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 GK satisfies dim H1(GK,Fp) and dim H 2(GK,Fp) = 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 Fp-cohomology algebra of GK is the quadratic dual of the graded algebra gr•Fp[GK], induced by the powers of the augmentation ideal of the group algebra Fp[GK], 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.