Right-angled Artin groups and enhanced Koszul properties

Alberto Cassella 1  and Claudio Quadrelli 2
  • 1 Department of Mathematics and Applications, University of Milano Bicocca, 20125, Milan, Italy
  • 2 Department of Mathematics and Applications, University of Milano Bicocca, 20125, Milan, Italy
Alberto Cassella and Claudio Quadrelli
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  • Department of Mathematics and Applications, University of Milano Bicocca, 20125, Milan, Italy
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Abstract

Let 𝔽 be a finite field. We prove that the cohomology algebra H(GΓ,F) with coefficients in 𝔽 of a right-angled Artin group GΓ is a strongly Koszul algebra for every finite graph Γ. Moreover, H(GΓ,F) is a universally Koszul algebra if, and only if, the graph Γ associated to the group GΓ has the diagonal property. From this, we obtain several new examples of pro-𝑝 groups, for a prime number 𝑝, whose continuous cochain cohomology algebra with coefficients in the field of 𝑝 elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-𝑝 Galois groups of fields formulated by J. Mináč et al.

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