One-relator maximal pro-p Galois groups and the Koszulity conjectures

Let p be a prime number and let K be a field containing a root of 1 of order p. If the absolute Galois group G_K satisfies dim H^1(G_K, F_p) < ∞ and dim H^2(G_K ,F_p) = 1, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for K. Also, under the above hypothesis we show that the F_p -cohomology algebra of G_K is the quadratic dual of the graded algebra gr_*F_p[G_K], induced by the powers of the augmentation ideal of the group algebra F_p[G_K], and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s Elementary Type Conjecture.