∼ = q − n H / • ( pC n ⊗ ( R n ,R n ) ( M ⊗ R n R x n n )) ∼ = p [ HHH ∂ q ( M ) x n . For the second isomorphism, again there is a short exact sequence of bicomplexes of ( R n +1 , R n +1 ) - arkovII 34 bimodules (cid:15) (cid:15) q R x n +1 n +1 [2] t∂ Id (cid:28) (cid:28) Id / / (cid:15) (cid:15) q R x n +1 n +1 [1] t∂ (cid:15) (cid:15) rb x x := pZ (cid:15) (cid:15) / / q R x n +1 n +1 [2] t∂ rb / / (cid:15) (cid:15) q ( x n +1 B − x n n )[1] t∂x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) ˜ br y y = pC ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) pT ′ n (cid:15) (cid:15) q R n +1 [1] t∂ rb / / Id (cid:28) (cid:28) qB − ( x n + x n +1 ) n Id z z / / (cid:15) (cid:15) q ˜ R n +1 [1] t∂x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) := pZ (cid:15) (cid:15) q R n +1 [1] t∂ rb / / qB − ( x n + x n +1 ) n (4.36)Getting rid of contractible summands, we see that pZ is homotopy equivalent to q R n +1 [1] t∂ x n +1 − x n −−−−−−→ qR − ( x n + x n +1 ) n +1 , (4.37)which is, in turn, quasi-isomorphic to qR − x n n . The result follows after accounting for the shift built into p [ HHH ∂ q . Remark 4.14.
A closer examination of the proof of Corollary 4.13 shows the necessity of collapsing t and a gradings for the construction of p [ HHH ∂ q . Indeed, a comparison between equations (4.20) and (4.35), (4.29)and (4.36) shows that, if there were an a and t bigrading as for [ HHH , then the grading shifts arising from thepositive and negative Markov II moves would not match. This is caused by the fact that the homologicalshift [1] t∂ and grading shift t functors are different when p > , and thus ( t [ ] ) ◦ = t = [1] t∂ . Consequently, there does not seem to exist an overall compensation factor such as a − n t n in Definition 4.1that would make a triply graded p [ HHH invariant under both Markov II moves.
Theorem 4.15.
Let β and β be two braids whose closures represent the same link L of r components up toframing. Suppose the framing numbers of the closures b β of β and b β of β differ by f i ( b β ) − f i ( b β ) = a i , i = 1 , . . . , r . Then [ HHH ∂ q ( β ) ∼ = [ HHH ∂ q ( β ) P ri =1 a i x i and p [ HHH ∂ q ( β ) ∼ = p [ HHH ∂ q ( β ) P ri =1 a i x i where the generator of the polynomial action for the i th component is denoted x i and [ HHH ∂ ( β ) x i meanswe twist the H q -module structure on the i th component by x i . nlinksandtwistings 35 Proof.
The topological invariance follows from the proof of the braid relations in Section 3 and the proof ofthe Markov moves.
In this section, we compute
HHH ∂ q and p HHH ∂ q for the identity element of the braid group Br n , and definean unframed link invariant in R .For the unknot, recall from the previous section the Koszul resolution C of k [ x ] , as a bimodule, is givenby q a k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x / / k [ x ] ⊗ k [ x ] . Tensoring this complex with k [ x ] as a bimodule yields q a k [ x ] x / / k [ x ] . Thus the homology of the unknot (up to shift) is identified with the bigraded H q -module k [ x ] ⊕ q a k [ x ] x . More generally, via the Koszul complex C n = C ⊗ n , we have that the homology of the n -componentunlink L is equal to [ HHH ∂ q ( L ) ∼ = a − n t n HH • ( R n ) ∼ = a − n t n n O i =1 (cid:0) k [ x i ] ⊕ q a k [ x i ] x i (cid:1) . (4.38)Alternatively, up to the grading shift a − n t n , we may identify [ HHH ∂ q ( L ) with the exterior algebra over R n generated by the differential forms dx i of bidegree aq , i = 1 , . . . , n , subject to the condition that each dx i accounts for a twisting of H q -module structure by x i .It follows that, as for the ordinary HOMFLYPT homology, given a framed link L of n components arisingas a braid closure b β , its untwisted HOMFLYPT H q -homology [ HHH ∂ q ( β ) is a module over [ HHH ∂ q , , • ( L ) ∼ = R n , and thus one may consider a twisting of the H q -module structure on [ HHH ∂ q ( β ) via the functor R fn ⊗ R n ( - ) ,where f is a linear polynomial in x , . . . , x n , (see Section 3.1). Definition 4.16.
Let L be an n -strand framed link arising from the closure of a braid β . Label the compo-nents of L by through k , and set the (linear) framing factor of β to be the linear polynomial f β = − k X i =1 f i x i . (1) The H q -HOMFLYPT homology of β is the triply graded H q -module HHH ∂ q ( β ) := [ HHH ∂ q ( β ) f β ∼ = R f β k ⊗ R k [ HHH ∂ q ( β ) . (2) Likewise, the H q -HOMFLYPT p -homology is the doubly graded H q -module p HHH ∂ q ( β ) := p [ HHH ∂ q ( β ) f β ∼ = R f β k ⊗ R k p [ HHH ∂ q ( β ) . Corollary 4.17.
Given a braid β , both HHH ∂ q ( β ) and p HHH ∂ q ( β ) are link invariants that only depend onthe closure of β as a link in R . (i) The slash homologies of HHH ∂ q ( β ) and p HHH ∂ q ( β ) are finite-dimensional.(ii) The Euler characteristic of HHH ∂ q ( β ) is equal to the HOMFLYPT polynomial of b β in the formal vari-ables q and a , while the Euler characteristic of p HHH ∂ q ( β ) is equal to the Jones polynomial of b β in aformal q -variable.(iii) The Euler characteristic of the slash homology of HHH ∂ q ( β ) is equal to the specialization of the HOM-FLYPT polynomial of b β at a root of unity q , while the Euler characteristic of the slash homology of p HHH ∂ q ( β ) is equal to the specialization of the Jones polynomial of b β at a root of unity q . Proof.
For the first statement, just notice that the twisting of the p -DG structure by the framing factor takescare of the Markov II move.Next, the finite-dimensionality of the homology theories follows, by construction, from the fact that f i B g i i ⊗ R · · · ⊗ R f ik B g ik i k is an H q -module with k -step filtration whose subquotients are isomorphic to R f as left R H q -modules, and thus Corollary 3.2 applies.The Poincar´e polynomial of HHH ∂ q ( β ) , which is independent of the H q -module structure on HHH ∂ q ( β ) ,is well-known to be a Laurent polynomial in a and t (i.e., in Z [ a ± , t ± ] ), and Laurent series in q (i.e., in Z [ q − , q ]] ). Specializing t = − recovers the HOMFLYPT polynomial (see, e.g., [KR08b, KR16]). On theother hand, in the construction of p HHH ∂ q ( β ) , we have categorically specialized the a and t grading shiftsaccording to the relation a = q t , and then transformed t into [1] t∂ . The Grothendieck ring of t and q bigraded p -complexes up to homotopy is equal to O p ⊗ Z Z [ q, q − ] (c.f. Corollary 2.8). In this ring, [1] t∂ descends to − ∈ O p . In turn, the a variable is then evaluated at − q . Hence the Poincare polynomial of p HHH ∂ q ( β ) ,taking value in Z [ q − , q ]] ∼ = Z ⊗ Z Z [ q − , q ]] ⊂ O p ⊗ Z Z [ q − , q ]] , is equal to the HOMFLYPT polynomial with a = − q . This is just the Jones polynomial in a formal variable q , and the second part follows.Finally, taking slash homology of the homology theories has the effect, on the level of Grothendieckgroups, of passing from Z [ q − , q ] onto O p (here we need part (i) showing that both HHH ∂ q ( β ) and p HHH ∂ q ( β ) are quasi-isomorphic to finite-dimensional H q -modules). Therefore, taking slash homology of HHH ∂ q ( β ) and p HHH ∂ q ( β ) is equivalent to categorically specializing q at a primitive p th root of unity. This finishes theproof of the corollary. Remark 4.18.
One of the open problems in the triply graded Khovanov-Rozansky theory is whether thetheory is (projectively) functorial with respect to link cobordisms. A fundamental obstruction lies in thefact that the conditions of the (projective) TQFT would require one to assign, to the unknot, a Frobeniusalgebra that is finite-dimensional (or rather, a compact Frobenius algebra object in a triangulated category).Therefore, the slash homology of
HHH ∂ q ( β ) and p HHH ∂ q ( β ) serve as candidates of potentially functoriallink homology theories. sl -homology theory From Corollary 4.17, one sees that the slash homology of p HHH ∂ q ( β ) categorifies the Jones polynomial ata root of unity q . However, the specialized Jones polynomial, as an element of O p , sits inside the ambientring O p ⊗ Z O p ⊃ Z ⊗ Z O p ∼ = O p . Following [Cau17] (see also [RW20] and [QRS18]), we define a p -differential on p HH ∂ • ( β ) of a braid β as acategorically specialized homology theory of links. The slash homology of this theory bypasses the ambientring construction, and directly constructs a singly graded finite-dimensional homology theory whose Eulercharacteristic lives in O p . singlygradedhomology 37 Consider the H q -Koszul complex in one-variable: C : 0 −→ aq k [ x ] x ⊗ k [ x ] x d C −→ k [ x ] ⊗ k [ x ] −→ (5.1)where d C is the map d C ( f ) = ( x ⊗ ⊗ x ) f . We regard the differential on the arrow as an endomorphsimof the Koszul complex, of bidegree ( − , . Lemma 5.1.
The commutator of the endomorphisms d C and ∂ q ∈ H q is null-homotopic on the Koszulcomplex C . Proof.
The commutator map φ := [ d C , ∂ q ] is given by / / k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x / / φ (cid:15) (cid:15) k [ x ] ⊗ k [ x ] / / / / k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x / / k [ x ] ⊗ k [ x ] / / where φ sends the bimodule generator ⊗ ∈ k [ x ] x ⊗ k [ x ] x to φ (1 ⊗
1) = d C ( ∂ q (1 ⊗ − ∂ q d C (1 ⊗
1) = d C ( x ⊗ ⊗ x ) − ∂ q ( x ⊗ ⊗ x )= ( x ⊗ ⊗ x )( x ⊗ ⊗ x ) − x ⊗ ⊗ x )= ( x ⊗ ⊗ x )( x ⊗ ⊗ x ) − x ⊗ ⊗ x )( x ⊗ − x ⊗ x + 1 ⊗ x )= ( x ⊗ ⊗ x )( − x ⊗ x ⊗ x + 1 ⊗ x )= − ( x ⊗ ⊗ x )( x ⊗ − ⊗ x ) . We may thus choose a null-homotopy to be / / k [ x ] x ⊗ k [ x ] xh u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ x ⊗ − ⊗ x / / φ (cid:15) (cid:15) k [ x ] ⊗ k [ x ] / / / / k [ x ] x ⊗ k [ x ] x x ⊗ − ⊗ x / / k [ x ] ⊗ k [ x ] / / where h is given by multiplication by the element ⊗ x − x ⊗ , and acts on the rest of the complex byzero. The result follows.The Koszul complex C n inherits the endomorphism d C by forming the n -fold tensor product from theone-variable case. It follows, that for a given p -DG bimodule M over R n , there is an induced differential,still denoted d C , given via the identification HH ∂ q • ( M ) ∼ = H • ( M ⊗ ( R n ,R n ) C n ) , (5.2)where the induced differential acts on the right-hand side by Id M ⊗ d C . By construction, d C has Hochschilddegree − and q -degree .Lemma 2.4 immediately implies the following. Corollary 5.2.
The induced differential d C on HH ∂ q • ( M ) commutes with the H q -action. (cid:3) Remark 5.3.
The differential, first observed by Cautis [Cau17], has the following more algebro-geometricmeaning. Identifying HH ( R n ) as vector fields on Spec( R n ) = A n , HH ( R n ) acts as differential operatorson HH • ( M ) for any R n -bimodule M , regarded as a coherent sheaf on A n × A n ∼ = T ∗ ( A n ) . Under thisidentification, d C is given by, up to scaling by a nonzero number, contraction with the vector field ζ C := n X i =1 x i ∂∂x i . singlygradedhomology 38 On the other hand, this is also the vector field that defines the p -DG structure on R n by derivation. There-fore, via the Gerstenhaber module structure on HH • ( M ) , the two actions naturally commute with eachother on HH • ( M ) .In a more general context, Hochschild homology is a Gerstenhaber module over Hochschild cohomol-ogy viewed as a Gerstenhaber algebra. We may view d C and ∂ q as the same element φ in Hochschildcohomology but the element d C acts on homology via cap product φ ∩ • and the element ∂ q acts via a Liealgebra action L φ ( • ) . The compatibility of these actions is given by the equation φ ∩ L φ ( x ) = [ φ, φ ] ∩ x + L φ ( φ ∩ x ) . Since [ φ, φ ] = 0 , these actions commute.Now we are ready to introduce a collapsed p -homology theory of a braid closure. Let β ∈ Br n be an n -stranded braid. We have associated to β a usual chain complex of H q -equivariant Soergel bimodules T β as in equation (4.2), of which we take p HH ∂ q • for each term:... ... ... . . . ∂ t / / p HH ∂ q i ( pT k +1 β ) ∂ C O O ∂ t / / p HH ∂ q i ( pT kβ ) ∂ C O O ∂ t / / p HH ∂ q i ( pT k − β ) ∂ C O O ∂ t / / . . .. . . ∂ t / / p HH ∂ q i +1 ( pT k +1 β ) ∂ t / / ∂ C O O p HH ∂ q i +1 ( pT kβ ) ∂ C O O ∂ t / / p HH ∂ q i +1 ( pT k − β ) ∂ C O O ∂ t / / . . . ... ∂ C O O ... ∂ C O O ... ∂ C O O (5.3)Here, ∂ C is a p -differential arising from d C as follows. By Proposition 4.8, the p -Hochschild homologygroups in a column above are identified with the terms in · · · d C / / HH ∂ q k +1 ( pT kβ ) · · · HH ∂ q k +1 ( pT kβ ) d C / / HH ∂ q k ( pT kβ ) d C / / HH ∂ q k − ( pT kβ ) · · · , (5.4)where each term in odd Hochschild degree is repeated p − times. Here the horizontal differential isthe p -Hochschild induced map of the topological differential, which we have denoted by ∂ t to indicate itsorigin. On the arrows connecting even and odd Hochschild degree terms, we put the map d C while keepingthe repeated terms connected by identity maps. This defines a p -complex structure, denoted ∂ C , in eachcolumn in diagram (5.3). The p -differential ∂ C commutes with the H q -action on each term by Corollary 5.2.It follows that, applying the totalization construction T of Lemma 2.5, we obtain a bigraded p -bicomplexof H q -modules, with a horizontal (topological) p -differential ∂ t , a vertical p -differential ∂ C and internal p -differential ∂ q . Denote the total p -differential ∂ T := ∂ t + ∂ C + ∂ q , which collapses the double grading into asingle q -grading. Definition 5.4.
Let β be an n stranded braid. The untwisted sl p -homology of β is the slash homology group p b H( β ) := q − n H / • ( p HH ∂ q • ( pT β ) , ∂ T ) , viewed as an object in C ( k , ∂ q ) .The homology group p b H( β ) is only singly graded as an object in C ( k , ∂ q ) . By construction, p b H( β ) is theslash homology with respect to the ∂ T action on ⊕ i,j p HH ∂ q i ( pT jβ ) (see diagram (5.4)). The latter space isdoubly graded by the topological degree and q -degree with values in Z × Z (the Hochschild a degree isalready forced to be collapsed with the q degree to make the Cautis differential ∂ C homogeneous). How-ever, as in the proof of Corollary 4.13, the Markov II invariance for the homology theory requires one to ramedtopologicalinvariance 39 collapse the t -grading onto the a -grading, thus also onto the q -grading. We will use p b H i ( β ) to stand for thehomogeneous subspace sitting in some q -degree i .Let us also emphasize an important point about the vertical grading collapsing as the following remark. Remark 5.5.
A special remark is needed here about the grading specialization. In order to p -extend theKoszul complex (5.1) into a p -Koszul complex with ∂ C of degree two, we are forced to make the functorspecialization from [1] ad = a into q [1] q∂ , so that the p -extended complex looks like pC : 0 −→ q k [ x ] x ⊗ k [ x ] x [1] q∂ d C −→ k [ x ] ⊗ k [ x ] −→ . (5.5)Taking tensor product of pC , and this determines the correct vertical q -degree shifts in each column ofdiagram (5.3) of the p -Hochschild homology groups.Notice that, on the level of Grothendieck groups, this has the effect of specializing the formal variable a into − q .When [1] t∂ = [1] q∂ and a = q [1] q∂ , the grading shifts in equation (4.7) translate into pT i := q − (cid:16) B i br i −−→ R [ − q∂ (cid:17) , pT ′ i := q (cid:16) R [1] q∂ rb i −−→ q − B − ( x i + x i +1 ) i (cid:17) . (5.6)This also explains the necessity of p -extension in the collapsed t and a direction in p HHH in the previoussection: the homological shift in that direction needs to be p -extended to agree with the homological shiftin the q -direction.Furthermore, the bigrading in diagram (5.3) is now interpreted as a single grading, with both ∂ C and ∂ t raising q -degree by two.This approach to a categorification of the Jones polynomial, at generic values of q , was first developedby Cautis [Cau17]. We follow the exposition of Robert and Wagner from [RW20] and the closely relatedapproach of Queffelec, Rose, and Sartori [QRS18]. In this subsection, we establish the topological invariance of the untwisted homology theory.
Theorem 5.6.
The homology p b H( β ) is a finite-dimensional framed link invariant depending only on thebraid closure of β . Proof.
The proof of the theorem will mostly be parallel to that of Proposition 4.12 and Corollary 4.13. Itamounts to showing that taking slash homology of p HH ∂ • ( β ) with respect to ∂ T satisfies the Markov IImove.We start by discussing the normal H q -equivariant Hochschild homology version. Let L be a link in R obtained as a braid closure b β , where β ∈ Br n is an n -stranded braid. Recall that the homology groups HH ∂ • ( L ) are defined by tensoring a complex of Soergel bimodules M determined by β with the Koszulcomplex C n and computing its termwise vertical (Hochschild) homology. The differential d C is defined onthe Koszul complex C n . To emphasize its dependence on n , we will write d C on C n as d n in this proof, andlikewise write ∂ n for the p -extended differential on pC n .Since C n +1 = C n ⊗ C ′ = C n ⊗ k [ x n +1 ] ⊗ Λ h dx n +1 i ⊗ k [ x n +1 ] , the vertical differential may be inductively defined as d n +1 = d n ⊗ Id + Id ⊗ d ′ . (5.7)Here we have set C ′ = k [ x n +1 ] ⊗ Λ h dx n +1 i ⊗ k [ x n +1 ] equipped with part of the Cautis differential d ′ := x n +1 ⊗ ι ∂∂xn +1 ⊗ ⊗ ι ∂∂xn +1 ⊗ x n +1 . The notation ι denotes the contraction of dx n +1 with ∂∂x n +1 . ramedtopologicalinvariance 40 From the proof of Proposition 4.12 (see equation (4.23)), we have a vector space decomposition HH ∂ q • (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) ∼ = H v • ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) ⊕ H v • ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) . Here Y and Y are the terms of C ′ ⊗ k [ x n +1 ] , k [ x n +1 ] T n (see equation (4.20) for the definition). We claim that,instead of a direct sum decomposition, we obtain a filtration of HH ∂ q • (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) as a moduleover k [ d C ] / ( d C ) : → H v • ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) → HH ∂ q • (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) → H v • ( C n ⊗ ( R n ,R n ) ( M ⊗ R n Y )) → . (5.8)Indeed, since d C acts on the Y and Y tensor factors via d ′ , it suffices to check that d ′ preserves the sub-module arising from Y and presents the part arising from Y as a quotient. To do this, we re-examine thesequence (4.20) under vertical (Hochschild) homology. The part Y , under vertical homotopy equivalence,contributes to the horizontal (topological) complex (see equation (4.26)) Y ′ := (cid:16) R n +1 [1] td d t =Id −−−−→ R n +1 (cid:17) (5.9a)sitting entirely in Hochschild degree . Likewise, the part Y contributes to the horizontal Y ′ := (cid:18) R x n +3 x n +1 n +1 [1] td [1] ad d t =2( x n +1 − x n ) −−−−−−−−−−→ R x n +1 n +1 [1] ad (cid:19) (5.9b)sitting entirely in Hochschild degree . Since d ′ decreases Hochschild degree by one, Y ′ must be preservedunder d ′ , acting upon it trivially, and Y ′ is equipped with the (zero) quotient action by d ′ .Similar behavior happens under p -extension and degree collapsing with respect to the ∂ C action (c.f. theproof and notation of Corollary 4.13). Write ∂ ′ as the p -extended differential of d ′ . Consider the degreecollapsed diagram obtained from tensoring equation (5.6) with equation (5.5) while taking vertical slashhomology: q ( x n +1 B x n +1 n )[1] q∂ br / / x n +1 ⊗ − ⊗ x n +1 (cid:15) (cid:15) qR x n +1 n +10 (cid:15) (cid:15) = pC ′ ⊗ ( k [ x n +1 ] , k [ x n +1 ]) pT n q − B n br / / q − R n +1 [ − q∂ . (5.10)After taking (vertical) p -Hochschild homology, the part of the term pY that is not killed arises from pY ′ := q − (cid:16) R n +1 Id −→ R n +1 [ − q∂ (cid:17) (5.11a)sitting entirely horizontally inside the lower horizontal arrow of (5.10). On the other hand, the part pY contributes to the horizontal pY ′ := q R x n +3 x n +1 n +1 [1] q∂ x n +1 − x n ) −−−−−−−−→ qR x n +1 n +1 (5.11b)sitting in the top horizontal line of the square (5.10). Since ∂ ′ acts vertically down, upon taking the slashhomology with respect to ∂ T , it now follows that the term arising from pY ′ , on which ∂ T acts just via ∂ t + ∂ q ,contributes nothing to the total slash homology, as this term is the cone of the identity map in the homotopycategory of p -complexes.Now, translating the exact sequence (5.8) as in the final step in the proof of Corollary 4.13, we obtainthat H / • ( p HH • (( M ⊗ k [ x n +1 ]) ⊗ R n +1 T n ) , ∂ T ) ∼ = H / • ( p HH • ( M ⊗ R n pY ′ ) , ∂ T ) ∼ = q H / • ( p HH • ( M ) , ∂ T ) x n . The q factor is cancelled out in the overall shift of p b H . This finishes the first part of Markov II move.The other case of the Markov II move is entirely similar, which we leave the reader as an exercise.Finally, the finite-dimensionality of p b H( β ) follows from Corollary 3.2. The theorem follows. ategoricalJonesnumber 41Remark 5.7. Note that there are more general super differentials d N considered in [Cau17, RW20, QRS18]which gives rise to an sl N link homology for general N when the quantum parameter q is assumed to begeneric. It is not clear how to recover general sl N link homologies when q is a root of unity because d N and ∂ q only commute for N = 2 . For the unknot, the p -Hochschild homology of k [ x ] (see Section 4.6) but now with the Cautis differentialincluded, is given by q − p ) k [ x ] x · · · q k [ x ] x x / / k [ x ] (5.12)whose slash homology is equal to k [ x ] / ( x ) equipped with the trivial p -differential. Likewise, using the p -extended Koszul complex pC k one obtains, for the k -component unlink L , that p b H( L ) = k O i =1 k [ x i ]( x i ) , (5.13)with the zero p -differential. Let us call this p -DG algebra A k .However, to correct the twisting factor as we have done for [ HHH and p [ HHH in Section 4.6, it is a bitmore subtle. As the following example would show.
Example 5.8.
Consider the two rank-one p -DG modules over k [ x ] , where ∂ ( x ) = x , k [ x ] and k [ x ] x . It isclear that twisting the second module by k [ x ] − x results in an isomorphism k [ x ] ∼ = k [ x ] x ⊗ k [ x ] k [ x ] − x . However, this can not be done after taking slash homology of k [ x ] and k [ x ] x , as the first module is quasi-isomorphic to the ground field, while the second module is acyclic.We therefore need to introduce a p -differential twisting to correct the framing factor occurring in Theo-rem 5.6 slightly differently from what we have done in Definition 4.16. For a braid β ∈ Br n whose closure isa k -stranded framed link. Choose for each framed component of b β in β a single strand in β that lies in thatcomponent after closure, say, the i r th strand is chosen for the r th component. Then define the polynomialring k [ x i , . . . , x i k ] as a subring of k [ x , . . . , x n ] generated by the chosen variables. Set k [ x i , . . . , x i k ] f β := k [ x i , . . . , x i k ] · β , ∂ (1 β ) := − k X r =1 f r x i r β . (5.14)Then we make the twisting of H q -modules on the p HH • -level, termwise on p HH • ( pT iβ ) : p HH f β • ( pT β ) := p HH • ( pT β ) ⊗ k [ x i ,...,x ik ] k [ x i , . . . , x i k ] f β . (5.15) Definition 5.9.
Given β ∈ Br n whose closure is a k stranded framed link, the sl p -homology is the object p H( β ) := H / • ( p HH f β • ( pT β ) , ∂ T ) in the homotopy category C ( k , ∂ q ) .The internal H q -structure shifts do not interfere with the proof of Theorem 5.6, and they correct theoverall twisting of H q -modules arising from Markov II moves in the theorem. It follows that we obtain thefirst part of the following. Theorem 5.10.
The sl p -homology p H( β ) is a singly graded, finite-dimensional link invariant dependingonly on the braid closure of β as a link in R . Furthermore its graded Euler characteristic χ ( p H( L )) := X i q i dim k ( p H i ( L )) is equal to the Jones polynomial at a prime root of unity. Proof.
As in the proof of Corollary 4.17, the twisting compensates for the linear factors appearing in MarkovII moves, thus establishing the topological invariance of p H( β ) .For the last statement, we will use the fact that the Euler characteristic does not change before or aftertaking slash homology. This is because, as with the ususal chain complexes, taking slash homology onlygets rid of acyclic summands whose Euler characteristics are zero.Let us revisit diagram (5.4). Before collapsing the t -grading onto q , the diagram arises by p -extending thesame Hochschild homology group diagram of T β in the t -direction. Let P β ( v, t ) be the Poincar´e polynomialof Cautis’s bigraded diagram for now, where v , t are treated as formal variables coming from q and t gradingshifts. As shown by Cautis [Cau17], P β ( v, − is the Jones polynomial of the link b β in the variable v .The p -extension in the topological direction is equivalent to categorically specializing [1] td to [1] q∂ . Ithas the effect, on the Euler characteristic level, of specializing t = − . Thus we obtain that the Eulercharacteristic of p H( β ) is equal to P β ( v = q, t = − . This the Jones polynomial evaluated at a p th root ofunity q . The result follows. In this section we compute the various homologies constructed earlier for (2 , n ) torus links. Note that thereare no framing factors to incorporate in this family of examples. We follow the exposition in [KR16] to compute variations of the HOMFLYPT homology of (2 , n ) torus linkswhile making the necessary p -DG modifications.We begin by reviewing the homology of unlinks. Recall the Koszul resolutions C and pC of k [ x ] asbimodules are written uniformly as q k [ x ] x ⊗ k [ x ] x [1] a ∗ / / k [ x ] ⊗ k [ x ] , where ∗ ∈ { d, ∂ } . Tensoring this complex with k [ x ] as a bimodule yields q k [ x ] x [1] a ∗ / / k [ x ] . Thus the homology of the unknot (up to shift) is given by: k [ x ] ⊕ q k [ x ] x [1] a ∗ . By the monoidal structure, the homology of the n -component unlink L is HHH ∂ q ( L ) ∼ = ( a − t ) n n O i =1 (cid:0) k [ x i ] ⊕ q a k [ x i ] x i (cid:1) , (6.1a) p HHH ∂ q ( L ) ∼ = n O i =1 q − (cid:0) k [ x i ] ⊕ q [1] t∂ k [ x i ] x i (cid:1) . (6.1b)Here, in the second equation (6.1b), the shift q [1] td arises from specializing the grading shift functor a = [1] ad in (6.1a) to q [1] t∂ . The slash homology of (6.1b) is (cid:0) q − k ⊕ q p +1 V qp − [1] t∂ (cid:1) ⊗ n and hence its Euler characteristic is ( q + q − ) n .Throughout the remainder of this subsection, let R = k [ x , x ] , B = B , and T = T . We begin with thecomputation of the two component unlink since it will play a role in the homology of the (2 , n ) torus link T ,n . OMFLYPThomology 43
By tensoring the Koszul resolution of k [ x ] by itself we obtain a resolution of k [ x , x ] as bimoduleshomotopic to q R x + x ⊗ R x + x [2] a ∗ (cid:16) x ⊗ − ⊗ x − x ⊗ ⊗ x (cid:17) / / q R x ⊗ R x ⊕ q R x ⊗ R x [1] a ∗ ( x ⊗ − ⊗ x , x ⊗ − ⊗ x ) / / R ⊗ R . (6.2)Tensoring this complex with R as a bimodule yields q R x + x ) [2] a ∗ / / ( q R x ⊕ q R x )[1] a ∗ / / R. Thus the homology of the two component unlink (up to shift) is given by the relative Hochschild ho-mology (assuming ∗ = d ): HH ∂ q i = R if i = 0 q R x ⊕ q R x if i = 1 q R x + x ) if i = 20 otherwise. (6.3)We write the relative Hochschild homology uniformly for ∗ ∈ { d, ∂ } as R ⊕ q ( R x ⊕ R x )[1] a ∗ ⊕ q R x + x ) [2] a ∗ . Now we return to the computation of the HOMFLYPT homology of the (2 , n ) torus link. The complex(6.2) is isomorphic by a change of basis in the middle term in the complex by the matrix (cid:18) − − (cid:19) , to q R x + x ⊗ R x + x [2] a ∗ / / ( q R ⊗ R ⊕ q R ⊗ R )[1] a ∗ / / R ⊗ R , (6.4)where now the H q -structure in the middle term is twisted by the matrix (cid:18) x ⊗ ⊗ x x ⊗ ⊗ x − x ⊗ − ⊗ x x ⊗ ⊗ x (cid:19) . (6.5)Now we determine the relative Hochschild homology of the bimodule B . Tensoring (6.4) on the left asa bimodule with B yields q ( x + x B x + x )[2] a ∗ α / / ( q B ⊕ q B )[1] a ∗ β / / B (6.6)where the middle term has an H q -structure twisted by (6.5) and α and β are given by α = (cid:18) − x ⊗ ⊗ x (cid:19) β = (cid:0) x ⊗ − ⊗ x (cid:1) . The kernel of α is generated as a left R -module by Γ = 12 ( x ⊗ ⊗ x − x ⊗ − ⊗ x ) . It is easy to verify that ∂ q (Γ) = ( x + x )Γ so that after accounting for the extra twist, ker α ∼ = R x + x ) .Note that in coke α , we have x ⊗ ⊗ x and x ⊗ ⊗ x , so coke α is generated as an R -moduleby ⊗ and thus coke α ∼ = R . Similarly, ker β ∼ = coke β ∼ = R . OMFLYPThomology 44
These observations combined with the H q -structure given in (6.6) yield the following result for therelative Hochschild homology of B : HH ∂ q i ( B ) ∼ = R if i = 0 q R ⊕ q R twisted by x x + 3 x ! if i = 1 q R x + x ) if i = 20 otherwise. (6.7) Lemma 6.1.
The braiding complex T ⊗ n simplifies in the following ways.(i) In C ∂ q ( R, R, d ) , one has T ⊗ n ∼ = ( atq ) − n (cid:18) q n − B ( n − e [ n ] td p n / / q n − B ( n − e [ n − td p n − / / · · · p / / q B e [2] td p / / B [1] td br / / R (cid:19) . (ii) In C ∂ q ( R, R, ∂ ) , one has T ⊗ n ∼ = ( q − [ − t∂ ) n (cid:18) q n − B ( n − e [ n ] t∂ p n / / q n − B ( n − e [ n − t∂ p n − / / · · · p / / q B e [2] t∂ p / / B [1] t∂ br / / R (cid:19) , where p i = 1 ⊗ ( x − x ) − ( x − x ) ⊗ p i +1 = 1 ⊗ ( x − x ) + ( x − x ) ⊗ . Proof.
This is proved by induction on n . One uses homotopy equivalences q i B ie ⊗ T ∼ = q i +1) B ( i +1) e and then determining the images of the maps br , p i , and p i +1 under these equivalences. Proposition 6.2.
The H q -HOMFLYPT homology of a (2 , n ) torus link, as an H q -module depends on theparity of n .(i) If n is odd: ( atq ) − n a − t (cid:0) q [1] ad k [ x ] x ⊕ q [2] ad k [ x ] x (cid:1) MM i ∈{ , ,...,n − } (cid:16) q i − k [ x ] i − x ⊕ [1] ad q i k [ x ] ⊕ q i +2 k [ x ] ⊕ q i +4 [2] ad k [ x ] i +1) x (cid:17) [ i ] td ! with the H q -structure on the middle object q i k [ x ] ⊕ q i +2 k [ x ] given by (cid:18) ix − i + 2) x (cid:19) .(ii) If n is even: ( atq ) − n a − t (cid:0) q [1] ad k [ x ] x ⊕ q [2] ad k [ x ] x (cid:1) MM i ∈{ , ,...,n − } (cid:16) q i − k [ x ] i − x ⊕ [1] ad q i k [ x ] ⊕ q i +2 k [ x ] ⊕ q i +4 [2] ad k [ x ] i +1) x (cid:17) [ i ] td M(cid:16) q n − k [ x , x ] ( n − x + x ) ⊕ [1] ad q n k [ x , x ] ⊕ q n +2 k [ x , x ] ⊕ q n +4 [2] ad k [ x , x ] ( n +2)( x + x ) (cid:17) [ n ] td ! OMFLYPThomology 45 with the H q -structure on the middle object q i k [ x ] ⊕ q i +2 k [ x ] given by (cid:18) ix − i + 2) x (cid:19) and the H q struc-ture on the middle object q n k [ x , x ] ⊕ q n +2 k [ x , x ] given by (cid:18) n ( x + x ) 0 − n ( x + x ) + 2 x (cid:19) . Proof.
We sketch some of the details. It is clear that HH ∂ q • ( p i ) = 0 and HH ∂ q • ( p i +1 ) = 2( x − x ) . Thentaking the Hochschild homology of the complex in Lemma 6.1 breaks up into a sum of pieces of the form HH ∂ q • ( q B ) HH ∂q • ( br ) / / HH ∂ q • ( R ) , HH ∂ q • ( q i B i ( x + x ) ) x − x ) / / HH ∂ q • ( q i − B (2 i − x + x ) ) and (if n is even) the left-most piece is HH ∂ q • ( q n − B ( n − x + x ) ) . The result follows using the Hochschild homologies of R calculated in (6.3) and of B calculated in (6.7).Recall that HH ∂ q • ( B ) is generated by elements of the form coke β, coke α, ker β, ker α .The morphism HH ∂ q • ( br ) maps coker β and coker α isomorphically onto their images. On the otherhand, under HH ∂ q • ( br ) , the image of ker β and ker α identify the variables x and x in HH ∂ q • ( R ) . We call thisidentified variable x .The map HH ∂ q • ( p i +1 ) : HH ∂ q • ( q i B i ( x + x ) ) x − x ) / / HH ∂ q • ( q i − B (2 i − x + x ) ) has no kernel and the image also identifies x and x as a common variable x .Similarly, one has the following analagous p -version of the previous result. Proposition 6.3.
The bigraded H q -HOMFLYPT p -homology of a (2 , n ) torus knot, as an H q -module dependson the parity of n .(i) If n is odd it is: q − n − [ − n ] t∂ (cid:0) q [1] t∂ k [ x ] x ⊕ q [2] t∂ k [ x ] x (cid:1) MM i ∈{ , ,...,n − } q − n − q i − k [ x ] i − x ⊕ q [1] t∂ q i k [ x ] ⊕ q i +2 k [ x ] ⊕ q i +8 [2] t∂ k [ x ] i +1) x [ i − n ] t∂ with the H q -structure on the middle object q i k [ x ] ⊕ q i +2 k [ x ] given by (cid:18) ix − i + 2) x (cid:19) .(ii) If n is even it is: q − n − [ − n ] t∂ (cid:0) q [1] t∂ k [ x ] x ⊕ q [2] t∂ k [ x ] x (cid:1) MM i ∈{ , ,...,n − } q − n − q i − k [ x ] i − x ⊕ q [1] t∂ q i k [ x ] ⊕ q i +2 k [ x ] ⊕ q i +8 [2] t∂ k [ x ] i +1) x [ i − n ] t∂ M q − n − q n − k [ x , x ] ( n − x + x ) ⊕ q [1] t∂ q n k [ x , x ] ⊕ q n +2 k [ x , x ] ⊕ q n +8 [2] t∂ k [ x , x ] ( n +2)( x + x ) xampleoftheJonesinvariantwhen q isgeneric 46 with the H q -structure on the middle object q i k [ x ] ⊕ q i +2 k [ x ] given by (cid:18) ix − i + 2) x (cid:19) and the H q struc-ture on the middle object q n k [ x , x ] ⊕ q n +2 k [ x , x ] given by (cid:18) n ( x + x ) 0 − n ( x + x ) + 2 x (cid:19) . Corollary 6.4.
In the stable category of H q -modules, the slash homology of the H q -HOMFLYPT p -homologyof a (2 , n ) torus link depends on the parity of n .(i) If n is odd it is: q − n − [ − n ] t∂ (cid:0) q p +2 V qp − [1] t∂ ⊕ q p +4 V qp − [2] t∂ (cid:1) MM i ∈{ , ,...,n − } q − n − q p V qp − i − ⊕ q p +2 V qp − i ⊕ q p +2 V qp − i − [1] t∂ ⊕ q p +6 V qp − i +1) [2] t∂ [ i − n ] t∂ . (ii) If n is even it is: q − n − [ − n ] t∂ (cid:0) q p +2 V qp − [1] t∂ ⊕ q p +4 V qp − [2] t∂ (cid:1) MM i ∈{ , ,...,n − } q − n − q p V qp − i − ⊕ q p +2 V qp − i ⊕ q p +2 V qp − i − [1] t∂ ⊕ q p +6 V qp − i − [2] t∂ [ i − n ] t∂ M q − n − q p V qp − ( n − ⊗ V qp − ( n − ⊕ (cid:0) q p +2 V qp − n ⊗ V qp − n ⊕ q p +2 V qp − n ⊗ V qp − n − (cid:1) [1] t∂ ⊕ q p +4 V qp − ( n +2) ⊗ V qp − n − [2] t∂ . q is generic We will compute the Jones homology of a (2 , n ) torus link when the quantum parameter q is generic so weassume ∂ q = 0 . We denote this homology by H • , • ( T ,n ) . As mentioned earlier, this more classical homologywas constructed from various perspectives in [Cau17], [QRS18], and [RW20] and its formulation is built intoDefinition 5.4. Setting ∂ q = 0 and not p -extending in the a or t directions, allows for a doubly graded theoryrather than the singly graded theory of Section 5. We thus use the grading shift conventions of Section 4.1.Recall the Koszul resolution of R in (6.2). Then HH • ( R ) with the induced Cautis differential d C is givenby Rq R x < < ②②②②②②②② q R x b b ❊❊❊❊❊❊❊❊ q R − x a a ❉❉❉❉❉❉❉❉ x = = ③③③③③③③③ (6.8)and HH • ( B ) with d C is given by Rq R x + x < < ①①①①①①①① q R x ( x − x ) b b ❋❋❋❋❋❋❋❋ q R x ( x − x ) b b ❊❊❊❊❊❊❊❊ x + x < < ②②②②②②②② . (6.9) xampleoftheJonesinvariantwhen q isgeneric 47 When n is even, the leftmost term in T ⊗ n maps by zero into the rest of the complex, so we need tounderstand the homology of HH ∗ ( B ) by itself in this case. All of the maps in (6.9) are injective so homologyis concentrated in R . Thus we need to find a basis of R/ ( x + x , x ( x − x )) . In the quotient, note that x = x x = − x x = − x . Since x = x x , we get x = x x = − x . But also x = x x = x . Thus x = x = 0 . Thus the homology is spanned by { , x , x , x , x x , x x } . In the rest of the complex for T ⊗ n , there are two types of maps we need to analyze. First we study HH • ( br ) : HH • ( B ) → HH ∗ ( R ) . R - - Rq R x + x < < ①①①①①①①① q R x ( x − x ) b b ❋❋❋❋❋❋❋❋ x − x + + q R x < < ②②②②②②②② q R x b b ❊❊❊❊❊❊❊❊ q R x ( x − x ) b b ❊❊❊❊❊❊❊❊ x + x < < ②②②②②②②② ( x − x ) q R − x a a ❉❉❉❉❉❉❉❉ x = = ③③③③③③③③ (6.10)While it is not very difficult to compute the total homology of this bicomplex, we use a fact from the proofof [RW20, Theorem 6.2]. According to this trick, we may calculate homology with respect to d C first andthen with respect to the topological differential d t . Thus the homology of the bicomplex with respect to thetotal differential is spanned by { tx , tx x } .Next we analyze HH • ( p i +1 ) : HH • ( q i B ) −→ HH • ( q i − B ) . q i R x − x ) . . q i − Rq i +4 R x + x ttttttttt q i +6 R x ( x − x ) e e ❏❏❏❏❏❏❏❏❏ (cid:0) x − x ) 00 2( x − x ) (cid:1) , , q i +2 R x + x : : ttttttttt q i +4 R x ( x − x ) d d ❏❏❏❏❏❏❏❏❏ q i +10 R x ( x − x ) e e ❏❏❏❏❏❏❏❏❏ x + x ttttttttt x − x ) q i +8 R x ( x − x ) d d ❏❏❏❏❏❏❏❏❏ x + x : : ttttttttt (6.11)Again using the proof of [RW20, Theorem 6.2], we compute the homology with respect to d C and then withrespect to d t to obtain the total homology of this bicomplex is spanned by { q i t i +1 x , q i t i +1 x x , q i − t i , q i − t i x } . We now assemble all of this information together to get the homology of the (2 , n ) torus link. Recall thatthe complex used for T ⊗ n comes with a shift of ( at ) − n q − n and the Hochschild homology functor comeswith a shift of a − t . Thus there is an overall shift of a − n − t − n +22 q − n . Specializing a = q t yields an overallshift of q − n − t − n . Thus we get the following homology in terms of Poincar´e series. • If n is odd, then the bigraded Poincar´e series of H • , • ( T ,n ) is equal to q − n − t − n · (cid:16) (1+ q ) q t (1+ q t + q t + · · · + q n − t n − )+(1+ q ) q t (1+ q t + q t + · · · + q n − t n − ) (cid:17) . • If n is even, then the bigraded Poincar´e series of H • , • ( T ,n ) is equal to q − n − t − n · (cid:16) (1 + q ) q t (1 + q t + · · · + q n − t n − ) + (1 + q ) q t (1 + q t + · · · + q n − t n − )+ t n q n − (1 + 2 q + 2 q + q ) (cid:17) . Remark 6.5.
This was also computed (for n = 2 , ) in [QRS18] and [RW20]. It is interesting to note that thisis different from the Khovanov homology of the Hopf link and trefoil. xampleoftheJonesinvariantwhen q -primerootofunity 48 q -prime root of unity To compute the p -DG Jones invariant, we will utilize the following auxiliary tool. Proposition 6.6.
Let M • = (cid:16) · · · ∂ t −→ M i +1 ∂ t −→ M i ∂ t −→ M i − ∂ t −→ · · · (cid:17) be a contractible p -complex of H q = k [ ∂ q ] / ( ∂ pq ) -modules. Then the totalized complex ( T ( M • ) , ∂ T = ∂ t + ∂ q ) is acyclic. Proof.
Since M • is contractible, there is an H q -linear map σ : M • −→ M • +1 such that [ σ, ∂ t ] = Id M by Lemma2.4. Thus we have [ σ, ∂ T ] = [ σ, ∂ t + ∂ q ] = [ σ, ∂ t ] + [ σ, ∂ q ] = Id M . The result follows again from Lemma 2.4.We will be applying Proposition 6.6 in the following situation. Suppose N • is a p -complex of H q -moduleswhose boundary maps preserve the H q -module structure. Further, let M • be a sub p -complex that is closedunder the H q -action, and there is a map σ on M • as in Proposition 6.6 that preserves the H q -module struc-ture. Then, when totalizing the p -complexes, we have T ( M • ) ⊂ T ( N • ) and the natural projection map T ( N • ) −→T ( N • ) / T ( M • ) is a quasi-isomorphism. Similarly, if M • is instead a quotient complex of N • that satisfies the condition ofProposition 6.6, and K • is the kernel of the natural projection map −→ K • −→ N • −→ M • −→ , then the inclusion map of totalized complexes T ( K • ) −→T ( N • ) is a quasi-isomorphism.We modify the the calculation of the homology in the previous section of the (2 , n ) torus link to accountfor the differential ∂ q . Recall that in this singly graded theory that a = tq and t = [1] q∂ . Consequently [1] a∂ = q [1] t∂ = q [1] q∂ .First we study p HH • ( br ) : p HH • ( B )[1] q∂ −→ p HH • ( R ) , R [1] q∂ - - Rq R [1] q∂ [1] q∂ x + x ♥♥♥♥♥♥♥♥♥♥♥♥ q R [1] q∂ [1] q∂x ( x − x ) g g PPPPPPPPPPPP x − x - - q R x [1] q∂ x ♣♣♣♣♣♣♣♣♣♣♣♣ q R x [1] q∂x f f ◆◆◆◆◆◆◆◆◆◆◆◆ q R e [1] q∂ [2] q∂x ( x − x ) g g PPPPPPPPPPPP x + x ♥♥♥♥♥♥♥♥♥♥♥♥ ( x − x ) q R e [2] q∂ − x f f ◆◆◆◆◆◆◆◆◆◆ x qqqqqqqqqqq (6.12)where the object q R [1] q∂ [1] q∂ ⊕ q R [1] q∂ [1] q∂ in the left square is twisted by the matrix (cid:18) x x + 3 x (cid:19) . (6.13)Filtering the total complex (6.12) and applying Proposition 6.6, we obtain that the total p -complex is quasi-isomorphic to k h , x , x , x x , x , x x i [1] q∂ / / k h , x , x , x x i . (6.14)Let us illustrate how this is obtained. For instance, the p -complex q R e [2] q∂ φ =( − x ,x ) −−−−−−−−→ Im( φ ) ⊂ (cid:0) q R x [1] q∂ ⊕ q R x [1] q∂ (cid:1) xampleoftheJonesinvariantwhen q -primerootofunity 49 is a quotient of the total p -complex of the right most square. The map φ is an H q -intertwining isomor-phism onto its image because of the H q -module twists imposed on the modules. Note that this p -complex(ignoring q -grading shifts) R e φ =( − x ,x ) −−−−−−−−→ ∼ = Im( φ ) = Im( φ ) = · · · = Im( φ ) , where Im( φ ) is repeated p − times, is a contractible p -complex of H q -modules. Proposition 6.6 then appliesto this quotient complex, and shows that it contributes nothing to the total slash homology.The p -complex in (6.14), in turn, is quasi-isomorphic to k h x , x x i [1] q∂ . This is quasi-isomorphic to q V [1] q∂ .Once again when n is even, the leftmost term in T ⊗ n maps by zero into the rest of the complex so wehave to understand the total homology of p HH • ( q n − B ( n − e [ n ] q∂ ) . Filtering q n − R ( n − e [ n ] q∂ q n +1) R ( n − e [ n + 1] q∂x + x ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ q n +2) R ( n − e [ n + 1] q∂x ( x − x ) j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ q n +4) R ( n +2) e [ n + 2] q∂x ( x − x ) j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ x + x ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (6.15)where the middle terms q n +1) R ( n − e [ n ] q∂ [1] q∂ ⊕ q n +2) R ( n − e [ n ] q∂ [1] q∂ are further twisted by the matrix(6.13), yields that (6.15) is quasi-isomorphic to q n − k h , x , x , x x , x , x x i [ n ] q∂ with a differential inherited from the polynomial algebra and twisted by ( n − e . Explicitly, the differentialacts on the basis by n − | | ②②②②②②②②② n − $ $ ❍❍❍❍❍❍❍❍❍❍ x n (cid:15) (cid:15) n − ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ x n − (cid:15) (cid:15) − n u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ x
21 2 n ! ! ❇❇❇❇❇❇❇❇ x x x x . (6.16)This is isomorphic to the direct sum of p -complexes n − (cid:15) (cid:15) x − x n (cid:15) (cid:15) x + x n − (cid:15) (cid:15) L x n (cid:15) (cid:15) x x x x . xampleoftheJonesinvariantwhen q -primerootofunity 50 Thus the total homology of p HH • ( q n − B ( n − e [ n ] q∂ ) is isomorphic to the following p -complex dependentof the characteristic of the ground field Y n := q n V [ n ] q∂ ⊕ q n +2 V [ n ] q∂ if p ∤ n − , nq n − ( V ⊕ q V ⊕ q V )[ n ] q∂ ⊕ q n +2 V [ n ] q∂ if p | n − , p ∤ nq n V [ n ] q∂ ⊕ q n ( V ⊕ q V ⊕ q V )[ n ] q∂ if p ∤ n − , p | n . (6.17)Finally we analyze p HH ∗ ( p i +1 ) : p HH ∗ ( q i B ie [2 i + 1] q∂ ) −→ HH ∗ ( q i − B (2 i − e [2 i ] q∂ ) where p HH • ( q i B ie [2 i + 1] q∂ ) = q i R ie [2 i + 1] q∂ q i +4 R [2 i + 2] q∂x + x ♠♠♠♠♠♠♠♠♠♠♠♠♠ q i +6 R [2 i + 2] q∂x ( x − x ) h h ◗◗◗◗◗◗◗◗◗◗◗◗◗ q i +10 R (2 i +3) e [2 i + 3] q∂x ( x − x ) h h ◗◗◗◗◗◗◗◗◗◗◗◗◗ x + x ♠♠♠♠♠♠♠♠♠♠♠♠♠ , (6.18)and p HH • ( q i − B (2 i − e [2 i ] q∂ ) = q i − R (2 i − e [2 i ] q∂ q i +2 R [2 i + 1] q∂ x + x ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ q i +4 R [2 i + 1] q∂x ( x − x ) i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ q i +8 R (2 i +2) e [2 i + 2] q∂x ( x − x ) i i ❘❘❘❘❘❘❘❘❘❘❘❘❘ x + x ❧❧❧❧❧❧❧❧❧❧❧❧❧ (6.19)where the differentials for both objects in the middle horizontal rows of (6.18) and (6.19) are twisted by(6.13) and p HH • ( p i +1 ) = 2( x − x ) (diagonal multiplication by x − x ) ). Filtering this total complexyields the total complex q i k h , x , x , x , x x , x x i [2 i + 1] q∂ x − x ) / / q i − k h , x , x , x , x x , x x i [2 i ] q∂ . (6.20)This is quasi-isomorphic to q i k h x , x x i [2 i + 1] q∂ M q i − k h , x i [2 i ] q∂ , (6.21)where the differential on the basis elements is given by x i +2 (cid:15) (cid:15) i − (cid:15) (cid:15) L x x x . Thus the total homology is isomorphic to the p -complex X i := q i +5 V [2 i + 1] q∂ ⊕ q i − V [2 i ] q∂ if p ∤ i + 1 , i − q i +4 ( V ⊕ q V )[2 i + 1] q∂ ⊕ q i − V [2 i ] q∂ if p | i + 1 , p ∤ i − q i +5 V [2 i + 1] q∂ ⊕ q i − ( V ⊕ q V )[2 i ] q∂ if p ∤ i + 1 , p | i − . (6.22) EFERENCES 51
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J. S.:
DepartmentofMathematics,CUNYMedgarEvers,Brooklyn,NY,11225,USA email: [email protected]