Connected Hopf corings and their Dieudonné counterparts

We define coring objects in the category of algebras over a perfect field of characteristic p (with connected underlying Hopf algebra) and the corresponding notion for Dieudonné modules, and prove the equivalence of the two resulting categories, extending thus the methods of Dieudonné theory for Hopf rings from Ravenel (Reunión Sobre Teoría de Homotopía, volume 1 of Serie notas de matemática y simposia, 177–194, 1975), Schoeller (Manusc Math 3:133–155, 1970), Goerss (Homotopy invariant algebraic structures: a conference in honor of J. Michael Boardman, 115–174, 1999) and Saramago (Dieudonné theory for ungraded and periodically graded Hopf rings, Ph.D. thesis, The Johns Hopkins University, 2000).


Introduction
Dieudonné theory permits the construction of functors giving equivalences between certain categories of Hopf algebras and corresponding categories of modules (over specific rings), called their Dieudonné modules. Such functors have been defined for some objects in the categories of graded Hopf algebras [4,10], ungraded Hopf algebras [1] and periodically graded Hopf algebras [6]. Additionally, functors from categories of Hopf rings have also been devised: for the graded case [3] and the ungraded and periodically graded cases [2,7,8].
To prove such equivalences, it is usual to give a universal bilinear product in the categories of Hopf algebras and also in the corresponding categories of Dieudonné modules. One proves that these two products correspond functorally. This then reflects on an equivalence between Hopf rings and Dieudonné rings.
In this paper, we define what such a correspondence should be for Hopf corings, that is, algebras furnished with two coproducts that are related by a form of codistributivity. We present universal cobilinear coproducts of connected Hopf algebras and the corresponding notion for Dieudonné modules. This allows us to extend Dieudonné theory for Hopf corings, giving the equivalence of these categories.

Bilinear products of Hopf algebras over a perfect field of characteristic p
Given a Hopf algebra H over a perfect field k of characteristic p, the coproduct of an element x ∈ H will be written as ψ(x) = x (1) ⊗ x (2) and its co-unit will be denoted by : H → k. Its unit map will be denoted by η : k → H . The Frobenius morphism F : H → H is given by F(x) = x p , and the Verschiebung will be denoted by V : H → H (dual to the Frobenius in the dual algebra). We will write * for the algebra product.
For the results in Sect. 4, we will need the definition of induced primitives from [9]. In characteristic 2, a first-order induced primitive element of H (relative to the primitive q) is any x ∈ H that does ψ(x) = 1 ⊗ x + x ⊗ 1 + q ⊗ q. The difference between two induced primitives (relative to the same q) is always a primitive element. Any first order induced primitive relative to q will be denoted byq (or q (1) ).
An induced primitive behaves, thus, as a q 2 /2 does whenever the characteristic of the base field is not 2.
A second-order induced primitive element of H (in characteristic 2, relative to the primitive q) is any x such that ψ(x) contains a termq ⊗q (for some first order primitiveq relative to q) and additionally just the extra terms that the definition of Hopf algebra imposes. Specifically, if ψ(x) containsq ⊗q, then (1 ⊗ ψ)(ψ(x)) containsq ⊗ q ⊗ q, so (ψ ⊗ 1)(ψ(x)) contains the same, and so ψ(x) contains a a ⊗ q with ψ(a) featurinĝ q ⊗ q. Following this reasoning through, one gets that any second-order induced primitive x relative to q must actually make This is the minimum number of terms necessary for ψ(x) to feature aq ⊗q, and we denote by q (2) any second-order induced primitive relative to q (and to theq chosen.) Denote by q (m) an induced primitive of order m, in characteristic 2, relative to the primitive q, which is defined as any x in H whose coproduct contains a term q (m−1) ⊗ q (m−1) plus all the minimum required terms that the definition of Hopf algebra imposes.
If the characteristic is an odd prime r , a first-order induced primitive elementq (or q (1) ) (relative to the primitive q) is any x ∈ H whose coproduct contains q ⊗ q r −1 and also just all the terms that the definition of Hopf algebra imposes, and an induced primitive of order m (relative to the primitive q), is as any x ∈ H whose coproduct contains q (m−1) ⊗ [q (m−1) ] r −1 and additionally just all the terms required by the definition of Hopf algebra.
In any case, the Verschiebung acts on induced primitives by V (q (m) ) = q (m−1) (where q (m−1) is the order m − 1 primitive relative to q that was chosen in the definition of the q (m) given.) Let H 1 , H 2 and K be Hopf algebras over a commutative ring R. A morphism of coalgebras is called a bilinear map of coalgebras if we have: (1) , y) φ(x (2) , z) These relations can be viewed as reflecting the commutativity of some corresponding diagrams. For example, (1) states the commutativity of (Here, sw is the switch map.) Hopf algebras have universal bilinear products [3]. This is defined, for each pair of Hopf algebras H 1 x x q q q q q q q q q q q K This universal bilinear product is constructed as a quotient of the symmetric algebra S( This symmetric algebra has a coproduct given by the requirement that the inclusion H 1 ⊗ H 2 → S(H 1 ⊗ H 2 ) be a map of coalgebras. That is, for a ∈ H 1 and b ∈ H 2 , we have Consider the ideal J in S(H 1 ⊗ H 2 ) generated by the elements The universal bilinear product γ is related to the Frobenius and Verschiebung in the following way [2]. Given a ∈ H 1 and b ∈ H 2 , we have: These are the properties used in the proof of the equivalence between categories of Hopf rings and Dieudonné rings.
The construction of H 1 H 2 permits us to give a straight definition of Hopf rings. A commutative Hopf ring over a commutative ring R is a Hopf algebra H over R together with a commutative map φ : H H → H that is associative: This definition is equivalent to saying there has to be a circle product • : H ⊗ H → H , which is a map of coalgebras, satisfying convenient distributivity properties with respect to the algebra product [5], as given by the following commutative diagrams (Here, * represents the algebra product and sw is the switch map.)

Dieudonné ring theory
Let HA be the category of ungraded connected Hopf algebras over F p , p a prime (Additional details for the concepts at hand can be found in [8]). Given a sequence of indeterminates {x i }, we consider the Hopf algebras H (n) = F p [x 0 , . . . , x n ] for n ≥ 0. We have Hopf algebra maps α : H (n) → H (n + 1) (given by inclusion) andV : For each ungraded connected Hopf algebra H , this sequence induces a sequence of F p -modules where eachV is given by composition withV on the left. Consider now the Since any H ∈ HA is connected, its coaugmentation filtration exhausts it and, moreover, if we write then all the x and x that appear in the expression are in those F q H that have q < q . Thus, the Verschiebung on such Hopf algebras is eventually zero. This carries over to D H, where we have that for each x ∈ D H there must exist an n ≥ 0 such that V n x = 0.  One defines a bilinear map for R-modules M, N and L as a map g : M ⊗ N → L that satisfies: for every m ∈ M and n ∈ N .
(This is a similar definition to that made for Hopf algebras.) Two Dieudonné modules M and N also have a universal bilinear product, given by a Dieudonné module M N together with a bilinear map M ⊗ N → M N that is universal with respect to all bilinear maps [3], [2].
The following result gives the fundamental equivalence between the category of ungraded connected Hopf rings and that of connected Dieudonné rings, which are ungraded connected Dieudonné modules together with a product compatible with V and F. Theorem 3.3 [2,3] The category DM together with the bilinear product is equivalent to the category of ungraded connected Hopf rings.

Cobilinear coproducts of connected Hopf algebras over a perfect field of characteristic p
Next, we build for algebras a theory corresponding to what was done for coalgebras, defining cobilinear maps, universal cobilinear maps and the notion of a Hopf coring.
We start with Hopf algebras over a commutative ring R. Let H 1 , H 2 and K be such Hopf algebras. A morphism of algebras is called a cobilinear map if the following diagrams commute: The diagrams reflect the inversion of the diagrams one can construct to define the bilinearity of maps between coalgebras.
As for bilinear products, we can also describe cobilinear maps in terms of relations, but these do not come up as neatly as the above diagrams. Given x ∈ K and φ : K → H 1 ⊗ H 2 cobilinear, denote φ(x) by x (1) ⊗ x (2) (recall that superscripts are reserved for coproducts.) Using this notation, we then get, for any x (1) ⊗ (x (2) ) (1) ⊗ (x (2) ) (2) = [(x (1) ) (1) (x (2) ) (1) ] ⊗ (x (1) ) (2) ⊗ (x (2) ) (2) (3) (2) )) The universal cobilinear coproduct is defined, for each pair of Hopf algebras H 1 and H 2 , when it exists, as the unique Hopf algebra H 1 H 2 together with a cobilinear map (of algebras) that is universal with respect to all cobilinear maps; that is, such that for any cobilinear map K → H 1 ⊗ H 2 there exists a unique Hopf algebra map K → H 1 H 2 that makes the following diagram commute: Proof Suppose first the field has characteristic zero. Any connected Hopf algebra over such a field is generated as an algebra by its primitive elements (see [9]). Thus, for any algebra map φ : K → H 1 ⊗ H 2 (given connected Hopf algebras H 1 , H 2 and K ), the polynomial algebra generated by all elements φ(q), where q is a primitive in K , has a Hopf algebra structure given by putting all these generators as primitives. This Hopf algebra is moreover a connected Hopf algebra by construction. Given two connected Hopf algebras H 1 and H 2 , consider then any Hopf algebra K and any cobilinear map φ : K → H 1 ⊗ H 2 . Write P(K , φ) for the polynomial algebra P({φ(q) : q is primitive}). Give it a Hopf algebra structure by declaring φ(q) a primitive whenever q is a primitive. Then we define the disjoint union of all the P (K , φ).
This has the obvious operations that make it a Hopf algebra, which is connected because it is generated by its primitives.
The map H 1 H 2 → H 1 ⊗ H 2 given on each generator φ(q) by φ(q) (viewed as an element in H 1 ⊗ H 2 ) is a cobilinear map that moreover will be universal by construction: Given a Hopf algebra K and a cobilinear map K → H 1 ⊗ H 2 , the very definition of implies the existence of a unique map K → H 1 H 2 making the corresponding diagram commute. Since the Hopf algebras are connected, the fact that the diagram commutes for primitives is enough to prove that it is truly a Hopf algebra map.
If the field has characteristic a prime, Saramago [9] says that any connected Hopf algebra over it is generated as an algebra by its primitive and induced primitives. In this case, write P(K , φ) for the polynomial algebra P({φ(q) : q is primitive or induced primitive}). Its Hopf algebra structure is now obtained by declaring, for each primitive q ∈ K , that φ(q) is primitive and [φ(q)] (n) = φ(q (n) ) whenever q (n) is an induced primitive of order n in K .
Build H 1 H 2 as before, and define the map H 1 H 2 → H 1 ⊗ H 2 similarly. Again, for each cobilinear K → H 1 ⊗ H 2 there exists a natural map K → H 1 H 2 . The corresponding diagram commutes for primitives and induced primitives of K , and the result follows from the fact that giving the values of a map on primitives and induced primitives is enough to define it completely whenever the Hopf algebra K is connected.
The definition of a universal cobilinear coproduct for connected Hopf algebras gives us a way to define what a connected Hopf coring should be. This is given as a connected Hopf k-algebra H (where k is a perfect field of characteristic p) together with a map φ : H → H H that is coassociative, that is, such that (φ 1) φ = (1 φ) φ as maps from H to H H H .
By the universal construction, this is equivalent to giving two maps of algebras (two "coproducts") H → H ⊗ H that are related by a codistributivity. This mapφ, together with the original coproduct ψ, satisfies a form of codistributivity. This comes from the inversion of the diagrams from Sect. 2 that deal with distributivity for Hopf rings.
We get then: We can write the relations coming from these diagrams. Subscripts refer to the proper coproduct in each case, and for instance the first diagram yields: (2) on page 6.) As for coassociativity, it comes from the following two diagrams.
The condition translates thus as where all q α and q β are primitive. (2) If x ∈ K is an induced primitive of order n (in characteristic 2), then where all q α and q β are primitive. (3) If x ∈ K is an induced primitive of order n (in odd characteristic r ), then where all q α and q β are primitive.
Proof (1) The first relation in the definition of cobilinear map (from page 6), whenever applied to a primitive x, gives and so ψ(x (1) (1) . The second relation from page 6 gives that any x (2) is also primitive. (2) We prove it by induction in n. If n = 1, then ψ(x) = 1 ⊗ x + x ⊗ 1 + q ⊗ q, with q a primitive, and so the first relation from page 6 now becomes From this one gets that each x (1) is either primitive or a first-order induced primitive relative to q (1) (which is primitive by the first part of this lemma), and in this case x (2) = q 2 (2) (which in characteristic 2 is also a primitive). Using the second relation from page 6, we get a similar result if one switches x (1) and x (2) , and so where all q α and q β are primitive. If the result is true for n − 1, let x be an induced primitive of order n relative to the primitive q. Then ψ(x) has a q (n−1) ⊗ q (n−1) , and so the right side in the first relation from page 6 has a (q (n−1) ) (1) We know by induction that (q (n−1) ) (1) ⊗ (q (n−1) ) (2) primitives q α and q β . We get then that a x (1) is either primitive or is a [q α (n−i+1) ] 2 i , and in this case Using the second relation from page 6 and collecting all terms together now gives the result.
(3) If the characteristic is an odd prime r , the definition of induced primitive of order n now states that the coproduct of any such x will have a term q (n−1) ⊗ [q (n−1) ] r −1 . This factor of r − 1 now carries over to the relations from page 6 and, following the reasoning from the previous item in this lemma with that added r − 1 throughout, we obtain the result.
This lemma now permits the following result, relating the universal cobilinear coproduct to the Verschiebung and Frobenius.

Proposition 4.3 For any x
with φ : K → H 1 ⊗ H 2 cobilinear (K connected) and t ∈ K , write φ i for the composition of φ with projection on the ith component. We get: (2) Suppose now x = φ(q (n) ), an induced primitive of order n (include here the case n = 0, viewing regular primitives as induced primitives of order 0) in characteristic r (including r = 2). Then vx = φ(q (n−1) ).
We get then: and the result follows. (3) The proof is similar to (2).

Cobilinear coproducts of Dieudonné modules
Next, we will do for connected Dieudonné modules a construction similar to the one carried out in the previous section. We define cobilinear maps for those Dieudonné modules, and introduce the notion of universal cobilinear coproducts in the same context. Dieudonné corings are defined, and we end with the equivalence of the categories of connected Hopf corings and connected Dieudonné corings.
As in Sect. 3, Hopf algebras will be considered in HA, the category of ungraded connected Hopf algebras over F p , p a prime.
Recall that DM was defined previously as the category of modules M over R = Z p [V, F]/(V F = F V = p) such that for each x ∈ M there exists an order n ≥ 0 with V n x = 0.
A cobilinear map for R-modules M, N and L is a map g : M → N ⊗ L satisfying: for every m ∈ M.
The universal cobilinear coproduct is defined (when it exists), for every pair of connected Dieudonné modules M 1 and M 2 , as the unique connected Dieudonné module M 1 M 2 together with a cobilinear map γ : M 1 M 2 → M 1 ⊗ M 2 that is universal with respect to all cobilinear maps. That is, such that for any cobilinear map K → M 1 ⊗ M 2 , there exists a unique Dieudonné module map K → M 1 M 2 making the following diagram commute. Proof Since H is connected, it is enough to define g on primitives and induced primitives [9]. Define then g (q) as α(ω r ) ⊗ β(ω s ).
If q (n) is an induced primitive (relative to the primitive q), we can still defineq and obtain α and β as before.
For the next result, we construct a cobilinear map γ : Note that this is a cobilinear map of Dieudonné modules, since γ was such a map for connected Hopf algebras and V and F are defined from the Verschiebung and the Frobenius by composition on the right. commutes.
Applying the functor D, we get that Dh : D H → D (H 1 H 2 ) is unique, and combining with γ on the right finishes the proof. A connected Diedudonné coring will then be an ungraded connected Dieudonné module M together with a coproduct M → M ⊗ M that is compatible with V and F.
From this last corollary we can get the following result, thus completing the picture on category equivalences in the case of connected Hopf corings.

Theorem 5.4 The category of connected Hopf corings is equivalent to the category of connected Dieudonné corings.
Proof This follows from the previous considerations, plus Theorem 3.3. and Proposition 4.3.