2016/01/31 by Claudio Quadrelli
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cohomology #Combinatorics #Commutative Algebra and Its Applications #Conjecture #Discrete mathematics #Field (mathematics) #Galois group #Galois module #Group (periodic table) #Ideal (ethics) #Mathematics #Order (exchange) #Physics #Prime (order theory) #Pure mathematics #math.GR #math.NT #msc:12G05 #msc:17A45 #msc:20F14 #msc:20F40
paper · pdf · doi:10.1093/qmath/haaa049
Updated version
arxiv created 2020/04/18 · openalex created_date 2020/04/24 · openalex publication_date 2020/10/16 · arxiv updated 2020/11/10 · openalex updated_date 2026/08/05
Abstract Let p be a prime number and let \mathbbK be a field containing a root of 1 of order p. If the absolute Galois group G_\mathbbK satisfies dim H1(G_\mathbbK,\mathbbFp)\lt∞ and dim H 2(G_\mathbbK,\mathbbFp)=1, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for \mathbbK. Also, under the above hypothesis, we show that the \mathbbFp-cohomology algebra of G_\mathbbK is the quadratic dual of the graded algebra \rm gr_\bullet\mathbbFp[G_\mathbbK], induced by the powers of the augmentation ideal of the group algebra \mathbbFp[G_\mathbbK], 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.