Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-18T19:38:29.472Z Has data issue: false hasContentIssue false
  • English
  • Français

QUILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS AND THE RESOLUTION PROPERTY

Part of: (Co)homology theory Homological algebra Abelian categories

Published online by Cambridge University Press:  09 March 2020

SERGIO ESTRADA  and
ALEXANDER SLÁVIK
SERGIO ESTRADA*
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100Murcia, Spain
ALEXANDER SLÁVIK
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Prague 8, Czech Republic e-mail: Slavik.Alexander@seznam.cz
*
e-mail: sestrada@um.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to ‘mock’ the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

Keywords

Resolution propertyvery flat sheafmodel categoryQuillen equivalencehomotopy categoryrestricted flat Mittag-Leffler

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 18E10: Exact categories, abelian categories 18E30: Derived categories, triangulated categories 18G55: Homotopical algebra
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 3 , June 2021 , pp. 302 - 320
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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.)

Footnotes

Communicated by A. Henderson

The first author is supported by the grant MTM2016-77445-P and FEDER funds and the grant 19880/GERM/15 by the Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia.

References

Alonso Tarrío, L., Jeremías López, A. and Lipman, J., ‘Local homology and cohomology of schemes’, Ann. Sci. Éc. Norm. Supér. 30(4) (1997), 1–39.Google Scholar
Benson, D. and Goodearl, K., ‘Periodic flat modules, and flat modules for finite groups’, Pacific J. Math. 196(1) (2000), 45–67.Google Scholar
Bravo, D., Gillespie, J. and Hovey, M., ‘The stable module category of a general ring’, Preprint 2014, arXiv:1405.5768v1.Google Scholar
Drinfeld, V., ‘Infinite-dimensional vector bundles in algebraic geometry: an introduction’, in: The Unity of Mathematics (Birkhäuser, Boston, 2006), 263–304.Google Scholar
Estrada, S., Fu, X. and Iacob, A., ‘Totally acyclic complexes’, J. Algebra 470 (2017), 300–319.Google Scholar
Estrada, S., Guil Asensio, P. A., Prest, M. and Trlifaj, J., ‘Model category structures arising from Drinfeld vector bundles’, Adv. Math. 231 (2012), 1417–1438.Google Scholar
Estrada, S., Guil Asensio, P. A. and Trlifaj, J., ‘Descent of restricted flat Mittag-Leffler modules and locality for generalized vector bundles’, Proc. Amer. Math. Soc. 142 (2014), 2973–2981.Google Scholar
Gillespie, J., ‘Cotorsion pairs and degreewise homological model structures’, Homology Homotopy Appl. 10(1) (2008), 283–304.Google Scholar
Gillespie, J., ‘Model structures on exact categories’, J. Pure Appl. Algebra 215 (2011), 2892–2902.Google Scholar
Gillespie, J., ‘How to construct a Hovey triple from two cotorsion pairs’, Fund. Math. 230(3) (2015), 281–289.Google Scholar
Gillespie, J., ‘Models for mock homotopy categories of projectives’, Homology Homotopy Appl. 18(1) (2016), 247–263.Google Scholar
Göbel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules, 2nd rev. ext. edn. (W. de Guyter, Berlin, 2012).Google Scholar
Gross, P., ‘Tensor generators on schemes and stacks’, Preprint, 2013, arXiv:1306.5418v2.Google Scholar
Herbera, D. and Trlifaj, J., ‘Almost free modules and Mittag-Leffler conditions’, Adv. Math. 229 (2012), 3436–3467.Google Scholar
Hovey, M., ‘Cotorsion pairs, model category structures, and representation theory’, Math. Z. 241 (2002), 553–592.Google Scholar
de Jong, J. et al., ‘The Stacks Project’, Version 7016ab5. Available athttp://stacks.math.columbia.edu/download/book.pdf.Google Scholar
Murfet, D., ‘The mock homotopy category of projectives and Grothendieck duality derived’, PhD Thesis. Available at www.therisingsea.org.Google Scholar
Murfet, D., ‘Ample sheaves and ample families’. Available at www.therisingsea.org.Google Scholar
Murfet, D. and Salarian, S., ‘Totally acyclic complexes over noetherian schemes’, Adv. Math. 226 (2011), 1096–1133.Google Scholar
Neeman, A., ‘The homotopy category of flat modules, and Grothendieck duality’, Invent. Math. 174 (2008), 255–308.Google Scholar
Neeman, A., ‘Some adjoints in homotopy categories’, Ann. Math. 171 (2010), 2142–2155.Google Scholar
Positselski, L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence’, Preprint, 2009, arXiv:0905.2621v12.Google Scholar
Positselski, L., ‘Contraherent cosheaves’, Preprint, 2012, arXiv:1209.2995v4.Google Scholar
Positselski, L. and Slávik, A., ‘Flat morphisms of finite presentation are very flat’, Preprint, 2017, arXiv:1708.00846v1.Google Scholar
Raynaud, M. and Gruson, L., ‘Critères de platitude et de projectivité’, Invent. Math. 13 (1971), 1–89.Google Scholar
Šaroch, J., ‘Approximations and Mittag-Leffler conditions – the tools’, Preprint, 2016, arXiv:1612.01138.Google Scholar
Št’ovíček, J., ‘Deconstructibility and the Hill lemma in Grothendieck categories’, Forum Math. 25 (2013), 193–219.Google Scholar
Totaro, B., ‘The resolution property for schemes and stacks’, J. reine angew. Math. 577 (2004), 1–22.Google Scholar
Vakil, R., ‘Math 216: foundations of algebraic geometry’, 2013. Available at http://math.stanford.edu/ vakil/216blog/FOAGjun1113public.pdf.Google Scholar