A Luna étale slice theorem for algebraic stacks

Pages 675-738 from Volume 191 (2020), Issue 3 by Jarod Alper, Jack Hall, David Rydh

Abstract

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin’s algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

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    @article {MR217083,
      author = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {III}. \'{E}tude cohomologique des faisceaux cohérents. {I}},
      JOURNAL = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      NUMBER = {11},
      YEAR = {1961},
      PAGES = {167},
      ISSN = {0073-8301},
      MRCLASS = {14.55},
      MRNUMBER = {217085},
      url = {http://www.numdam.org/item/PMIHES_1961__11__5_0/},
      zblnumber = {0118.36206},
      }
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    @article {MR163911,
      author = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {III}. \'{E}tude cohomologique des faisceaux cohérents. {II}},
      JOURNAL = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      NUMBER = {17},
      YEAR = {1963},
      PAGES = {91},
      ISSN = {0073-8301},
      MRCLASS = {14.05},
      MRNUMBER = {163911},
      MRREVIEWER = {H. Hironaka},
      URL = {http://www.numdam.org/item?id=PMIHES_1963__17__91_0},
      }
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    @article {MR173675,
      author = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {I}},
      JOURNAL = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      NUMBER = {20},
      YEAR = {1964},
      PAGES = {259},
      ISSN = {0073-8301},
      MRCLASS = {14.05 (13.10)},
      MRNUMBER = {173675},
      MRREVIEWER = {H. Hironaka},
      URL = {http://www.numdam.org/item/PMIHES_1964__20__5_0},
      }
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      author = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {II}},
      JOURNAL = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      NUMBER = {24},
      YEAR = {1965},
      PAGES = {231},
      ISSN = {0073-8301},
      MRCLASS = {14.00},
      MRNUMBER = {199181},
      MRREVIEWER = {H. Hironaka},
      URL = {http://www.numdam.org/item?id=PMIHES_1965__24__231_0},
      }
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      author = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {III}},
      JOURNAL = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      NUMBER = {28},
      YEAR = {1966},
      PAGES = {255},
      ISSN = {0073-8301},
      MRCLASS = {14.55},
      MRNUMBER = {217086},
      MRREVIEWER = {J. P. Murre},
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      }
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      NUMBER = {32},
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      PAGES = {361},
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      MRNUMBER = {238860},
      MRREVIEWER = {J. P. Murre},
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      author={Grothendieck, Alexander},
      title={Rev\^{e}tements étales et groupe fondamental},
      series={Lecture Notes in Math.},
      note={Séminaire de Géométrie Algébrique du Bois Marie 1960--1961 (SGA 1); dirigé par Alexandre Grotherndieck; augmenté de deux exposés de M. Raynaud},
      volume={224},
      publisher={Springer-Verlag, New York},
      year={1971},
      pages={xxii+447},
      mrclass={14-06 (14E20)},
      mrnumber={0354651},
     doi = {10.1007/BFb0058656},
      url = {https://doi.org/10.1007/BFb0058656},
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      }
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      journal = {Math. Z.},
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      volume = {278},
      year = {2014},
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      issn = {0025-5874},
      mrclass = {14F05 (13D09 14A20 14D20 14D23)},
      mrnumber = {3267585},
      mrreviewer = {Jon Eivind Vatne},
      doi = {10.1007/s00209-014-1321-7},
      url = {https://doi.org/10.1007/s00209-014-1321-7},
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      author = {Hall, Jack},
      title = {Openness of versality via coherent functors},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {722},
      year = {2017},
      pages = {137--182},
      issn = {0075-4102},
      mrclass = {14A20 (14F05)},
      mrnumber = {3589351},
      mrreviewer = {Stefan Schröer},
      doi = {10.1515/crelle-2014-0057},
      url = {https://doi.org/10.1515/crelle-2014-0057},
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      }
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      author = {Hall, Jack and Neeman, Amnon and Rydh, David},
      title = {One positive and two negative results for derived categories of algebraic stacks},
      journal = {J. Inst. Math. Jussieu},
      fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      volume = {18},
      year = {2019},
      number = {5},
      pages = {1087--1111},
      issn = {1474-7480},
      mrclass = {14F05 (14A20 14G17 18G10)},
      mrnumber = {3995721},
      doi = {10.1017/s1474748017000366},
      url = {https://doi.org/10.1017/s1474748017000366},
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      }
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      title = {Algebraic groups and compact generation of their derived categories of representations},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana University Mathematics Journal},
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      year = {2015},
      number = {6},
      pages = {1903--1923},
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      mrclass = {14L15 (14F05)},
      mrnumber = {3436239},
      mrreviewer = {Scott R. Nollet},
      doi = {10.1512/iumj.2015.64.5719},
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      }
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      title = {Perfect complexes on algebraic stacks},
      journal = {Compos. Math.},
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      year = {2017},
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      pages = {2318--2367},
      issn = {0010-437X},
      mrclass = {14F05 (14A20 18E30)},
      mrnumber = {3705292},
      mrreviewer = {Stefan Schröer},
      doi = {10.1112/S0010437X17007394},
      url = {https://doi.org/10.1112/S0010437X17007394},
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      }
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      title = {Coherent {T}annaka duality and algebraicity of {H}om-stacks},
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      mrclass = {14A20 (14D23 18D10)},
      mrnumber = {4009673},
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      author={Orgogozo, Fabrice},
      title={Exposé {IV}. {L}e théoréme de {C}ohen-{G}abber},
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      title = {A pathological example of an action of {$k\sp{\ast}$}},
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      venue = {{V}ancouver, {B}.{C}., 1981},
      series = {Lecture Notes in Math.},
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      publisher = {Springer-Verlag, Berlin},
      year = {1982},
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      mrnumber = {0704987},
      mrreviewer = {E. Aky\i ld\i z},
      doi = {10.1007/BFb0101509},
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      mrnumber = {2026723},
      doi = {10.1023/B:ALGE.0000006492.02381.df},
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      doi = {10.1007/978-3-540-24899-6},
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      doi = {10.1017/9781316711736},
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      fjournal = {Journal of Mathematics of Kyoto University},
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      title = {The {G}rothendieck duality theorem via {B}ousfield's techniques and {B}rown representability},
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Keywords
, , ,
DOI
https://doi.org/10.4007/annals.2020.191.3.1
MR
4088350
zbMATH
07190303
Milestones
Received: 14 May 2015
Revised: 14 November 2019
Accepted: 15 November 2019
Published online: 15 April 2020

Authors

Jarod Alper

Department of Mathematics, University of Washington, Seattle, WA, USA

Jack Hall

University of Arizona, Tucson, AZ and School of Mathematics & Statistics, The University of Melbourne, Parkville, Australia

David Rydh

Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden