Pro-$p$ groups with few relations and universal Koszulity
DOI:
https://doi.org/10.7146/math.scand.a-123644Abstract
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.
References
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. https://doi.org/10.1515/jgth-2020-0049
Conca, A., Universally Koszul algebras, Math. Ann. 317 (2000), no. 2, 329–346. https://doi.org/10.1007/s002080000100
Conca, A., Koszul algebras and their syzygies, in “Combinatorial algebraic geometry”, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 1–31. https://doi.org/10.1007/978-3-319-04870-3_1
Conca, A., Trung, N. V., and Valla, G., Koszul property for points in projective spaces, Math. Scand. 89 (2001), no. 2, 201–216. https://doi.org/10.7146/math.scand.a-14338
Droms, C., Subgroups of graph groups, J. Algebra 110 (1987), no. 2, 519–522. https://doi.org/10.1016/0021-8693(87)90063-9
Efrat, I., Valuations, orderings, and Milnor $K$-theory, Mathematical Surveys and Monographs, vol. 124, American Mathematical Society, Providence, RI, 2006. https://doi.org/10.1090/surv/124
Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, vol. 346, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-30362-3
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. https://doi.org/10.1007/s40687-020-00208-5
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. https://doi.org/10.1016/j.aim.2021.107569
Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of number fields, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 323, Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-37889-1
Palaisti, M., Enhanced Koszulity in Galois cohomology, Ph.D. thesis, University of Western Ontario, 2019, Electronic Thesis and Dissertation Repository, no. 6038, https://ir.lib.uwo.ca/etd/6038/.
Papadima, S. and Suciu, A. I., Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), no. 3, 533–555. https://doi.org/10.1007/s00208-005-0704-9
Piontkovskiu ı, D. I., Koszul algebras and their ideals, Funktsional. Anal. i Prilozhen. 39 (2005), no. 2, 47–60. https://doi.org/10.1007/s10688-005-0024-6
Polishchuk, A. and Positselski, L., Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. https://doi.org/10.1090/ulect/037
Positselski, L., Koszul property and Bogomolov's conjecture, Int. Math. Res. Not. (2005), no. 31, 1901–1936. https://doi.org/10.1155/IMRN.2005.1901
Positselski, L., Galois cohomology of a number field is Koszul, J. Number Theory 145 (2014), 126–152. https://doi.org/10.1016/j.jnt.2014.05.024
Positselski, L. and Vishik, A., Koszul duality and Galois cohomology, Math. Res. Lett. 2 (1995), no. 6, 771–781. https://doi.org/10.4310/MRL.1995.v2.n6.a8
Priddy, S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. https://doi.org/10.2307/1995637
Quadrelli, C., One-relator maximal pro-$p$ Galois groups and the Koszulity conjectures, Q. J. Math. (2020), paper id. haaa049, online ahead of print. https://doi.org/10.1093/qmath/haaa049
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. https://doi.org/10.4007/annals.2011.174.1.11
Vogel, D., Massey products in the Galois cohomology of number fields, Ph.D. thesis, University of Heidelberg, 2004, http://www.ub.uni-heidelberg.de/archiv/4418. https://doi.org/10.11588/heidok.00004418
Weibel, C., The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372. https://doi.org/10.1112/jtopol/jtp013
Weigel, T., Graded Lie algebras of type FP, Israel J. Math. 205 (2015), no. 1, 185–209. https://doi.org/10.1007/s11856-014-1131-y
