GALOIS H-OBJECTS WITH A NORMAL BASIS IN CLOSED CATEGORIES. A COHOMOLOGICAL INTERPRETATION Abstract

such that J . N . ALONSO ALVAREZ AND J . M. FERNÁNDEZ VILABOA In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2 (H, K) of H with coefficients in K . Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action : BMinn(C, H) B(C) ® 12 (H, K) In particular, if C is the symmetric closed category of K-modules with K a field, H2 (H, K) is the second cohomology group introduced by Sweedler in [21] . Moreover, if H is a finitely generated projective, commutative and cocommutative Hopf algebra over a commutative ring with unit K, then the above decomposition theorem is the one obtained by Beattie [5] for the Brauer group of H-module algebras . Preliminary A monoidal category (C, ®, K) consists of a category C with a bifunctor ® : C x C -> C and a basic object K, and with natural isomorphisms : aABC :A®(B®C)=(A(9 B)®C lA :K®A-A rA :A®K-A (A (9 aBCD) o aA(B(DC)D 0 (aABC ® D) = aAB(C®D) ° a(A®B)CD (A(9 1B)oaAKB=rA®B 272 J . N. ALONSO ALVAREZ, J. M. FERNÁNDEZ VILABOA If there is a natural isomorphism TB : A ® B = B ® A such that TAB o TB = A ® B, TB®c = (B (9 7-C) o (TB ® C), then C is called a symmetric monoidal category. A closed category is a symmetric monoidal category in which each functor ® A : C , C has a specified right adjoint [A, -] : C -> C ([12], [18]) . Examples : 1) The category of sets and mappings . 2) The category of R-modules over a commutative ring R . 3) The category of chain complexes of R-modules and morphisms of degree 0, with R a commutative ring . 4) The category of sheaves of B-modules over a topological space, with 0 a sheaf of commutative rings . 5) The category of coherent sheaves of modules over a scheme . 6) The category of all R-graded modules with morphisms of degree 0 (R is a commutative graded ring) . 7) (R, u)-Mod, with R a commutative ring and o, an idempotent kernel functor in R-Mod. In what follows, C denotes a symmetric closed category with equalizers, co-equalizers and projective basic objec K. We denote by am and OM the unit and the co-unit, respectively, of the C-adjuntion M ® -1 [M, -] C ---> C which exists for each object M of C . 1 . An object M of C is called profinite in C if the morphism [M Om(K) ®M] o am(M ®M) : M ®M , [M, M] = E(M) ¡san isomorphism, where M = [M, K] . If, moreover, the factorization of Qm(K) : M ® 1V1 ---> K through the co-equalizer of the morphisms Qm(M)® M and M ® ([M, (3m (K) o (Om(M) ® M)] oam (E(M) ® M)) M ® E(M) ® M -> M ® M is an isomorphism, we say that M is a progenerator in C . 2 . A monoid in C is a triple A = (A, r7A, PA) where A is an object in C and MA : A ® A -> A, r7A : K -~ A are morphisms in C such that MA o (A (9 r7A) = A = UA o (?1A (9 A), MA 0 (UA ® A) = pA (A (9 PA) If MA o Tá = PA, then we will say that A is a commutative monoid . Given two monoids A = (A, 97A, PA) and B = (B, ?7B, [LB) in C, f : A -> B is a monoid morphism if MB o (f ® f) = f o PA and f o r7A = ?7B . A comonoid (cocommutative), D = (D, ED, 5D) is an object D in C together with two morphisms ED : D --> K, 5D : D --> D ® D, such that (6D ® D) o 5D = (D ® 6D) o 6D and (ED ® D) o 6D = 1D = (D ED) o 6D(TD o 6D = 6D) . If D = (D, ED, 6D) and E = (E, EE, SE) are comonoids, f : D ---> E is a comonoid morphism if (f ® f ) o SD = 5E o f and EE 0 f = ED . NORMAL BASIS IN CLOSED CATEGORIES AND COHOMOLOGY 273 3. For a monoid A = (A, T7A, PA) and a comonoid D = (D, ED, SD) in C, we denote by Reg(D, A) the group of invertible elements in C(D, A) (morphisms in C from D to A) with the operation "convolution" given by : f * g = PA o (f ® g) o SD . The unit element is ED ® ?1A . Observe that Reg(D, A) is aü abelian group when D is cocommutative and A is commutative . 4 . Definition . Let II = (C, qc, pc) be a monoid and C = (C, EC, SC) a comonoid in C and let A : C ---> C be a morphism . Then H = (C = (C, EC, 6C), II = (C, 77c, 7¿C), rC , A) is a Hopf algebra in C with respect to the comonoid C if Ec and Sc are monoid morphisms (equivalently, r7c and pc are comonoid morphisms) and A is the inverse of 1c : C ---> C in Reg(C, C) . We say that H is a finite Hopf algebra if C is profinite in C . 5 . Definition . (A, WA) = (A, ?7A, PA; WA) is a left H-module monoid i) A = (A, ?7A, PA) is a monoid in C . ii) (A, c0A) is a left H-module (WA o (C ® WA) = WA o (pc ® A), cOA o (r7c ® A) = A) . iii) r7A, PA are morphisms of left H-modules (WA0(C077A) = r7A0Ec and EPA o (C ® PA) = PA 0 IPA®A, where (PA®A = (WA (9 WA) o (C ,rA 0A)o(Sc®A®A)) . We say that the action WA of H in A is inner if there exists a morphism f in Reg(C, A) such that WA = PA 0 (A ® GIA o -rA )) 0 (f ® f-1 ® A) o (Sc ® A) : C ® A SA, where f-1 is the convolution inverse of f . 6. Definition . If H is a cocommutative Hopf algebra and (A, WA) is a commutative H-module monoid, then, we say that a morphism u in Reg(C ® C, A) is a 2-cocycle if al (Q) * 03 (v) = 02 (o,) * 04 (o,), where 191 (U) = EPA 0 (00 Q), 192 (U) = Q o (PC ® C), a3(Or) = o, o (C ® pc) and a4(Q) = Q ® EC. Two 2-cocycles o, and y are said to be cohomologous, written v -y, if there exists a morphism v E Reg(C, A) such that o, * a2 (v) = al(v) * 03 (v)*y, where al (v) = WA0(C®v), a2 (v) = vopc and 03 (v) = v0ec. Trivially, "-" is an equivalence relation . The set of equivalence clases shall be called the second cohomology group of the cocommutative Hopf algebra H with the coefficients in the left H-module monoid (A, WA), and will be denoted by H2 (H, A). If o, is a 2-cocycle in Reg(C ® C, A), then the morphism v = u*al(7r) * a2(7r-1 ) *a3(7r) is a 2-cocycle in Reg(C®C, A) cohomologous with Q such that Q o (r7c ® C) = Ec ®r7A = Q o (C ® r7c), where ir = or-1 0 (C ® 77C) is a morphism in Reg(C, A) with inverse 7r -1 = u o (C ® rlc) . Moreover, if 274 J . N. ALONSO ALVAREZ, J . M. FERNÁNDEZ VILABOA ,y is cohomologous with u, then there exists a morphism í9 E Reg(C, A) such that i9 o r)c = 91A and Q * c92 (~) = al (19) * o93(19) * 5 . Remark. Let C the category of K-modules over a field . In this case, H2(H,A) is the second cohomology group of the Sweedler's complex q {Reg(®C, A) ; Aq}q>0 Reg(K, A) °o . Reg(C, A) °1 . . . . ~' Reg(®C, A) where Aq := al * 82I * . . * aq+2)4+1 and for each f E Reg(®C, A), aI(f) =WA-(C®f) az(f)= fo(C®i-20C®uc (9 C®q-i+l0 C) Oq+2(f) = f ® EC A ~ Reg(q® 1 C, A) ~~ . . . ([211) . 7. Definition . (B,PB) = (B,''IB,MB ; PB) is a right H-comodule monoid if: i) B = (B, nB, MB) is a monoid in C ii) (B, PB) is a right H-comodule ((PB®C)OPB = (B(9SC)OPB ; (B® -c) o PB = B) . iii) PB : B --> B ® C is a monoid morphism from (B, ?7B, N-B) to the product monoid BH = (B ® C, ?7B ® ?1c, (AB ® pc) o(B ®TB ® C)) (that is, PB o ?1B = 71B ® ?1c and PB o P,B = (MB (9 ¡tc) o (B ® TB C) o (PB (9 PB)) From now en we assume that H is a finite cocommutative and commutative Hopf algebra . 8 . Definition . A right H-comodule monoid (B, PB) is said to be a Galois H-object if and only if : i) The morphism -YB := (MB (9 C) o (B (9 PB) : B (9 B -> B ® C is an isomorphism . ii) B is a progenerator in C . For example, in the case of (R, u)-Mod, a commutative H-comodule monoid is a couple (B, PB), where B is a commutative (R, u)-algebra and PB : B -, B L H := Q, (B (9 R H) is a morphism of algebas and it defines a right H-comodule structure over B. where NORMAL BASIS IN CLOSED CATEGORIES AND COHOMOLOGY 275 (B, PB) is a Galois H-object if and only if B is a (R, u)-progenerator and the mapping pB : B#H -> Hom(B, B) arising from the left B#H0 module structure on B is an isomorphism ([15, (1 .3.17)]) . If a Galois H-object is isomorphic to H as an H-comodule then we say that it has a normal basis . If B 1 and B2 are Galois H-objects, f : BI -> B2 is a morphism of Galois H-objects if it is a morphism of H-comodules (PB,, o f = (f ®C) o PBl ) and of monoids . If (A, PA) and (B, PB) are H-comodule monoids, then A o B, defined by the following equalizer diagram alAB AoB -> A®B i:i A®B®C 02 AB aAB = (A (9 TB) o (PA (9 B), and aAB =A® PB is an H-comodule monoid to be denoted by (A o B, pAB) . If moreover (A, PA) and (B, PB) are Galois H-objects, then (A o B, pAB) is also a Galois H-object, where pAB is the factorization of the morphism aAB o iAB (or AB oiAB) through the equalizer iAB ®C. The set of isomorphism classes of Galois H-objects (with a normal basis), with the operation induced by the one given above, is an abelian group to be denoted by Galc(H)(Nc(H)) . The unit element is the class of (II, SC) and the opposite of [(B, PB)] is [(B°P , (B (9 A) o PB)] where B'P = (B, r7B, MB o 7-B )_ Remark. In the case of a finitely generated projective, commutative and cocommutative Hopf algebra H over a commutative ring R, Galc(H) is the group of Galois H-objects in the sense of S . Chase and M. Sweedler in [9] . 9 . Proposition. ff [(B,PB)] E NC(H), then, there is a 2-cocycle o, in Reg(C ® C, K) satisfying a o (r7C ® C) = EC = Q o (C ® ?1C) . Proof. Let (B, PB) a Galois H-object with a normal basis . Then we have an isomorphism -YB : B ®B -> B ®C and an H-comodule isomorphism r : C ---~ B . Therefore the morphism of H-comodules f = (EC ® B) o (r-1 ® r) o (?7B (9 C) : C -> B is in Reg(C, B) with inverso .Í-1 =MBo(B®EC®B)o(B®r -1 ®r)o(-yBl (S ?7C)o(?7B®C)

If there is a natural isomorphism TB : A ® B = B ® A such that TAB o TB = A ® B, TB®c = (B (9 7-C) o (TB ® C), then C is called a symmetric monoidal category.
A closed category is a symmetric monoidal category in which each functor -® A : C , C has a specified right adjoint [A, -] : C -> C ( [12], [18]) .
Examples : 1) The category of sets and mappings .
2) The category of R-modules over a commutative ring R.
3) The category of chain complexes of R-modules and morphisms of degree 0, with R a commutative ring.4) The category of sheaves of B-modules over a topological space, with 0 a sheaf of commutative rings .5) The category of coherent sheaves of modules over a scheme.6) The category of all R-graded modules with morphisms of degree 0 (R is a commutative graded ring) .7) (R, u)-Mod, with R a commutative ring and o, an idempotent kernel functor in R-Mod.
In what follows, C denotes a symmetric closed category with equalizers, co-equalizers and projective basic objec K.We denote by am and OM the unit and the co-unit, respectively, of the C-adjuntion M ® --1 [M, -] C ---> C which exists for each object M of C. 1 3. For a monoid A = (A, T7A, PA) and a comonoid D = (D, ED, SD) in C, we denote by Reg(D, A) the group of invertible elements in C(D, A) (morphisms in C from D to A) with the operation "convolution" given by: f * g = PA o (f ® g) o SD .The unit element is ED ® ?1A .
Observe that Reg(D, A) is aü abelian group when D is cocommutative and A is commutative .We say that H is a finite Hopf algebra if C is profinite in C.

Definition. (A, WA)
and EPA o (C ® PA) = PA 0 IPA®A, where (PA®A = (WA (9 WA) o (C ,rA 0A)o(Sc®A®A)) .We say that the action WA of H in A is inner if there exists a morphism f in Reg(C, A) such that WA = PA 0 (A ® GIA o -rA )) 0 (f ® f -1 ® A) o (Sc ® A) : C ® A S A, where f -1 is the convolution inverse of f .6. Definition.If H is a cocommutative Hopf algebra and (A, WA) is a commutative H-module monoid, then, we say that a morphism u in Reg Two 2-cocycles o, and y are said to be cohomologous, written v --y, Trivially, "-" is an equivalence relation .
The set of equivalence clases shall be called the second cohomology group of the cocommutative Hopf algebra H with the coefficients in the left H-module monoid (A, WA), and will be denoted by H2 (H, A).
Remark .Let C the category of K-modules over a field.In this case, H2 (H, A) is the second cohomology group of the Sweedler's complex q {Reg(®C, A) ; Aq}q>0 Reg (K, A) °o .Reg(C, A) °1 . . . .~' Reg(®C, A) where A q := al * 82I * . .* aq+2)4+1 and for each f E Reg(®C, A), From now en we assume that H is a finite cocommutative and commutative Hopf algebra .

Definition. A right H-comodule monoid (B, PB) is said to be a Galois H-object if and only if:
i) The morphism -YB := (MB (9 C) o (B (9 PB) : B (9 B -> B ® C is an isomorphism .ii) B is a progenerator in C .For example, in the case of (R, u)-Mod, a commutative H-comodule monoid is a couple (B, PB), where B is a commutative (R, u)-algebra and PB : B -, B L H := Q, (B (9 R H) is a morphism of algebas and it defines a right H-comodule structure over B .where (B, PB) is a Galois H-object if and only if B is a (R, u)-progenerator and the mapping pB : B#H -> Hom(B, B) arising from the left B#H-0 module structure on B is an isomorphism ([15, (1 .3.17)]) .
If a Galois H-object is isomorphic to H as an H-comodule then we say that it has a normal basis .
If   Definition.We denote by BMinn(C, H) the subgroup of BM(C, H) built up with the equivalence classes that can be represented by an H-module Azumaya monoid with inner action .
If H is a finitely generated projective, commutative and cocommutative Hopf algebra over a commutative ring K, this result generalizes the one obtained by Beattie in [5], and if the action of H over A is inner, the description of H(A) is due to Beattie and Ulbricht ( [6]) .

4 .
Definition.Let II = (C, qc, pc) be a monoid and C = (C, EC, SC) a comonoid in C and let A : C ---> C be a morphism .Then H = (C = (C, EC, 6C), II = (C, 77c, 7¿C), rC , A) is a Hopf algebra in C with respect to the comonoid C if Ec and Sc are monoid morphisms (equivalently, r7c and pc are comonoid morphisms) and A is the inverse of 1c : C ---> C in Reg(C, C).