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Torsors and the Quillen-Barr-Beck cohomology for restricted Lie algebras

Given an exact sequence of restricted Lie algebras using Duskin's torsors theory, we establish an eight term exact sequence for Quillen-Barr-Beck cohomology of restricted Lie algebras. As an application we obtain for any extension of algebraic groups over an algebraic closed field of prime characteristic an eight term exact sequence for the corresponding restricted Lie algebras extension.


Introduction
Let 0 → H → G → A → 0 be an extension of commutative algebraic groups over an algebraic closed field. If B is another commutative algebraic group, then J.-P. Serre in [28] and M. Rosenlicht in [25] proved that there is an exact sequence If the ground field has prime characteristic then the extension of commutative algebraic groups induces a short exact sequence of abelian restricted Lie algebras. In this paper we associate to this exact sequence of abelian restricted algebras, an eight term exact sequence for the Quillen-Barr-Beck cohomology of restricted Lie algebras. Using Duskin's torsors theory we construct in Theorem 4.4, for any short exact sequence 0 → N → g → b → 0 of restricted Lie algebras, an eight term exact sequence for Quillen-Barr-Beck cohomology with coefficients any Beck b-module.
Here we use the cohomology defined in [10] following the general scheme of Quillen-Barr-Beck cohomology theory for universal algebras (see [2], [31], [30]). In contrast to the cases of Groups, Associative algebras, Lie algebras, where we obtain the classical cohomologies theories, we do not obtain Hochschild (co)-homology for restricted Lie algebras [17]. We obtain different cohomology which classifies more general abelian extensions of restricted Lie algebras and not strongly abelian extensions which are classified by Hochschild cohomology.
The Hochschild-Serre spectral sequence for (co)-homology in various algebraic categories gives rise to exact sequences of terms of low degree. For the category of restricted Lie algebras it is proved by Eckmann and Stammbach in [13] that there is a five-term exact sequence for Hochschild (co)-homology of restricted Lie algebras.
In Quillen-Barr-Beck cohomology this is replaced by the eight-term exact sequence of Theorem 4.4.
We now give an outline of the methods used in our proof. Duskin (cf. [11]), develops the theory of n-torsors in order to give an interpretation of the n-th cotriple cohomology group. Later, Glenn in [16] defines torsor cohomology groups and gives a slightly different definition of n-th torsor which coincides with the one given by Duskin in the context of a monadic category over Sets whose objects have an underlying group structure (cf. Corollary 7.2.4 in [16]). Cegarra and Aznar in [7] proved in the framework of a Barr-exact category that there is an eight term exact sequence for torsor cohomology. We use this result in order to obtain exact sequences for Quillen-Barr-Beck cohomology for restricted Lie algebras.
In Lemma 2.1 of Section 2 we compute the 0-torsors group for the category of restricted Lie algebras. Thus, we obtain a 5-term exact sequence for Quillen-Barr-Beck cohomology of restricted Lie algebras.
In section 3 in order to study 2-torsors groups we are led to consider crossed modules and internal groupoids in the category of restricted Lie algebras. In Theorem 3.3 is obtained that the notions of crossed modules and internal groupoids are equivalent.
In section 4 we study 2-fold extensions for the category of restricted Lie algebras. Two fold extensions in certain categories of interest are studied by M. Gerstenhaber, in [15] and is given cohomological interpretation. Besides, J.-L. Loday and C. Kassel in [21] consider 2-fold extensions of Lie algebras in connection to the study of relative cohomology of Lie algebras. In Lemma 4.2 we prove that the 2-torsors cohomology group is isomorphic to the group of equivalence classes of 2-fold extensions of restricted Lie algebras. Thus we obtain an interpretation of the second Quillen-Barr-Beck cohomology group in terms of 2-fold extensions. As a consequence using Cegarra-Aznar's eight term exact sequence we establish in Theorem 4.4 an eight-term exact sequence for Quilen-Barr-Beck cohomology of restricted Lie algebras which extends the 5-term exact sequence of Section 2. In the last subsection 4.4 we apply Theorem 4.4 to extensions of algebraic groups in prime characteristic.
1.1. Restricted Lie algebras. In modular Lie theory in order to extent theorems which are valid in characteristic zero we are led to consider restricted Lie algebras introduced by N. Jacobson in [19]. Let k denote a field of characteristic p = 0 and Lie the category of Lie algebras over k.
) over k is a Lie algebra L ∈ Lie together with a map (−) [pL] : L → L called the p-map such that the following relations hold: where ad x : L → L denotes the adjoint representation given by ad x (y) := [y, x] and x, y ∈ L, α ∈ k. A Lie algebra homomorphism f : [p] . We denote by RLie the category of restricted Lie algebras over k.
Remark 1.2. Let L ∈ RLie be a restricted Lie algebra and x, y ∈ L. If L x,y is the Lie algebra generated by the elements x, y then s i (x, y) ∈ L p x,y where L 1 x,y := L x,y and L n x,y := [L n−1 x,y , L x,y ] for 0 ≤ i ≤ p − 1.
There is a notion of free restricted Lie algebra over a set. Therefore, the category RLie of restricted Lie algebras is a monadic category over Sets. It follows that RLie is a Barr exact category.
Let L and L ′ be restricted Lie algebras. The direct product of L and L ′ is as a Lie algebra their direct product in the category of Lie algebras If f : L → R and f ′ : L ′ → R are restricted Lie homomorphisms, then the pullback of L and L ′ over R is given by the following commutative diagram in RLie Let L ∈ RLie be a restricted Lie algebra and U (L) its enveloping algebra. We denote by u(L) := U (L)/ < x p − x [p] >, x ∈ L the restricted enveloping algebra of L. A L-module A is called restricted if x [p] a = (x · · · (x(x p a) · · · ). The category of restricted Lie L-modules is equivalent to the category of u(L)-modules. Example 1.4. Let A be any associative algebra over a field k with characteristic p = 0. We denote by A Lie the induced Lie algebra with the bracket given by [x, y] := xy − yx, for all x, y ∈ A. Then (A, (−) p ) is a restricted Lie algebra where (−) p is the Frobenious map given by x → x p . Thus, there is a functor (−) RLie : As → RLie from the category of associative algebras to the category of restricted Lie algebras. Example 1.5. Let A be an associative algebra over k. Then gl n (A) the Lie algebra of n × n matrices with coefficients in A is a restricted Lie algebra. Moreover, one proves that sl n (A) is a restricted Lie subalgebra of gl n (A). Example 1.6. Let V be a k-vector space then a map f : V → V such that f (x + y) = f (x) + f (y) and f (αx) = α p f (x) for all x, y ∈ V and α ∈ k is called p-semi-linear. Any pair (V, f ) where V is k-vector space and f : V → V a p-semilinear map is an abelian restricted Lie algebra. Example 1.7. Let B be an k-algebra not necessarily associative then the set of k-derivations Der(B) is endowed with the structure of restricted Lie algebra. In particular, if D ∈ Der(B) by the Leibniz formula we one has for all x, y ∈ B. Since the char k = p we have p i = 0 for 1 ≤ i ≤ p − 1, therefore D p ∈ Der(B).
Let (L, [p]) ∈ RLie be a restricted Lie algebra then a derivation D ∈ Der(L) is Example 1.8. Let G be an algebraic group over k. The associated Lie algebra Lie(G) of G is endowed with the structure of restricted Lie algebra (cf. [4]).
Let (L, (−) [p] ), (N, (−) [p ′ ] ) be two restricted Lie algebras, and η : L → Der(N ) a restricted homomorphism such that η(x) is a restricted derivation for every x ∈ L. We recall that the Lie product of the semi-direct product of L and N is given by Then it follows from Jacobson's Theorem 11 in [19] that the semi-direct product of L and N is endowed with the a p-map extending the p-maps on L and N (cf. Theorem 2.5 in [14]). We call this restricted Lie algebra the semi-direct product of L and N in the category of restricted Lie algebras and we denote it by L ⋊ N .
Let us recall some definitions and results which we use in the next section. We refer the reader to [10] for details.
In his doctoral dissertation, Beck incorporates the various notions of module over enveloping algebras to the general notion of a Beck module.
1.2. Beck-modules for RLie. Let L ∈ RLie be a restricted Lie algebra, A a restricted L-module and f : A → A L a p-semi-linear map from A to the invariants A L = {a ∈ M : xa = 0 for all x ∈ L}. We denote by L× f A the semi-direct product in RLie of L and A. In particular, L × f A is the semi-direct product in Lie together with the p-map (l, a) [p] = (l [p] , l · · · l p−1 a+ f (a)) where x ∈ L and a ∈ A. For L ∈ RLie a restricted Lie algebra the next theorem gives a characterization of the category of abelian group objects of the slice category ab(RLie/L) i.e the category Beck L-modules in RLie. Theorem 1.9. The category of abelian group objects ab(RLie/L) is equivalent to the category A whose objects are pairs (A, f ) where A is a restricted L-module and f : A → A L is a p-semi-linear map from A into its submodule of invariants A L and whose morphisms ( Proof. See Theorem 1.7 in [10]. Let R f be the polynomial ring consisting of the set of polynomials i=m i=0 a i f i where a i ∈ k, f an indeterminate and f a = a p f . We denote by w(L) the ring which as k-vector space is w(L) := R f ⊗ k u(L) and such that R f → w(L) and u L → w(L) are algebra homomorphisms and (P ⊗ 1)(1 ⊗ l) := P ⊗ l and (l ⊗ 1)(P ⊗ 1) := 0 for all P ∈ R f and l ∈ L. The following theorem gives another characterization of the category of abelian group objects ab(RLie/L). Theorem 1.10. The category of abelian group objects ab(RLie/L) is equivalent to the category of w(L)-modules.
Remark 1.11. By Theorems 1.9, 1.10 we note that if L ∈ RLie, then each w(L)module A is associated to a couple (Ā, f ) whereĀ is a restricted L-module and f a p-semi-linear map such that l.f (ā) = 0 for all l ∈ L andā ∈Ā.
1.3. Beck-derivations for RLie. Let g ∈ RLie/L be a restricted Lie algebra and A an w(L)-module. The group of Beck derivations is defined as follows given by where pr A denotes the canonical projection. Cartan and Eilenberg in their book [6] describe a general context for the definition of (co)-homology groups for various algebraic structures. In each case is used an appropriate notion of enveloping algebra and the definition is given in terms of Ext and T or functors. In this context Hochschild in [17] and B. Pareigis in [23] defined (co)-homology groups for the category of restricted Lie algebras. In the paper of Barr and Beck [2] is given the relation between cotriple cohomology groups and the cohomology theories described in the Cartan-Eilenberg context. It is proved that for the categories of Groups, Associative algebras, Lie algebras the two theories coincide (considering a shift in dimension). Also, cohomology as a derived functor of derivations are studied for several algebraic categories by Barr and Rinehart in [3]. Besides, D. Quillen in [31], [30] develops an axiomatic homotopy theory and cohomology groups are defined in the context of model categories. Through this development are defined (co)-homology groups for universal algebras (see section 2 in [30]). In [10] is defined Quillen-Barr-Beck cohomology for the category of restricted Lie algebras.
1.4. Quillen-Barr-Beck cohomology for RLie. Let L ∈ RLie be a restricted Lie algebra and A a w(L)-module. Let F : Sets → RLie be the free functor, left adjoint to the forgetful functor U : RLie → Sets. The adjunction (F, U ) give rise to a cotriple G in RLie/L. In [10] are defined cotriple cohomology groups H n G (L, A) := H n (Der p (G * (L), A)) Bellow we recall the definition of a n-torsor, for details and terminology we refer the reader to [11], [12] and [16].
1.5. Torsors, interpretation of cotriple cohomology. Duskin in [11] gave an interpretation of the cotriple cohomology in terms of n-dimensional torsors generalizing to any dimension Beck's interpretation of dimension 1. Let E be a monadic category over Sets and G = F U the associated cotriple. The n-truncating functor from the category of simplicial objects of E to the category of n-truncated simplicial objects admits a right adjoint called Verdier's coskeleton functor. The coskeleton functor is constructed by iterating simplicial kernels. We denote by Cosk n the composition functor Let A ∈ ab(E) be an abelian group object, then the simplicial object K(A, n) is defined as the (n + 1)-coskeleton of the following (n + 1)-truncated simplicial object where 1 denotes the terminal object and k n = (−1) n i=n i=0 (−1) i pr i for all n ≥ 1. Definition 1.12. Let X ∈ E, then a K(A, n)-torsor in E over X relative to U is defined as an augmented over X simplicial object (X . , d i , s i ) together with a simplicial morphism χ : X . → K(A, n) such that (1) X . → X is a U -split augmented simplicial object (2) χ : X . → K(A, n) is a simplicial morphism such that the following squares are pullbacks, for each m ≥ n and all where Λ i (m−1) denotes the i-horn simplicial object of the (m−1)-truncated simplicial object (3) the canonical map X . → Cos n−1 aug (X . ) is an isomorphism, where Cosk n aug denotes augmented n-coskeleton functor.
If (X, s . , χ) and (X ′ , s . , χ ′ ) are two K(A, n)-torsors over X then a morphism of . of augmented simplicial objects such that χ = χ ′ f . The set of the connected components of n-torsors is denoted by T ors n (X, A). Let H * G (X, A) be the cotriple cohomology groups, then Duskin in [11] proved the following theorem. Theorem 1.13. If X ∈ E and A an abelian group object then there is a bijection between the set T ors n (X, A) of equivalence classes of n-torsors and the n-th cotriple cohomology H n G (X, A) for n ≥ 1. Proof. Theorem (8.9) in [11]. Later Glenn in [16] defined the notion of n-dimensional hypergroupoids and gave a slightly different definition of n-torsor. If E is a monadic category over Sets whose objects have an underlying group structure, then by Corollary 7.2.4 in [16] the two notions of n-torsors coincide. Let p : B → R be a regular epimorphism in E and B p * the simplicial object (with augmentation p), obtained by iterating the simplicial kernel construction. Then, A.M Cegarra and E. R. Aznar in [7] define (Definition 1.1) the abelian groups T or 0 (p, A) := Hom simpl(E) (B p * , K(A, 1)) and T ors 1 (p, A) of connected components of 2-torsors over R with fixed augmentation p (Definition 2.1). Moreover, it is proved in [7,Theorem 7.3] that there is an eight term exact sequence

A 5-term exact sequence for Quillen-Barr-Beck cohomology
The Hochschild-Serre spectral sequence for (co)-homology of Lie algebras gives rise to exact sequences of terms of low degree. In particular, if is an exact sequence of Lie algebras and if A, is a left (resp. right) b-module, then we have the following exact sequences and Proof. Let g * be the simplicial restricted Lie algebra which is obtained by iterating the simplicial kernel construction i.e If we denote by g ⋊ N the semi-direct product of g and N in RLie, we get an isomorphism of restricted Lie algebras given by (x, n) → (x, x + n) Therefore we get the following simplicial object for all x ∈ g and n, n ′ ∈ N . Since N = ker p we get Therefore is defined a mapφ : N ab → A given bȳ for all n ∈ N . Moreover for x ∈ L and n ∈ N we havē It follows thatφ is a morphism of w(L)-modules. Conversely, letφ : Hom w(L) (N ab , A) be a w(L)-homomorphism. We define φ : g ⋊ N → L × fĀ given by Then the associated derivation d φ (x, n) :=φ(n+ < [N, N ] > p ) satisfies the relation (2.1) and it follows that φ ∈ T or 0 (p, M ). Therefore the mapφ → φ is a well defined homomorphism, inverse to the homomorphism φ →φ.
Theorem 2.2. Let 0 → N → g → b → 0 be an exact sequence of restricted Lie algebras and A a w(b)-module. Then the following sequence is exact Proof. Since RLie is a monadic category over Sets whose objects have an underlying group structure, it follows from the above lemma and the exact sequence (1.5).

Internal groupoids and crossed modules
Crossed modules in groups were introduced by Whitehead [32] in the study of relative homotopy groups. Brown and Spencer in [5] noted that internal categories within the category of groups are equivalent to crossed modules. In the more general context of categories of groups with operations, crossed modules and internal categories are studied by Porter in [24]. Moreover, internal categories in a Mal'tsev variety are studied by Janelidze in [20]. Bellow we consider the case of the category of restricted Lie algebras.
Let as recall the definition of internal category in a category C with pullbacks. An internal category in C is a diagram in C satisfying: tθ = tp 2 and sθ = sp 1 , the associative low relation The morphisms t, s, e are called the target, source and unit morphisms respectively. An internal category is called internal groupoid if for any c ∈ C there is a c ′ ∈ C such that θ(c, c ′ ) = es(c) and θ(c ′ , c) = et(c).
J.-L. Loday and C. Kassel define in [21] the notion of crossed module for the category of Lie algebras. In the same way are defined crossed modules for the category of restricted Lie algebras (cf. [9]).   and in the same way we see that θ(((0, n), (0, n)) [p] ) = (θ((0, n), (0, n))) [p] and θ(((0, 0), (m ′ , 0)) [p] ) = (θ((0, 0), (m ′ , 0))) [p] Since θ is a Lie algebra homomorphism we deduce that is actually a restricted Lie homomorphism. Then one can easily check that the conditions of associativity and unit and right lows are satisfied. Besides, if we define (m, n) ′ := (−m, n + µ(m)) for all m ∈ M and n ∈ N then we see that the above diagram in RLie defines an internal groupoid in RLie. Let = n.a + 0 one sees that A is also a restricted R-module where R := coker µ. Therefore (A, p A ) becomes a Beck R-module.

2-fold extensions.
Let (A, p A ) be a Beck R-module. We denote by E 2 (R, A) the category whose objects are exact sequences in RLie such that (M, N, µ) is a crossed module and the induced Beck R-module structure is the given one. The morphisms are: where the morphisms f and g respect the actions. Two objects E 1 , E 2 ∈ E 2 (R, A) are called equivalent if there is a morphism in E 2 (R, A) from one to the other. We will denote by E 2 (R, A) the set of equivalence classes in E 2 (R, A).
For p : E → R a fixed epimorphism in RLie we consider the set E 2 (p, A) the set of 2-term extensions Two objects E 1 , E 2 ∈ E 2 (p, A) are called equivalent if there is a morphism from one to the other. We denote by E 1 (p, A) the set of equivalence classes.

4.2.
Baer sum. The set E 2 (R, A) can be endowed with the structure of abelian group. The Baer sum of two restricted Lie algebra extensions is induced from their Baer sum viewed as Lie algebra extensions. In particular, let E and E ′ be 2-fold extensions of restricted Lie algebras The Baer sum of (E) and (E ′ ) is defined as the extension where K := ker f and f : A × A → A given by f (a, a ′ ) := a + a ′ for all a, a ′ ∈ A. The restricted ideal K is consisting of the elements (a, −a).
Next we give an interpretation of 2-torsors over a restricted Lie algebra R in terms of 2-fold extensions. We note that 2-torsors relative to U in RLie are 2torsors relative to U in Lie viewed as simplicial objects in Lie. Proof. It suffices to construct maps from T ors 2 (R, A) to E 2 (R, A) inverse to each other. Let (E.) be a 2-torsor over R with simplicial morphism We consider the Moore complex M (E.) of (E.) given by M (E.) 0 = R and M (E.) n = ∩ n i ker d i , for all i ≥ 1 and with differential δ n := d 0 | M(E.)n . By conditions (1) and (2) of the Definition 1.12 the associated Moore complex M (E.) is given by the following exact sequence in RLie It follows from condition (2) of the Definition 1.12 that We have the following commutative diagram TORSORS AND THE QUILLEN-BARR-BECK COHOMOLOGY FOR RESTRICTED LIE ALGEBRAS 15 / / R Therefore we obtain Since 2-torsors in RLie are 2-torsors in Lie, it follows from [26] that the Lie action of R on A coincides with the Lie action induced from the above exact sequence. Moreover we have where s denotes a section of the surjection p : E 0 → R. Therefore the induced Beck R-module structure by the two-term exact sequence, coincides with the initial structure and we obtain a two term exact sequence in RLie be a two-term extension in E 2 (R, A). By Theorem 3.3 is associated an augmented over R groupoid Γ: where x 1 , y 1 , z 1 ∈ M and e 0 ∈ E 0 then x 1 + z 1 − y 1 ∈ A. As in the case of Lie algebras is defined a Lie algebra homomorphism δ 2 : E 2 → K(A × pA R, 2) given by δ 2 (µ(z 1 ) + e 0 , x 1 ), (e 0 , y 1 ), (e 0 , z 1 ) = (x 1 + z 1 − y 1 , p(e 0 )) We notice that z 1 + x 1 = y 1 + a for some a ∈ A, thus the Lie algebra L (z1+x1),y1 generated by (z 1 + x 1 ) and y 1 is zero. By Remark 1.2 we obtain Also, p is a restricted Lie homomorphism so Since δ 2 is a Lie algebra homomorphism we see that δ 2 is actually a restricted Lie homomorphism. Besides, as for the case of Lie algebras (see [26]) δ is a normalized cocycle. It follows from Section 4.1 in [11] that E . is a K(A × pA R, 2) torsor over R. Also, one can see that the Moore complex M (E . ) of E . is the exact sequence Consequently, we defined maps from and E 2 (R, A) → T ors 2 (R, A) which are inverse to each other. Moreover we can see that equivalent two-term extensions correspond to the same connected component in the category of 2-torsors and conversely elements in the same component correspond to equivalent two-term extensions.
Besides, by definition of the group structure on the set of torsors (see [16], [12]), the group structure on 2-torsors RLie is induced from the group structure on 2torsors in Lie. Moreover, the Baer sum of two restricted Lie algebra extensions is induced from their Baer sum viewed as Lie algebra extensions. It follows that above bijections are actually group isomorphisms. Therefore the theorem follows.
Proof. The theorem follows from the above Lemma 4.1 and the Theorem 1.13.
Remark 4.3. We note that in the Cartan-Eilenberg context crossed modules in various algebraic categories are associated with the third cohomology group. In the Quillen-Barr-Beck context crossed modules are associated to the second cohomology group since there is a shift by 1 in the notation. Besides, G. Hochschild in [18] gives an interpretation of the third Hochschild cohomology in terms of space of restricted kernel classes.

4.3.
Eight term exact sequence. The five term exact sequence of Theorem 2.2 for Quillen-Barr-Beck cohomology for restricted Lie algebras can be extended to an eight term exact sequence by the following theorem.
Proof. It follows from the Theorem 4.2 and the eight term exact sequence (1.5).

4.4.
Application to extensions of algebraic groups. Let k be algebraic closed field of prime characteristic and G an algebraic group over k. Since char k = p we have that the associated Lie algebra Lie(G) is actually a restricted Lie algebra. In fact in this way is defined a functor Lie : Gr → RLie from the category of algebraic groups to the category of restricted Lie algebras (see [4]). Let H, A be algebraic groups then M. Rosenlicht in [25] and J.-P. Serre in [28] define as an extension of H by A a short exact sequence of groups such that α, γ are separable rational homomorphisms. Thus H can be identified with a normal algebraic subgroup of G and A with G/A. Let be an other extension of H by A then (E), (E ′ ) are called equivalent if there is a rational homomorphism ψ : G → G ′ such that the following diagram is commutative The class of equivalent extensions of H by A is denoted by Ext(H, A).