Quotients in super-symmetry: formal supergroup case

We describe the structure of the quotient $\mathfrak{G}/\mathfrak{H}$ of a formal supergroup $\mathfrak{G}$ by its formal sub-supergroup $\mathfrak{H}$. This is a consequence which arises as a continuation of the authors' work (partly with M. Hashi) on algebraic/analytic supergoups.The results are presented and proved in terms of super-cocommutative Hopf superalgebras. The notion of co-free super-coalgebras plays a role, in particular.

1. Introduction 1.1.Background and objective.The quotient G/H of a group G by its subgroup H is the set of all H-orbits in G.This simple fact for abstract groups immediately turns into a difficult question if we replace abstract groups with groups with geometric structure, such as algebraic or analytic groups.Does the question become even more difficult when we consider those geometric groups in the generalized supersymmetric context, or namely, supergroups?The authors want to answer that it is not so difficult as was supposed, not so much more than that for geometric (non-super) groups.For we can describe the structure of the quotient G/H for supergroups in terms of the non-super quotient G ev /H ev , where G ev (resp., H ev ) is the geometric (non-super) group naturally associated with the supergroup G (resp., H).Indeed, such a description has been given by the authors [12,Theorem 4.12,Remark 4.13] for affine algebraic supergroups (see also [14,Sect. 14.2] for non-affine case) and by Hoshi and the authors [5, Theorem 7.3, Remark 7.4 (1)] for analytic supergroups.
A main objective of this paper is to give such a description for formal supergroups.Those formal supergroups (or resp., formal superschemes) form a category which is equivalent to the category of super-cocommutative Hopf superalgebras (resp., super-coalgebras), as is shown in Appendix below.
In the text the results all are presented in terms of such Hopf superalgebras and super-coalgebras, some of which will be translated into the language of formal super-objects in the Appendix.1.2.Main results.Throughout in this paper we work over a fixed base field k whose characteristic char k is not 2.The unadorned ⊗ denotes the tensor product over k.Except at the beginning part of the following Section 2 we always assume super-coalgebras and Hopf superalgebras to be super-cocommutative.Compare this situation with that in [12], where the authors discussed super-commutative Hopf superalgebras in order to investigate affine algebraic supergroups; some arguments of the present paper are in fact dual to those of the cited article.
Let G = Sp * (J) be a formal superscheme; the notation means that it corresponds to a super-coalgebra J.It is known that J includes the largest ordinary subcoalgebra, say, J.The formal scheme G ev naturally associated to G is the one corresponding to J, or in notation, G ev = Sp * (J).Suppose that G is a formal supergroup, which means that J is a Hopf superalgebra.Then J is a Hopf subalgebra of J. Let H = Sp * (K) be a formal sub-supergroup of G, or equivalently, let K be a Hopf sub-superalgebra of J.The associated formal group H ev = Sp * (K) is then a formal subgroup of G ev , or equivalently, K is a Hopf subalgebra of J.The quotient formal superscheme G/H corresponds to the quotient super-coalgebra J//K := J/JK + of J by the super-coideal JK + which is generated, as a left super-ideal, by the augmentation super-ideal K + of K; or in notation, G/H = Sp * (J//K).
The associated formal scheme (G/H) ev is proved to be naturally isomorphic to G ev /H ev (see Corollary 4.7), whence we have (G/H) ev = Sp * (J//K).
Let V J = P (J) 1 denote the vector space consisting of all odd primitive elements of J, which is in fact a left J-module under the adjoint J-action.Similarly we have the left K-module V K = P (K) 1 consisting of all odd primitive elements of K, which is seen to be a K-submodule of V J , so that we have the quotient K-module Q := V J /V K of V J by V K .A main result of ours (Theorem 4.6 (1)) states that there is an isomprphism (1.1) J ⊗ K ∧(Q) ≃ J//K of super-coalgebras, which turns into (1.2) J//K ⊗ ∧(Q) ≃ J//K, in many special cases including the case where J is pointed; see Proposition 4.4.Notice that the exterior algebra ∧(Q) is naturally a Hopf superalgebra in which every element of Q is odd primitive, and it is regarded naturally as a left K-module super-coalgebra for (1.1), and simply as a super-coalgebra for (1.2).A description of J//K which looks similar to (1.2) is given by [9, Proposition 3.14 (1)] under the assumption that J is irreducible, or namely, the coradical of J is k1.But, our method of the proof is more conceptual, which uses the notion of co-free super-coalgebras (see Definition 3.4 for precise definition) and proves on the course that J ⊗ K ∧(Q) and J//K ⊗∧(Q) are co-free; see Proposition 4.3 and Example 3.8.Furthermore, we obtain from (1.1) the smoothness criteria (Theorem 4.8): (1) J//K is smooth if and only if so is J//K, (2) The equivalent conditions are necessarily satisfied if char k = 0.
Throughout we will use Hopf-algebraic techniques; in particular, a key step to prove (1.1) is to show a Maschke-type result such as found in [3]; see Lemma 4.1.The techniques are so useful that we will work mostly in the Hopf-algebra language, not in the group-theoretical one.They also make it easier to investigate the associated graded object.Indeed, we obtain a description of the graded coalgebra gr(J//K) associated to J//K; see Theorem 4.6 (2).1.3.Organization of the paper.Section 2 is devoted to preliminaries, which contain basic definitions and results on supersymmetry and Hopf superalgebras.Section 3 discusses co-free super-coalgebras.Section 4 presents main results and their proofs.Appendix discusses formal superschemes and supergroups from scratch, and translates some of the results obtained in the text into the language of those formal super-objects.

Preliminaries
2.1.Supersymmetry.A super-vector space is a vector space V = V 0 ⊕ V 1 (over k) graded by the order-2 group Z 2 = {0, 1}.It is said to be purely even (resp., purely odd) if V = V 0 (resp., V = V 1 ).The super-vector spaces V, W, . . .form a monoidal category, with respect to the natural tensor product ⊗ over k, and the unit object k which is supposed to be purely even.This monoidal category is symmetric with respect to the supersymmetry where v ∈ V and w ∈ W are homogeneous elements with the parities (or degrees) |v|, |w|, respectively.Objects, such as (co)algebra, Hopf algebra or Lie algebra, constructed in SMod k are called so as super-(co)algebra, Hopf superalgebra or Lie superalgebra, with "super" thus attached.Ordinary objects, such as ordinary (co)algebra, . . ., are regarded as purely even super-objects.
If char k = 3, a Lie superalgebra g is required to satisfy in addition to the alternativity and the Jacobi identity formulated in the super context.Useful references for algebra and geometry in the supersymmetric context include Carmeli et al. [1] and Manin [6].

Super-coalgebras and Hopf superalgebras.
In what follows supercoalgebras are assumed to be super-cocommutative.Accordingly, Hopf superalgebras are so, as well.
For a super-coalgebra J, the structure is denoted by ∆ J : J → J ⊗ J, ε J : J → k, or simply by ∆, ε.The coproduct ∆ is presented so as ∆(a) = a (1) ⊗ a (2) , a ∈ J by this variant of the Heynemann-Sweedler notation.The super-cocommutativity assumption is presented by ∆ = c J,J • ∆.In case J is a Hopf superalgebra, the antipode is denoted by S J or S.
A graded coalgebra is a non-negatively graded coalgebta J = n≥0 J (n) which, regarded as a super-coalgebra J = J 0 ⊕ J 1 with respect to the parity is super-cocommutative.A graded Hopf superalgebra is a graded supercoalgebra, equipped with a graded-algebra structure, which is a Hopf superalgebra with respect to the parity as above.
2.3.The exterior algebra.Let V be a purely odd super-vector space.
The exterior algebra ∧(V) = n≥0 ∧ n (V) on V is a graded algebra, and is in fact a graded Hopf superalgebra in which every element v ∈ In what follows, ∧(V) will be often regarded only as a graded coalgebra or a super-coalgebra.To give an alternative description of that structure, first notice that given an integer n > 1, the symmetric group S n of degree n acts on the n-fold tensor product V ⊗n of V, so that the ith fundamental transposition (i, i + 1) acts as id , where 1 ≤ i < n.We suppose that S n acts on the right, so that the action is given explicitly by A n (V) := (V ⊗n ) Sn denote the subspace of V ⊗n consisting of all S n -invariants.The symmetrizer gives a linear isomorphism; this is to be called the anti-symmetrizer ordinarily in the non-super situation.
, and set For n ≥ 0 and 0 ≤ i ≤ n, we let denote the injection restricted from the canonical linear isomorphism V ⊗n ≃ −→ V ⊗i ⊗V ⊗(n−i) .The graded-coalgebra structure of ∧(V) is transferred through sym n to A(V) as follows: the coproduct is transferred to ∆ : while the counit is to the projection A(V) → A 0 (V) = k.Here, as for ∆ one should notice the formula where σ runs over the (i, n − i)-shuffles, i.e., the permutations such that σ(1) < • • • < σ(i) and σ(i + 1) < • • • < σ(n).We can thus identify so as as graded coalgebras.
2.4.The associated graded coalgebra.Let J be a super-coalgebra.The pullback (2.6) of J 0 ⊗ J 0 (⊂ J ⊗ J) along the coproduct is the largest purely even sub-supercoalgebra of J; see [9,Sect. 3].Let F −1 J = 0, F 0 J = J.For every integer n > 0, let ) be the kernel of the composite of the n-iterated coproduct of J with the natural projection onto (J/J) ⊗(n+1) .Then J is filtered (see [15,Sect. 11.1]), or more precisely, we have an ascending chain The associated graded coalgebra Suppose that J is a Hopf superalgebra.Then J is a purely even Hopf superalgebra of J, and we have (F n J)(F m J) ⊂ F n+m J, n, m ≥ 0.Moreover, gr J turns into a graded Hopf superalgebra.The super-vector space (2.7) consisting of all primitive elements in J form a Lie superalgebra with respect to the super-commutator [v, w] := vw − (−1) |v||w| wv, where v and w are homogeneous elements of P(J).Define (2.8) the odd component of the Lie superalgebra.This is stable under the right (resp., left) adjoint J-action where a ∈ J and v ∈ V J , whence it turns into a right (resp., left) purely odd J-supermodule.Notice that J is a J-ring, by which we mean an algebra equipped with an algebra map from J. With an arbitrarily chosen, totally ordered basis X = (x λ ) λ∈Λ of V J , J is, as a J-ring, generated by X, and defined by the relations (2.10) where a ∈ J and λ > µ in Λ.Moreover, the linear map where n > 0 and Remark 2.1.Theorem 10 of [10] proves a category equivalence between (super-cocommutative) Hopf superalgebras and dual Harish-Chandra pairs.We can make φ X into a natural isomorphism of Hopf superalgebras, giving to J ⊗∧(V J ) some additional structures that are obtained from the dual Harish-Chandra pair corresponding to J. But, such a more precise description of J will not be needed in the sequel.
Keep the notation as above.The category of right J-supermodules is a monoidal category, which is symmetric with respect to the supersymmetry (2.1) due to the cocommutativity of J.The Hopf superalgebra ∧(V J ) is in fact a Hopf algebra in SMod J with respect to the J-action arising from the right adjoint action.The associated smash product J ⋉ ∧(V J ) is identified with gr J, since the second relation of (2.10) is reduced in gr J to the super-commutativity x λ x µ = −x µ x λ .To be more precise, V J is naturally isomorphic to P(gr J) 1 , and the isomorphism together with the inclusion J = (gr J)(0) ֒→ gr J uniquely extend to a canonical isomorphism of graded Hopf superalgebras, which in fact coincides with gr(φ X ).
2.5.Quick review of structure of Hopf superalgebras.Let J be a super-coalgebra.Every simple subcoalgebra of the J, which is regarded as an ordinary coalgebra, is purely even.The coradical Corad J of J is by definition the (necessarily, direct) sum of all simple subcoalgebras; it is seen to be included in the J in (2.6).Suppose that J is a Hopf superalgebra.We say that J is irreducible if k1 is a unique simple subcoalgebra, or namely, k1 = Corad J.It is known that J includes the largest irreducible Hopf sub-superalgebra denoted by J 1 .If char k = 0, J 1 coincides with the universal envelope U (P(J)) of the Lie superalgebra P(J) in (2.7).We say that J is pointed if the simple subcoalgebras of J are all 1-dimensional; this is the case if k is algebraically closed.Suppose that J is pointed.Then the group of the grouplike elements, which all are necessarily even for J being over a field, spans Corad J.Moreover, J 1 is stable under the conjugation by G(J) a ⊳ g = g −1 ag, a ∈ J 1 , g ∈ G(J), and the resulting smash product kG(J) ⋉ J 1 is naturally identified with J.

Co-free super-coalgebras
Let C be a super-coalgebra; we remark this symbol shall not express the complex field, which does not appear in this paper.Let V be a super-vector space.Due to the super-cocommutativity of C, every left Therefore, given a C-super-comodule, we may regard it as any of left, right and bi-super-comodules, and do not specify which unless it is needed.
Given C-super-comodules V and W , the cotensor product V C W is defined by the equalizer diagram where ρ V (resp., λ V ) denotes the right (resp., left) C-coaction on V (resp., W ). The left C-coaction on V and the right C-coaction on W give rise to the same (in the super sense as given by (3.1)) coaction onto V C W , with which we regard V C W as a C-super-comodule.
We let SMod C denote the category of C-super-comodules; we thus take "right" for this notation.It is easy to see the following.Lemma 3.1.SMod C forms a monoidal category, whose tensor product is given by the cotensor product C , and whose unit object is C.This is in fact symmetric with respect to the supersymmetry restricted to the cotensor products.Giving a C-super-coalgebra is the same as giving a coalgebra in (SMod C , C , C).To be more precise we have the following.− → k composed with the counit of C make D into a super-coalgebra.This supercoalgebra, paired with the counit D → C in the category, is made into a C-super-coalgebra.This construction is seen to be an inverse procedure of the construction given in the lemma.
Let C be a (cocommutative) coalgebra, and let V be a purely odd Csuper-comodule.We emphasize that C is supposed to be purely even, and V purely odd.Definition 3.4.A C-super-coalgebra on V is a pair (D, π) of a C-supercoalgebra D and a C-super-colinear map π : D → V.The C-super-coalgebras on V form a category, whose morphisms are C-colinear super-coalgebra maps compatible with the maps to V. A C-super-coalgebra (D, π) on V is said to be co-free if it is a terminal object of the category, or more explicitly, if given a C-super-coalgebra (E, ̟) on V, there exists a unique morphism (E, ̟) → (D, π) of C-super-coalgebras on V. Proposition 3.5.There exists uniquely (up to isomorphism) a co-free Csuper-coalgebra on V.
Proof.The uniqueness is obvious.Let us construct explicitly a desired (D, π), which is in fact graded, D = n≥0 D(n), as a coalgebra.We write simply for C .Let Let n > 1.Notice that the cotensor product which we denote by ∆ i,n−i ; cf.(2.4) in the case where 0 < i < n.Define a C-super-colinear map ∆ : Let ε : D → D(0) = C be the projection.Then we see that (D, ∆, ε) is a (graded) C-super-coalgebra; cf. the graded-coalgebra structure of A(V) given in Section 2.3.
We wish to prove that this D, paired with the projection π : be the super-coalgebra map equipped to E, and be ̟, respectively.For n > 1, define f n : E → D(n) to be the composite Since the coradical Corad E of E is purely even, and is, therefore, killed by ̟, it follows that for every a ∈ E, we have f n (a) = 0 for sufficiently large n.
Thus we can define f : One sees that this f is a unique morphism E → D of C-super-coalgebras on V. Indeed, it is easy to see that f is compatible with the counits.Let n > 1.To see that f is compatible with the coproducts, one should use the formula where 0 ≤ i ≤ n.For the uniqueness one should use the fact that the composite coincides with the projection D → D(n).
We denote the thus constructed, co-free C-super-coalgebra on V by coF C (V).
We emphasize that this is graded so that and the associated map to V is the projection onto the first component.In addition, the largest purely even sub-super-coalgebra of coF C (V) is C, as is seen from (2.6) and the construction above.
Example 3.6.Let us be in the special case where C is the trivial coalgebra k spanned by a grouplike element, and V is, therefore, a purely odd supervector space.Then we have Remark 3.7.This fact (3.4) is essentially shown in the second half of the proof of [5,Propsition 3.11] by the authors joint with Hoshi, which, however, contains an error; the composite given on Page 41, line -9, should read Example 3.8.Suppose that C is an arbitrary (cocommutative) coalgebra, and V is co-free, or namely, V = C ⊗W, where W is a purely odd super-vector space.We see from the construction in the last proof that Notice that (3.4) and (3.5) are identifications of graded coalgebras, as well.The relevant following observation is simple, but will be used in the next section.Recall from (3.3) that coF C (V) is such a graded coalgebra D with the property It may be understood, as a C-super-coalgebra on V, to be the one which arises, as above, from its graded-coalgebra structure with the property (3.7).Moreover, it is a terminal object in the category of those graded coalgebras with that property; morphisms in the category are supposed to be identical in degrees 0 and 1.
An inclusion D ֒→ E of super-coalgebras is said to be essential if Corad E ⊂ D; see [11,Sect. 1].A super-coalgebra D is said to be smooth if every essential inclusion D ֒→ E of D into another super-coalgebra E splits.The condition is equivalent to saying that given an essential inclusion E ′ ֒→ E of super-coalgebras, every super-coalgebra map E ′ → D extends to some super-coalgebra map E → D.
A coalgebra, or a purely even super-coalgebra, is smooth as a coalgebra (in the sense of [16], [4]) if and only if it is smooth as a super-coalgebra (in the sense just defined).To see 'only if', one should notice that every super-coalgebra map from a super-coalgebra, say, E to a coalgebra uniquely factors through the quotient purely even super-coalgebra E/E 1 (= E 0 ) of E.
If char k = 0, then every Hopf algebra is smooth, as is well known, and this fact, combined with the isomorphism given by φ X in (2.11), proves that every Hopf superalgebra is then smooth; see (the proof of) Theorem 4.8 (2).Proposition 3.10.If C is a smooth coalgebra and V is an injective purely odd C-super-comodule, then coF C (V) is smooth.
Proof.Write D for coF C (V).We are going to prove that an arbitrarily given essential inclusion D ֒→ E splits under the assumptions.By the smoothness assumption for C, the projection D → D(0) = C extends to a supercoalgebra map E → C, with which we regard E as a C-super-coalgebra.By the injectivity assumption for V, the projection D → D(1) = V extends to a C-super-colinear map E → V, which gives rise to a unique C-super-coalgebra map E → D on V by the co-freeness of D. The map must be a retraction of the inclusion D ֒→ E, regarded as a C-super-coalgebra map on V, again by the co-freeness.
Remark 3.11.If D is a smooth super-coalgebra, then the largest purely even sub-super-coalgebra, say, D of D is smooth, as is easily seen using the fact that every super-coalgebra map from a purely even super-coalgebra to D maps into D. In particular, if coF C (V) is smooth, then C is; see the remark preceding Example 3.6.

The quotient J//K
Let J be a Hopf superalgebra, and let K be a Hopf sub-superalgebra of J. Then the left super-ideal JK + of J generated by the augmentation superideal K + = Ker(ε K ) of K is a super-coideal; see [17,Proposition 1] or [9, p.286].We denote the resulting quotient super-coalgebra of J by (4.1) J//K (= J/JK + ).
If K is a normal Hopf sub-superalgebra, or namely, if JK + = K + J (or equivalently, K is stable under the adjoint J-action, see [9, Theorem 3.10 (3)]), then J//K is a quotient Hopf superalgebra of J, in which case the structure of J//K is rather clear from the results obtained in [10,Sect. 3].So, in what follows, main interest of ours will be in the case where K is not normal.
We are going to investigate the structures of this super-coalgebra and of the associated graded coalgebra gr(J//K); the results will be then applied to show smoothness criteria for J//K.One will see easily that parallel results hold for the objects J\\K (= J/K + J), gr(J\\K) analogously constructed with the side switched.
Let J and K be the largest purely even Hopf sub-superalgebras of J and of K (see Section 2.4), and set as in (2.8).These are purely odd right (and left) supermodules over J and over K, respectively; see (2.9).For a while we regard these as ordinary modules, keeping their parity in mind.The natural maps K → J and V K → V J induced from the inclusion K ֒→ J are injections, through which we can regard K as a Hopf subalgebra of J, and V K as a K-submodule of V J .Define (4.2) We let Mod K denote the category of right K-modules, which is in fact a symmetric monoidal category, (Mod K , ⊗, K).Since J is a coalgebra in the category, we have the category (4.3) Mod J K (= (Mod K ) J ) of right J-comodules in Mod K .We have the short exact sequence Tensored with J, this gives rise to the short exact sequence in Mod J K , in which K acts diagonally on each tensor product, while J coacts on the single tensor factor J.
Lemma 4.1.The short exact sequence (4.4) in Mod J K splits.Proof.Clearly, the surjection J ⊗ V J → J ⊗ Q in (4.4) splits J-colinearly.Therefore, it splits, as desired, as a morphism in Mod J K , as is seen from the following.

Fact. A surjective morphism
This fact is proved by dualizing the argument proving [3, Theorem 1], as follows.Since J is projective as a right K-module by [7, Theorem 1.3], we have a right K-linear map ξ : as a section.To see, for example, that σ is J-colinear, we compute the J-coaction on σ(n) so that = σ(n (0) ) ⊗ n (1) , where n (0) ⊗ n (1) ⊗ n (2)  ) )); notice that the second and the last equalities hold since K and J are cocommutative.Choose a J-colinear section s : N → M of p. Then the composite where the last arrow indicates the K-action on M, is seen to be a desired section.
By Lemma 4.1 we can choose a section γ : Notice that it is necessarily of the form γ(a ⊗ q) = a (1) ⊗ g(a (2) ⊗ q), where a ∈ J, q ∈ Q and g = (ε J ⊗id Q )•γ.Recall here that we are discussing purely odd super-objects.
In addition, recall from Example 3.8 that J ⊗ ∧(Q) and J ⊗ ∧(V J ) are the co-free J-super-coalgebras on the purely odd J-super-comodules J ⊗ Q and J ⊗ V J , respectively.Then we see that the J-colinear map γ between those J-super-comodules uniquely extends to a graded coalgebra map in degrees 0 and 1, and in degree n ≥ 2, where a ∈ J and As a super analogue of (4.3) we have the category SMod J K of J-comodules in SMod K , which is indeed a symmetric monoidal category, (4.5) (SMod J K , J , J), just as (SMod J , J , J) is; see Lemma 3.1.Notice that J ⊗∧(Q) and J ⊗∧(V J ) are co-free coalgebras in (SMod J K , J , J) in a generalized sense (defined in an obvious manner), and γ, arising from the morphism γ in the category, is a coalgebra morphism in the category.
Define a morphism in SMod J K by (4.6) where a ∈ J, u ∈ ∧(V J ).Here, on the target ∧(V J ) ⊗ J, K acts and J coacts on the tensor factor J. We see that τ is in fact a graded coalgebra isomorphism with inverse Choosing a totally ordered basis X = (x λ ) λ∈Λ of V J , one obtains the oppositesided version φ ′ X : ∧(V J ) ⊗ J → J of the φ X in (2.11), which is a coalgebra isomorphism in SMod J .Define Γ : J ⊗ ∧(Q) → J to be the composite This is a coalgebra morphism in SMod K , which sends a⊗1 ∧(Q) to a for every a ∈ J.
Proof.Clearly, Γ K is a coalgebra morphism in SMod K , which sends (a ⊗ 1 ∧(Q) ) ⊗ K 1 K to a for every a ∈ J.To see that it is an isomorphism, we aim to prove that gr(Γ K ) is bijective.Notice that coincides with the canonical isomorphism J ⋉∧(V J ) ≃ −→ gr J in (2.13).Using the similar isomorphism K ⋉ ∧(V K ) ≃ −→ gr K, one sees that gr(Γ K ) coincides with the composite where prod : Regard ∧(V K ) ⊂ ∧(V J ) as Hopf superalgebras, naturally as in the beginning of Section 2.3.Clearly, gr we can regard gr(Γ K ) as a ∧(V J )-coalgebra morphism in SMod ∧(V K ) (in a sense slightly generalized from Definition 3.2), or in other words (see Lemma 3.3), as a coalgebra morphism in (SMod which is indeed a symmetric monoidal category just as the one in (4.5).Since ∧(V J )//∧(V K ) is naturally isomorphic to ∧(Q), it follows essentially by [17, Theorem 1] (see also [9,Proposition 1.1]) that the symmetric monoidal category above is equivalent to through the functor which assigns to an object M in the former category, where one should notice (∧(V K )) + = n>0 ∧ n (V K ).Therefore, for our aim, it suffices to prove that is bijective, or is indeed the identity map.One sees that this is a graded coalgebra endomorphism which is identical clearly in degree 0, and also in degree 1 by choice of γ.By co-freeness of J ⊗ ∧(Q) (see Example 3.8) it must be the identity map in view of Remark 3.9.Now, we regard V K ⊂ V J as left K-modules with respect to the left adjoint action (see (2.9)), so that Q is a left K-module, and whence it is a J//K-super-coalgebra with respect to the projection onto the neutral component; notice that the associated J//K-coaction on J ⊗ K Q (see (3.6)) is the natural one on the tensor factor J.Moreover, we have the following.
This proposition and the following will be proved below together.

Proposition 4.4. Assume one of the following (i)-(iv):
( which is the identity map of J//K in degree 0, and is the previous isomorphism in degree 1. Proof of Propositions 4.3 and 4.4.By (4.7) we have a canonical morphism (4.9) of graded coalgebras, which is identical in degrees 0 and 1.We wish to show that this is an isomorphism, which will prove Proposition 4.3.Replacing everything with its base extension to the algebraic closure k of k, we may suppose that k is algebraically closed.Then J is pointed, whence we have a J//K-colinear and right K-linear isomorphism (4.10) and an isomorphism such as (4.8).The morphism (4.9) is then identified, through the isomorphisms just obtained, with a graded coalgebra map which is identical in degrees 0 and 1.By co-freeness of J//K ⊗ ∧(Q) (see Example 3.8) this must be an isomorphism in view of Remark 3.9.
Let us turn to Proposition 4.4.The argument above proves it in Cases (i) and (ii), since we then have an isomorphism such as (4.10) by [7,Theorem 1.3 (4)], again.In Case (iv), as well, we have such an isomorphism.To see this, assume (iv).By modifying the proof of [8, Theorem 4.1, Lemma 4.2] into the cocommutative situation, we see that the inclusion K ֒→ J has a right K-linear coalgebra retraction, say, r : J → K, and gives a desired isomorphism.Here and below we let a → a present the natural projection J → J//K.
Finally, assume (iii).Then Q is naturally a left J-module.We see that Theorem 4.6.We have the following.
(1) There is an isomorphism of super-coalgebras, which, restricted to is natural in the sense that it is induced from the natural projection J → J//K restricted to J and J ⊗ V J (see (2.12)).
(2) gr K is naturally regarded as a graded Hopf sub-superalgebra of gr J.
The natural projection gr J → gr(J//K) induces an isomorphism (4.12) gr J// gr K ≃ −→ gr(J//K) of graded coalgebras.These graded coalgebras are naturally isomorphic to the co-free Proof.
(1) Recall from Proposition 4.2 the isomorphism Γ K : (J ⊗ ∧(Q)) ⊗ K K ≃ −→ J, which has the property, among others, that it sends (a ⊗ 1 ∧(Q) ) ⊗ K 1 K to a for every a ∈ J.With ⊗ K K/K + applied, it induces a desired isomorphism.To verify the prescribed naturality, one uses the property above and the fact that the composite of the first component τ (1) of the τ (see (4.6)) with the product in J coincides with the product map J ⊗ V J → J, a ⊗ v → av.
(2) We have the following commutative diagram in which the bottom horizontal arrow indicates the isomorphism just obtained, This, with gr applied, induces Here one should notice gr((J ⊗ ∧(Q)) ⊗ K K) = (J ⊗ ∧(Q)) ⊗ K gr K, using an isomorphism (gr K =) K⊗∧(V K ) ≃ −→ K analogous to (2.11).We see from the horizontal isomorphism at the top, which sends (1 J ⊗1 ∧(Q) )⊗ K 1 gr K to 1 gr J , that gr K is a graded Hopf sub-superalgebra of gr J; see also [9,Remark 3.8].Moreover, the projection indicated by the vertical arrow on the RHS induces the isomorphism (4.12).The horizontal isomorphism at the bottom proves the last statement.
There is given in [9, Proposition 3.14 (1)] a description of J//K that looks similar to the one obtained in Theorem 4.6 (1) above.But the cited result assumes that J is irreducible, and indeed, the down-to-earth proof given there is not valid in the present general case.The description of ours gives a simpler and more natural proof of the following result reproduced from [9].  2), (3)]).We have the following.
(1) J//K is naturally isomorphic to the largest purely even sub-supercoalgebra of J//K.(2) Q is naturally isomorphic to the purely odd super-vector space of odd primitive elements of J//K, where 1 = 1 J mod(JK + ).
Proof.(1) Notice from Proposition 4.3 and the remark preceding Example 3.1 that J//K (= J ⊗ K ∧ 0 (Q)) is the largest purely even sub-super-coalgebra of J ⊗ K ∧(Q).Then the desired result follows from the isomorphism proved by Theorem 4.5 (1) and its partial naturality on J//K.
(2) Notice that taking P ( ) is compatible with base extension.Then by using the isomorphism (4.8) after base extension to the algebraic closure of k, we see The desired result follows from the isomorphism proved by Theorem 4.5 (1) and its partial naturality on J ⊗ As another advantage (from [9]) of our more conceptual treatment that uses the notion of co-free super-coalgebras, we have the following smoothness criteria for J//K.Theorem 4.8.We have the following.
(1) J//K is smooth if and only if so is J//K.
Proof.(1) By Theorem 4.6 we may suppose J//K = coF J/ /K (J ⊗ K Q).We claim that the J//K-comodule J ⊗ K Q is injective, or equivalently, co-flat; see [16,Proposition A.2.1].Indeed, the co-flatness (that is, the right exactness of the associated co-tensor product functor) follows, since the comodule turns into the co-free comodule J//K ⊗ Q after base extension to the algebraic closure of k; see the proof of Proposition 4.3.The claim, combined with Proposition 3.10 and Remark 3.11, proves the desired result.
(2) By Part 1 it suffices to prove that the coalgebra J//K is smooth in case char k = 0.In view of [4, Proposition 1.5] we may suppose by base extension such as above that k is algebraically closed, in which case J and K are smash products (see Section 2.5) where we have set being spanned by the grouplike elements of all right cosets of G L by G K , is a smooth coalgebra.So is U L //U K , since it is a a pointed irreducible coalgebra of Birchkoff-Witt type, or more explicitly, it is isomorphic to the tensor product where (x λ ) λ∈Λ are arbitrarily chosen elements of P (L) such that they modulo P (K) form a k-basis of P (L)/P (K).

Appendix A. On formal superschemes
Let us translate some of the results obtained in the text into the language of formal superschemes.The authors do not know any literature which discusses in detail theory of formal schemes over a field such as developed by [16], in the generalized super context.But, at least basic definitions and results found in part of [16] are directly generalized as will be seen below.We continue to suppose char k = 2 and that super-coalgebras and Hopf superalgebras are super-cocommutative.
Let C be a super-coalgebra (over k).
denote the set of all even grouplike elements of R ⊗ C.This is naturally a group if C is a Hopf superalgebra.One sees that R → G R (R ⊗ C) gives rise to a functor, Sp * (C), which is called the formal superscheme or formal supergroup corresponding to C. The formal superscheme Sp * (C) is said to be smooth if C is a smooth super-coalgebra.Every formal supergroup is smooth if char k = 0. See the paragraphs following Remark 3.9.
The assignment C → Sp * (C) gives rise to a category equivalence from the category of super-coalgebras (resp., Hopf superalgebras) to that full subcategory of the category of super k-functors (resp., super k-groups) which consists of all formal superschemes (resp., formal supergroups).The source category has (possibly, infinite) direct products given by the tensor product ⊗, and the target full subcategory is closed under the direct product.By this fact on the category of super-coalgebras (resp., formal superschemes) one can define group objects in the respective categories, which are precisely Hopf superalgebras (resp., formal supergroups); see [2, Appendix B], for example.
We remark that the category of formal superschemes consists precisely of those super k-functors X such that X is isomorphic to the inductive limit associated to some filtered inductive system (X λ , u λµ ) of finite affine superschemes, where each X λ is thus represented by a finite-dimensional supercommutative superalgebra, say, A λ , and the morphism u λµ : X λ → X µ for λ < µ is supposed to be a closed immersion, or namely, to arise from a surjective superalgebra map A µ → A λ .It follows that the category is included in the category of sheaves in the fppf topology (faisceaux).Also, a monomorphism Sp * (C) → Sp * (D) of formal superschemes uniquely arises from an injective super-coalgebra map C → D. Therefore, we may and we do call Sp * (C) as a formal sub-superscheme (resp., formal sub-supergroup) of Sp * (D) if C is a sub-super-coalgebra (resp., Hopf sub-superalgebra) of a super-coalgebra (resp., Hopf superalgebra) D.
Given a super k-functor X, we have a super k-functor, X ev , defined by If X is a formal superscheme or a formal supergroup, so is X ev .For if X = Sp * (C), then X ev = Sp * (C), where C is the largest purely even sub-supercoalgebra of C; see (2.6).In this case, X ev can be regarded as an ordinary formal scheme or group.In general, every (ordinary) formal scheme, say, Sp * (D) is regarded as a formal superscheme, with D regarded as a (purely even) super-coalgebra.It is smooth at the same time as a formal scheme and as a formal superscheme.Let G = Sp * (J) be a formal supergroup corresponding to a Hopf superalgebra J.As in (2.8), let V J = P (J) 1 be the purely odd super-vector space of odd primitive elements of J.One sees that G ev is a formal sub-supergroup of G.The following results from the isomorphism φ X in (2.11).
Proposition A.1.We have a G ev -equivariant isomorphism A group-equivariant object in the category of super-coalgebras is a coalgebra in the monoidal category J SMod (or SMod J ) of left (or right, according to the side on which J acts) supermodules.On the other hand, such an object in the category of formal superschemes is called a left or right formal G-superscheme.Obviously, those objects in the respective categories are in a category equivalence.In Proposition A.1 above, G is regarded as a left formal G ev -superscheme by multiplication.
Retain G = Sp * (J) as above, and let H = Sp * (K) be a formal subsupergroup of G; thus, K is a Hopf sub-superalgebra of J. Then for every R ∈ SAlg k , H(R) is a subgroup of G(R).It holds that H is normal, that is, for every R, H(R) is a normal subgroup of G(R), if and only if K is a normal Remark A.4.As analogues to Theorem A.3 above, Proposition 4.8 and Corollary 4.10 (see also Remark 4.13) of [12] prove results on affine algebraic supergroups, for which the situation is more complicated, so that the analogous isomorphisms hold only locally.

Definition 3 . 2 .
A super-coalgebra over C or C-super-coalgebra is a pair (D, ω) of a super-coalgebra D and a super-coalgebra map ω : D → C.
regarded to be mapping into D C D, and whose counit is ω : D → C. Every coalgebra in the category arises uniquely in this way.Proof.Given a coalgebra D in the category, its coproduct D → D C D ֒→ D ⊗ D composed with the embedding into D ⊗ D, and its counit D → C ε C

Corollary 4 . 7 (
[9, Theorem 3.13 ( Remark 4.5.It is known that there exist (cocommutative) Hopf algebras J ⊃ K over a non-algebraically closed field, for which there does not exist any isomorphism such as (4.10), or J is not even free as a left or right K-module; see [17, Sect.5], for example.