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Koszul property and Bogomolov's conjecture

This is an enhanced version of the author's 1998 Harvard Ph.D. thesis, as published by IMRN in 2005. We propose an extension of Bogomolov's conjecture about commutator subgroups of Galois groups to arbitrary fields. A somewhat weaker conjecture is then shown to follow, together with the famous Milnor-Bloch-Kato conjecture, from a certain hypothesis about homological properties of the Milnor K-theory algebras.

Theorem 1. Let l be a prime number and F be a field containing a primitive l-root of unity. Suppose that (1) the map K M n (F ) ⊗ Z/l −→ H n (G F , µ ⊗n l ) is an isomorphism for n = 2 and a monomorphism for n = 3; (2) the algebra K M * (F ) ⊗ Z/l is Koszul. Let G (l) F be the maximal quotient pro-l-group of the group G F . Then the natural homomorphism K M * (F ) ⊗ Z/l −→ H * (G (l) F , µ ⊗ * l ) is an isomorphism. Of course, the first part of the condition (1) is the theorem of Merkurjev and Suslin; the second part of (1) is long known to be true, at least, for l = 2. It is not difficult to show, using the transfer operations on the Milnor K-theory, that it suffices to verify the Milnor-Kato conjecture in the pro-l-group case [7,20]. Thus our Koszulity hypothesis essentially implies the Milnor-Kato.
1.6. The main goal of this paper is to show that a certain much stronger hypothesis about the Milnor ring would imply Conjecture 2 (Bogomolov's conjecture) as well.
Under the conditions of Conjecture 2, the Milnor K-theory algebra modulo l is a quotient algebra of the exterior algebra of the vector space F * /F * l over the prime field Z/l, so we can consider the natural homomorphism Λ * l (F ) = * Z/l (F * /F * l ) −−→ K M * (F ) ⊗ Z/l. Notice that the exterior algebra Λ * l (F ) is isomorphic to the cohomology algebra of the Galois group Gal(F [ l ∞ √ F ]/F ) with coefficients Z/l (see section 9). The stronger Koszulity hypothesis states that the kernel J * l (F ) of this map should be a Koszul module over the exterior algebra (see section 3 for the definition). Our main result can be formulated as follows.
Theorem 2. Let l be a prime number and F be a field containing the l-roots of unity if l is odd, and containing the 4-roots of unity if l = 2. Suppose that (1) the condition (1) of Theorem 1 holds; (2) the ideal J * l (F ) generated by the Steinberg relations in the exterior algebra Λ * l (F ) is a Koszul module (in the grading shifted by 2) over the algebra Λ * l (F ). Then the maximal quotient pro-l-group of the absolute Galois group G K of the field K = F [ l ∞ √ F ] is a free pro-l-group.
It is shown in section 6 that the condition (2) of Theorem 1 follows from the condition (2) of Theorem 2. Therefore, in the assumptions of Theorem 2 both the Milnor-Kato and Bogomolov's conjectures are true. 1.7. Two proofs of Theorem 2 are given in this paper. The first one is rather computational; it is based on the Serre-Hochschild spectral sequence for pro-l-groups and a result extending the techniques of [14] to cohomology of conilpotent comodules. The latter is proven in section 4. The second argument uses the results of [14] in order to pass to the associated graded coalgebras and then applies directly a general principle claiming that Koszulity-type conditions on morphisms of Koszul algebras correspond to freeness-type conditions under duality. This is explained in section 7.
1.8. Finally, let us point to a simple application of our results to a different area of motivic theory-that of mixed Tate motives with rational coefficients over a field of finite characteristic. For such a field F , the standard conjectures [4] imply that the Milnor K-theory algebra with rational coefficients K M * (F ) ⊗ Q should coincide with the algebraic K-theory K * (F ) ⊗ Q and be a Koszul algebra. Furthermore, in this case the motivic pro-Lie algebra L * (F ) = ∞ i=1 L i (F ) describing the abelian tensor category of mixed Tate motives with rational coefficients over a field F is isomorphic to the quadratic pro-Lie algebra dual to K M * (F ) ⊗ Q. (The duality here connects the quadratic pro-Lie algebras with the skew-commutative quadratic algebras.) A conjecture of A. Goncharov [9] claims that for any field F the subalgebra L 2 (F ) = ∞ i=2 L i (F ) of the motivic pro-Lie algebra L * (F ) is a free graded pro-Lie algebra. It follows from our result that this conjecture together with the Koszulity conjecture above is equivalent to the following hypothesis about the Milnor K-theory algebra. Let J * Q (F ) be the kernel of the homomorphism from the exterior algebra * Q (F * ⊗Q) to K M * (F ) ⊗ Q; then the ideal J * Q (F ) should be a Koszul module over the exterior algebra. Note that in this setting, unlike in the Bogomolov case, the analogue of the statement converse to Theorem 2 also holds: the algebra K M * (F ) ⊗ Q is Koszul and the pro-Lie algebra L 2 (F ) is free if and only if the ideal J * Q (F ) is a Koszul module. The basic reason is that L * (F ) is a graded pro-Lie algebra, while in the Bogomolov case we deal with an ungraded Galois group. In this sense, the hypothesis of Koszulity of the Steinberg ideal J * l (F ) can be thought of as a stronger graded version of Conjecture 2. This is explained in section 8.
1.9. Coalgebras, comodules, and their cohomology are considered in section 2. Definitions and basic properties of Koszul (co)algebras and (co)modules are presented in section 3. The cohomology of conilpotent comodules are studied in section 4, the cohomology of nonconilpotent coalgebras in section 5. We discuss Koszul properties related to morphisms of graded algebras in section 6. The duality between Koszulity and freeness conditions is explained in section 7. Koszulity and freeness for Lie coalgebras are considered in section 8. We prove the main theorem in section 9. Evidence related to Conjecture 1 is presented in the appendix.
The author wishes to express his gratitude to V. Voevodsky for posing the problem and to A. Beilinson, F. Bogomolov, A. Goncharov, F. Pop, and A. Yakovlev for very helpful discussions. I owe to my advisor B. Mazur the idea of the counterexample in the appendix, from which the above formulation of Conjectures 1-2 eventually grew. Most of the mathematical content of this paper was invented when the author was a graduate student at Harvard University (the paper is an enhanced version of my Ph. D. thesis). The rest was done when I was working at the Independent University of Moscow and visiting the Max-Planck-Institut für Mathematik in Bonn. The author wishes to thank all the mentioned institutions.

Homology of Algebras and Cohomology of Coalgebras
2.1. The cohomology of comodules. Recall that a coalgebra C over a field k is a vector space equipped with a comultiplication map ∆ : C −→ C ⊗ C and a counit map ε : C −→ k satisfying the conventional coassociativity and counit axioms. A coaugmented coalgebra is a coalgebra C endowed with a coalgebra homomorphism γ : k −→ C. The cohomology algebra H * (C) of a coaugmented coalgebra C is the opposite multiplication algebra to the Ext-algebra Ext * C (k, k), where k is endowed with the left C-comodule structure by means of γ.
A left comodule P over a coalgebra C is a vector space over k together with a coaction map ∆ ′ : P −→ C ⊗ P satisfying the usual coaction axiom. The cohomology module of a comodule P over a coaugmented coalgebra C is the left H * (C)-module H * (C, P ) = Ext * C (k, P ). This cohomology can be calculated in terms of the following explicit cobar-resolution where C + = coker γ and the the cobar-differential is given by the well-known formula d(c 0 ⊗· · ·⊗c n ⊗p) = n i=0 (−1) i−1 c 0 ⊗· · ·⊗∆(c i )⊗· · ·⊗c n ⊗p+(−1) n c 0 ⊗· · ·⊗c n ⊗∆ ′ (p). It is easy to check that this is an injective comodule resolution of P . Applying the functor Hom C (k, −) to it, we get and in particular The multiplication on H * (C) and the action of H * (C) on H * (C, P ) are induced by the obvious multiplication and action on the level of cobar-complexes.
with A + = ker α and the standard bar-differential. Therefore we have and in particular The comultiplication on H * (A) and the coaction of H * (A) on H * (A, M) are induced by the obvious comultiplication and coaction ∆(a 1 ⊗ · · · ⊗ a n ) = n i=0 (a 1 ⊗ · · · ⊗ a i ) ⊗ (a i+1 ⊗ · · · ⊗ a n ) ∆ ′ (a 1 ⊗ · · · ⊗ a n ⊗ m) = n i=0 (a 1 ⊗ · · · ⊗ a i ) ⊗ (a i+1 ⊗ · · · ⊗ a n ⊗ m) on the level of bar-complexes. 5 2.3. Graded modules and comodules. By a graded coalgebra we mean a nonnegatively graded vector space C = ∞ n=0 C n equipped with a coalgebra structure which respects the grading, i. e., ∆(C n ) ⊂ i+j=n C i ⊗C j and ε(C >0 ) = 0. Moreover, we will always assume the map ε : C 0 −→ k to be an isomorphism. Then a graded coalgebra structure induces a coaugmented coalgebra structure in the obvious way. A graded comodule over C is a graded vector space P = ∞ i=0 P i equipped with a C-comodule structure which respects the grading, i. e., ∆ ′ (P n ) ⊂ i+j=n C i ⊗P j . We always consider the nonnegatively graded comodules only, that is, P i = 0 for i < 0.
We leave it to the reader to formulate the dual definitions of graded algebras and modules; as above, we assume that A i = 0 for i < 0, that A 0 is a one-dimensional vector space generated by the unit, and that M i = 0 for i < 0.
One can immediately see from the bar-complexes above that the homology and cohomology of graded objects are endowed with a natural second grading: for a graded algebra A and a graded A-module M we have and analogously, for a graded coalgebra C and a graded C-comodule P

Opposite multiplications and comultiplications.
For algebras A and A op with opposite multiplication there is a canonical isomorphism of graded vector spaces H * (A) ≃ H * (A op ) given by the tautological isomorphism Tor A * (k, k) ≃ Tor A op * (k, k). It is easy to see from the bar-complexes above that this isomorphism makes H * (A) and H * (A op ) coalgebras with opposite comultiplication.
On the other hand, for coalgebras C and C op with opposite comultiplication there is no self-obvious isomorphism between Ext * C (k, k) and Ext * C op (k, k). Nevertheless, one can use the cobar-complexes to construct an isomorphism of graded vector spaces H * (C) ≃ H * (C op ) which will make them algebras with opposite multiplication.
To define this isomorphism without mentioning explicit resolutions, one has to introduce the cotensor product and cotorsion functors, as explained in the next subsection. This will complete the symmetry between the algebras and coalgebras.
2.5. The cotorsion functors and cohomology of comodules. Let P and Q be a right and a left comodule over a coalgebra C. The cotensor product vector space P ⊙ C Q is defined as the kernel of the map ∆ ′ ⊗id−id⊗∆ ′ : P ⊗Q −→ P ⊗C ⊗Q. It is not difficult to check that this operation satisfies the associativity and unit identities analogous to those for the tensor product of modules over algebras: The cotensor product is a left-exact functor on both arguments; according to the above identities, the functors of cotensor product with cofree comodules are exact. As usual, it follows that the right derived functors of the functor of cotensor product along the first and the second arguments coincide; we will call them the cotorsion functors and denote by Cot i C (P, Q). Using the cobar-resolution from subsection 1.1, one can compute those derived functors explicitly: , where C can be replaced with C + = ker ε.
For any left C-comodule P and finite-dimensional left C-comodule Q there are natural isomorphisms Ext i C (Q, P ) ≃ Cot i C (Q * , P ), just as for any left A-module M and right A-module N one has Ext i A (M, N * ) ≃ Tor A i (N, M) * . Therefore, we may alternatively define the cohomology of a coaugmented coalgebra as H * (C) = Cot * C (k, k) and the cohomology of a comodule as H * (C, P ) = Cot * C (k, P ). Remark: All the definitions of this section can be given for the (co)algebra and (co)module structures on (graded) objects of an arbitrary (not necessarily semisimple; associative, but not necessarily commmutative) abelian tensor category (with an exact functor of tensor product). The same holds for all the definitions and results of section 3, 6 and 7; all of them are duality-symmetric, in particular. For the results of section 8 mentioning commutative and Lie structures, the tensor category has to be commutative, of course. All of this does not apply to the results of sections 4 and 5, however, which depend essentially on the standard properties of the functor of direct limit and therefore are not duality-symmetric at all. But we prefer to deal with vector spaces in this paper.

Koszul Property for Modules and Comodules
Different accounts of the basic properties of quadratic and Koszul algebras can be found in the papers [16,2,5,6]. The notion of a Koszul module and related results first appeared in [5]. In the infinite-dimensional case one has to define the quadratic duality as a correspondence between algebras and coalgebras [14]. The purpose of this section is to present the analogous results for modules and comodules.

Quadratic modules and comodules.
A graded algebra A is called onegenerated if it is multiplicatively generated by A 1 . A graded algebra A is called quadratic if it is isomorphic to the quotient algebra {V, R} = T(V )/(R) of the tensor algebra T(V ) = n V ⊗n of the vector space V = A 1 by the ideal generated by a subspace R ⊂ V ⊗2 . For any graded algebra A there exists a unique quadratic algebra qA together with a morphism of graded algebras r A : qA −→ A which is an isomorphism on A 1 and a monomorphism on A 2 . To construct the algebra qA, one can take A 1 for V and the kernel of the multiplication map A 1 ⊗ A 1 −→ A 2 for R.
A graded module M over a one-generated algebra A is called one-generated if it is generated by M 0 , that is the map A ⊗ A graded coalgebra C is called one-cogenerated if the iterated comultiplication maps ∆ (n) : C n −→ C ⊗n 1 are injective, or equivalently, all the maps ∆ : C i+j −→ C i ⊗ C j for i, j 0 are injective. A graded coalgebra is called quadratic if it is isomorphic to the subcoalgebra of the tensor coalgebra n V ⊗n of the form for the vector space V = C 1 and a certain subspace R ⊂ V ⊗2 . For any graded coalgebra C there exists a unique quadratic coalgebra qC together with a morphism of graded coalgebras r C : C −→ qC which is an isomorphism on C 1 and an epimorphism on C 2 . Indeed, it suffices to take C 1 for V and the image of the comultiplication map A graded comodule P over a one-cogenerated coalgebra C is called one-cogenerated (cogenerated by P 0 ) if all the coaction maps ∆ ′ : P i+j −→ C i ⊗ P j for i, j 0 are injective, or equivalently, the maps ∆ ′ : P i −→ C i ⊗ P 0 are injective. A graded comodule P over a quadratic coalgebra C = V, R is called quadratic if it is isomorphic to the subcomodule of the cofree C-comodule C ⊗ U of the form for the vector space U = P 0 and a certain subspace S ⊂ V ⊗ U. For any graded comodule P over a graded coalgebra C there exists a unique quadratic comodule q C P over the quadratic coalgebra qC together with a morphism of graded qC-comodules r C,P : P −→ q C P which is an isomorphism on P 0 and an epimorphism on P 1 . Namely, one can take P 0 for U and the image of the coaction map P 1 −→ C 1 ⊗ P 0 for S.

Quadratic duality.
The quadratic algebra A = {V, R} and the quadratic coalgebra C = V, R are called dual to each other; we denote this as A = C ! and C = A ? . This rule defines an equivalence between the category of quadratic algebras and the category of quadratic coalgebras. Furthermore, the quadratic A-module M = {U, S} A and the quadratic C-comodule P = U, S C are called dual to each other, too; the notation: M = P ! C and P = M ? A . This defines an equivalence between the category of quadratic A-modules and the category of quadratic C-comodules.

Proposition 1.
A graded coalgebra C is one-cogenerated if and only if H 1,j (C) = 0 for all j > 1. A one-cogenerated coalgebra C is quadratic if and only if H 2,j (C) = 0 for all j > 2. More precisely, for any graded coalgebra C the morphism r C : C −→ qC is an isomorphism in degree n if and only if H i,j (C) = 0 for all i < j n and i = 1, 2. The analogous statements are true for graded algebras and their homology.
Proposition 2. A graded comodule P over a one-cogenerated coalgebra C is onecogenerated if and only if H 0,j (C, P ) = 0 for all j > 0. A one-cogenerated comodule P over a quadratic coalgebra C is quadratic iff H 1,j (C, P ) = 0 for all j > 1. More precisely, for any graded comodule P over a quadratic coalgebra C the morphism r C,P : P −→ q C P is an isomorphism in degree n if and only if H i,j (C, P ) = 0 for all i < j n and i = 0, 1. The analogous statements are true for graded modules.
Proof : The argument is based on the explicit form of the cobar-complex computing the cohomology in question. The space H 0 (C, P ) is isomorphic to the kernel of the map ∆ ′ : P −→ C + ⊗ P ; if the coalgebra C is one-cogenerated, it follows easily that the morphism r C,P : P −→ q C P ⊂ C ⊗ P 0 is injective in degree n if and only if H 0,j (C, P ) = 0 for all 0 < j n. Now assume that the coalgebra C is quadratic and the map r C,P is injective in degree < n. Let z ∈ C + ⊗ P be a homogeneous cocycle of degree n; then z = s+t=n s 1,t 0 z st with z st ∈ C s ⊗ P t . The cocycle condition means that the images of (∆ ⊗ id)(z u+v,w ) and Since the comultiplication and coaction maps are injective, the latter equivalently means that the images of z st in C ⊗n 1 ⊗ P 0 coinside for all s, t. So we get an element of C ⊗n 1 ⊗ P 0 ; it is easy to see that it represents an element of q C P . Besides, this element belongs to the image of r C,P iff the cocycle z is a coboundary. Finally, if the map r C,P is an isomorphism in degree < n, then any element of q C P of degree n corresponds to a cocycle z in this way.
Proposition 3. For any graded coalgebra C, the diagonal subalgebra i H i,i (C) of the cohomology algebra H * (C) is a quadratic algebra isomorphic to (qC) ! . For any graded C-comodule P , the diagonal part i H i,i (C, P ) of the cohomology module H * (C, P ) is a quadratic module over the diagonal subalgebra of H * (C) isomorphic to the (qC) ! -module (q C P ) ! qC . Analogously, for any graded algebra A the diagonal homology coalgebra i H i,i (A) is isomorphic to the quadratic coalgebra (qA) ? . For any graded A-module M, the diagonal homology qA over (qA) ? . Proof : Straightforward computation with the bar and cobar-complexes.

Koszul modules and comodules.
The notion of a Koszul algebra and the construction of the Koszul complex were invented by S. Priddy in [16]. The result relating Koszulity with distributivity of collections of vector subspaces is due to J. Backelin [1]. Detailed expositions can be found in the papers [5,6]; see also [2].
Note that all Koszul algebras and coalgebras, as well as modules and comodules, are quadratic. This follows from Propositions 1 and 2.
It is clear from subsection 1.4 that algebras with opposite multiplication, as well as coalgebras with opposite comultiplication, are quadratic or Koszul simultaneously. A graded right module over a Koszul algebra A is called Koszul if it is a Koszul module over A op ; analogously for right comodules.
Let A be a quadratic algebra and A ? be the dual coalgebra; suppose we are given a right A-module M and a left A ? -comodule P . The Koszul complex K(M, P ) is defined as the vector space M ⊗ P with the differential given by the formula Applying this result to the trivial A-module M = k, we conclude that a quadratic algebra A is Koszul if and only if the Koszul complex K(A, A ? ) is a resolution of k. Besides, we proved that a quadratic module M over a Koszul algebra A is Koszul

Koszulity and distributivity. A collection of subspaces
satisfied for any triple of subspaces X, Y , Z which can be obtained from the subspaces X k using the operations of sum and intersection.
Proof : The first statement is proven in [14]; the proof of the second one is analogous and based on the same lemma.
Remark: Many natural Koszulity-type homological conditions on data involving Koszul algebras and modules (including most of the properties that will be considered in section 5) can be simply rewritten in terms of distributivity of various collections of subspaces. It is not at all clear why this so happens. The related results often can be proven by both homological and lattice-theoretical methods, though the former are generally more powerful. Here are two of the various examples.
Let A = {V, R} be a Koszul algebra and E ⊂ V be a vector subspace. Then the subalgebra generated by E in the algebra A is Koszul if and only if the collection of subspaces E ⊗n and V ⊗k−1 ⊗ R ⊗ V n−k−1 in the vector space V ⊗n is distributive for all n 3. Note that in general this subalgebra does not have to be quadratic.
The latter observation was communicated to me by R. Bezrukavnikov.

Conilpotent Comodules and Koszulity
4.1. Coaugmentation filtrations. Let C be a coaugmented coalgebra with the coaugmentation map γ : k −→ C. Then the coaugmentation filtration N on the coalgebra C is an increasing filtration defined by the formula [14] Furthermore, let P be a comodule over a coaugmented coalgebra C. Then the coaugmentation filtration N on the comodule P is defined as A coaugmented coalgebra C is called conilpotent if the coaugmentation filtration N is full, that is C = n N n C. Note that in this case for any C-comodule P one has P = n N n P ; in other words, any comodule over a conilpotent coalgebra is conilpotent.
For any coaugmented coalgebra C, we denote by Nilp C the maximal conilpotent subcoalgebra n N n C of the coalgebra C. The following simple result is needed here. Proposition 6. The coaugmentation filtrations respect the coalgebra and comodule structures on a coalgebra C and comodule P , that is Furthermore, the associated graded coalgebra gr N C = ∞ n=0 N n C/N n−1 C and the comodule gr N P = ∞ n=0 N n P/N n−1 P over it are one-cogenerated. Proof : The statements concerning the coalgebras only are proven in [14]. Let us show that for any comultiplicative filtration N on a coalgebra C the filtration on a C-comodule P given by the above formula is compatible with the coaction. Let ψ : C −→ k be a linear function annihilating N k−1 C; we have to prove that (ψ ⊗ id)∆ ′ (N n P ) ⊂ N n−k P . The latter inclusion, by the definition, can be rewritten Since one has ∆ ′ (p) / ∈ N n−1 C ⊗ P for any p / ∈ N n−1 P , the last assertion of the proposition immediately follows.

4.2.
Pro-finite groups. Let G be a pro-finite group and k be a field. Consider the coalgebra k(G) of locally constant k-valued functions on G with respect to the convolution; in other words, let k(G) = lim − → k(G/U), where the direct limit is taken over all open normal subgroups U ⊂ G and the finite-dimensional coalgebra k(G/U) is the dual vector space to the group algebra k[G/U]. Then the category of discrete G-modules over k is equivalent to the category of k(G)-comodules. (Of course, the same is true over Z or an arbitrary commutative coefficient ring.) Now let γ : k −→ k(G) be the coaugmentation map that takes a constant from k to the corresponding constant function on G. Then for any discrete G-module P over k one has H * (G, P ) = H * (k(G), P ), because Ext * G (k, P ) = Ext * k(G) (k, P ). Suppose k is a field of characteristic l. Then one can see that the coaugmented coalgebra k(G) is conilpotent if and only if G is a pro-l-group. Moreover, for any profinite group G the coalgebra Nilp k(G) is isomorphic to the group coalgebra k(G (l) ) of the maximal quotient pro-l-group G (l) of G.

Nilpotency and Koszulity.
The following theorem is the main result of my paper with A. Vishik [14].
Theorem 3. Let A = H * (C) be the cohomology algebra of a conilpotent coalgebra C. Assume that (1) the quadratic algebra qA is Koszul; (2) the morphism of graded algebras qA −→ A is an isomorphism in degree 2 and a monomorphism in degree 3. Then the algebra A is quadratic (and therefore, Koszul). In addition, the graded coalgebra gr N C is Koszul and there is an isomorphism A ≃ (gr N C) ! .
Our next theorem is a module version of the above one. Their proofs are completely parallel; still we prefer to give the full argument here.
Theorem 4. Let C be a conilpotent coalgebra such that the cohomology algebra A = H * (C) is Koszul (equivalently, a coalgebra C satisfies the conditions of Theorem 3). Let P be a comodule over C. Consider the cohomology module M = H * (C, P ) over the algebra A. Assume that (1) the quadratic A-module q A M is Koszul; (2) the morphism of graded A-modules q A M −→ M is an isomorphism in degree 1 and a monomorphism in degree 2. Then the A-module M is quadratic (and therefore, Koszul). In addition, the graded gr N C-module gr N P is Koszul and there is an isomorphism M ≃ (gr N P ) ! gr N C of graded modules over the graded algebra A ≃ (gr N C) ! .
Proof : The coaugmentation filtrations on the coalgebra C and comodule P induce multiplicative filtrations on their cobar-complexes by the standard rule where N j C + = N j C/γ(k); so the filtrations on the terms C +⊗i and C +⊗i ⊗ P start with N i . Clearly, the associated graded complexes coincide with the cobar-complexes 13 of gr N C and gr N P . Therefore, we obtain a multiplicative spectral sequence Since the filtrations on the cobar-complexes are increasing ones, these spectral sequences converge in the direct limit. The differentials have the form . Furthermore, there are induced increasing filtrations N on H * (C) and H * (C, P ) compatible with the multiplication and the action and such that gr j N H i (C) = E i,j ∞ and gr j N H i (C, P ) = ′ E i,j ∞ . It was shown in the proof of Theorem 3 given in [14] that the spectral sequence . Besides, by Theorem 3 the graded algebra gr N C is Koszul and H * (gr N C) ≃ (gr N C) ! ≃ A.
From the spectral sequence ′ E i,j r we see that the submodule by the images of the differentials. According to Proposition 6, the graded gr N C-comodule gr N P is one-cogenerated, hence (by Proposition 2) we have ′ E 0,j 1 = H 0,j (gr N P, gr N C) = 0 for j > 0, which implies H 0 (C, P ) = N 0 H 0 (C, P ) ≃ H 0,0 (gr N C, gr N P ). Now the spectral sequence shows that E 1,1 ∞ = E 1,1 1 and therefore N 1 H 1 (C, P ) ≃ H 1,1 (gr N C, gr N P ). Since (by Proposition 3) the diagonal cohomology A-module ∞ i=0 H i,i (gr N C, gr N P ) is quadratic, we conclude that it is isomorphic to q A H * (C, P ).
By Proposition 3 again, the A-module ∞ i=0 H i,i (gr N C, gr N P ) is quadratic dual to the gr N C-comodule q gr N C gr N P ; since we suppose that the A-module q A H * (C, P ) is Koszul, the dual gr N C-comodule q gr N C gr N P is Koszul, too (Proposition 4). Besides, we have assumed that the morphism q A H * (C, P ) −→ H * (C, P ) is an isomorphism in degree 1, hence H 1 (C, P ) = N 1 H 1 (C, P ) and ′ E 1,n ∞ = 0. Furthermore, we have supposed that the morphism q A H * (C, P ) −→ H * (C, P ) is a monomorphism in degree 2.
∞ is a monomorphism and therefore all the differentials vanish. Now let us establish by induction on j that H 1,j (gr N C, gr N P ) = 0 for all j > 1. Assume that this is true for 1 < j n − 1. By Proposition 2, it follows that the map r gr N C, gr N P : gr N P −→ q gr N C gr N P is an isomorphism in degree n − 1. Therefore, the induced map of the cobar-complexes is also an isomorphism in these degrees, hence in particular H 1,j (gr N C, gr N P ) = H 1,j (gr N C, q gr N C gr N P ) for j n − 1 (and actually even for j n). Since the gr N C-comodule q gr N C gr N P is Koszul, it follows that ′ E 2,j 1 = H 2,j (gr N C, gr N P ) = 0 for all 2 < j n − 1. From the latter it is clear that ′ E 1,n 1 = ′ E 1,n ∞ , since the differentials targeting at ′ E 2,2 r also vanish. It was noticed above that ′ E 1,n ∞ = 0; so H 1,n (gr N C, gr N P ) = ′ E 1,n 1 = 0 and we are done. 14 We have shown that the gr N C-comodule gr N P is quadratic and the comodule q gr N C gr N P is Koszul, hence the comodule gr N P is Koszul. It follows that ′ E i,j 1 = 0 for i = j, so the spectral sequence degenerates and H * (C, P ) = H * (gr N C, gr N P ) = (gr N P ) ! gr N C . Therefore, H * (C, P ) is a Koszul H * (C)-module, too.

Maximal Conilpotent Subcoalgebra and Koszulity
The results of this section connect the cohomology of a coaugmented coalgebra with that of its maximal conilpotent subcoalgebra (and, in particular, the cohomology of a pro-finite group with the cohomology of its maximal quotient pro-l-group) under certain Koszulity conditions. Let us start with the following important lemma. Lemma 1. Let C be a coaugmented coalgebra and Nilp C be its maximal conilpotent subcoalgebra; then the natural map H * (Nilp C) −→ H * (C) is an isomorphism in degree 1 and a monomorphism in degree 2. In particular, the quadratic algebras qH * (C) and qH * (Nilp C) are naturally isomorphic to each other.
Proof : Any comodule over the coalgebra Nilp C can be considered as a comodule over C in the most obvious way. Furthermore, a C-comodule comes from a comodule over Nilp C if and only if it is a direct limit of successive extensions of the trivial C-comodule k (which is defined in terms of the coaugmentation). This is not difficult to prove, in the "if" direction by induction, using the coaugmentation filtrations on comodules and Proposition 6. It is helpful to remember that every comodule over a coassociative coalgebra is a direct limit of finite-dimensional comodules.
For any two comodules P and Q over the subcoalgebra Nilp C there is a natural morphism of graded vector spaces Ext * Nilp C (P, Q) −→ Ext * C (P, Q). Now it follows from the above that this map is always an isomorphism in degree 1. It turns out that one can derive purely formally from the latter that this is also a monomorphism in degree 2. Namely, it suffices to compute the spaces Ext 2 in question in terms of Ext 1 using a one-step injective resolution of the comodule Q over the coalgebra Nilp C, for The next result can be considered as a direct generalization of Theorem 3.
Theorem 5. Let C be a coaugmented coalgebra such that the cohomology algebra H * (C) satisfies the conditions of Theorem 3. Then the cohomology algebra of the maximal conilpotent subcoalgebra Nilp C is isomorphic to the quadratic part of the cohomology of C, that is H * (Nilp C) ≃ qH * (C).
Proof : Let us show that the cohomology algebra of the coalgebra Nilp C satisfies the conditions of Theorem 3, too. Indeed, the second statement of Lemma 1 provides the condition (1). To check the condition (2), one should consider the composition qH * (C) ≃ qH * (Nilp C) −→ H * (Nilp C) −→ H * (C) and use the first statement of Lemma 1. Applying Theorem 3, we conclude that the algebra H * (Nilp C) is quadratic. The latter implies the desired isomorphism of graded algebras. Corollary 1. Let C be a coaugmented coalgebra and Nilp C be its maximal conilpotent subcoalgebra. In this setting, if the cohomology algebra H * (C) is Koszul, then the natural homomorphism H * (Nilp C) −→ H * (C) is an isomorphism.
The analogous results comparing the cohomology of a pro-finite group with that of its maximal quotient pro-l-group can be either deduced from the results for coalgebras using the observations made in subsection 3.2, or they can be proven independently in the very parallel way. The proof of the analogue of Lemma 1 can be done differently (and even simpler) for pro-finite groups.
Lemma 2. If G is a pro-finite group and G (l) is its maximal quotient pro-l-group, then the natural map H i (G (l) , Z/l) −→ H i (G, Z/l) is an isomorphism for i = 1 and a monomorphism for i = 2. In particular, the quadratic algebras qH * (G, Z/l) and qH * (G (l) , Z/l) are naturally isomorphic to each other.
Proof : Let G (l) = G/G ′ ; then by the definition the subgroup G ′ has no nontrivial quotient pro-l-groups, so H 1 (G ′ , Z/l) = 0. It remains to apply the Serre-Hochschild spectral sequence E p,q 2 = H p (G (l) , H q (G ′ , Z/l)) =⇒ H p+q (G, Z/l). Corollary 2. Let G be a pro-finite group and G (l) be its maximal quotient pro-lgroup. If the cohomology algebra H * (G, Z/l) satisfies the conditions of Theorem 3, then the cohomology algebra of the group G (l) is isomorphic to the quadratic part of the cohomology of G, H * (G (l) , Z/l) ≃ qH * (G, Z/l). Furthermore, if the cohomology algebra H * (G, Z/l) is Koszul, then the natural homomorphism H * (G (l) , Z/l) −−→ H * (G, Z/l) is an isomorphism.
Remark: All the results of this section actually have a rather general categorical nature: they can be formulated for an arbitrary abelian category (or, better yet, a triangulated category) in place of the category of C-comodules with a "nilpotent" abelian subcategory in place of the subcategory of comodules over Nilp C; the nilpotency can be generalized to mean that all the objects have finite length. The proof is based on the generalization of Theorem 3 to nilpotent abelian categories. We wouldn't go so far here, but will only mention several elementary applications.
Let A be an augmented algebra. The coalgebra of pro-nilpotent completion C = A ∧ of the algebra A is defined as C = lim − →I (A/I) * , where the limit goes over all the ideals I ⊂ A + such that the algebra A/I is a finite-dimensional nilpotent algebra (meaning that there exists such n that (A + /I) n = 0) and (A/I) * is the dual vector space to A/I with its natural coalgebra structure. Then the following analogue of Theorem 5 is true: if the cohomology algebra H * (A) = Ext * A (k, k) satisfies the conditions of Theorem 3, then the algebra H * (C) is isomorphic to the quadratic part of H * (A).
Furthermore, the analogue of Corollary 2 is valid for a discrete group Γ in place of G and its pro-l-completion Γ (l) or pro-unipotent completion Γ (Q) in place of G (l) , where one considers the cohomology with the constant coefficients Z/l or Q, respectively.
One can reduce the last two statements to the one about the cohomology of an augmented algebra using the results of Quillen's paper [17], or prove all the three assertions independently just in the same way as it was done for coalgebras above.

Morphisms of Graded Algebras and Koszulity
The results of this and the next sections were found in an attempt to generalize and clarify some of the statements from the paper of J. Backelin and R. Fröberg [2]. . If A and B are graded algebras (M and N are graded modules) and the homomorphism f preserves the grading, then all the terms of the spectral sequence E r p,q bear the corresponding additional grading which is preserved by all the differentials.
We will be interested in two specific cases, which can be thought of as two kinds of Koszulity condition on a morphism f : A −→ B. The first case is when one has H i,j (A, B) = 0 unless i = j. A bit more general, but nevertheless interesting, second case is when H i,j (A, B) = 0 unless j − i 1. Note that the first condition implies surjectivity of the morphism f ; the second one doesn't.
For a nonnegatively graded vector space N = ∞ i=0 N i and an integer n 0 let us denote by N(n) the graded vector space with the components N(n) i = N i−n . We will say that a graded module M over a Koszul algebra A is a Koszul module in the grading shifted by n if there exists a Koszul A-module N such that M = N(n). Proof : We have to prove that H p,j (B) = 0 for all j = p. Proceeding by induction, assume that this is true for all 0 p < n. Since H q,j (A, B) = 0 for all q = j, j +1 and the graded right B-module H q (A, B) can be presented as an extension of its grading components endowed with the trivial B-module structures, it follows immediately that for any p < n the term E 2 p,q = H p (B op , H q (A, B)) of the above spectral sequence has no components of degrees other than p + q and p + q + 1. Now let us consider the term E 2 n,0 . The graded components of the terms E r n,0 can only cancel in the spectral sequence with those of the terms E r n−r,r−1 ; according to the above, the latter are concentrated in the degrees n − 1 and n. The limit term E ∞ n,0 = gr n H n (A) is known to be only nonzero in degree n; it follows easily that the graded vector space H n (B, H 0 (A, B)) = E 2 n,0 is concentrated in degree n. By assumption, the graded vector space H 0 (A, B) = B/f (A) is concentrated in the degrees 0 and 1. Thus we have an exact triple of graded right B-modules where the vector spaces B 1 /f (A 1 ) and k have the trivial B-module structures. Consider the corresponding long exact sequence of homology: the term H n (B) is placed between H n (B op , H 0 (A, B)) and H n−1 (B) ⊗ B 1 /f (A 1 )(1) and so it follows from the induction hypothesis that H n,j (B) = 0 for all j = n.
Conversely, if the graded spaces H p (B) are concentrated in degree p and the spaces H q (A, B) are concentrated in degree q, then all the terms E 2 p,q = H p (B op , H q (A, B)) are concentrated in degree p + q. In this case the same holds for the terms E ∞ p,q = gr p H p+q (A), hence the spaces H n (A) are concentrated in degree n.
The next corollary lists the most typical specific situations (see also [2]).
Corollary 3. Let f : A −→ B be a homomorphism of graded algebras. Suppose that the algebra A is Koszul and one of the following three conditions holds: (a) the A-module B is Koszul (equivalently, the morphism f is surjective and its kernel J is a Koszul A-module in the grading shifted by 1), or (b) the morphism f is injective and its cokernel B/f (A) is a Koszul A-module in the grading shifted by 1, or (c) the morphism f is surjective and its kernel J is a Koszul A-module in the grading shifted by 2. Then the algebra B is Koszul, too.
The proof of the following result is completely analogous to that of Theorem 6.

Graded Lie Coalgebras and Koszulity
It turns out somewhat unexpectedly that the results Theorem 7 implies for morphisms of Koszul commutative and Lie (co)algebras are stronger than in general. On the other hand, Lie coalgebras involve some troubles [11], which disappear in the positively graded case.
By a graded Lie algebra we mean here a positively graded vector space L = ∞ i=1 L n equipped with a Lie algebra structure such that [L i , L j ] ⊂ L i+j for all i, j. The universal enveloping algebra U(L) of a graded Lie algebra is a graded associative algebra. A graded Lie algebra is called quadratic if it isomorphic to the quotient algebra of the free Lie algebra generated by L 1 by an ideal generated by a set of elements of degree 2. A graded Lie algebra L is quadratic if and only if the algebra U(L) is quadratic. The homology coalgebra of a Lie algebra L can be defined as the homology of its enveloping algebra, H * (L) = H * (U(L)), or, more explicitly, as the homology coalgebra of the standard complex From the second definition it is obvious that the homology coalgebra of a Lie algebra is always skew-cocommutative. By a graded super-Lie algebra we mean a positively graded vector space equipped with a super-Lie algebra structure with respect to the parity induced by the grading such that the bracket respects the grading. There is a similar standard complex computing the homology of a super-Lie algebra.
Proposition 9. A quadratic algebra A is isomorphic to the enveloping algebra of a quadratic Lie algebra if and only if the dual quadratic coalgebra A ? is skewcocommutative. A quadratic algebra A is isomorphic to the enveloping algebra of a quadratic super-Lie algebra iff the coalgebra A ? is cocommutative. Recall that a Lie coalgebra is a vector space Λ together with a cobracket map δ : Λ −→ 2 Λ satisfying the equation dual to the Jacobi identity. The cohomology algebra of a Lie coalgebra Λ is defined as the cohomology of the standard complex The notion of the coenveloping coalgebra of a Lie coalgebra involves troubles, which may be overcome by restricting to the following condition. A Lie coalgebra is called conilpotent if it is a direct limit of finite-dimensional Lie coalgebras dual to nilpotent Lie algebras. The conilpotent enveloping coalgebra C(Λ) of a conilpotent Lie coalgebra Λ is defined as the universal object in the category of morphisms of conilpotent Lie coalgebras C −→ Λ from the Lie coalgebra associated with a conilpotent coassociative coalgebra C to the Lie coalgebra Λ. One can show that for any conilpotent Lie coalgebra Λ the cohomology algebras H * (Λ) and H * (C(Λ)) are naturally isomorphic.
By a graded Lie coalgebra we mean a positively graded vector space Λ = ∞ n=1 Λ n endowed with a Lie coalgebra structure such that δ(Λ n ) ⊂ i+j=n Λ i ∧Λ j . Obviously, any positively graded Lie coalgebra is conilpotent. The conilpotent coenveloping coalgebra of a graded Lie coalgebra is naturally a graded coassociative coalgebra and coincides with the coenveloping coalgebra defined with respect to the category of graded coalgebras. There are analogous notions of a conilpotent super-Lie coalgebra, its coenveloping coalgebra, and its cohomology algebra. A graded (super)-Lie coalgebra Λ is called Koszul if one has H i,j (Λ) = 0 for all i = j. Proof : This follows from the appropriate version of Proposition 9 and the analogue of Corollary 6 for morphisms of Koszul Lie or super-Lie coalgebras. An important difference is that the computation with enveloping algebras from the proof of the latter Corollary wouldn't work for coenveloping coalgebras in general, since the dual version of Poincare-Birkhoff-Witt theorem does not always hold. It holds for conilpotent Lie coalgebras, though.
By the definition, a pro-Lie algebra is a projective limit of finite-dimensional Lie algebras and a pro-nilpotent Lie algebra is a projective limit of finite-dimensional nilpotent Lie algebras. Note that the category of pro-Lie algebras is anti-equivalent to the category of locally finite Lie coalgebras and the category of pro-nilpotent Lie algebras is anti-equivalent to the category of conilpotent Lie coalgebras. Any positively graded pro-Lie algebras is, of course, pro-nilpotent. The cohomology algebra of a pro-nilpotent Lie algebra is defined as the cohomology of the corresponding Lie coalgebra; a graded pro-nilpotent (super)-Lie algebra is called Koszul if the corresponding (super)-Lie coalgebra is Koszul. A graded pro-nilpotent (super)-Lie algebra is free iff the corresponding (super)-Lie coalgebra is conilpotent cofree. This is the dual version of the second statement of Corollary 6. The result about freeness of a graded pro-Lie algebra mentioned in subsection 1.8 of the Introduction now follows immediately from Theorem 6 and Corollary 8.

Main Theorem
Two different proofs of Theorem 2 are given below, one based on Theorem 4 and the other on Theorem 7. Both approaches actually lead to somewhat stronger results, each in its own direction; they are formulated here as Theorem 9 and Corollary 9. Proof : Note that any morphism of conilpotent coalgebras g : C −→ D commutes with the coaugmentations. For the coaugmentation filtrations N on the coalgebras C and D one obviously has g(N i C) ⊂ N i D and it follows that these filtrations are compatible with the D-comodule structure on C. Thus there is a spectral sequence where the structure of graded gr N D-comodule on gr N C coincides with the one induced by the morphism of graded coalgebras gr N g : gr N C −→ gr N D. Therefore, we have H t (D, C) = 0 provided that H t (gr N D, gr N C) = 0. According to Theorem 3, there are natural isomorphisms gr N C ≃ A ? and gr N D ≃ B ? . Now the first statement follows from Theorem 7; it only remains to notice that for any comodule P over a conilpotent coalgebra C one has H i (C, P ) = 0 for all i t whenever H t (C, P ) = 0. The latter property is deduced from the fact that H 0 (C, P ) = 0 implies P = 0. In general, the coaugmentation filtration on the coalgebra C can differ from the coaugmentation filtration defined on the space C as a D-comodule. (2) the kernel J of the morphism f is isomorphic to the shift K(2) of a graded A-module K for which the morphism r A,K : q A K −→ K is an isomorphism in degree 1 and a monomorphism in degree 2; (3) the algebra A is Koszul and the A-module q A K is Koszul. Then the subgroup G is a free pro-l-group.

23
The operator d 2 : E p−2, 1 2 −→ E p,0 2 together with the projection of E p,0 2 into E p,0 ∞ define a 3-term complex of graded modules over E * ,0 Let us denote by P the G ′′ -module H 1 (G, Z/l). Taking the kernel of the second arrow of this complex, we get a morphism of graded A-modules H * (G ′′ , P ) −→ K.
According to Theorem 4, it follows from the latter that the A-module H * (G ′′ , P ) is quadratic, since the algebra A and the A-module q A H * (G ′′ , P ) ≃ q A K are Koszul. The above composition now shows that the map H 2 (G ′′ , P ) −→ K 2 is injective. Returning back to the spectral sequence, we conclude that the morphism d 0,2 2 : E 0,2 2 −→ E 2,1 2 is zero because the map H 2 (G ′′ , P ) −→ K 2 is injective and that the morphism d 0,2 3 : E 0,2 is zero because the map H 1 (G ′′ , P ) −→ K 1 is surjective.
Lemma 3. Let F be a field containing a primitive l-root of unity if l is odd and containing a square root of −1 if l = 2. Then the cohomology algebra of the Galois group Gal(F [ l ∞ √ F ]/F ) with constant coefficients Z/l is naturally isomorphic to the exterior algebra Λ * l (F ) = * Z/l (F * /F * l ). Proof : By Kummer's theory, it is clear that the first cohomology group of The square root of −1 condition guarantees that the cohomology algebra is skew-commutative. It suffices to show that the cohomology algebra is the exterior algebra over the first cohomology group.
Introduce the notation for fields . By Kummer's theory, the Galois group Gal(K/E) is naturally a subgroup of the group of homomorphisms from F * to the projective limit of the groops of l N -roots of unity, lim ← − µ l N . The group Hom(F * , lim ← − µ l N ) is an abelian pro-l-group without torsion, hence so is the group Gal(K/E). It follows that Gal(K/E) is a free abelian pro-l-group and its cohomology ring is an exterior algebra.
If the field F contains all the l N -roots of unity, then E = F and we are done. Otherwise, the group Gal(E/F ) is isomorphic to Z l and it acts trivially in the cohomology algebra of Gal(K/E). The Serre-Hochschild spectral sequence degenerates 24 into a series of exact sequences of the form It follows easily that H * (Gal(K/F ), Z/l) is an exterior algebra.
Proof of Theorem 2 : It follows from the condition (1) of Theorem 2 that the quadratic part qH * (G F , Z/l) of the Galois cohomology algebra is isomorphic to the Milnor algebra K M * (F ) ⊗ Z/l. By Theorem 6, it follows from the condition (2) that the latter algebra is Koszul. Therefore, Corollary 2 yields the isomorphisms . Now it remains to apply Theorem 9 or Corollary 9 to the pro-l-group G ′ = G (l) F with the quotient group G ′′ = Gal(K/F ) and the corresponding subgroup G = Gal(F (l) /K) (where F (l) is the maximal Galois pro-l-extension of F ). It is obvious that G is isomorphic to the maximal quotient pro-l-group of the absolute Galois group of K that we are interested in.

Appendix. Supporting Evidence for Conjecture 1
In this section we present a counterexample showing that the most straightforward generalization of Bogomolov's conjecture to arbitrary fields fails and prove several results providing evidence in support of our Conjecture 1.
Counterexample. There exist fields F such that the commutator subgroup of the maximal quotient pro-l-group of the absolute Galois group of F is not a free pro-lgroup. Moreover, there are fields such that the commutator subgroup of their Sylow pro-l-subgroup of the absolute Galois group is not free.
Proof : Let F be a field of characteristic not equal to l which contains a primitive root of unity of degree l but no primitive roots of unity of degree l N for some N > 1 (assumed to be the minimal such integer). Suppose that the maximal quotient prol-group of G F is not abelian. It is very easy to find a number field, l-adic field, or power series field satisfying the listed properties.
Consider the field F ((z)) of formal Laurent power series in the variable z with coefficients in F . Then the maximal quotient pro-l-group of the absolute Galois group of the field of power series G F with l N Z l . This is clearly not a free pro-l-group. The Sylow pro-l-subgroup of the absolute Galois group of F ((z)) is isomorphic to the semidirect product of the Sylow subgroup of G F with Z l . The commutator subgroup of this group is described as above; it is not a free pro-l-group either.
In particular, for any field F containing a primitive l-root of unity but no primitive l N -roots of unity for some N > 1, the field of doubly iterated formal Laurent power series F ((w))((z)) provides the desired counterexample.
Proposition A.1. Conjecture 1 holds for number fields and their functional analogues, i. e., algebraic extensions of Q and F q (x). Moreover, in these cases it suffices to add roots of unity only, i. e., take K = F [ l ∞ √ 1].
Proof : The assertion is (essentially) a particular case of the result of [18,II.3.3]. It suffices to prove that H 2 (G M , Z/l) = 0 for any field M between K and F . For any field L containing a primitive l-root of unity, the group H 2 (G L , Z/l) is isomorphic to the subgroup of the Brauer group Br(L) annihilated by l. Obviously, the Brauer group functor preserves direct limits of fields. So it suffices to show that for any field E finite over Q or F q (x) and any element α ∈ Br(E) there exists a field L finite over E and contained in the composite EK such that the image of α in Br(L) vanishes.
It remains to use the description of Brauer groups given by class field theory. The Brauer group of a global field is a subgroup of the direct sum of the Brauer groups of local completions. For any class α ∈ Br(E) there is a finite number of completions E v of the field E where the local class α v is nonzero. Adjoining l N -roots of unity with large enough N to a nonarchimedean complete field E v , one can obtain extensions of E v of degrees divisible by arbitrary high powers of l (since E v is finite over a field Q p or F q [[z]], which has this property). And by local class field theory an l-torsion element of Br(E v ) dies in any finite extension of E v of degree divisible by l. In the archimedean case, the local Brauer group is zero (for E v = C) or Z/2 (for E v = R), and it suffices to adjoin the square root of −1 in order to kill it.
Proposition A.2. Let F be a Henselian discrete valuation field and f be its residue field. Then the field F satisfies Conjecture 1 whenever f does (for a given l). In particular, if char f = l, then F satisfies Conjecture 1.
Proof : As in the proof of Proposition A.1, we will argue in terms of the Brauer groups. The Brauer groups of Henselian discrete valuation fields with perfect residue fields were computed in [19, chapitre XII] and [13, §2].
Consider the two cases char f = l and char f = l separately. It is easy to verify following the arguments of [19] and [13] that for char f = l there is always a natural exact sequence where v(F ) denotes the group of values of the discrete valuation, G f is the absolute Galois group of f , and the subindex l in parentheses signifies the l-primary components of the corresponding abelian groups. Let E be a finite extension of the field F and e be its residue field. We would like to prove that any element of Br(E) (l) can be killed by a radical extension of F . Let α ∈ Br(E) (l) ; for compactness reasons, the corresponding element of Hom(G e , v(E) ⊗ Q/Z) is a homomorphism with finite image. We kill this class by adjoining a root of an appropriate degree from a uniformizing element of the valuation v in F . This extends our field E to a larger field D, still finite over F ; let d denote its residue field. Applying the above exact sequence to the valued field D, we get an element α ′ ∈ Br(d) (l) . By our assumption, this element can be killed by adjoining l-power roots of elements of f . To make the original element α ∈ Br(E) (l) vanish, it now suffices to adjoin the corresponding roots of any preimages in F of those residue elements, together with the root of the uniformizing element in F that we adjoined already.
In the case char f = l the main step is to reduce the problem to the situation of a perfect residue field. Let us repeat transfinitely the following process of constructing purely inseparable extensions e of the field f . Choose any element of f which is not an l-power in f and adjoin to f all roots of this element of degrees l N ; this is the first step. On each of the subsequent steps we choose an element of f which is still not an l-power in e and adjoin all its l N -degree roots to e. When the transfinite induction terminates, we have a purely inseparable extension d of the field f which is easily found to be its perfect closure. Indeed, any x ∈ f is an l-power in d by construction; let us assume that it is an l n−1 -power and show that it is also an l n -power. Let x = y l n−1 , y ∈ d. The element y can be expressed in terms of elements of f and roots of elements of f that we adjoined in the process. But both kinds of elements are l-powers in d by the construction. Now we repeat this transfinite process on the level of valued fields. At each step we choose a compatible family of roots of all degrees l N , one root of each degree, taken from an arbitrary element of F lifting x ∈ f . Notice that we avoid any extension of the group of values of the valuation; indeed, at each step the total degree of the extention coincides with the degree of extension of the residue fields. So the valuation remains discrete. The transfinite process results in the construction of a Henselian discrete valuation field D with a perfect residue field. According to [19] and [13] In addition, the l-primary component of the Brauer group of a perfect field of characteristic l vanishes. By construction, we have F ⊂ D ⊂ F [ l ∞ √ F ]. Finally, let M be a finite extension of the field D and m be its residue field. It remains to annihilate the class from Hom(G m , v(M) ⊗ Q/Z) (l) . As above, we do this by adjoining a root of high enough l-power degree from a uniformizing element of the valuation v of the field F .

Remark 1:
Here is another partial result in support of Conjecture 1, provided by a version of an argument which I learned from F. Bogomolov sometime in 1995-96. Let F be a field of characteristic not equal to l; set K = F [ l ∞ √ F ]. The conjecture can be rephrased by saying that for any finite extension E of the field F the image of the group K M 2 (E)/l in K M 2 (KE)/l vanishes. Let us show that this is true for any finite Galois extension E/F with the Galois group isomorphic to a direct sum of copies of Z/2, or a direct sum of copies of Z/3, or, more generally, for any abelian field extension with the Galois group annihilated by a number q such that the field F has no nontrivial finite extensions of degrees less or equal to q/2. In particular, the assertion applies to any extension E/F with the Galois group Gal(E/F ) ≃ Z/l ⊕ · · · ⊕ Z/l of a field F having no finite extensions of the degrees prime to l.