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Perfect complexes on algebraic stacks

Part of: (Co)homology theory Commutative algebra: Homological methods Abelian categories Algebraic geometry: Foundations

Published online by Cambridge University Press:  17 August 2017

Jack Hall  and
David Rydh
Jack Hall
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA email jackhall@math.arizona.edu
David Rydh
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden email dary@math.kth.se
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Abstract

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

Keywords

derived categoriesalgebraic stackscompact generationperfect complexes

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 13D09: Derived categories 14A20: Generalizations (algebraic spaces, stacks) 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 11 , November 2017 , pp. 2318 - 2367
Copyright
© The Authors 2017 

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References

Alonso Tarrío, L., Jeremías López, A., Pérez Rodríguez, M. and Vale Gonsalves, M. J., On the derived category of quasi-coherent sheaves on an Adams geometric stack , J. Pure Appl. Algebra (2017), doi:10.1016/j.jpaa.2017.05.009, to appear, available online.Google Scholar
Alper, J., Good moduli spaces for Artin stacks , Ann. Inst. Fourier (Grenoble) 63 (2013), 2349–2402.Google Scholar
Alper, J., Hall, J. and Rydh, D., A Luna étale slice theorem for algebraic stacks, Preprint (2015), arXiv:1504.06467.Google Scholar
Antieau, B., A local-global principle for the telescope conjecture , Adv. Math. 254 (2014), 280–299.Google Scholar
Antieau, B. and Gepner, D., Brauer groups and étale cohomology in derived algebraic geometry , Geom. Topol. 18 (2014), 1149–1244.Google Scholar
Auslander, M., Coherent functors , in Proc. conf. on categorical algebra (La Jolla, CA, 1965) (Springer, New York, 1966), 189–231.CrossRefGoogle Scholar
Behrend, K., Derived l-adic categories for algebraic stacks , Mem. Amer. Math. Soc. 163 (2003).Google Scholar
Ben-Zvi, D., Francis, J. and Nadler, D., Integral transforms and Drinfeld centers in derived algebraic geometry , J. Amer. Math. Soc. 23 (2010), 909–966.CrossRefGoogle Scholar
Bergh, D., Lunts, V. A. and Schnürer, O. M., Geometricity for derived categories of algebraic stacks , Selecta Math. (N.S.) 22 (2016), 2535–2568.Google Scholar
Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories , Compos. Math. 86 (1993), 209–234.Google Scholar
Bondal, A. and Van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry , Mosc. Math. J. 3 (2003), 1–36; 258.CrossRefGoogle Scholar
Brandenburg, M., Tensor categorical foundations of algebraic geometry, PhD thesis, Wilhelms-Universität Münster (2014), p. 243.Google Scholar
Brion, M., On linearization of line bundles , J. Math. Sci. Univ. Tokyo 22 (2015), 113–147.Google Scholar
Canonaco, A. and Stellari, P., Uniqueness of dg enhancements for the derived category of a Grothendieck category, J. Eur. Math. Soc. (2016), to appear.Google Scholar
de Jong, A. J., A result of Gabber, Preprint (2003), p. 9, available at http://www.math.columbia.edu/∼dejong/.Google Scholar
Drinfeld, V. and Gaitsgory, D., On some finiteness questions for algebraic stacks , Geom. Funct. Anal. 23 (2013), 149–294.CrossRefGoogle Scholar
Dubey, U. V. and Mallick, V. M., Spectrum of some triangulated categories , J. Algebra 364 (2012), 90–118.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci. 4, 8, 11, 17, 20, 24, 28, 32 (1960, 1961, 1961, 1963, 1964, 1965, 1966, 1967).Google Scholar
Fulton, W. and Lang, S., Riemann–Roch algebra, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 277 (Springer, New York, 1985).Google Scholar
Gabber, O., Some theorems on Azumaya algebras , in The Brauer group (séminaire, Les Plans-sur-Bex, 1980), Lecture Notes in Mathematics, vol. 844 (Springer, Berlin–New York, 1981), 129–209.Google Scholar
Gaitsgory, D. and Rozenblyum, N., A study in derived algebraic geometry: Volume I: correspondences and duality, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, Providence, RI, 2017).Google Scholar
Gross, P., Vector bundles as generators on schemes and stacks, PhD thesis, Heinrich-Heine-Universität Düsseldorf (2010).Google Scholar
Gross, P., Tensor generators on schemes and stacks , Algebr. Geom. 4 (2017), arXiv:1306.5418, to appear.Google Scholar
Hall, J., Cohomology and base change for algebraic stacks , Math. Z. 278 (2014), 401–429.Google Scholar
Hall, J., The Balmer spectrum of a tame stack , Ann. K-Theory 1 (2016), 259–274.Google Scholar
Hall, J., Openness of versality via coherent functors , J. reine angew. Math. 722 (2017), 137–182.Google Scholar
Hall, J., Neeman, A. and Rydh, D., One positive and two negative results for derived categories of algebraic stacks, Preprint (2014), arXiv:1405.1888v2.Google Scholar
Hall, J. and Rydh, D., Algebraic groups and compact generation of their derived categories of representations , Indiana Univ. Math. J. 64 (2015), 1903–1923.Google Scholar
Hall, J. and Rydh, D., Mayer–Vietoris squares in algebraic geometry, Preprint (2016), arXiv:1606.08517.Google Scholar
Hall, J. and Rydh, D., The telescope conjecture for algebraic stacks , J. Topol. 10 (2017), 776–794.CrossRefGoogle Scholar
Hartshorne, R., Coherent functors , Adv. Math. 140 (1998), 44–94.Google Scholar
Hovey, M., Palmieri, J. H. and Strickland, N. P., Axiomatic stable homotopy theory , Mem. Amer. Math. Soc. 128 (1997).Google Scholar
Kashiwara, M. and Schapira, P., Categories and sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332 (Springer, Berlin, 2006).Google Scholar
Krishna, A., Perfect complexes on Deligne–Mumford stacks and applications , J. K-Theory 4 (2009), 559–603.Google Scholar
Laszlo, Y. and Olsson, M., The six operations for sheaves on Artin stacks. I. Finite coefficients , Publ. Math. Inst. Hautes Études Sci. 107 (2008), 109–168.Google Scholar
Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 39 (Springer, Berlin, 2000).Google Scholar
Lieblich, M., Moduli of twisted sheaves and generalized Azumaya algebras, PhD thesis, Massachusetts Institute of Technology, ProQuest LLC, Ann Arbor, MI (2004).Google Scholar
Lipman, J., Notes on derived functors and Grothendieck duality , in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960 (Springer, Berlin, 2009), 1–259.Google Scholar
Lipman, J. and Neeman, A., Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor , Illinois J. Math. 51 (2007), 209–236.Google Scholar
Liu, Y. and Zheng, W., Enhanced six operations and base change theorem for Artin stacks, Preprint (2012), arXiv:1211.5948.Google Scholar
Lurie, J., Tannaka duality for geometric stacks, Preprint (2004), arXiv:math/0412266, p. 14.Google Scholar
Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170 (Princeton University Press, Princeton, NJ, 2009).Google Scholar
Lurie, J., Derived algebraic geometry XI: descent theorems, Preprint (2011),http://www.math.harvard.edu/∼lurie/.Google Scholar
Lurie, J., Derived algebraic geometry XII: proper morphisms, completions, and the Grothendieck existence theorem, Preprint (2011), http://www.math.harvard.edu/∼lurie/.Google Scholar
Lurie, J., Higher algebra, Preprint (2016), http://www.math.harvard.edu/∼lurie/.Google Scholar
Lurie, J., Spectral algebraic geometry, Preprint (2016), http://www.math.harvard.edu/∼lurie/.Google Scholar
Moret-Bailly, L., Un problème de descente , Bull. Soc. Math. France 124 (1996), 559–585.Google Scholar
Neeman, A., The chromatic tower for D (R) , Topology 31 (1992), 519–532; with an appendix by Marcel Bökstedt.CrossRefGoogle Scholar
Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel , Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 547–566.Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , J. Amer. Math. Soc. 9 (1996), 205–236.Google Scholar
Neeman, A., On the derived category of sheaves on a manifold , Doc. Math. 6 (2001), 483–488; (electronic).Google Scholar
Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Neeman, A., Non-left-complete derived categories , Math. Res. Lett. 18 (2011), 827–832.CrossRefGoogle Scholar
Neeman, A., An improvement on the base-change theorem and the functor $f^{!}$ , Preprint (2014), arXiv:1406.7599.Google Scholar
Olsson, M., Sheaves on Artin stacks , J. reine angew. Math. 603 (2007), 55–112.Google Scholar
Rydh, D., Étale dévissage, descent and pushouts of stacks , J. Algebra 331 (2011), 194–223.Google Scholar
Rydh, D., Noetherian approximation of algebraic spaces and stacks , J. Algebra 422 (2015), 105–147.Google Scholar
Berthelot, P., Grothendieck, A. and Illusie, L. (eds), Théorie des intersections et théorème de Riemann–Roch , inSéminaire de géométrie algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, 1971); with the collaboration of D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud, and J. P. Serre.Google Scholar
The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu.Google Scholar
Sumihiro, H., Equivariant completion , J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
Sumihiro, H., Equivariant completion. II , J. Math. Kyoto Univ. 15 (1975), 573–605.Google Scholar
Thomason, R. W., Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes , Adv. Math. 65 (1987), 16–34.Google Scholar
Thomason, R. W., The classification of triangulated subcategories , Compos. Math. 105 (1997), 1–27.CrossRefGoogle Scholar
Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories , in The Grothendieck festschrift, Vol. III, Progress in Mathematics, vol. 88 (Birkhäuser, Boston, MA, 1990), 247–435.Google Scholar
Toën, B., Derived Azumaya algebras and generators for twisted derived categories , Invent. Math. 189 (2012), 581–652.Google Scholar
Toën, B. and Vezzosi, G., Homotopical algebraic geometry. II. Geometric stacks and applications , Mem. Amer. Math. Soc. 193 (2008).Google Scholar
Totaro, B., The resolution property for schemes and stacks , J. reine angew. Math. 577 (2004), 1–22.Google Scholar