TT H E CO H O M O LO G Y O F T W I S T E D COA LG E B R A S an invitation to twisted Koszul duality theory
Pedro Tamaroff16th June 2020
Abstract
In this paper, which is based on the author’s MSc thesis, we study in detail the co-homology theory for twisted coalgebras introduced in [2] by M. Aguiar and S. Mahajan.We compute it completely in various examples, including those proposed by Aguiar andMahajan, and obtain structural results: in particular, we study its multiplicative struc-ture, provide a Künneth formula, and succeed in giving an alternative description of thiscohomology theory which, in particular, allows for its effective computation.At the very end of the paper, we briefly outline how all the computations done in thispaper can be swiftly explained and extended to an arbitrary Koszul twisted coalgebrathrough their corresponding Koszul duality theory. While doing so, we work out the ex-ample of the species of linear orders: we show it is twisted Koszul, compute its dual andthe (doubly) graded dimensions of its components, which turn out to be the unsignedStirling numbers of the first kind.
Contents
1. Algebras and coalgebras in species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. The cohomology of twisted coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. An alternative description of cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 E -page — 3.3. Convergence — 3.4. The small complex — 3.5. Some computations — 3.6. Multiplicative matters
4. Coda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 a r X i v : . [ m a t h . K T ] J un T HE COHOMOLOGY OF TWISTED COALGEBRAS
Introduction
The present paper serves two purposes. First, it is an illustration of the possibility to com-pletely understand a new cohomology theory, that of coalgebras in the combinatorial spe-cies of A. Joyal [11], which are usually known as twisted coalgebras. This theory originatedin the work of M. Aguiar and S. Mahajan on deformations of coalgebras in the category ofspecies: their objective, among other, was constructing certain Hopf algebras in this cat-egory that encode the combinatorics of structures such as linear orders, graphs and posets,for example.Second, it is intended to show how homological algebra can complement the field ofenumerative and analytic combinatorics: using homological tools, we completely solve theproblem of computing the cohomology groups of a coalgebra in the category species and inparticular the second cohomology group of it, thereby completely solving a problem posedoriginally by M. Aguiar and S. Mahajan; our solution is effective and can be implementedin a computer, being analogous to the computation of the cohomology groups of a locallyfinite CW-complex through the use of the cellular cochain complex.To explain our results, we use the language of representation theory and homological al-gebra. Concretely, let E be the species of singletons, which is a Hopf algebra in the categoryof species. The cohomology theory of Aguiar and Mahajan is then encoded by the derivedfunctor X (cid:55)→ Ext( X , E ) in the category of E -bicomodules: it turns out that the full subcat-egory of the coalgebras that Aguiar and Mahajan are interested in corresponds to the fullsubcategory of certain “linearized” bicomodules.It is useful to think of the left bicomodule structure as the data of an operation of restric-tion on the combinatorial objects encoded by X , and of the right bicomodule structure asthe data of an operation on such objects, the compatibility axiom of a bicomodule encodinga compatiblity relation between these two operations. The datum of such a bicomodule X includes, in particular, the sequence ( X [0], X [1], . . .) where for each p ∈ (cid:78) the k -vector space X [ p ] is a kS p -module. We say X is weakly projective if for each p ∈ (cid:78) the module X [ p ] is aprojective kS p -module, and write sgn p for the one dimensional sign representation of S p ,its appearance which we now explain.Although the machinery we use in the paper is mainly homological, the appearance ofthe species of the sign representation is due to a rich interplay between homological al-gebra and the combinatorics of hyperplane arrangements: the cobar construction of thetwisted coalgebra E is the simplicial cochain complex of the triangulation of the sphere bythe Coxeter complex for the braid arrangement, and its cohomology is concentrated in topdegree, where it is the sign representation of the symmetric group. The main result of thispaper is the following: EDRO T AMAROFF Theorem.
Suppose that X is a weakly projective E -bicomodule. There is a complex S ∗ ( X ) so that for each p ∈ (cid:78) we have S p ( X ) = Hom S p ( X [ p ], sgn p ) that computes the cohomologygroups H ∗ ( X ) . If the ground ring contains (cid:81) , the differential δ p : S p ( X ) −→ S p + ( X ) is suchthat for z ∈ X [ p + we have ( δ p φ )( z ) = p + (cid:88) j = ( − j (cid:179) φ ( z (cid:48) j ) − φ ( z (cid:48)(cid:48) j ) (cid:180) , where the assignments z (cid:55)→ z (cid:48) j , z (cid:48)(cid:48) j restrict and contract, respectively, the “ j th label” of z. In particular, one can compute the second cohomology group H ( X ) using the three rep-resentations X [1], X [2] and X [3] and the contraction and restriction operations of X on suchspaces.We have also addressed the problem of determining when products exist in this cohomo-logy theory and describing them using the complex above. Our result comes paired with aKünneth isomorphism in the category of E -bicomodules: since E is a Hopf algebra, there isan internal product ⊗ in its category of bicomodules, and our result is the following: Theorem.
Let X and Y be E -bicomodules, and assume that k is a field and at least one of X or Y is locally finite. There is an isomorphism of complexes, natural in X and Y , of the formS ∗ ( X ) ⊗ S ∗ ( Y ) −→ S ∗ ( X ⊗ Y ) .Moreover, every morphism of E -bicomodules X −→ X ⊗ X induces a product in cohomo-logy and, in particular, if the bicomodule X is induced from a coalgebra structure on X , thecomultiplication of X is a morphism of E -bicomodules and induces a cup product in H ∗ ( X ) . To conclude this paper and set the stage for future applications, we interpret the main res-ults above in terms of Koszul duality for twisted (co)algebras. As it turns out, the twistes coal-gebra E is Koszul, and the complex S ∗ ( X ) that we discovered through a spectral sequencemethod is, in fact, the Koszul complex K ∗ ( X , E ) of E .After explaining this, we conclude with an example where we show that the species L oflinear orders considered by Aguiar and Mahajan is Koszul, compute its Koszul dual and itsdoubly graded Betti numbers: we show these are precisely the unsigned Stirling numbers ofthe first kind. More, precisely, we have the following result, which follows immediately fromthe twisted version of the Milnor–Moore theorem over a field of characteristic zero. Theorem.
The twisted Koszul dual algebra to L is L ¡ = S ( s − Lie ) , the free twisted commutat-ive algebra on the desuspension of the symmetric sequence Lie . In particular, for each j , n ∈ (cid:78) ,the component of weight j of the S n -module L ¡ ( n ) is in bijection with the permutations of nconsisting of exactly j disjoint cycles, which are enumerated by the unsigned Stirling numbersof the first kind. (cid:206) T HE COHOMOLOGY OF TWISTED COALGEBRAS
Useful references.
We refer the reader to [22] for an introduction to homological al-gebra and recommend coupling it with [17] for a comprehensive exposition on spectral se-quences. As a reference on combinatorial species, we use the seminal article of A. Joyal [11]and the book of Labelle, Leroux y Bergeron [13]. Finally, our reference for the formalism ofmonoidal categories is C. Kassel’s book [12], for the basics on abelian categories the book ofFreyd [9], and for the simplicial formalism, the book [22] and that of S. MacLane [16].
Running conventions.
Throughout, k is a unital commutative ring, and when we write ⊗ y Hom, we will be considering the usual functors on k -modules, unless stated otherwise; animportant exception is our use of ⊗ for the Cauchy product of species. Since we will writethem with lower-case boldfaced letters, while k -modules will always be written in capitalitalics, no confusion should arise.A decomposition S of length q of a set I is an ordered tuple ( S , . . . , S q ) of possible emptysubsets of I , which we call the blocks of S , that are pairwise disjoint and whose union is I . We say S is a composition of I if every block of S is nonempty. It is clear that if I has n elements, every composition of I has at most n blocks. We will write S (cid:96) I to mean that S is a decomposition of I , and if necessary will write S (cid:96) q I to specify that the length of S is q . Notice the empty set has exactly one composition which has length zero, the emptycomposition, and exactly one decomposition of each length n ∈ (cid:78) . If T is a subset of I and σ : I −→ J is a bijection, we let σ T : T −→ σ ( T ) be the bijection induced by σ . Acknowledgements.
I wholeheartedly thank my former MSc advisor M. Suárez–Alvarezfor the countless mathematical conversations we had during my time in the University ofBuenos Aires (2013–2017) —which I remember fondly— and for his invaluable input duringthe period of time when the thesis that this article is based on was written.
EDRO T AMAROFF Denote by
Set × the category of finite sets and bijections. Definition 1.1. A combinatorial species over a category C is a functor X : Set × −→ C . Con-cretely, a combinatorial species X is obtained by assigning S1. to each finite set I an object X ( I ) in C , S2. to each bijection σ : I −→ J an arrow X ( σ ) : X ( I ) −→ X [ j ],in such a way that S3. for every pair of composable bijections σ and τ , we have X ( τσ ) = X ( τ ) X ( σ ) and, S4. for every finite set I , it holds that X (id I ) = id X ( I ) .In particular, for every finite set I we have a map σ ∈ Aut( I ) (cid:55)−→ X ( σ ) ∈ Aut( X ( I )) whichgives an action of the symmetric group with letters in I on X ( I ). The category Set × is agrupoid, and it has as skeleton the full subcategory spanned by the sets [ n ] = {1, . . . , n } (inparticular, [0] = ∅ ) , and a species is determined, up to isomorphism, by declaring its valueson the finite sets [ n ] and on every σ ∈ S n . In view of this, one can think of a combinatorialspecies as a sequence ( X ( n )) n (cid:202) of objects in C endowed with S n actions ( S n × X ( n ) −→ X ( n )) n (cid:202) .We denote by Sp ( C ) the category Fun( Set × , C ) of species over C , whose morphisms are nat-ural transformations: explicitly, an arrow η : X −→ Y is an assignment of a map η I : X ( I ) −→ Y ( I ) to each finite set I , in such a way that for any bijection I σ −→ J the following diagramcommutes X ( I ) Y ( I ) X [ j ] Y ( J ) η I X ( σ ) Y ( σ ) η J This says that we must specify, for each finite set I , an Aut( I )-equivariant map η I : X ( I ) −→ Y ( I ). If we view species as sequences of objects on which the symmetric grupoid acts, amorphism of species X −→ Y is simply a sequence of equivariant maps ( η n : X n −→ Y n ) n (cid:202) .Our main interest will lie on species over sets or vector spaces. We write Sp for the categoryof species over Set , the category of sets and functions, and call its objects set species . If aspecies takes values on the subcategory
FinSet of finite sets we call it a finite set species, andif X ( ∅ ) is a singleton, we say it is connected . We write Sp k for the category of species over k Mod, the category of modules over k , and call its objects linear species . If a species takes T HE COHOMOLOGY OF TWISTED COALGEBRAS values on the subcategory k mod of finite generated modules we call it a linear species of finite type , and we say it is connected if X ( ∅ ) is k -free of rank one.Denote by k [ − ] the functor Set −→ k Mod that sends a set X to the free k -module withbasis X , which we will denote by k X , and call it the linearization of X . By postcomposition,we obtain a functor L : Sp −→ Sp k that sends a set species X to the linear species k X . Thespecies in Sp k that are in the image of k [ − ] are called linearized species . Thus, a linearizedspecies X = k X is such that, for every finite set I , the vector space X ( I ) has a chosen basis X ( I ), the morphisms X ( I ) −→ X [ j ] map basis elements to basis elements, and the actionof Aut( I ) on X ( I ) is by permutation of the basis elements. Definition 1.2.
Given a species X : Set × −→ Set and a finite set I , we call X ( I ) the set ofstructures of species X over I . If s ∈ X ( I ), we call I the underlying set of s , and call s an element of X or an X -structure . If I σ −→ J is a bijection, the element X ( σ )( s ) = t is the structure over J obtained by transporting s along σ , which we will usually denote, for simplicity, by σ s . Definition 1.3.
Two X structures s and t over respective sets I and J are said to be iso-morphic if there is a bijection σ : I −→ J that transports s to t , and we say σ is a structureisomorphism from s to t . A permutation that transports a structure s to itself is said to be an automorphism of s .In most cases, if X is a species and I is a set, X ( I ) consists of a collection of combinatorialstructures of some kind labelled in some way by the elements of I . For example, there isa species Pos that assigns to every finite set I the set Pos ( I ) of partial orders on I , and toevery bijection σ : I −→ J the function Pos ( σ ) : Pos ( I ) −→ Pos ( J ) which assigns to every orderon I the unique order on J that makes σ an order isomorphism: in concrete terms, Pos ( σ )“relabels” a poset on I according to σ . Examples
To understand all that follows it useful to have a list of examples in mind. We collect in thissection such a list. For a comprehensive treatment of combinatorial species, we refer thereader to [13].
E1.
The exponential or uniform species E : Set × −→ FinSet is the species that assigns toevery finite set I the singleton set { I }, and to any bijection σ : I −→ J the unique bijec-tion E ( σ ) : E ( I ) −→ E ( J ). Remark that E is the unique species, up to isomorphism, thathas exactly one structure over each finite set. For ease of notation, we will write ∗ I for{ I }. E2.
The species of partitions P assigns to each finite set I the collection of partitions of I :sets T = { T , . . . , T s } of nonempty disjoint subsets of I whose union is I . If σ : I −→ J is a EDRO T AMAROFF bijection and T is a partition of I , P ( σ )( T ) = { σ T , . . . , σ T s } is the partition of J obtainedby transporting T along σ . E3.
The species of compositions C assigns to each finite set I the collection of compositionof I : ordered tuples ( F , . . . , F t ) of nonempty disjoint subsets of I whose union is I . If σ : I −→ J is a bijection and F is a composition of I , C ( σ )( F ) = ( σ F , . . . , σ F t ) is thecomposition of J obtained by transporting F along σ . E4.
There is a species
Simp that assigns to each set I the collection of simplicial structureson I , this is, collections of finite subsets S ⊆ I that contain all singleton sets of ele-ments of I , and such that whenever ∆ ∈ S and ∆ (cid:48) ⊆ ∆ , then ∆ (cid:48) ∈ S . We call the elementsof S simplices . E5.
Again, let X be a topological space. There is a species F X that assigns to each finiteset I the configuration space F X ( I ) ⊆ X I of X with coordinates on I : F X ( I ) consistsof tuples ( x i ) i ∈ I with x i (cid:54)= x j whenever i and j are distinct elements of I . As in theprevious example, there is an obvious action of any bijection σ : I −→ J that permutesthe coordinates. For each fixed finite set I , the set of types of structures over I is usuallycalled the unordered configuration space E X ( I ). E6.
There is a species of parts ℘ that sends each finite set I to the collection 2 I of parts of I ,and sends each bijection σ : I −→ J to the induced bijection σ ∗ : 2 I −→ J . In a similarway, if n is a positive integer, there is a species ℘ n which sends each finite set I to theset ℘ n of its subsets of cardinality n ; notice that ℘ n ( I ) is empty if I has less than n elements, and that ℘ n is a subspecies of ℘ for each n . E7.
A graph with vertices on a set I is a pair ( I , E ) where E is a collection of 2-subsets of I .For each finite set I , let Gr ( I ) be the collection of graphs on I . If σ : I −→ J is a bijectionand ( I , E ) is a graph on I , we set Gr ( σ )( I , E ) = ( J , σ ( E )). This defines the species Gr ofgraphs . E8.
For each finite set I , let L ( I ) be the collection of linear orders on I . If σ : I −→ J isa bijection, we let L ( σ ) send a linear order i i · · · i t on the set I to the linear order σ ( i ) · · · σ ( i t ) on J . This defines the species L of linear orders. The category Sp k of species over k Mod is abelian and monoidal with respect to the “point-wise” Hadamard product given for each pair of species X and Y and each finite set I by theformula( X ⊗ H Y )( I ) = X ( I ) ⊗ Y ( I ).It turns out that algebras for this tensor product are rather simple: endowing a species X with the structure of an algebra for ⊗ H amounts to endowing each individual space X ( I ) T HE COHOMOLOGY OF TWISTED COALGEBRAS with an algebra structure and, in particular, does not combine in any interesting the se-quence of spaces defined by X .There is another product in Sp k , called the Cauchy product which we will denote by ⊗ ,which will play a central role in all that follows, and which categorifies the usual (Cauchy)product of power series. In particular, it will intertwine into a single object the variouspieces of X , and provide us with a richer product and, hence, with a more interesting classof (co)algebras. Definition 1.4.
Let X and Y be linear species over k . The Cauchy product X ⊗ Y is the linearspecies such that for every finite set I ( X ⊗ Y )( I ) = (cid:77) ( S , T ) (cid:96) I X ( S ) ⊗ Y ( T ),the direct sum running through all decompositions of I of length two, and for every bijection σ : I −→ J ( X ⊗ Y )( σ ) = (cid:77) ( S , T ) (cid:96) I X ( σ S ) ⊗ Y ( σ T ).As it happens with the Hadamard product, the Cauchy product is better understood whenviewing species as the product of representations of the various symmetric groups. Indeed,for each n and each pair ( p , q ) with p + q = n , there is an isomorphism (cid:77) S ⊆ I , S = p X ( S ) ⊗ Y ( T ) (cid:39) Ind S p + q S p × S q ( X [ p ] ⊗ Y [ q ]),and these collect to give an isomorphism( X ⊗ Y )([ n ]) (cid:39) (cid:77) p + q = n Ind S p + q S p × S q ( X [ p ] ⊗ Y [ q ]).This construction extends to produce a bifunctor ⊗ : Sp k × Sp k −→ Sp k . In what follows,whenever we speak of the category Sp k , we will view it as a monoidal category with themonoidal structure given by the Cauchy product.It is important to notice the construction of the Cauchy product in Sp k carries over tothe category Sp ( C ) when C is any monoidal category with finite coproducts which commutewith its tensor product. The main example of this phenomenon happens when C is thecategory Set . If X and Y are set species, the species X ⊗ Y has( X ⊗ Y )( I ) = (cid:71) ( S , T ) (cid:96) I X ( S ) × Y ( T ), EDRO T AMAROFF so that a structure z of species X ⊗ Y over a set I is determined by a decomposition ( S , T ) of I and a pair of structures ( z , z ) of species X and Y over S and T , respectively.The linearization functor L : Sp −→ Sp k preserves the monoidal structures we have definedon these categories, in the sense there is a natural isomorphism L ( X ⊗ Y ) −→ L X ⊗ L Y for each pair of objects X , Y in Sp . For details on such monoidal functors see [12, Chapter XI§4]. The following will be useful, and we record it for future reference: Proposition 1.1. If X , Y , . . . , Y r are linear species, a map of species α : X −→ Y ⊗ · · · ⊗ Y r determines and is determined by a choice of equivariant k-module maps α I : X ( I ) −→ (cid:77) Y ( S ) ⊗ · · · ⊗ Y r ( S r ), one for each finite set I , the direct sum running through decompositions ( S , . . . , S r ) of lengthr of I . (cid:206) The map α I is specified uniquely by its components at each decomposition S = ( S , . . . , S r ),which we denote α ( S , . . . , S r ) without further mention to the set I which is implicit, for (cid:83) S equals I . Moreover, it suffices to specify α I for I the sets (cid:130) n (cid:131) with n ∈ (cid:78) . This said,we will usually define a map α : X −→ Y ⊗ · · · ⊗ Y r by specifying its components at eachdecomposition of length r of I . An associative algebra ( X , µ , η ) in the category Sp k , which we will call simply a twisted al-gebra , is determined by a multiplication µ : X ⊗ X −→ X and a unit η : −→ X . Specifying the first amounts to giving its components µ ( S , T ) : X ( S ) ⊗ X ( T ) −→ X ( I )at each decomposition ( S , T ) of every finite set I , and specifying the latter amounts to achoice of the element η ( ∅ )(1) ∈ X ( ∅ ), which we will denote by 1 if no confusion shouldarise. We think of the multiplication as an operation that glues partial structures on I , andof the unit as an “empty” structure.For example, the species of graphs admits a multiplication k Gr ⊗ k Gr −→ k Gr which is thelinear extension of the map that takes a pair of graphs ( g , g ) ∈ Gr ( S ) × Gr ( T ) and constructs T HE COHOMOLOGY OF TWISTED COALGEBRAS the disjoint union g (cid:116) g on I . The unit for this multiplication is the empty graph ∅ ∈ Gr ( ∅ ).One can readily check µ is associative and unital with respect to η , so we indeed have aalgebra k Gr .Dually, a coalgebra ( X , ∆ , ε ) in Sp k , which we call a twisted coalgebra , is determined by acomultiplication ∆ : X −→ X ⊗ X and a counit ε : X −→ . The comultiplication has, at each decomposition ( S , T ) of I , acomponent ∆ ( S , T ) : X ( I ) −→ X ( S ) ⊗ X ( T ), which we think of as breaking up a combinator-ial structure on I into substructures on S and T , while the counit is a map of k -modules X ( ∅ ) −→ k .To continue with our example, the linearization of the species of graphs admits a comulti-plication k Gr −→ k Gr ⊗ k Gr that sends a graph g on a set I to g S ⊗ g T ∈ k Gr ( S ) ⊗ k Gr ( T ), where g S and g T are the subgraphs induced by g on S and T , respectively. This comultiplicationadmits as counit the morphism ε : k Gr −→ that assigns 1 ∈ k to the empty graph. In thisway, we obtain a coalgebra structure on k Gr which is, in fact, compatible with the algebrastructure we described in the previous paragraph: we therefore have a bialgebra structureon k Gr .Our main example of a bialgebra in Sp k is the provided by the following proposition. Proposition 1.2.
The linearized exponential species E is a twisted bialgebra with multiplica-tion and comultiplication with components µ ( S , T ) : E ( S ) ⊗ E ( T ) −→ E ( I ), ∆ ( S , T ) : E ( I ) −→ E ( S ) ⊗ E ( T ) at each decomposition ( S , T ) of a finite set I such that µ ( S , T )( ∗ S ⊗ ∗ T ) = ∗ I , ∆ ( S , T )( ∗ I ) = ∗ S ⊗ ∗ T and with unit and counit the morphisms ε : E −→ and η : −→ E such that ε ( ∗ ∅ ) = and η (1) = ∗ ∅ .Proof. The verifications needed to prove this follow immediately from the fact that E ( I ) is asingleton for every finite set I . (cid:206) The exponential species plays a central role in the category of bialgebras, as evinced bythe following proposition.
Proposition 1.3.
1. The exponential species E admits a unique structure of set- theoretic bialgebra. EDRO T AMAROFF
2. If X is a set theoretical coalgebra in Sp , the linearization of the unique morphism ofspecies X −→ E is a morphism of coalgebras.3. In particular, every twisted coalgebra coming from a set theoretic coalgebra is canonic-ally an E -bicomodule.Proof. If s is a singleton set and x is any set, there is a unique function x −→ s , and it followsfrom this, first, that the bialgebras structure defined on E is the only linearized bialgebrastructure, and, second, that if X is a species in Sp , there is a unique morphism of species X −→ E . If X is a pre-coalgebra in Sp , the following square commutes because E ( S ) × E ( T )has one element: E ( I ) E ( S ) × E ( T ) X ( I ) X ( S ) × X ( T ), ∆∆ and, by the same reason, X −→ E is pre-counital. All this shows that the exponential spe-cies E is terminal in the category of linearized coalgebras. This completes the proof of theproposition. (cid:206) We will fix some useful notation to deal with coalgebras. Let X = k X be a linearizedspecies that is a coalgebra in Sp k . If z is an element of X ( I ), we write ∆ ( I )( z ) = (cid:88) z (cid:13) S ⊗ z (cid:12) T with z (cid:13) S ⊗ z (cid:12) T denoting an element of X ( S ) ⊗ X ( T ) (not necessarily an elementary tensor,à la Sweedler).Consider now a left E -comodule X with coaction λ : X −→ E ⊗ X . Since E ( S ) = k { ∗ S }, thecomponent X ( I ) −→ E ( S ) ⊗ X ( T ) can canonically be viewed as map X ( I ) −→ X ( T ) which wedenote by λ IT , and call the it the restriction from I to T to the right .In these terms, that λ be counital means λ II is the identity for all finite sets I , and theequality 1 ⊗ λ ◦ λ = ∆ ⊗ ◦ λ , which expresses the coassociativity of λ , translates to the condi-tion that we have λ IA = λ BA ◦ λ IB for any chain of finite sets A ⊆ B ⊆ I . It follows that, if FinSet inc is the category of finite sets and inclusions, a left E -comodule X in Sp k can be viewed as apre-sheaf FinSet inc −→ k Mod .These are usually called FI-modules in the literature, see for example [7]. When convenient,we will write z (cid:12) S for λ IS ( z ) without explicit mention to I , which will usually be understood T HE COHOMOLOGY OF TWISTED COALGEBRAS from context. Using this notation, we can write the coaction on X as λ ( I )( z ) = (cid:88) e S ⊗ z (cid:12) T .Of course the same consideration apply to a right E -comodule, and we write z (cid:13) T for ρ IT ( z ). If X is both a left and a right E -comodule with coactions λ and ρ , the compatilibitycondition for it to be an E -bicomodule is that, for any finite set I and pair of non-necessarilydisjoint subsets S , T of I , we have ρ SS ∩ T λ IS = λ TS ∩ T ρ IT . Schematically, we can picture this asfollows: S ∩ TS TI ρ λρλ
There is a category
FinSet binc such that an E -bicomodule is exactly the same as a pre-sheaf FinSet binc −→ Sp k ; we leave its construction to the categorically inclined reader. If the struc-ture on X is cosymmetric, we will write z ∥ S for the common value of z (cid:13) S and z (cid:12) S . There isa close relation between linearized coalgebras and linearized E -bicomodules, as describedin the following proposition. Proposition 1.4.
Let ( X , ∆ ) be a linearized coalgebra, and let f X : X −→ E be the uniquemorphism of linearized coalgebras described in Proposition 1.3. There is on X an E -bicomodulestructure so that the coactions λ : X −→ E ⊗ X and ρ : X −→ X ⊗ E are obtained from postcom-position of ∆ with f X ⊗ and ⊗ f X , respectively. (cid:206) We refer the reader to [2, Chapter 8, §3, Proposition 29]. Remark that, with this proposi-tion at hand, the notation introduced for bicomodules and that introduced for coalgebrasis consistent.
Let X be a twisted bialgebra with structure maps ∆ and µ . Recall that a species X is connected if X ( ∅ ) is free of rank one. The following result in [2] states every connected twisted bial-gebra is automatically a Hopf algebra, and this automatically endows the various categoriesof representations of X with extra structure.More generally, a twisted bialgebra X is a Hopf algebra precisely when X ( ∅ ) is a Hopf k -algebra, and the antipode of X can, in that case, be explicitly constructed from the antipodeof X ( ∅ ) —this is a variant of what is known as Takeuchi’s theorem , see the monograph [3,Proposition 9] for more details.
EDRO T AMAROFF Theorem 1.1.
Let ( X , µ , ∆ ) be a twisted bialgebra.1. If X is a Hopf algebra with antipode s, then X ( ∅ ) is a Hopf k-algebra with antipode s ( ∅ ) .2. If X ( ∅ ) is a Hopf k-algebra with antipode s , then X is a Hopf algebra, and s can beiteratively constructed from s , µ and ∆ .3. In particular, if X is a connected bialgebra, X is a Hopf algebra.Proof. For a proof and an explicit formula for s in terms of s , we refer the reader to [2,Chapter 8, §3.2, Proposition 8.10, and §4.2, Proposition 8.13]. The third part follows fromthe second since k is, in a unique way, a Hopf k -algebra. (cid:206) We define some connected bialgebras that will be of interest in Section 2. In view of theprevious result, they are all Hopf algebras in the category of linear species. Remark that,since the algebraal category Sp k is symmetric, the tensor product of two twisted Hopf al-gebras is again a twisted Hopf algebra, so the following examples provide further ones bycombining them into products. In all cases the unit and counit are the projection and theinclusion of the unit in the component of ∅ . H1.
Fix a finite set I and a decomposition ( S , T ) of I . If (cid:96) and (cid:96) are linear orders on S and T respectively, their concatenation (cid:96) · (cid:96) is the unique linear order on I that restricts to (cid:96) in S and to (cid:96) in T , and such that s < t if s ∈ S and t ∈ T ; this operation is in generalnot commutative. If (cid:96) is a linear order on I , write (cid:96) | S for the restriction of (cid:96) to S , and (cid:96) for the reverse order to (cid:96) . The species of linear orders L admits a bialgebra structuresuch that• multiplication is given by concatenation: µ ( S , T )( (cid:96) , (cid:96) ) = (cid:96) · (cid:96) ,• comultiplication is given by restriction: ∆ ( S , T )( (cid:96) ) = (cid:96) | S ⊗ (cid:96) | T .In particular, this endows the linearization L with a cosymmetric bicomodule structureover E . The map L −→ E that sends a linear order on a finite set I to ∗ I = { I } is a mapof bialgebras. The antipode is given, up to sign, by taking the reverse of a linear order: s ( I )( (cid:96) ) = ( − I (cid:96) . H2.
If ( S , T ) is a decomposition of a finite set I , and F = ( F , . . . , F s ) and G = ( G , . . . , G t ) arecompositions of S and of T , respectively, the concatenation F · G is the composition( F , . . . , F s , G , . . . , G t ) of I . If F = ( F , . . . , F t ) is a composition of I , the restriction of F to S is the composition F | S of S obtained from the decomposition ( F ∩ S , . . . , F t ∩ S ) of S by deleting empty blocks, which usually has shorter length than that of F . Finally, thereverse of a composition F is the composition F whose blocks are listed in the reverseorder of those in F . The species C of compositions has a bialgebra structure such that• multiplication is given by concatenation: µ ( S , T )( F , G ) = F · G ,• comultiplication is given by restriction: ∆ ( S , T )( F ) = F | S ⊗ F | T . T HE COHOMOLOGY OF TWISTED COALGEBRAS
This is cocommutative but not commutative. The morphism L −→ C that sends a linearorder i · · · i t on a set I to the composition ({ i }, . . . , { i t }) is a map of bialgebras. Theformula for the antipode is not as immediate as the previous ones. For details, see[3, Section 11]. H3.
If ( S , T ) is a decomposition of a finite set I , and X and Y are partitions of S and T ,respectively, the union X ∪ Y is a partition of I . If X is a partition of I , then X | S = { x ∩ S : x ∈ X } (cid:224) { ∅ } is a partition of S , which we call the restriction of X to S . The species P ofpartitions admits a bialgebra structure such that• multiplication is given by the union of partitions: µ ( S , T )( X , Y ) = X ∪ Y ,• comultiplication is given by restriction: ∆ ( S , T )( X ) = X | S ⊗ Y | T .This is both commutative and cocommutative. The map C −→ P that sends a decom-position F of a set I to the partition X of I consisting of the blocks of F is a bialgebramap. The morphism E −→ P that sends ∗ I = { I } to the partition of I into singletons isalso a map of bialgebras. See [3, Theorem 33] for a formula for the antipode of P . H4. If p is a poset with underlying set I , and ( S , T ) is a decomposition of I , we say S is a lowerset of T with respect to p and write S ≺ p T if no element of T is less than an element of S for the order p , and we write p S for p ∩ ( S × S ), the restriction of p to S . The linearizedspecies Pos of posets admits a bialgebra structure so that• multiplication is given by the disjoint union of posets: if p and p are posets withunderlying sets S and T , respectively, µ ( S , T )( p , p ) = p (cid:116) p ,• comultiplication is obtained by lower sets and by restriction: for p a poset definedon S (cid:116) T , we set ∆ ( S , T )( p ) = p S ⊗ p T if S ≺ p T , and set ∆ ( S , T )( p ) = p and p are posets on disjoint sets S and T , respectively, let p ∗ p be the usual join ofposets. This is associative and has unit the empty poset, and the inclusion of linear ordersinto posets L −→ P is a morphisms of algebras if L is given the concatenation product. Wealso remark that the maps described above fit into a commutative diagram of Hopf algebrasas illustrated in the figure CL PE and we will analyse the resulting maps in cohomology in Section 2. For more examples ofHopf algebras in species, and their relation to classical combinatorial results, we refer thereader to [2, Chapter 13].
EDRO T AMAROFF Let H be a twisted coalgebra and X a -bicomodule, and let us define a cosimplicial k -module C ∗ ( X , H ) as follows. For each n ∈ (cid:78) :1. Define C n ( X , H ) to be Hom Sp k ( X , H ⊗ n ) the set of maps in Sp k from X to the iteratedtensor product H ⊗ n .2. For 0 < i < n +
1, consider the map d i : C n ( X , H ) −→ C n + ( X , H ) induced by post-composition with the coproduct of H at the i th position.3. For i = n +
1, let d , d n + : C n ( X , H ) −→ C n + ( X , H ) be the maps obtained post com-posing with the left and right comodule maps of X , respectively.It is straightforward to check that, by virtue of the coassociativity of H and the bicomoduleaxioms, the maps above satisfy the usual cosimplicial identities. It follows that if for each n ∈ (cid:78) we define the alternated sum δ n = (cid:80) n + i = ( − i d i , we obtain a cohomologically gradedcomplex ( C ∗ ( X , H ), δ ∗ ). Definition 2.1.
The cohomology of X with values in H is the cohomology of the complex( C ∗ ( X , H ), δ ∗ ), and we denote it by H ∗ ( X , H ).The homologically inclined reader will notice that these cohomology groups are equal toExt ∗ ( X , H ) with the Ext taken in the category of H -bicomodules. In the following we willmainly consider the case in which H is the exponential species, but will make it clear whena certain result can be extended to other twisted coalgebras. Usually, it will be necessary that H is linearized and with a linearized bimonoid structure, and we will usually require that H be connected. Because of the plethora of relevant examples of such bimonoids found in [2]and other articles by the same authors, such as [3], there is no harm in restricting ourselvesto such species.Fix an E -bicomodule X . The complex C ∗ ( X , E ), which we will denote more simply by C ∗ ( X ), has in degree q the collection of morphisms of species α : X −→ E ⊗ q . Such a morph-ism is determined by a collection of k -linear maps α ( I ) : X ( I ) −→ E ⊗ q ( I ), one for each finiteset I , which is equivariant, in the sense that for each bijection σ : I −→ J between finite sets,and every z ∈ X ( I ), the equality σ ( α ( I )( z )) = α ( J )( σ z ) holds. Remark 2.1.
For each finite set I , the space E ⊗ q ( I ) is a free k -module with basis the tensorsof the form F ⊗ · · · ⊗ F q T HE COHOMOLOGY OF TWISTED COALGEBRAS with F = ( F , . . . , F q ) a decomposition of I ; for simplicity, we use the latter notation for suchbasis elements. In terms of this basis, we can write α ( I )( z ) = (cid:88) F (cid:96) q I α ( F )( z ) F where α ( F )( z ) ∈ k .We recall that the cochain α is completely determined by an equivariant collection offunctionals α ( F ) : X ( I ) −→ k , the components of α , one for each decomposition F of a finiteset I . The equivariance condition is now that, for a bijection σ : I −→ J , and ( F , . . . , F q ) adecomposition of I , we have α ( F , . . . , F q )( z ) = α ( σ ( F ), . . . , σ ( F q ))( σ z )for each z ∈ X ( I ). Recall that when writting α ( F )( z ) we omit I , recalling that it is always thecase I = ∪ F .Now fix a q -cochain α : X −→ E ⊗ q in C ∗ ( X ). By the remarks in the last paragraph, todetermine the ( q + δα : X −→ E ⊗ ( q + it is enough to determine its components. Lemma 2.1.
For each decomposition F = ( F , . . . , F q ) of a set I , then the component of the i thcoface d i α at F is given, for z ∈ X ( I ) , by ( d i α )( F , . . . , F q )( z ) = α ( F , . . . , F q )( z (cid:12) F c ) if i = , α ( F , . . . , F i ∪ F i + , . . . , F q + )( z ) if < i < q + , α ( F , . . . , F q − )( z (cid:13) F cq ) if i = q +
1. (1)
Proof.
Indeed, let us follow the prescription above and compute each coface map explicitly.If z ∈ X ( I ), to compute d α ( z ), we must coact on z to the left and evaluate the result at α ,that is(1 ⊗ α ◦ λ )( I )( z ) = (cid:88) ( S , T ) (cid:96) I ∗ S ⊗ α ( T )( z (cid:12) T ),and the coefficient at a decomposition F = ( F , . . . , F q ) is α ( F , . . . , F q )( z (cid:12) F c ). The sameargument gives the last coface map. Now consider 0 < i < q +
1, so that we must take z ∈ X ( I ),apply α , and then comultiply the result at coordinate i . Concretely, write α ( I )( z ) = (cid:88) F (cid:96) q I α ( I )( F )( z ) F and pick a decomposition F (cid:48) = ( F , . . . , F q ) into q + I . There exists then a unique EDRO T AMAROFF F (cid:96) q I such that 1 i − ⊗ ∆ ⊗ q − i ( F ) = F (cid:48) , to wit, F = ( F , . . . , F i ∪ F i + , . . . , F q ), and in this waywe obtain the formulas of Equation (1). (cid:206) Since E is counital, the complex above admits codegeneracy maps, which are much easierto describe: they are obtained by inserting an empty block into a decomposition. Con-cretely, for each j ∈ {0, . . . , q + σ j α )( F , . . . , F q )( z ) = α ( F , . . . , F j , ∅ , F j + , . . . , F q ).As a consequence of this, a cochain α : X −→ E ⊗ q in C ∗ ( X ) is in the normalized subcomplex C ∗ ( X ) if its components are such that α ( F )( z ) = ∈ k whenever F contains an empty block.Alternatively, we can construct a (non-unital) coalgebra E with E ( ∅ ) = E ( I ) = E ( I )whenever I is nonempty, and describe the normalized complex C ∗ ( X ) as the complex ofmaps X −→ E ⊗∗ with differential induced by the alternating sum of the coface maps we justdescribed. Remark 2.2.
For each finite set I the space E ⊗ q ( I ) has basis the compositions of I into q blocks, while E ⊗ q ( I ) has basis the decompositions of I into q blocks. In particular, E ⊗ q ( I ) = q > I , while E ⊗ q ( I ) is always nonzero. This observation will be crucial in Section 3. Since the twisted coalgebra E is, in fact, a twisted commutative Hopf algebra, we can endowthe complex C ∗ ( X ) with the structure of a dga algebra and hence produce on the cohomo-logy groups H ∗ ( X , E ) a structure of a commutative associative algebra, as follows.First, let us give an alternative way of constructing the complex C ∗ ( X ). Let Ω ∗ ( E ) denotethe cobar construction on the coalgebra E . This is a dg twisted algebra which is freely gener-ated by s − E , the shift of the species E without the counit, and whose differential is inducedfrom the coproduct of E : it is the unique coderivation extending the map ∆ : s − E −→ ( s − E ) ⊗ ⊆ Ω ∗ ( E ).We can then form the spaceHom Sp k ( X , Ω ∗ ( E ))which, as a graded vector space, coincides with the normalized complex for C ∗ ( X , E ): theway we shifted E makes sure that maps X −→ E ⊗ n live in degree n . Observe, moreover, thatthat hom-set above inherits a differential δ by postcomposition with the differential d : Ω ∗ ( E ) −→ Ω ∗+ ( E ), T HE COHOMOLOGY OF TWISTED COALGEBRAS and this coincides in fact with the internal sum of the coface maps above, omitting the end-points 0 and n +
1. To obtain the full differential δ , we consider the canonical degree − τ : E −→ Ω ∗ ( E ) and the differential δ obtained by the following composition where p is the degree of ϕ : X −→ E ⊗ p : δ ( ϕ ) = µ Ω ∗ ( E ) ( ϕ ⊗ τ ◦ λ + ( − p τ ⊗ ϕ ◦ ρ ).A perhaps involved but straightforward computation shows that δ − δ coincides with δ ,so that we obtain a new description of the complex C ∗ ( X , E ) a complex twisted by τ (thesummand δ is the twist determined by τ ): C ∗ ( X , E ) = (cid:161) Hom τ ( X , Ω ∗ ( E )), δ − δ (cid:162) . Proposition 2.1.
The dg coalgebra Ω ∗ ( E ) is in fact a dg bialgebra if we endow it with the shuffle coproduct induced from the commutative product of E , which we will denote by ∆ Ω ∗ ( E ) .Proof. This statement is completely dual to the classical statement (see for example Chapter8 in [17]) that if A is a commutative algebra then the bar construction B A is a commutativealgebra with the shuffle product induced from the commutative product of A . We remindthe reader that it is crucial that A be commutative (and hence, in our case, that E be cocom-mutative) for this product to be compatible with the differential of B A . (cid:206) Definition 2.2.
We define the external product − × − : C ∗ ( X , E ) ⊗ C ∗ ( X , E ) −→ C ∗ ( X ⊗ Y , E )so that for two cochains ϕ , ψ ∈ C ∗ ( X , E ) we have ϕ × ψ = µ Ω ∗ ( E ) ◦ ( ϕ ⊗ ψ ).Note that we use the fact E is a twisted Hopf algebra, which implies that the category of E -bicomodules admits an internal tensor product. Concretely, if X and Y are E -bicomodules,we endow the tensor product X ⊗ Y with the left and right diagonal actions coming from theproduct of E . Remark 2.3.
In case we have a comultiplication map ∆ : X −→ X ⊗ X making X into a coal-gebra in the category of E -bicomodules, we can use this external product to obtain a cupproduct in C ∗ ( X ), which we will write − (cid:94) − : C ∗ ( X , E ) ⊗ C ∗ ( X , E ) −→ C ∗ ( X , E ). Remark 2.4.
In general, the algebra H ∗ ( X ) will be non-commutative: for example, if X isconcentrated in cardinal 0, then the datum of X really amounts to that of the coalgebra X [0], EDRO T AMAROFF a coalgebra in k -modules, and H ∗ ( X ) is the algebra dual to it, which may well be non-commutative.If X is a E -bicomodule and ∆ : X → X ⊗ X a morphism of E -bicomodules, we write, foreach I and each z ∈ X [ I ], ∆ [ I ]( z ) = (cid:88) ( S , T ) (cid:96) I z ( S ) ⊗ z ( T ) à la Sweedler, with each summand z ( S ) ⊗ z ( T ) appearing here standing for an element —notnecessarily an elementary tensor— of the submodule X [ S ] ⊗ X [ T ] of ( X ⊗ X )[ I ]. If α : X → E ⊗ p and β : X → E ⊗ q are a p - and a q -cochain in the complex C ∗ ( X ), then their product α (cid:94) β ∈ C p + q ( X ) has coefficients given by α (cid:94) β ( F )( z ) = α ( F p )( z ( F p ) ) · β ( F p + p + q )( z ( F p + p + q ) )for all I , all decompositions F = ( F , . . . , F p + q ) of I and all z ∈ X [ I ]. Here we are being suc-cinct and writing F i , i + j for both the decomposition ( F i , . . . , F i + j ) obtained from F and for theunion of this decomposition. Our main source of examples of coalgebras in E -bicomodulescomes from the following simple observation: Proposition 2.2.
Let X (cid:48) be a nonempty set-valued species with left and right restrictions andlet X be the E -bicomodule obtained by linearization from X (cid:48) . There is a morphism of E -bicomodules ∆ : X → X ⊗ X such that ∆ [ I ]( z ) = (cid:88) ( S , T ) (cid:96) I z (cid:13) S ⊗ z (cid:12) Tfor each finite set I and each z ∈ X (cid:48) [ I ] . (cid:206) In what follows, we will usually consider every E -bicomodule whose underlying speciesis a linearization of a twisted coalgebra in the way described in this proposition. The objective of this chapter is to obtain an alternative and more useful description of thecohomology groups of an E -bicomodule X . We show that for every E -bicomodule X thereis a filtration on the complex C ∗ ( X ) giving rise to a spectral sequence of algebras whichconverges to H ∗ ( X ). If X is weakly projective , that is, if for each non-negative integer j , X [ j ] is a projective kS j -module, this collapses at the E -page. Because we can completelydescribe this page, this provides us with a complex that calculates H ∗ ( X ), and which can beused for effective computations. T HE COHOMOLOGY OF TWISTED COALGEBRAS
To be explicit, by this we mean each component of this complex is finitely generatedwhenever X has finitely many structures on each finite set, and in that case the differentialof an element depends on finite data obtained from it —this is in contrast with the situationof C ∗ ( X ). Moreover, the spectral sequence is one of algebras whenever we endow C ∗ ( X )with a cup product arising from a diagonal map ∆ : X −→ X ⊗ X , so these remarks applyto the computation of the cup product structure of H ∗ ( X ), and we exploit this for the cupproduct we defined in Section 2. Some more running conventions . Let X be a species. The support of X is the set of non-negative integers j for which X [ j ] is nontrivial. We say X is finitely supported if is has finitesupport, and that it is concentrated in cardinal j if the support of X is exactly { j }. The sup-port of a nontrivial species X is contained in a smallest interval of non-negative integers,whose length we call the length of X . The species X is of finite type if X [ j ] is a finitely gen-erated k -module for each nonnegative integer j , and it is finite if it is both of finite type andfinitely supported. Let X be a species in Sp k and let j be a non-negative integer. We define species τ j X and τ j X ,which we call the upper truncation of X at j and the lower truncation of X after j as follows.For every finite set I , we put τ j X ( I ) = X ( I ) if I (cid:201) j ,0 else, τ j X ( I ) = X ( I ) if I (cid:202) j ,0 else.If σ : I −→ J is a bijection then ( τ j X )( σ ) = X ( σ ) whenever I has at most j elements, while( τ j X )( σ ) is the unique isomorphism 0 −→ τ j X )( σ ) = X ( σ ) whenever I has at least j elements, while ( τ j X )( σ ) is the unique isomorphism 0 −→ X , andthat there is a short exact sequence0 −→ τ j X −→ X −→ τ j + X −→
0. (2)By convention, τ j X = τ j X = X if j is negative. We will write τ ji for the composition τ i ◦ τ j , which is the same as τ j ◦ τ i , and X [ j ] instead of τ jj ; this species is concentrated in cardinal j . This will be of use in Section 3.2. Observe that we can carry out these constructions in thecategories of H -(bi)comodules for any twisted coalgebra H . Precisely, we have the followingproposition: EDRO T AMAROFF Proposition 3.1.
Let H be a twisted coalgebra, let X be a left H -comodule, and fix a non-negative integer j . T1.
The truncated species τ j X is an H -subcomodule of X in such a way that the inclusion τ j X −→ X is a morphism of H -comodules, and T2. the truncated species τ j X is uniquely an H -comodule in such a way that the morphismsin the short exact sequence (2) in Sp k are in fact of H -comodules. It is clear the above can, first, be extended to H -bicomodules, and second, be dualizedto H -modules, and then extended to H -bimodules. This provides a spectral sequence formonoids and modules, which we will not discuss. Proof.
Denote by λ the coaction of X . To see T1 , we have to show that λ ( τ j X ) ⊆ H ⊗ τ j X ,which is immediate, and T2 is deduced from this: we identify τ j X with the quotient X / τ j X ,which inherits an H -comodule structure making the maps in the short exact sequence (2)maps of H -comodules. (cid:206) In what follows, we will need to identify the comodule structure of X [ j ]. This is done inthe following lemma. Lemma 3.1.
Let H be a connected twisted coalgebra. An H -(bi)comodule concentrated inone cardinal necessarily has the trivial H -coaction. (cid:206) Proof.
For trivial reasons, the comodule map(s) land only on the summand that is com-pletely determined by the counitality axioms, meaning that the action is trivial: for any x ∈ X we have that λ ( x ) = x ⊗ ρ ( x ) = ⊗ x . (cid:206) Let X be an E -bicomodule. For each integer p , let F p C ∗ ( X ) be the collection of chains thatvanish on τ p − X . This is a subcomplex because τ p X is a E -subbicomodule of X , so we havea descending filtration of the complex C ∗ ( X ). When there is no danger of confusion, we willwrite F p C ∗ instead of F p C ∗ ( X ). This filtration in C ∗ ( X ) induces a filtration on H ∗ = H ∗ ( X )with F p H ∗ ( X ) = im( H ∗ ( F p C ∗ ) −→ H ∗ ), and we write E ( H ) for the bigraded object with E p , q ( H ) = F p H p + q F p + H p + q .As explained in detail in [17, Chapter 2, §2], this filtration gives rise to a cohomology spec-tral sequence ( E r , d r ) r (cid:202) . According to the construction carried out there, the E -page has E p , q = F p C p + q F p + C p + q T HE COHOMOLOGY OF TWISTED COALGEBRAS and differential d pq : E pq −→ E p , q + induced by that of C ∗ ( X ), and, in particular, we have E pq = p < p + q <
0. Moreover:
Proposition 3.2.
Let p be an integer.1. There is a natural isomorphism F p C ∗ ( X ) −→ C ∗ ( τ p X ) that induces, in turn, an iso-morphism ( E p ∗ , d p ∗ ) −→ C p +∗ ( X ( p )) , so that2. for every integer q, there are isomorphisms E pq −→ H p + q ( X ( p )) , and, viewing this asan identification,3. the differential d pq : E pq −→ E p + q is the composition of the connecting homomorph-ism H p + q ( X ( p )) −→ H p + q + ( τ p + X ) of the long exact sequence corresponding to theshort exact sequence −→ X ( p ) −→ τ p X −→ τ p + X −→ and the map H ∗ ( ι ) induced bythe inclusion ι : X ( p + −→ τ p + X .Proof. The exact sequence of E -bicomodules0 −→ τ p − X −→ X π −→ τ p X −→ Sp k , so applying the functor C ∗ ( − ) gives an exact sequence0 −→ C ∗ ( τ p X ) −→ C ∗ ( X ) −→ C ∗ ( τ p − X ) −→ F p C ∗ ( X ) with C ∗ ( τ p X ), since the injective map C ∗ ( π )has image the kernel of C ∗ ( ι ), which is, by definition, F p C ∗ ( X ). This proves the first claim ofthe proposition.Similarly, we have a short exact sequence of bicomodules0 −→ X ( p ) −→ τ p X −→ τ p + X −→ Sp k , and which gives us the exactness of the second row of the following com-mutative diagram:0 F p + C ∗ F p C ∗ E p ∗ C ∗ ( τ p + X ) C ∗ ( τ p X ) C ∗ ( X ( p )) 0The desired natural isomorphism E p , ∗ −→ C ∗ ( X ( p )) is the unique dashed arrow that ex-tends the commutative diagram, and this proves the second claim of the proposition. Toprove the last one, we note the diagram above can be viewed as an isomorphism of exactsequences , and so the connecting morphisms are also identified. Moreover, the differential EDRO T AMAROFF at the E -page is induced from the connecting morphism of long exact sequence associatedto the exact sequence0 −→ F p + C ∗ −→ F p C ∗ −→ F p C ∗ / F p + C ∗ −→ F p + C ∗ −→ F p + C ∗ / F p + C ∗ which correspond, under our isomorph-isms, to the connecting morphism of the short exact sequence0 −→ X ( p ) −→ τ p X −→ τ p + X −→ C ∗ ( ι ) : C ∗ ( τ p + X ) −→ C ∗ ( X ( p + (cid:206) We will prove in the next section that the spectral sequence just constructed converges to H ∗ ( X ). A first step towards this is the following result: Proposition 3.3.
The filtration is bounded above and complete.Proof.
Using the identification provided by the isomorphisms F p C ∗ −→ C ∗ ( τ p , X ) of Pro-position 3.2 and the split exact sequences 0 −→ τ p X −→ X −→ τ p + X −→ C ∗ ( X )/ F p + C ∗ ( X ) with C ∗ ( τ p X ). In these terms, what the propositionclaims is that the canonical map C ∗ ( X ) −→ lim ←−− C ∗ ( τ p X )is an isomorphism, and this is clear: if a cochain vanishes on every τ p X then it is zero, sothe map is injective, and if we have cochains α p : τ p X −→ E ⊗∗ that glue correctly, we obtaina globally defined cochain α : X −→ E ⊗∗ , so the map is surjective. (cid:206) Proposition 3.4. If X vanishes in cardinals above N , then the normalized complex C ∗ ( X ) vanishes in degrees above N , and, a fortiori, the same is true for H ∗ ( X ) .Proof. Let p > N , consider a p -cochain α in the normalized complex C ∗ ( X ), and let us showthat α vanishes identically. Indeed, if I is a finite set, the map α ( I ) : X ( I ) −→ E ⊗ p ( I ) is zero: if I has more than p elements, its domain is zero because X vanishes on I , and if I has at most p elements, then its codomain is zero, since there are no compositions of length p of I . (cid:206) This has two important consequences, the first of which will be thoroughly exploited inthe next sections.
Corollary 3.1.
Fix an integer j .1. We have H q ( τ j X ) = if q > j .2. The E -page of the spectral sequence lies in a cone in the fourth quadrant. (cid:206) T HE COHOMOLOGY OF TWISTED COALGEBRAS
Because the E -page of the spectral sequence involves the cohomology of the species X ( p ) for p (cid:202)
0, we turn our attention to the cohomology of species concentrated in a car-dinal. E -page This section is devoted to describing the E -page of the spectral sequence, and showing itconcentrated in one row —so that the spectral sequence degenerates at the E -page— when X is weakly projective in Sp k . Recall that by this we mean that, for each non-negative integer j , X [ j ] is a projective kS j -module.For j (cid:202) p (cid:202) −
1, let Σ p ( j ) be the collection of compositions of length p + j ]. We will identify the elements of Σ j − ( j ) with permutations of [ j ] in the obviousway. There are face maps ∂ i : Σ p ( j ) −→ Σ p − ( j ) for i ∈ {0, . . . , p } given by ∂ i ( F , . . . , F i , F i + , . . . , F p + ) = ( F , . . . , F i ∪ F i + , . . . , F p + )that make the sequence of sets Σ ∗ ( j ) = ( Σ p ( j )) p (cid:202)− into an augmented semisimplicial set.We write k Σ ∗ ( j ) for the augmented semisimplicial k -module obtained by linearizing Σ ∗ ( j ),and k Σ ∗ ( j ) (cid:48) for the dual semicosimplicial augmented k -module.There is an action of S j on each Σ p ( j ) by permutation, so that if τ ∈ S j and if ( F , . . . , F t ) isa composition of [ j ], then τ ( F , . . . , F t ) = ( τ ( F ), . . . , τ ( F t )).It is straightforward to check the coface maps are equivariant with respect to this action,so Σ ∗ ( j ) is, in fact, an augmented semisimplicial S j -set. Consequently, k Σ ∗ ( j ) and k Σ ∗ ( j ) (cid:48) have corresponding S j -actions compatible with their semi(co)simplicial structures.This complex Σ ∗ ( j ) is known in the literature as the Coxeter complex for the braid arrange-ment , and its cohomology can be completely described.
Proposition 3.5.
The complex associated to k Σ ∗ ( j ) (cid:48) computes the reduced cohomology of a ( j − -sphere with coefficients in k, that is,H p ( k Σ ( j ) (cid:48) ) = if p (cid:54)= j − k (cid:130) ξ j (cid:131) if p = j − The non-trivial term is the k-module freely generated by the class of the map ξ j : k Σ ∗ ( j ) −→ ksuch that ξ j ( σ ) = δ σ = and the action of kS j on H j − ( k Σ ∗ ( j ) (cid:48) ) is the sign representation. EDRO T AMAROFF Remark 3.1.
In what follows, sgn j will denote the sign representation of kS j just described.Note that, when j = S j − = ∅ , and the reduced cohomology of such space is concentratedin degree −
1, where it has value k . Proof.
We sketch a proof, and refer the reader to [1] and [5] for details. The braid arrange-ment B j of dimension j in (cid:82) j is the collection of hyperplanes{ H s , t : 1 (cid:201) s < t (cid:201) j }, with H s , t defined by the equation x t = x s . This arrangement has rank j − H with equation x + · · · + x j = K ofthe unit sphere S j − ⊆ H . Concretely, the r -dimensional simplices of K are in bijectionwith compositions of [ j ] into r + F = ( F , . . . , F r + ) corres-ponds to the r -simplex obtained by intersecting S j − with the subset defined by the equal-ities x s = x t whenever s , t are in the same block of F and the inequalities x s (cid:202) x t whenever t > s relative to the order of the blocks of F . It follows that k Σ ∗ ( j ) (cid:48) computes the reducedsimplicial cohomology of S j − , and the generator of the top cohomology group is the func-tional ξ j : k Σ ∗ ( j ) −→ k described in the statement of the proposition. More generally, if ξ σ : k Σ j − ( j ) −→ k is the functional that assigns σ to 1 and every other simplex to zero, then (cid:130) ξ σ (cid:131) = ( − σ (cid:130) ξ j (cid:131) . Because the action of S j on k Σ j − ( j ) (cid:48) is such that σξ j = ξ σ , this proves H j − ( k Σ ∗ ( j ) (cid:48) ) is the sign representation of kS j . (cid:206) We can describe the complex that calculates the cohomology of a species concentrated incardinal j in terms of the Coxeter complex Σ ∗ ( j ): Proposition 3.6.
Fix a non-negative integer j (cid:202) , and let X be an E -bicomodule concen-trated in cardinal j . There is an isomorphism of semi-cosimplicial k-modules Ψ ∗ : C ∗ ( X ) −→ Hom S j ( X [ j ], k Σ ∗ ( j ) (cid:48) [2]). In particular, if X [ j ] is a projective kS j -module, then H p ( X ) = when p (cid:54)= j and there is anisomorphism ξ : H j ( X ) −→ Hom S j ( X [ j ], sgn j ). This isomorphism is such that if α : X −→ E ⊗ j is a normalized j -cocycle, then ξ ( (cid:130) α (cid:131) )( z ) = (cid:88) σ ∈ S j ( − σ α ( σ )( z ) (cid:130) ξ j (cid:131) , (3) for each z ∈ X [ j ] . T HE COHOMOLOGY OF TWISTED COALGEBRAS If k is a field of characteristic coprime to j ! then every kS j -module is projective by virtueof Maschke’s theorem, so the above applies. If k is a field of characteristic zero, then everyspecies X is weakly projective, and conversely. Proof.
Since X is concentrated in cardinal j , a normalized p -cochain α : X −→ E ⊗ p is com-pletely determined by an equivariant k -linear map ˜ α : X [ j ] −→ E ⊗ p ( j ). Moreover, E ⊗ p ( j ) isa free k -module with basis the tensors F ⊗· · ·⊗ F p with ( F , . . . , F p ) a composition of [ j ], thatis, E ⊗ p ( j ) can be equivariantly identified with k Σ p − ( j ). Because E ⊗ p ( j ) is a free k -module,every k -linear map β : X [ j ] −→ E ⊗ p ( j ) corresponds uniquely to a map β t : X [ j ] −→ E ⊗ p ( j ) (cid:48) so that β t ( z )( F , . . . , F p ) = β ( F , . . . , F p )( z ).In this way we obtain a map Ψ ∗ : C ∗ ( X ) −→ Hom S j ( X [ j ], k Σ ∗ ( j ) (cid:48) [2]),which is clearly an isomorphism of graded k -modules, and this map is compatible withthe semicosimplicial structure and S j -equivariant. The non-trivial observation needed tocheck this is that the first and last coface maps of C ∗ ( X ) are zero: this follows from Lemma 3.1,which states X has trivial coactions, so these maps vanish upon normalization.Assume now that X [ j ] is kS j -projective, so that the functor Hom S j ( X [ j ], − ) is exact. Thecanonical map θ : H ∗ (Hom S j ( X [ j ], k Σ ∗ ( j ) (cid:48) [2])) −→ Hom S j ( X [ j ], H ∗ ( k Σ ∗ ( j ) (cid:48) [2]))is then an isomorphism, and we can conclude by Lemma 3.5 that H p ( X ) is zero except for p = j , and that we have a canonical isomorphism induced by Ψ and θξ : H j ( X ) −→ Hom S j ( X [ j ], sgn j ).It remains to prove the last formula. To this end, consider a j -cocycle α : X [ j ] −→ E ⊗ j ( j ).This corresponds under Ψ to the map X ( p ) −→ k Σ j − ( j ) (cid:48) that assigns to z the functional (cid:80) σ α ( σ )( z ) ξ σ . Passing to cohomology and using the equality (cid:130) ξ σ (cid:131) = ( − σ (cid:130) ξ j (cid:131) valid in viewof Lemma 3.5 for all σ ∈ S j , we obtain (3). (cid:206) Corollary 3.2. If X is weakly projective, then E is concentrated in the p-axis, whereE p ,01 (cid:39) Hom S p ( X ( p ), sgn p ), so that, in particular, the spectral sequence degenerates at E . EDRO T AMAROFF This motivates us to consider, independently of convergence matters, the complex S ∗ ( X )that has S p ( X ) = Hom S p ( X ( p ), sgn p ) and differentials induced from that of the E -page. Al-though this may not compute H ∗ ( X ), it provides us with another invariant for X . We call S ∗ ( X ) the small complex of X . We will give an explicit formula for its differential in The-orem 3.2. Proof.
The above follows for p (cid:202) p = E -page. (cid:206) The description of the inverse arrow to ξ will be useful for computations. Lemma 3.2.
With the hypotheses of Proposition 3.6, the inverse arrow to ξ is the map Θ : Hom S j ( X [ j ], sgn j ) −→ H j ( X ) that assigns to an S j -equivariant map f : X [ j ] −→ sgn j the class of any lift F of f accordingto the diagram X [ j ] k Σ j − ( j ) (cid:48) sgn j fF π In particular, if k is a field of characteristic coprime to j ! , we can choose F to be the composi-tion of f with the S j equivariant map Λ : sgn j −→ k Σ j − ( j ) (cid:48) such that Λ ( (cid:130) ξ j (cid:131) ) = j ! (cid:88) σ ∈ S j ( − σ ξ σ .We can now prove, by an easy inductive argument, that the support of the cohomologygroups of a weakly projective species of finite length X is no bigger than the support of X .This is a second step toward proving the convergence of our spectral sequence, which wewill completely address in the next section. Concretely: Proposition 3.7.
Let X be an E -bicomodule of finite length, which is weakly projective in Sp k ,and let q be a non-negative integer.1. If X is zero in cardinalities below q, then H i ( X ) = for i < q.2. In particular, it follows that H p ( τ q X ) = for p < q.Proof. Assume X is a species that vanishes in cardinalities below q , and proceed by induc-tion on the length (cid:96) of X . The base case in which (cid:96) = X has lenght 1 it is concentrated in some degree p larger than q , and thatproposition says H j ( X ) = j (cid:54)= p . T HE COHOMOLOGY OF TWISTED COALGEBRAS
For the inductive step, suppose (cid:96) >
1, and let j be the largest element of the support of X .The long exact sequence corresponding to0 −→ τ j − X −→ X −→ τ j X −→ H q ( τ j X ) (cid:124) (cid:123)(cid:122) (cid:125) −→ H q ( X ) −→ H q ( τ j − X ) (cid:124) (cid:123)(cid:122) (cid:125) . (4)The choice of the integer j implies τ j X is of length one, and τ j − X is of length smaller thanthat of X , so by induction the cohomology groups appearing at the ends of (4) vanish. Thisproves the first claim, and the second claim is an immediate consequence of it. (cid:206) Proposition 3.8.
Let X be an E -bicomodule. For every non-negative integer j , the projection X −→ τ j + X induces1. a surjection H j + ( τ j + X ) −→ H j + ( X ) , and2. isomorphisms H q ( τ j + X ) −→ H q ( X ) for q > j + . In terms of the filtration on H ∗ ( X ), this means that F p H p + q = H p + q for q (cid:202) Proof.
Fix a non-negative integer j and consider the exact sequence0 −→ τ j X −→ X −→ τ j + X −→ H j + ( τ j + X ) −→ H j + ( X ) −→ H j + ( τ j X ) (cid:124) (cid:123)(cid:122) (cid:125) ,and exact sequences H q − ( τ j X ) (cid:124) (cid:123)(cid:122) (cid:125) δ −→ H q ( τ j + X ) −→ H q ( X ) −→ H q ( τ j X ) (cid:124) (cid:123)(cid:122) (cid:125) ,for q > j +
1, with the zeroes explained by Proposition 3.4. This proves both claims andfinishes our proof. (cid:206)
The filtration defined on C ∗ ( X ) is bounded above, and we have shown it is complete, so itsuffices to check the spectral sequence is regular to obtain convergence —see the Complete
EDRO T AMAROFF Convergence Theorem in [22, Theorem 5.5.10]. We have proven the spectral sequences de-generates at the E -page when X is weakly projective, and this implies the spectral sequenceis regular, so the cited theorem can be applied. We give a mildly more accessible argumentto justify convergence, which the reader can compare with the exposition in [10, pp. 137-140] and [17, pp. 99-102]. Proposition 3.9. If X is an E -bicomodule that is weakly projective in Sp k , then the groupH p ( τ q + X ) vanishes for every integer p < q. In other words, the filtration on C ∗ ( X ) is regular , that is, for each integer n , we have that H n ( F p C ∗ ) = p depending on n ; in this case p > n works. This guarantees thespectral sequence is regular, see [6, Chapter XV, §4]. Proof.
Let X be as in the statement. The sequence of inclusions · · · −→ τ j X −→ τ j + X −→ · · · (5)gives a tower of cochain complexes C = { C ( τ j X )} j (cid:202) of k -modules. We noted, in the proofof Proposition 3.3, that the canonical map C ∗ ( X ) −→ lim ←−− j C ∗ ( τ j X ) is an isomorphism, andfurnishes a map η : H ∗ ( X ) −→ lim ←−− j H ∗ ( τ j X ).Let us show that this is an isomorphism. Fix r (cid:202)
0. The tower of cochain complexes C satisfies the Mittag-Leffler condition since every arrow in it is onto: every inclusion in (5) issplit in Sp k , so there is a short exact sequence0 −→ lim ←−− j H r − ( τ j X ) −→ H r ( X ) η −→ lim ←−− j H r ( τ j X ) −→ ←−− j H r − ( τ j X ) =
0, and, to do this, that the tower of abelian groups{ H r − ( τ i X )} i (cid:202) satisfies the Mittag-Leffler condition: let ι ( k , j ) : H r ( τ k X ) −→ H r ( τ j X )) bethe arrow induced by the inclusion for k (cid:202) j , and let us show that for each j there is some i such that image( ι ( k , j )) = image( ι ( i , j )) for every k (cid:202) i . Fix j , and let us show i = r + j < r , then for every k (cid:202) j the map ι ( k , j ) is zero because its codomain is zero, so theclaim is true.• If j (cid:202) r +
1, then for every k (cid:202) j , the map ι ( k , j ) is an isomorphism. In this case, wehave the exact sequence0 −→ τ j X i −→ τ k X π −→ τ kj + X −→ T HE COHOMOLOGY OF TWISTED COALGEBRAS whose corresponding long exact sequence includes the segment H r ( τ kj + X ) (cid:124) (cid:123)(cid:122) (cid:125) −→ H r ( τ k X ) ι ( k , j ) −−−−→ H r ( τ j X ) −→ H r + ( τ kj + X ) (cid:124) (cid:123)(cid:122) (cid:125) ,with the zeroes explained by Proposition 3.7 and the fact τ kj + X is zero at cardinals r and r + j = r , and fix k (cid:202) j . If k (cid:202) r +
2, the map ι ( k , j +
1) is an isomorphism,and ι ( k , j ) factors as ι ( j + j ) ◦ ι ( k , j + ι ( k , j ) equals the imageof ι ( j + j ).Fix non-negative integers p and q with p < q as in the statement. For every integer j ,the double truncation τ jq + X is of finite length and begins in degrees greater than q , so that H p ( τ jq + X ) = η : H p ( τ q + X ) −→ lim ←−− H p ( τ jq + X )is an isomorphism, we can conclude that H p ( τ q + X ) =
0, as we wanted. (cid:206)
Proposition 3.10.
Suppose X is a weakly projective E -bicomodule. There is an isomorphismof bigraded objects E ∞ −→ E ( H ) , so that the spectral sequence converges to H , and, as itcollapses at the E -page, this gives an isomorphism E p ,02 −→ H p .Proof. We have already shown that E = E ∞ . Moreover, as we observed after Proposition 3.8,we have F p H p + q = H p + q if q (cid:202) H p + q ( τ p X ) = q <
0, sothat F p H p + q = E p ,00 ( H ) = H p , and that there is an isomorphism E p ,0 ∞ = E p ,02 −→ E p ,00 ( H )which can be explicitly described as follows. Consider the diagram in Figure 1, built fromportions of long exact sequences coming from the split exact sequences0 −→ X ( i ) −→ τ i X −→ τ i + X −→ i ∈ { q − q , q + d of the E -page of our spectral sequence. The maps labelled ι ∗ in the diagram are injective becausethe diagonals are exact and there are zeros where indicated, and π ∗ is surjective by the same EDRO T AMAROFF · · · · · · H q ( X ( q )) H q − ( X ( q − H q + ( X ( q + H q ( τ q X ) H q ( τ q − X ) H q ( X ( q − = = H q − ( τ q X ) H q + ( τ q + X )0 = H q ( τ q + X ) d d δ ι ∗ δ ι ∗ π ∗ Figure 1: The diagram used in the proof of Proposition 3.10. T HE COHOMOLOGY OF TWISTED COALGEBRAS reason. We now calculate: E p ,0 ∞ = E p ,02 = ker d im d = ker δ im ι ∗ δ = ι ∗ ( H q ( τ q X )) ι ∗ im δ (cid:39) H q ( τ q X )im δ = H q ( τ q X )ker π ∗ (cid:39) H q ( τ q − X ) = E p ,00 ( H ) = H q ( X ).This is what we wanted. (cid:206) We can summarize the above in the following theorem.
Theorem 3.1. If X is an E -bicomodule, weakly projective in Sp k , the small complex S ∗ ( X ) computes H ∗ ( X ) . (cid:206) A useful corollary of this is what follows.
Corollary 3.3. If X is an E -bicomodule over a field of characteristic zero, then for every integerq, the dimension of H q ( X ) is bounded above by the multiplicity of the irreducible representa-tion sgn q in X ( q ) . In particular, the support of H ∗ ( X ) is contained in that of X . (cid:206) Observation 3.1.
Fix a nonnegative integer q and a linearized species X . It is useful to notethat an element f ∈ Hom S q ( X ( q ), sgn q ) vanishes on every basis structure z ∈ X ( q ) that isfixed by an odd permutation. This improves the last bound on dim k H q ( X ) and significantlysimplifies computations. The purpose of this section is to give an explicit formula for the differential of the E -pageof the spectral sequence, equivalently, for the differential of the combinatorial complex,corresponding to a weakly projective E -bicomodule X . Once this is addressed, we showhow to use it to calculate H ∗ ( X ) for the species considered in Section 2. Throughout thesection, we fix a weakly projective E -bicomodule X . Lemma 3.3.
The connecting morphism δ : H j ( X [ j ]) −→ H j + ( τ j + X ) corresponding to theshort exact sequence X [ j ] τ j X τ j + X is such that, for a cocycle α : X [ j ] −→ E ⊗ j , we have δ (cid:130) α (cid:131) = (cid:130) d ˜ α (cid:131) where ˜ α : τ j X −→ E ⊗ j EDRO T AMAROFF is the cochain that extends α by zero away from cardinal j . Therefore, the differential of theE -page is such thatd (cid:130) α (cid:131) = (cid:130) d ˜ α ◦ ι (cid:131) , that is, d (cid:130) α (cid:131) is the class of the restriction of d ˜ α to X ( j + .Proof. One follows the construction of the connecting morphism for the diagram of nor-malized complexes... ... ...Hom Sp k ( τ j + X , E ⊗ j ) Hom Sp k ( τ j X , E ⊗ j ) Hom Sp k ( X [ j ], E ⊗ j )Hom Sp k ( τ j + X , E ⊗ ( j + ) Hom Sp k ( τ j X , E ⊗ ( j + ) Hom Sp k ( X [ j ], E ⊗ ( j + )... ... ... ι ∗ d π ∗ If α : X [ j ] −→ E ⊗ j is a normalized cocycle, and if ˜ α : τ j X −→ E ⊗ j extends α by zero thencertainly ι ∗ ˜ α = α , and ˜ α is normalized, and its restriction to X [ j ] is zero. So in fact d ˜ α is acochain d ˜ α : τ j + X −→ E ⊗ ( j + and it is then its own lift for the map π ∗ . The lemma follows. (cid:206) Corollary 3.4.
Suppose that c ∈ H j ( X [ j ]) is represented by the class of a normalized cocycle α : X [ j ] −→ E ⊗ j . Then d ( c ) ∈ H j + ( X ( j + is represented by the class normalized cocycle γ : X ( j + −→ E ⊗ ( j + such that for a permutation σ of a finite set I of j + elements andz ∈ X ( I ) , γ ( σ )( z ) = α ( σ (2), . . . , σ ( j + z (cid:12) ( I (cid:224) σ (1))) + ( − j + α ( σ (1), . . . , σ ( j ))( z (cid:13) ( I (cid:224) σ ( j + T HE COHOMOLOGY OF TWISTED COALGEBRAS
Proof.
We compute: d ˜ α ( σ (1), . . . , σ ( j + z ) = ˜ α ( σ (2), . . . , σ ( j + z (cid:12) ( I (cid:224) σ (1))) + j (cid:88) i = ( − i ˜ α ( σ (1), . . . , σ ( i ) ∪ σ ( i + σ ( j + z ) + ( − j + ˜ α ( σ (1), . . . , σ ( j ))( z (cid:13) ( I (cid:224) σ ( j + α equals α on sets of cardinality j so the first and last summands are those of thestatement of the corollary, while the sum vanishes, since ˜ α vanishes on sets of cardinalitydifferent from j . (cid:206) We have a commutative diagram H p ( X ( p )) S p ( X ) H p + ( X ( p + S p + ( X ) d Ψ ∂ Ψ and we have already identified d . We now carefully follow the horizontal isomorphismsto obtain the formula for the differential ∂ of the combinatorial complex. The followingnotation will be useful. Definition 3.1. If j ∈ [ p + λ j be the unique order preserving bijection[ p + (cid:224) j −→ [ p ],and, given a permutation σ ∈ S p + , we write σ (cid:224) σ ( j ) for the permutation λ σ ( j ) σλ − j in S p . In simple terms, this permutation is obtained by applying λ σ j to numbers of the list σ · · · ˆ σ ( j ) · · · σ ( p + Lemma 3.4.
With the notation above,1. the sign of σ (cid:224) σ (1) is ( − σ − σ (1) − , and2. the sign of σ (cid:224) σ ( p + is ( − σ + p + − σ ( p + .Proof. We may obtain the sign of a permutation by counting inversions, that is, if m is thenumber of inversions in σ , then the sign of σ is ( − m . By deleting the first number σ (1) in σ , we lose σ (1) − σ (1), and by deletingthe last number in σ , we lose p + − σ ( p +
1) invesions, coming from those numbers largerthan σ ( p + (cid:206) EDRO T AMAROFF Definition 3.2.
Fix a finite set I and a structure z ∈ X ( I ). The left deck of z is the set ldk( z ) = { z (cid:13) ( I (cid:224) i ) : i ∈ I }, while the right deck of z is the set rdk( z ) = { z (cid:12) ( I (cid:224) i ) : i ∈ I }. If z ∈ X ( p ) and j ∈ [ p ], we will write z (cid:48) j ∈ X ( p −
1) for λ j ( z (cid:13) ([ p ] (cid:224) j )) and z (cid:48)(cid:48) j ∈ X ( p −
1) for λ j ( z (cid:12) ([ p ] (cid:224) j )).We now assume k is a field of characteristic zero. With this at hand, we have the followingcomputational result: Theorem 3.2.
The differential of the small complex S ∗ ( X ) is such that iff : X ( p ) −→ sgn p is S p -equivariant, then d f : X ( p + −→ sgn p + is the S p + -equivariant map so that for everyz ∈ X ( p ) ,d f ( z ) = p + (cid:88) j = ( − j − (cid:179) f ( z (cid:48) j ) − f ( z (cid:48)(cid:48) j ) (cid:180) . It follows that if X is a linearization k X , the value of d f ( z ) for f ∈ S p ( X ) and an elementz ∈ X ( p + depends only on the left and right decks of z, and that this data is degree-wisefinite if X is of finite type.Proof. Fix f ∈ S p ( X ). Following the correspondence described in Lemma 3.2, the normal-ized cochain α : X ( p ) −→ E ⊗ p representing f is such that α ( σ )( z ) = ( − σ p ! f ( z ) for each σ ∈ S p and each z ∈ X ( p ). By Lemma 3.3 and its corollary, the differential of α is represented by thecochain γ : X ( p + −→ E ⊗ ( p + such that for z ∈ X ( p +
1) and σ ∈ S p + , γ ( σ )( z ) = α ( σ − σ (1))( z (cid:48) σ ,1 ) + ( − p + α ( σ − σ ( p + z (cid:48)(cid:48) σ , p + ).For brevity, we are writing z (cid:48) σ , i for z (cid:12) ([ p + (cid:224) σ ( i )) and z (cid:48)(cid:48) σ , i for z (cid:13) ([ p + (cid:224) σ ( i )). We arealso writing F − F t to denote the composition obtained from F by deleting block F t . Goingback to S p + ( X ) via Proposition 3.6, we obtain that d f ( z ) = p ! (cid:88) σ ∈ S p + ( − σ (cid:179) α ( σ − σ (1))( z (cid:48) σ ,1 ) + ( − p + α ( σ − σ ( p + z (cid:48)(cid:48) σ , p + ) (cid:180) and we now split the sum according to the value of σ (1) and σ ( p +
1) as follows. If σ (1) = j ,then z (cid:48) σ ,1 ∈ X ([ p + (cid:224) j ), so we may transport this to [ p ] by means of λ j : using the notationprevious to the statement of the theorem, we have α ( σ − σ (1))( z (cid:48) σ ,1 ) = α ( λ j ( σ − σ (1)))( z (cid:48) j ).Now the sign of the permutation corresponding to the composition λ j ( σ − σ (1)), which cor- T HE COHOMOLOGY OF TWISTED COALGEBRAS responds to the permutation σ (cid:224) σ (1), is ( − σ − ( j − by Lemma 3.4, so that α ( σ − σ (1))( z (cid:48) σ ,1 ) = ( − σ − ( j − f ( z (cid:48) j ).Because there are p ! permutations σ such that σ (1) = j for each j ∈ [ p + p ! (cid:88) σ ∈ S p + ( − σ α ( σ − σ (1))( z (cid:48) σ ,1 ) = p ! p ! p + (cid:88) j = ( − σ + σ − ( j − f ( z (cid:48) j ) = p + (cid:88) j = ( − j − f ( z (cid:48) j )and this gives the first half of the formula. The second half is completely analogous: the sign( − σ + p + partially cancels with ( − σ + p + − j where j = σ ( p +
1) and we obtain the chain ofequalities:1 p ! (cid:88) σ ∈ S p + ( − σ ( − p + α ( σ − σ ( p + z (cid:48)(cid:48) σ , p + ) = p ! p ! p + (cid:88) j = ( − j f ( z (cid:48)(cid:48) j ) = − p + (cid:88) j = ( − j − f ( z (cid:48)(cid:48) j ).This completes the proof of the theorem. (cid:206) As a consequence of this last theorem, we obtain the following immediate corollaries,which address the structure of the differential of the combinatorial complex for bicomod-ules that are symmetric or trivial to one side. There is an analogous statement for for bico-modules with trivial right structure, and we denote the corresponding differential by d (cid:48)(cid:48) . Corollary 3.5.
For every symmetric bicomodule X and every nonnegative integer q there is anisomorphism H q ( X ) (cid:39) Hom S q ( X ( q ), sgn q ) . On the other hand, if X has a trivial left structure,then the differential in S ∗ ( X ) is such that for every functional f ∈ S p ( X ) , we have d (cid:48) f ( z ) = (cid:80) p + j = ( − j − f ( z (cid:48) j ) . (cid:206) To illustrate the use of the combinatorial complex we compute the cohomology groups ofsome of the twisted coalgebras introduced in Section 4.1 and, in doing so try to convincethe reader of the usefulness of the results of this section.To begin with, we include a new computation that is greatly simplified with the use ofthe small complex. We remark that, as far as the author knows, the only computation of
EDRO T AMAROFF such cohomology groups that was known previously before the methods in this paper wereintroduced, are H ∗ ( E ) and the first two cohomology groups of H ∗ ( L ). The species of singletons and suspension.
Define the species s of singletons so that forevery finite set I , s ( I ) is trivial whenever I is not a singleton, and is k -free with basis I if I is asingleton. By Lemma 3.1, the species s admits unique right and left E -comodule structures,and thus a unique E -bicomodule structure. By induction, it is easy to check that, for eachinteger q (cid:202)
1, the species s ⊗ q , which we write more simply by s q , is such that s q ( I ) is k -freeof dimension q ! if I has q elements with basis the linear orders on I , and the action of thesymmetric group on I is the regular representation, while s q ( I ) is trivial in any other case. Byconvention, set s = (cid:49) , the unit species. It follows that the sequence of species S = ( s n ) n (cid:202) consists of weakly projective species, and we can completely describe their cohomologygroups. They are the analogues of spheres for species, its first property consisting of havingcohomology concentrated in the right dimension: Proposition 3.11.
For each integer n (cid:202) , the species s n hasH q ( s n ) = k if q = n ,0 else . Proof.
Fix n (cid:202)
0. By the remarks preceding the proposition, it follows that S q ( s n ) alwaysvanishes except when q = n , where it equals Hom S n ( kS n , sgn n ), and this is one dimensional.Because each s n is weakly projective, S ∗ ( s n ) calculates H ∗ ( s n ), and the claim follows. (cid:206) The above motivates us to check whether s ⊗ − acts as a suspension for H ∗ ( − ). Assumethat X is weakly projective, so we may use S ∗ ( X ) to compute H ∗ ( X ). We claim that S ∗ ( s X )identifies with S ∗ ( X )[ − s X )( n ) is isomorphic,as an kS n -module, to the induced representation k ⊗ X ( n −
1) from the inclusion S × S n − (cid:44) → S n , and second, that the restriction of the sign representation of S n under this inclusion isthe sign representation of S n − , so that:Hom S n (( s X )( n ), sgn n ) = Hom S n (Ind S n S × S n − ( k ⊗ X ( n − n )) (cid:39) Hom S × S n − ( k ⊗ X ( n − S n S × S n − sgn n ) (cid:39) Hom S n − ( X ( n − n − ).A bit more of a calculation shows the differentials are the correct ones. By induction, ofcourse, we obtain that s j X has the cohomology of X , only moved j places up. Proposition 3.12.
Assume X is weakly projective. For each j , the suspension s j X is also T HE COHOMOLOGY OF TWISTED COALGEBRAS weakly projective, and there is a natural suspension isomorphismH ∗ ( s j X ) −→ s j H ∗ ( X ) in cohomology groups. (cid:206) The exponential species.
Every structure on a set of cardinal larger than 1 over the ex-ponential species E is fixed by an odd permutation: if I is a finite set with more than oneelement, there is a transposition I −→ I , and it fixes ∗ I . It follows that S q ( E ) is zero for q > S ( E ) and S ( E ) are one dimensional, while we already know d = H q ( E , E ) is zero for q > k for q ∈ {0, 1}. The cup product is thencompletely determined. This is in line with the computations done in the thesis [8] of J.Coppola: Proposition 3.13 (J. Coppola) . The cohomology algebra of E is isomorphic to an exterioralgebra k [ s ]/( s ) in one generator. (cid:206) The species of linear orders.
Recall the species of linear orders L from Section 3; we en-dowed its linearization L with the E -bicomodule structure obtained by restricting a linearorder to a subset. The kS j -module L ( j ) is free of rank one for every j (cid:202)
0, because S j actsfreely and transitively on the set L ( j ) . It follows that the k -module Hom S j ( L ( j ), sgn j ) is freeof rank one and, by virtue of Theorem 3.2, the computation ends here: the differential onthis combinatorial complex is identically zero. We thus deduce that: Proposition 3.14.
For every integer j (cid:202) the k-module H j ( L ) is free of rank one. (cid:206) We will address the multiplicative structure of H ∗ ( L ) below. The species of partitions.
The species of partitions P assigns to each finite set I the col-lection P ( I ) of partitions X of I , that is, families { X , . . . , X t } of disjoint non-empty subsetsof I whose union is I . There is a left E -comodule structure on P defined as follows: if X isa partition of I and S ⊂ I , X (cid:13) S is the partition of S obtained from the non-empty blocks of{ x ∩ S : x ∈ X }. We already noted there is an inclusion E −→ P . Proposition 3.15.
The cohomology group H ( P ) is free of rank one, and H ( P ) is free of rankone generated by the cardinality cocycle. In fact, the inclusion E −→ P induces an isomorph-ism of commutative algebras S ∗ ( P ) −→ S ∗ ( E ) .Proof. A partition of a set with at least two elements is fixed by a transposition, and this im-plies, in view of Observation 3.1, that S j ( P ) = j (cid:202)
2. On the other hand, S ( P ) and S ( P )are both k -free of rank one, and we already know from Proposition 3.2 that the differentialof S ∗ ( P ) is zero. This proves both claims. (cid:206) EDRO T AMAROFF The species of compositions.
The species of compositions C is the non-abelian analogueof the species of partitions P . Let us recall its construction: the species of compositions C assigns to each finite set I the set C ( I ) of compositions of I , that is, ordered tuples ( F , . . . , F t )of disjoint non-empty subsets of I whose union is I . This has a standard left E -comodulestructure such that if F = ( F , . . . , F t ) is a composition of I and S ⊆ I , F (cid:13) S is the compositionof S obtained from the tuple ( F ∩ S , . . . , F t ∩ S ) by deleting empty blocks. We view C as an E -bicomodule with its cosymmetric structure. Proposition 3.16.
The morphism L −→ C induces an isomorphism H ∗ ( C ) −→ H ∗ ( L ) and, infact, an isomorphism of commutative algebras S ∗ ( C ) −→ S ∗ ( L ) .Proof. It suffices that we prove the second claim, and, since S ∗ ( − ) is a functor, that for a fixedinteger q , the map S q ( C ) −→ S q ( L ) is an isomorphism of modules. This follows from Obser-vation 3.1: a decomposition F of a set I is fixed by a transposition as soon as it has a blockwith at least two elements, and therefore an element of S q ( C ) vanishes on every compositionof [ q ], except possibly on those into singletons. Thus the surjective map S ∗ ( C ) −→ S ∗ ( L ) isinjective and it is thus an isomorphism of commutative algebras. (cid:206) The species of graphs.
We have already defined the species Gr of graphs along with its E -bicomodule structure obtained by restriction. We have the following result concerning thecohomology groups of Gr : Theorem 3.3.
If k is of characteristic zero then, for each non-negative integer p (cid:202) , dim k H p ( Gr ) equals the number of isomorphism classes of graphs on p vertices with no odd automorph-isms, namely,
1, 1, 0, 0, 1, 6, 28, 252, 4726, 150324, . . .This sequence is [20, A281003].
Proof.
Since the structure on Gr is cosymmetric, the differential of S ∗ ( Gr ) vanishes, and Ob-servation 3.1 tells us S q ( Gr ) has dimension as in the statement of the proposition. The tab-ulation of the isomorphism classes of graphs in low cardinalities can be done with the aidof a computer —we refer to Brendan McKay’s calculation [18] for the final result— and thenfilter out those graphs with odd automorphisms. (cid:206) We can exhibit cocycles whose cohomology classes generate H ( Gr ) and H ( Gr ): in de-gree one, we have the cardinality cocyle κ , and in degree four, the normalized cochain p : Gr −→ E ⊗ such that for a decomposition F (cid:96) I , and a graph g with vertices on I , p ( F , F , F , F )( g ) is the number of inclusions ζ : p −→ g , where p is the graph T HE COHOMOLOGY OF TWISTED COALGEBRAS and ζ ( i ) ∈ F i for i ∈ [4]. One can check this cochain is in fact a cocycle, and it is normalizedby construction. We now describe how to exploit the small complex to deduce a Künneth theorem for theCauchy product. Later, we will address the multiplcative structure of the spectral sequence.
Proposition 3.17.
For each such p , q ∈ (cid:78) there is an isomorphism φ p : Hom S p × S q ( X [ p ] ⊗ Y [ q ], sgn p ⊗ sgn q ) −→ Hom S p + q (( X ⊗ Y p [ p + q ], sgn p + q ) where ( X ⊗ Y ) p [ p + q ] is the space of summands X [ S ] ⊗ Y [ T ] with S of cardinality p.Proof. This is readily described as follows. For each decomposition ( S , T ) of n , let u = u S , T be the unique bijection that assigns S to [ p ] and T to [ p + p + q ] in a monotone fashion,and given an element f ∈ Hom p , q ( X [ p ] ⊗ Y [ q ], sgn p ⊗ sgn q ), set φ p ( f )( z ⊗ w ) = ( − sch( S , T ) f ( z (cid:48) ⊗ w (cid:48) )where uz = z (cid:48) , uw = w (cid:48) and z ⊗ w ∈ X ( S ) ⊗ Y ( T ). We claim this is S p + q -equivariant. Note thatthe sign of u is sch( S , T ). Indeed, if τ is a permutation of n and ( S , T ) is a decomposition of n , we can write τ = ξρ where ρ = τ × τ is a shuffle of ( S , T ) and ξ is monotone over S andover T . It is clear that if ( S (cid:48) , T (cid:48) ) is the image of ( S , T ) under τ and if u (cid:48) = u (cid:48) S (cid:48) , T (cid:48) then u = u (cid:48) ξ .Moreover, note that u ( τ z ⊗ τ w ) is transported to u ( z ⊗ w ) by u ρ − u − , which belongs to S p × S q , and we now compute( − τ φ p ( f )( τ ( z ⊗ w )) = ( − τ + u (cid:48) f ( u (cid:48) τ ( z ⊗ w )) (6) = ( − τ + u (cid:48) f ( u ( τ z ⊗ τ w )) (7) = ( − τ + u (cid:48) + ρ f ( u ( z ⊗ w )) (8) = ( − τ + u (cid:48) + ρ + u φ p ( f )( z ⊗ w ) (9) = φ p ( f )( z ⊗ w ) (10)where the signs cancel by virtue of the identities ξρ = τ and u (cid:48) ξ = u . (cid:206) For each p , q ∈ (cid:78) there are canonical mapsHom S p ( X [ p ], sgn p ) ⊗ Hom S q ( Y [ q ], sgn q ) −→ Hom S p , q ( X [ p ] ⊗ Y [ q ], sgn p ⊗ sgn q ) EDRO T AMAROFF that are all isomorphisms if k is a field and X or Y is finite in each arity, and they collectalong with the maps φ to define a map − × − : S ∗ ( X ) ⊗ S ∗ ( Y ) −→ S ∗ ( X ⊗ Y ).Explicitly, given maps f p : X ( p ) −→ sgn p and g q : Y ( q ) −→ sgn q , we have for each decom-position ( S , T ) and z ⊗ w ∈ X ( S ) ⊗ Y ( T )( f p ∨ g q )( z ⊗ w ) = ( − sch( S , T ) f p ( u S ( z )) ⊗ g q ( u T ( w )),where u = u S , T . We obtain now the main result of this section, a Künneth formula thatallows us to compute the cohomology groups of a product in terms of its factors. Theorem 3.4 (Künneth formula) . Suppose that k is a field of characteristic zero and X or Y islocally finite. The map − × − : S ∗ ( X ) ⊗ S ∗ ( Y ) → S ∗ ( X ⊗ Y ) is an isomorphism of complexes.Proof. The only detail to check is that this is a morphism of complexes, since we havealready observed it is an equivariant bijection. To see this, we observe that following thedefinition reveals that this map is exactly the map induced by the external product − × − : C ∗ ( X ) ⊗ C ∗ ( Y ) −→ C ∗ ( X ⊗ Y )on the E -pages of the corresponding spectral sequences for the complexes C ∗ ( X ⊗ Y ) and C ∗ ( X ) ⊗ C ∗ ( Y ), which is what we wanted. (cid:206) The reader can find details for the computation suggested in the last proof in the nextsection, where we consider the case of the (interior) cup product (cid:94) . Multiplicative structure of the spectral sequence
We have defined a complete descending filtration S ∗ ( X ) ⊇ F C ∗ ( X ) ⊇ · · · ⊇ F p C ∗ ( X ) ⊇ F p + S ∗ ( X ) ⊇ · · · on S ∗ ( X ) where F p C ∗ ( X ) consists of those cochains that vanish on τ p X . Assume now that X is a linearized coalgebra of the form k X , so that there is a cup product defined on S ∗ ( X ),as detailed in Subsection 2.2. Remark that the proof of the following proposition adaptsimmediately to any cup product on S ∗ ( X ) induced from a diagonal map X −→ X ⊗ X . Proposition 3.18.
The cup product on S ∗ ( X ) is compatible with the filtration, in the sensethat, for every two non-negative integers p and p (cid:48) , we have that F p (cid:94) F p (cid:48) ⊆ F p + p (cid:48) . T HE COHOMOLOGY OF TWISTED COALGEBRAS
Proof.
Consider cochains α ∈ F p and β ∈ F p (cid:48) . Then α (cid:94) β ∈ F p + p (cid:48) by a pigeonhole argu-ment: if F = ( F (cid:48) , F (cid:48)(cid:48) ) is a decomposition of a finite set with p + p (cid:48) elements, then F (cid:48) is adecomposition of a set with at most p elements or F (cid:48)(cid:48) is a decomposition of a set with atmost p (cid:48) elements, and the formula( α (cid:94) β )( F )( z ) = α ( F (cid:48) )( z (cid:13) F (cid:48) ) β ( F (cid:48)(cid:48) )( z (cid:12) F (cid:48)(cid:48) )then makes it evident that α (cid:94) β is an element of F p + p (cid:48) . (cid:206) It follows from this proposition that the cup product descends to a product F p CF p + C ⊗ F p (cid:48) CF p (cid:48) + C −→ F p + p (cid:48) CF p + p (cid:48) + C so we obtain a multiplicative structure − (cid:94) − : E pq × E p (cid:48) q (cid:48) −→ E p + p (cid:48) , q + q (cid:48) induced on the E -page of the spectral sequence. This induces in turn a multiplicative structure on our spectralsequence ( E r , d r ) r (cid:202) . Because this spectral sequence degenerates at E , we can compute thecup product in H ∗ ( X ) from the combinatorial complex S ∗ ( X ). We describe how to do so inexplicit terms.If S is a subset of [ n ] = {1, . . . , n } with m (cid:201) n elements, and if σ is a permutation of S , weregard σ as a permutation of [ m ] by means of the unique order preserving bijection λ S : S −→ [ m ]. We say ( σ , σ ) is a ( p , q ) -shuffle of a finite set I with p + q elements whenever σ is a permutation of a p -subset S of I , σ is a permutation of a q -subset T of I , and S ∩ T = ∅ .Call ( S , T ) the associated composition of such a shuffle. If ( S , T ) is a composition of [ n ],we will write sch( S , T ) for the Schubert statistic of ( S , T ), which counts the number of pairs( s , t ) ∈ S × T such that s < t according to the canonical ordering of [ n ]. Our result is thefollowing Theorem 3.5.
The cup product induced by the diagonal X −→ X ⊗ X − (cid:94) − : S p ( X ) ⊗ C q ( X ) −→ S p + q ( X ) is such that for equivariant maps f : X ( p ) −→ sgn p and g : X ( q ) −→ sgn q , and z ∈ X ( p + q ) , ( f (cid:94) g )( z ) = (cid:88) ( S , T ) (cid:96) [ p + q ] ( − sch( S , T ) f ( λ S ( z (cid:13) S )) g ( λ T ( z (cid:12) T )) where the sum runs through decompositions of [ p + q ] with S = p and T = q. Before giving the proof, we begin with a few preliminary considerations. First, considera ( p , q )-shuffle ( σ , σ ) of [ p + q ], with associated composition ( S , T ), and let σ be the per-mutation of [ p + q ] obtained by concatenating σ and σ . EDRO T AMAROFF Lemma 3.5.
For any σ ∈ S p + q and any ( p , q ) -composition ( S , T ) of [ p + q ] ,1. the sign of σ is ( − σ + σ + sch( S , T ) , and2. ( − sch( S , T ) = ( − sch( T , S ) + pq .Proof. Indeed, by counting inversions, it follows that the number of inversions in σ is pre-cisely inv σ + inv σ + sch( S , T ), which proves the first assertion. The second claims followsfrom the first and the fact σ σ and σ σ differ by p q transpositions. (cid:206) Recall that if α : X ( p ) −→ E ⊗ p is a cochain, we associate to it the equivariant map f : X ( p ) −→ sgn p such that f ( z ) = α ( ν p )( z ) where ν p = (cid:80) σ ∈ S p ( − σ σ is the antisymmetriza-tion element. Conversely, given such an equivariant map, we associate to it the cochain α : X ( p ) −→ E ⊗ p such that α ( σ )( z ) = ( − σ p ! f ( z ). We now proceed to the proof of Theorem 3.5. Proof.
To calculate a representative of the class of f (cid:94) g , we lift first lift the maps f : X ( p ) −→ sgn p and g : X ( q ) −→ sgn q to cochains α : X −→ E ⊗ p , β : X −→ E ⊗ q that are supported in X ( p ) and X ( q ) respectively, and represent f and g according to the correspondence in theprevious paragraph. We compute for any decomposition ( F , F ) of a finite set I and any z ∈ X ( I ) that( α (cid:94) β )( F , F )( z ) = α ( F )( z (cid:13) F ) β ( F )( z (cid:12) F ).Now consider z ∈ X ( p + q ). If σ is a permutation of [ p + q ], write ( σ , σ ) for the ( p , q )-shuffleobtained by reading σ (1) · · · σ ( p ) as a permutation of S σ = { σ (1), . . . , σ ( p )} and by reading σ ( p + · · · σ ( p + q ) as a permutation of T σ = { σ ( p + σ ( p + q )}. Then( f (cid:94) g )( z ) = (cid:88) σ ∈ S p + q ( − σ ( α (cid:94) β )( σ )( z ) = (cid:88) σ ∈ S p + q ( − σ α ( σ )( z (cid:13) S σ ) β ( σ )( z (cid:12) T σ )Fix a composition ( S , T ) of [ p + q ]. In the sum above, the permutations σ with ( S σ , T σ ) = ( S , T ) are the ( p , q )-shuffles with associated composition ( S , T ). We may then replace thesum throughout S p + q with the sum throughout ( p , q )-compositions ( S , T ) of [ p + q ] and inturn with the sum throughout shuffles ( σ , σ ) of ( S , T ). This reads( f (cid:94) g )( z ) = (cid:88) ( S , T ) (cid:96) [ p + q ] (cid:88) ( σ , σ ) ( − σ σ α ( σ )( z (cid:13) S ) β ( σ )( z (cid:12) T ).We now note that α ( σ )( z (cid:13) S ) = α ( λ S ( σ ))( λ S ( z (cid:13) S )), that the sign of λ S ( σ ) ∈ S p is ( − σ , T HE COHOMOLOGY OF TWISTED COALGEBRAS and that the same considerations apply to β , so we obtain that( f (cid:94) g )( z ) = p ! q ! (cid:88) ( S , T ) (cid:96) [ p + q ] (cid:88) ( σ , σ ) ( − σ σ + σ + σ f ( λ S ( z (cid:13) S )) g ( λ T ( z (cid:12) T )).Using Lemma 3.5 finishes the proof: the sum (cid:80) ( σ , σ ) ( − σ σ + σ + σ consists of p ! q ! in-stances of ( − sch( S , T ) . (cid:206) Suppose now that X is a symmetric E -bicomodule. Then Theorem 3.2 proves the dif-ferential in S ∗ ( X ) is trivial, while Lemma 3.5 along with Proposition 3.5 prove that the cupproduct in S ∗ ( X ) is graded commutative. We obtain the Theorem 3.6.
Suppose that X is a cosymmetric E -bicomodule. Then S ∗ ( X ) is isomorphicto the cohomology algebra H = H ∗ ( X ) via the isomorphism of algebras E −→ E ( H ) . Inparticular, H ∗ ( X ) is graded commutative. (cid:206) To illustrate, take X to be the species of linear orders. Each S j ( X ) is one dimensionalgenerated by the map f j : L ( j ) −→ k that assigns σ (cid:55)−→ ( − σ . A calculation, which we omit,shows Proposition 3.19.
The algebra S ∗ ( L ) is generated by the elements f and f , so that if f p isthe generator of S p ( L ) , we havef p (cid:94) f q = (cid:195) p + qp (cid:33) f p + q ) , f (cid:94) f p = f p + , f (cid:94) f p + = These relations exhibit H ∗ ( L ) as a tensor product of a divided power algebra and an exterioralgebra. (cid:206) For a second example, consider Gr with its cosymmetric E -bicomodule structure. Wealready know H is one dimensional, and the functional p : Gr (4) −→ sgn that assigns the4-path to 1 and every other graph on four vertices to zero is a generator of S . Even morecan be said: our formula for the cup product and induction shows that for each n (cid:202)
1, theproduct f n is nonzero on the graph that is the disjoint union of n paths p , so that H n is always nonvanishing for n (cid:202)
1. Hence the cohomology algebra H ∗ ( Gr ) contains both anexterior algebra in degree 1 and a polynomial algebra in degree 4. EDRO T AMAROFF One can extend the work done in the first two sections of Section 3 by replacing E withany linearized connected twisted Hopf algebra H along the following lines. Let X be an H -bicomodule. The filtration F p C ∗ ( X , H ) of C ∗ ( X , H ) by the subcomplexes{ α : X −→ H ⊗∗ : α vanishes on τ p − X }is natural with respect to H , and it is complete and bounded above. This yields a spec-tral sequence starting at E p , ∗ = C p +∗ ( X ( p ), H ) with first page concentrated in a cone in thefourth quadrant, and to prove this is convergent in the sense of [22], it suffices we show thisspectral sequence is regular. It should be possible to prove the filtration giving rise to suchspectral sequence is regular, which is equivalent to the statement that H p ( τ j X , H ) = j . This is trivially true if X is of finite length because in such case τ j X = j . Independent of convergence matters, we can identify its first page. Indeed,for each natural number q , write 〈 H ; q 〉 for the cosimplicial k -module0 −→ H ⊗ ([ p ]) −→ H ⊗ ([ p ]) −→ · · · −→ H ⊗ j ([ p ]) −→ · · · with coface maps and codegeneracies induced by ∆ and ε , and write 〈 H ; p 〉 for the cor-responding normalized complex of 〈 H ; p 〉 . Often we can find a topological space 〈 H ; p 〉 whose cohomology coincides with that of 〈 H ; p 〉 . Denote by H p , q the cohomology groups H p + q ( 〈 H ; p 〉 ), which are all S p -modules. If X is weakly projective, the arguments outlinedin Section 3.2 of Section 3 show that the E -page of the spectral sequence has E p , q (cid:39) Hom S p ( X ( p ), H p , q ).To illustrate this, we observe that the key point of Section 3, which is the case in which H = E ,is that we may take 〈 H ; p 〉 to be a sphere S p − , and H p , q = q (cid:54)=
0, while H p ,0 is the signrepresentation sgn p of S p . In the general case, one must understand the various modules H p , q , hopefully via a geometric construction.When writing the MSc thesis that then resulted in this paper, we obtained preliminaryresults for this in the case H is the species of linear orders. With the aid of a computer, weobtained the rank of H p , q for 0 (cid:201) p (cid:201)
5, which we list in Figure 2. The attentive reader mightnotice this table is nothing else than that of the unsigned Stirling numbers of the first kind .We will prove this is the case in the next subsection. Read “ H evaluated at p ”. T HE COHOMOLOGY OF TWISTED COALGEBRAS Ω ∗ ( L ). We include here a brief way to summarize this paper intended for the readers familiar withthe theory of Koszul duality between algebras and coalgebras which works equally well inthe symmetric monoidal category of species over k . The two main observations to make arethe following:1. The coalgebra E is the free cocommutative conilpotent coalgebra in one generator (cid:49) —the unit of Sp k — and as such is trivially Koszul with Koszul dual algebra E ¡ the freecommutative algebra on the desuspension of (cid:49) : it has the same dimension in eachcardinality as E , but the suspension accounts for a change of the trivial representationin each cardinality to the sign representation sgn ∗ that we used so heavily above.2. The small complex S ∗ ( X ) is nothing but the Koszul complex K ( X , E ) = Hom( X , E ¡ ) andthe resulting cup product in the spectral sequence (that collapses since E is Koszul)is induced by the comultiplication of E ¡ . The technical requirement that X be weaklyprojective guarantees that homology commutes with the functor Hom( X , − ) in eachcardinality, which allows us to replace the twisted dg coalgebra Ω ∗ ( E ) by its homology,the twisted algebra E ¡ .Naturally, this observation was done post hoc by the author some time later after finishingwriting thesis; we have decided to preserve the work done there to illustrate how one can,without the “heavy machinery” of Koszul duality theory, obtain the complex S ∗ ( X ) throughthe combinatorics of hyperplane arrangements and Coxeter complexes.It is interesting to observe that this shows us how the purely algebraic theory of Koszulduality can shed light into combinatorics: one can see the observation above implies im-mediately that the Coxeter complex has the homology of a sphere, for example, and thatthe representation of this top homology group is the sign representation without doing any computation at all. EDRO T AMAROFF In particular, the above implies that whenever H is a Koszul twisted coalgebra, for every H -bicomodule we have available the Koszul complex K ∗ ( X , H ) = Hom Sp k ( X , H ¡ )to compute its cohomology groups. Moreover, it often happens that the structure of twisted(co)algebras arising from combinatorial objects can be nicely understood through combin-atorial methods. We also remark that in the book [4], V. Dotsenko and M. Bremner explainhow apply methods of Gröbner bases to twisted algebras, which can then be effectively usedto obtain results on the Koszulness of these. Remark 4.1.
It very often happens that H is Koszul for trivial reasons: if H is a cocommut-ative connected twisted bialgebra, then by the Milnor–Moore theorem [3, Theorem 118] theunderlying coalgebra of H is cocommutative cofree over the collection of its primitives.We can apply this to the species L of linear orders to deduce the following result whichshould aid us in computing H ∗ ( − , L ) in the category of L -bicomodules. Corollary 4.1.
The twisted coalgebra L is cofree conilpotent over the species underlying thefree Lie algebra functor and, in particular, it is twisted Koszul with Koszul dual twisted algebra L ¡ = S ( s − Lie ) , the free twisted commutative algebra over the desuspension of Lie .Proof.
The fact that the primitives of L is equal to Lie contained in Corollary 121, and by theMilnor–Moore theorem in the remark above we have have isomorphism S c ( Lie ) −→ L of twisted cocommutative conilpotent coalgebras. Since twisted cocommutative coalgebrasare twisted Koszul, the result follows. (cid:206) With this at hand, we can prove the following result.
Theorem 4.1.
The twisted Koszul dual algebra of the coalgebra L is the free commutativetwisted algebra S ( s − Lie ) over the desuspesion of Lie . In particular, for each p , q ∈ (cid:78) we havethat H p − q ( Ω ∗ ( L )[ p ]) = S ( s − Lie )[ p ] p − q has basis in correspondence with the permutations of p with p − q disjoint cycles and, hence, dim k S ( s − Lie )[ p ] p − q = (cid:34) pp − q (cid:35) T HE COHOMOLOGY OF TWISTED COALGEBRAS an unsigned Stirling number of the first kind.Proof.
The first equality follows by Koszul duality, since the algebra S ( s − Lie ) is Koszul dualto L and, as such equal to the homology of the cobar construction on L . For the secondequality, all that we need to do is observe that for a finite set I of size p , an elementarytensor z (cid:175) · · · (cid:175) z p − q ∈ S ( s − Lie )[ p ] p − q of S ( s − Lie ) is in homological degree p − q and corresponds to the datum of an unordered partition of I into subsets ( F , . . . , F p − q ) with z i ∈ Lie ( F i ). On the other hand, for a finite set[ n ] we have a basis of Lie [ n ] indexed by permutations of n that fix 1. In this way, such anelementary tensor is indeed in bijection with a permutation of I with p − q disjoint cycles,which is what we wanted. (cid:206) References [1] Marcelo Aguiar and Swapneel Mahajan,
Coxeter groups and Hopf algebras , Fields Institute Monographs,vol. 23, American Mathematical Society, Providence, RI, 2006. With a foreword by Nantel Bergeron.MR2225808 ↑ Monoidal Functors, Species and Hopf Algebras , American Mathematical Society and Centre deRecherches Mathématiques, 2010. ↑
1, 12, 13, 14, 15[3] ,
Hopf monoids in the category of species , Hopf algebras and tensor categories, Contemp. Math.,vol. 585, Amer. Math. Soc., Providence, RI, 2013, pp. 17–124. MR3077234 ↑
12, 14, 15, 47[4] Murray R. Bremner and Vladimir Dotsenko,
Algebraic operads , CRC Press, Boca Raton, FL, 2016. An al-gorithmic companion. MR3642294 ↑ Buildings , Springer-Verlag, New York, 1989. MR969123 ↑ Homological algebra , Princeton University Press, Princeton, N. J.,1956. MR0077480 ↑ [7] Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representationsof symmetric groups , Duke Math. J. (2015), no. 9, 1833–1910, DOI 10.1215/00127094-3120274.MR3357185 ↑ Cohomología en especies , MSc Thesis, UdelaR, Facultad de Ingeniería - PEDECIBA, 2015.Available at . ↑ Abelian categories. An introduction to the theory of functors , Harper’s Series in Modern Math-ematics, Harper & Row, Publishers, New York, 1964. MR0166240 ↑ [10] Allen Hatcher, Algebraic topology , Cambridge University Press, Cambridge, 2002. MR1867354 ↑ Une théorie combinatoire des séries formelles , Adv. in Math. (1981), no. 1, 1–82. MR633783 ↑
2, 4
EDRO T AMAROFF [12] Christian Kassel, Quantum groups , Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York,1995. MR1321145 ↑
4, 9[13] Gilbert Labelle, François Bergeron, and Pierre Leroux,
Combinatorial Species and Tree-like Structures ,Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge University Press, 1998. Translatedby Margaret Readdy. ↑
4, 6[14] Jean-Louis Loday,
Cyclic homology , 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 301,Springer-Verlag, Berlin, 1998. Appendix E by María O. Ronco; Chapter 13 by the author in collaborationwith Teimuraz Pirashvili. MR1600246 ↑ [15] Jean-Louis Loday and Daniel Quillen, Cyclic homology and the Lie algebra homology of matrices , Com-ment. Math. Helv. (1984), no. 4, 569–591. MR780077 ↑ [16] Saunders MacLane, Categories for the working mathematician , Springer-Verlag, New York-Berlin, 1971.Graduate Texts in Mathematics, Vol. 5. MR0354798 ↑ A user’s guide to spectral sequences , 2nd ed., Cambridge Studies in Advanced Mathematics,vol. 58, Cambridge University Press, Cambridge, 2001. MR1793722 ↑
4, 18, 21, 29 [18] Brendan McKay,
Graphs (2017), http://users.cecs.anu.edu.au/~ddm/data/graphs.html . ↑ Homologie singulière des espaces fibrés. Applications , Ann. of Math. (2) (1951), 425–505. MR0045386 ↑ [20] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (2017), http://oeis.org . ↑ Enumerative combinatorics. Vol. 1 , Cambridge Studies in Advanced Mathematics,vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Correctedreprint of the 1986 original. MR1442260 ↑ [22] Charles A. Weibel, An introduction to homological algebra , Cambridge Studies in Advanced Mathematics,vol. 38, Cambridge University Press, Cambridge, 1994. MR1269324 ↑↑