Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T01:24:30.718Z Has data issue: false hasContentIssue false
  • English
  • Français

The inertia operator on the motivic Hall algebra

Part of: Families, fibrations

Published online by Cambridge University Press:  14 March 2019

Kai Behrend  and
Pooya Ronagh
Kai Behrend
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2, Canada email [email protected]
Pooya Ronagh
Affiliation:
Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the action of the inertia operator on the motivic Hall algebra and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.

Keywords

algebraic stacksinertia stackmotivic Hall algebraDonaldson–Thomas theory

MSC classification

Primary: 14D23: Stacks and moduli problems
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 3 , March 2019 , pp. 528 - 598
Copyright
© The Authors 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramovich, D., Olsson, M. and Vistoli, A., Tame stacks in positive characteristic , Ann. Inst. Fourier (Grenoble) 58 (2008), 1057–1091; MR 2427954.Google Scholar
Behrend, K. and Fantechi, B., The intrinsic normal cone , Invent. Math. 128 (1997), 45–88.Google Scholar
Bridgeland, T., An introduction to motivic Hall algebras , Adv. Math. 229 (2012), 102–138; MR 2854172.Google Scholar
Bühler, T., Exact categories , Expo. Math. 28 (2010), 1–69; MR 2606234.Google Scholar
Cartier, P., A primer of Hopf algebras , in Frontiers in number theory, physics, and geometry. II (Springer, Berlin, 2007), 537–615; MR 2290769.Google Scholar
D’Agnolo, A. and Polesello, P., Deformation quantization of complex involutive submanifolds , in Noncommutative geometry and physics (World Scientific Publishing, Hackensack, NJ, 2005), 127–137; MR 2186385.Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique, EGA, Publications Mathématiques, vols. 4, 8, 11, 17, 20, 24, 28, and 32 (Institut des Hautes Études Scientifiques, 1960–1967).Google Scholar
Joyce, D., Configurations in abelian categories. II. Ringel-Hall algebras , Adv. Math. 210 (2007), 635–706; MR 2303235 (2008f:14022).Google Scholar
Joyce, D., Motivic invariants of Artin stacks and ‘stack functions’ , Q. J. Math. 58 (2007), 345–392; MR 2354923 (2010b:14004).Google Scholar
Joyce, D. and Song, Y., A theory of generalized Donaldson–Thomas invariants , Mem. Amer. Math. Soc. 217 (2012); MR 2951762.Google Scholar
Kontsevich, M., Deformation quantization of algebraic varieties , Lett. Math. Phys. 56 (2001), 271–294; EuroConférence Moshé Flato 2000, Part III (Dijon); MR 1855264 (2002j:53117).Google Scholar
Kresch, A., Cycle groups for Artin stacks , Invent. Math. 138 (1999), 495–536.Google Scholar
Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 39 (Springer, Berlin, 2000); MR 1771927 (2001f:14006).Google Scholar
Noohi, B., Fundamental groups of algebraic stacks , J. Inst. Math. Jussieu 3 (2004), 69–103; MR 2036598.Google Scholar
Oesterlé, J., Schémas en groupes de type multiplicatif , in Autour des schémas en groupes. Vol. I, Panoramas et Synthèses, vol. 42/43 (Soc. Math. France, Paris, 2014), 63–91; MR 3362640.Google Scholar
Schedler, T., Deformations of algebras in noncommutative geometry , in Noncommutative algebraic geometry, Mathematical Sciences Research Institute Publications, vol. 64 (Cambridge University Press, New York, 2016), 71–165; MR 3618473.Google Scholar
Grothendieck, A., Revêtements etales et groupe fondamental, SGA1, Lecture Notes in Mathematics, vol. 224 (Springer, 1971).Google Scholar
Artin, M., Bertin, J. E., Demazure, M., Grothendieck, A., Gabriel, P., Raynaud, M. and Serre, J.-P., Schémas en groupes, SGA3 (Institut des Hautes Études Scientifiques, Paris, 1963/1966).Google Scholar
Artin, M., Grothendieck, A. and Verdier, J. L., Théorie des topos et cohomologie etale des schémas, SGA4, Lecture Notes in Mathematics, vols. 269, 270, 305 (Springer, Berlin, Heidelberg, New York, 1972/1973).Google Scholar
The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2015, retrieved December 2016.Google Scholar