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The resolution property via Azumaya algebras

  • Siddharth Mathur


Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.

Funding statement: A part of this research was conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the DFG.


A large part of this paper comes from the authors Ph.D thesis. I would like to thank my advisor, Max Lieblich, for his generous support and guidance. I am also indebted to Jarod Alper and Aise Johan de Jong for their encouragement and suggestions. I also benefited from helpful discussions with Dan Abramovich, Dan Edidin, Lucas Braune, Jack Hall, Ariyan Javanpeykar, Andrew Kresch, Minseon Shin, David Rydh, Stefan Schröer, Jason Starr, and Angelo Vistoli. I would also like to thank the referee for carefully reading an earlier version of this paper and providing many helpful comments.


[1] D. Abramovich, M. Olsson and A. Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 4, 1057–1091. 10.5802/aif.2378Search in Google Scholar

[2] J. Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402. 10.5802/aif.2833Search in Google Scholar

[3] K. Behrend and B. Noohi, Uniformization of Deligne–Mumford curves, J. reine angew. Math. 599 (2006), 111–153. 10.1515/CRELLE.2006.080Search in Google Scholar

[4] B. Bhatt, Algebraization and Tannaka duality, Camb. J. Math. 4 (2016), no. 4, 403–461. 10.4310/CJM.2016.v4.n4.a1Search in Google Scholar

[5] B. Conrad, Smooth linear algebraic groups over the dual numbers, MathOverflow, Search in Google Scholar

[6] B. Conrad, M. Lieblich and M. Olsson, Nagata compactification for algebraic spaces, J. Inst. Math. Jussieu 11 (2012), no. 4, 747–814. 10.1017/S1474748011000223Search in Google Scholar

[7] N. Deshmukh, A. Hogadi and S. Mathur, Quasi-affineness and the 1-resolution property, Int. Math. Res. Not. IMRN 2020 (2020), 10.1093/imrn/rnaa125. 10.1093/imrn/rnaa125Search in Google Scholar

[8] D. Edidin, B. Hassett, A. Kresch and A. Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001), no. 4, 761–777. 10.1353/ajm.2001.0024Search in Google Scholar

[9] O. Gabber, Some theorems on Azumaya algebras, The Brauer group, Lecture Notes in Math. 844, Springer, Berlin (1981), 129–209. 10.1007/BFb0090480Search in Google Scholar

[10] J. Giraud, Cohomologie non abélienne, Grundlehren Math. Wiss. 179, Springer, Berlin 1971. 10.1007/978-3-662-62103-5Search in Google Scholar

[11] P. Gross, Vector bundles as generators on schemes and stacks, Ph.D. thesis, Heinrich-Heine-Universität Düsseldorf, 2010. Search in Google Scholar

[12] P. Gross, The resolution property of algebraic surfaces, Compos. Math. 148 (2012), no. 1, 209–226. 10.1112/S0010437X11005628Search in Google Scholar

[13] P. Gross, Tensor generators on schemes and stacks, Algebr. Geom. 4 (2017), no. 4, 501–522. 10.14231/AG-2017-026Search in Google Scholar

[14] A. Grothendieck, Le groupe de Brauer. II. Théorie cohomologique, Séminaire Bourbaki. Vol. 9, Société Mathématique de France, Paris (1995), 287–307, Exp. No. 297. Search in Google Scholar

[15] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique. I. Le langage des schémas, Publ. Math. Inst. Hautes Études Sci. 4 (1960), 5–214. 10.1007/BF02684778Search in Google Scholar

[16] A. Grothendieck, M. Raynaud and D. S. Rim, Séminaire de Géométrie Algébrique Du Bois-Marie 1967-1969. Groupes de monodromie en géométrie algébrique (SGA 7 I), Lecture Notes in Math. 288, Springer, Berlin 2006. Search in Google Scholar

[17] J. Hall, Cohomology and base change for algebraic stacks, Math. Z. 278 (2014), no. 1–2, 401–429. 10.1007/s00209-014-1321-7Search in Google Scholar

[18] J. Hall and D. Rydh, Mayer–Vietoris squares in algebraic geometry, preprint (2016), Search in Google Scholar

[19] J. Hall and D. Rydh, Coherent Tannaka duality and algebraicity of Hom-stacks, Algebra Number Theory 13 (2019), no. 7, 1633–1675. 10.2140/ant.2019.13.1633Search in Google Scholar

[20] J. C. Jantzen, Representations of algebraic groups, 2nd ed., Math. Surveys Monogr. 107, American Mathematical Society, Providence 2003. Search in Google Scholar

[21] A. Johan de Jong, A result of Gabber, preprint (2005). Search in Google Scholar

[22] A. Kresch and A. Vistoli, On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map, Bull. Lond. Math. Soc. 36 (2004), no. 2, 188–192. 10.1112/S0024609303002728Search in Google Scholar

[23] S. Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563. 10.2307/2372673Search in Google Scholar

[24] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin 2000. 10.1007/978-3-540-24899-6Search in Google Scholar

[25] M. Lieblich, Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), no. 1, 1–31. 10.1112/S0010437X07003144Search in Google Scholar

[26] J. Lurie, Tannaka duality for geometric stacks, preprint (2004), Search in Google Scholar

[27] H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge University, Cambridge 1986. Search in Google Scholar

[28] M. Olsson, A boundedness theorem for Hom-stacks, Math. Res. Lett. 14 (2007), no. 6, 1009–1021. 10.4310/MRL.2007.v14.n6.a9Search in Google Scholar

[29] M. Olsson, Algebraic spaces and stacks, Amer. Math. Soc. Colloq. Publ. 62, American Mathematical Society, Providence 2016. 10.1090/coll/062Search in Google Scholar

[30] M. C. Olsson, On proper coverings of Artin stacks, Adv. Math. 198 (2005), no. 1, 93–106. 10.1016/j.aim.2004.08.017Search in Google Scholar

[31] M. C. Olsson, Deformation theory of representable morphisms of algebraic stacks, Math. Z. 253 (2006), no. 1, 25–62. 10.1007/s00209-005-0875-9Search in Google Scholar

[32] D. Rydh, Do line-bundles descend to coarse moduli spaces of artin stacks with finite inertia (answer), MathOverflow, Search in Google Scholar

[33] S. Schröer, There are enough Azumaya algebras on surfaces, Math. Ann. 321 (2001), no. 2, 439–454. 10.1007/s002080100236Search in Google Scholar

[34] S. Schröer and G. Vezzosi, Existence of vector bundles and global resolutions for singular surfaces, Compos. Math. 140 (2004), no. 3, 717–728. 10.1112/S0010437X0300071XSearch in Google Scholar

[35] J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, Secrétariat mathématique, Paris (1958), 1–18. 10.1007/978-3-642-39816-2_39Search in Google Scholar

[36] C. S. Seshadri, Geometric reductivity over arbitrary base, Adv. Math. 26 (1977), no. 3, 225–274. 10.1016/0001-8708(77)90041-XSearch in Google Scholar

[37] J. Tate, Finite flat group schemes, Modular forms and Fermat’s last theorem (Boston 1995), Springer, New York (1997), 121–154. 10.1007/978-1-4612-1974-3_5Search in Google Scholar

[38] R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), no. 1, 16–34. 10.1016/0001-8708(87)90016-8Search in Google Scholar

[39] B. Totaro, The resolution property for schemes and stacks, J. reine angew. Math. 577 (2004), 1–22. 10.1515/crll.2004.2004.577.1Search in Google Scholar

[40] The Stacks Project Authors, Stacks project,, 2017. Search in Google Scholar

Received: 2020-02-24
Revised: 2020-10-29
Published Online: 2021-02-23
Published in Print: 2021-05-01

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