Pro-$p$ groups with few relations and universal Koszulity


  • Claudio Quadrelli



Let $p$ be a prime. We show that if a pro-$p$ group with at most $2$ defining relations has quadratic $\mathbb{F}_p$-cohomology algebra, then this algebra is universally Koszul. This proves the “Universal Koszulity Conjecture” formulated by J. Miná{č} et al. in the case of maximal pro-$p$ Galois groups of fields with at most $2$ defining relations.


Bush, M. R., Gärtner, J., Labute, J., and Vogel, D., Mild $2$-relator pro-$p$-groups, New York J. Math. 17 (2011), 281–294.

Cassella, A. and Quadrelli, C., Right-angled Artin groups and enhanced Koszul properties, J. Group Theory 24 (2021), no. 1, 17–38.

Conca, A., Universally Koszul algebras, Math. Ann. 317 (2000), no. 2, 329–346.

Conca, A., Koszul algebras and their syzygies, in “Combinatorial algebraic geometry”, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 1–31.

Conca, A., Trung, N. V., and Valla, G., Koszul property for points in projective spaces, Math. Scand. 89 (2001), no. 2, 201–216.

Droms, C., Subgroups of graph groups, J. Algebra 110 (1987), no. 2, 519–522.

Efrat, I., Valuations, orderings, and Milnor $K$-theory, Mathematical Surveys and Monographs, vol. 124, American Mathematical Society, Providence, RI, 2006.

Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, vol. 346, Springer, Heidelberg, 2012.

Mináč, J., Palaisti, M., Pasini, F. W., and Tân, N. D., Enhanced Koszul properties in Galois cohomology, Res. Math. Sci. 7 (2020), no. 2, paper no. 10, 34 pp.

Mináč, J., Pasini, F. W., Quadrelli, C., and Tan, N. D., Koszul algebras and quadratic duals in Galois cohomology, Adv. Math. (2021), paper id. 107569, online ahead of print.

Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of number fields, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 323, Springer-Verlag, Berlin, 2008.

Palaisti, M., Enhanced Koszulity in Galois cohomology, Ph.D. thesis, University of Western Ontario, 2019, Electronic Thesis and Dissertation Repository, no. 6038,

Papadima, S. and Suciu, A. I., Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), no. 3, 533–555.

Piontkovskiu ı, D. I., Koszul algebras and their ideals, Funktsional. Anal. i Prilozhen. 39 (2005), no. 2, 47–60.

Polishchuk, A. and Positselski, L., Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005.

Positselski, L., Koszul property and Bogomolov's conjecture, Int. Math. Res. Not. (2005), no. 31, 1901–1936.

Positselski, L., Galois cohomology of a number field is Koszul, J. Number Theory 145 (2014), 126–152.

Positselski, L. and Vishik, A., Koszul duality and Galois cohomology, Math. Res. Lett. 2 (1995), no. 6, 771–781.

Priddy, S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60.

Quadrelli, C., One-relator maximal pro-$p$ Galois groups and the Koszulity conjectures, Q. J. Math. (2020), paper id. haaa049, online ahead of print.

Quadrelli, C., Snopce, I., and Vannacci, M., On pro-$p$ groups with quadratic cohomology, eprint arXiv:1906.04789 [math.GR], 2019.

Snopce, I. and Zaleskii, P., Right-angled Artin pro-$p$ groups, eprint arXiv:2005.01685 [math.GR], 2020.

Voevodsky, V., On motivic cohomology with $mathbf Z/l$-coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438.

Vogel, D., Massey products in the Galois cohomology of number fields, Ph.D. thesis, University of Heidelberg, 2004,

Weibel, C., The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372.

Weigel, T., Graded Lie algebras of type FP, Israel J. Math. 205 (2015), no. 1, 185–209.



How to Cite

Quadrelli, C. (2021). Pro-$p$ groups with few relations and universal Koszulity. MATHEMATICA SCANDINAVICA, 127(1), 28–42.