Koszul duality and Galois cohomology

Abstract

It it shown that the Bloch-Kato conjecture on the norm residue homomorphism $K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is Koszul. This conclusion is a case of a general result on the cohomology of nilpotent (co-)algebras and Koszulity.Comment: AMS-LaTeX v.1.1, 10 pages, no figures. Replaced for tex code correction (%&amslplain added) by request of www-admi

Koszulity of cohomology = K(π,1)-ness + quasi-formality

“…The alternative between considering nonhomogeneous quadratic relations of the types (1) and (2) roughly leads to a division of the Koszul duality theory into two streams, the former of them going back to the classical paper [41] and the present author's work [31], and the latter one originating in the paper [32]. The former theory, invented originally for the purposes of computing the cohomology of associative algebras generally and the Steenrod algebra in particular, eventually found its applications in semi-infinite homological algebra.…”

“…It was conjectured in the papers [32,37] that the cohomology algebra H * (G F , Z/l) is Koszul for the absolute Galois group G F of any field F containing a primitive l-root of unity; and the question was asked in the paper [16] whether the cochain DG-algebra of the group G F with coefficients in Z/l is formal. As the group H in our counterexample is the maximal quotient pro-l-group of the absolute Galois group G F of an appropriate p-adic field F containing a primitive l-root of unity, our results provide a negative answer to this question of Hopkins and Wickelgren.…”

“…Its subject can be roughly described as cohomological characterization of the coalgebras C defined by the relations (2) with the quadratic principal parts (3) q 2 (x) = 0 of the relations defining a Koszul graded coalgebra. In fact, according to the main theorem of [32] (see also [23]) a conilpotent coalgebra C is defined by a self-consistent system of relations (2) with Koszul quadratic principal part (3) if and only if its cohomology algebra H * (C) = Ext * C (k, k) is Koszul. Moreover, a certain seemingly weaker set of conditions on the algebra H * = H * (C) is sufficient, and implies Koszulity of algebras of the form H * (C).…”

“…In other words, a positively graded algebra A = k ⊕ A 1 ⊕ A 2 ⊕ · · · over a field k is called quadratic if it is generated by its first-degree component A 1 with relations in degree 2. A positively graded associative algebra A is called Koszul [41,4,32] if one has Tor A ij (k, k) = 0 for all i = j, where the first grading i on the Tor spaces is the usual homological grading and the second grading j, called the internal grading, is induced by the grading of A. In particular, this condition for i = 1 means that the algebra A is generated by A 1 , and the conditions for i = 1 and 2 taken together mean that A is quadratic.…”

“…When the algebra A is quadratic, the two quadratic algebras A and n Ext n,n A (k, k) are called quadratic dual to each other. Without the finite-dimensionality assumption on the grading components, the quadratic duality connects quadratic graded algebras with quadratic graded coalgebras [32,33].…”