Abstract
In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some associative ring R and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we construct derived versions of Grassmannians and flag varieties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Artin, M.: Versal deformations and algebraic stacks. Invent. Math. 27, 165–189 (1974)
Avramov, L., Foxby, H., Halperin, S.: Differential Graded Homological Algebra, in preparation
Cegarra, A.M., Remedios, J.: The relationship between the diagonal and the bar constructions on a bisimplicial set. Topology and its Applications 153(1), 21–51 (2005)
Di Natale, C.: A Period Map for Global Derived Stacks, arXiv:1407.5906 (2014)
Bressler, P., Nest, R., Tsygan, B.: Riemann-Roch Theorems via deformation quantization i. Adv. Math. 167, 1–25 (2002)
Dwyer, W.G., Spalińsky, J.: Homotopy Theories and Model Categories, pp 73–126. Handbook of Algebraic Topology, North Holland (1995)
Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Birkäuser (1999)
Goerss, P.G., Schemmerhon, K.: Model Categories and Simplicial Methods. In: Interactions Between Homotopy Theory and Algebra: 3–49, Contemporary Mathematics, 436. American Mathematical Society, Providence (2007)
Halpern-Leistner, D., Preygel, A.: Mapping Stacks and Categorical Notions of Properness, arXiv:1402.3204 (2014)
Hirschhorn, P.S.: Model categories and their localizations american mathematical society (2003)
Hovey, M.: Model categories american mathematical society (1999)
Hilton, P.J., Stammbach, U.: A course in homological algebra. Springer (1970)
Hinich, V.: DG Coalgebras as formal stacks. Journal of Pure and Applied Algebra 162, 209–250 (2001)
Hirschovitz, A., Simpson, C.: Descente pour les n-Champs, arXiv:math/9807049v3 (2001)
Hütterman, T.: On the derived category of a regular toric scheme. Geom. Dedicata. 148, 175–203 (2010)
Illusie, L.: Complexe Cotangent et Déformations I–II. Springer (1972)
Lowrey, P.: The Moduli Stack and Motivic Hall Algebra for the Bounded Derived Category, arXiv:1110.5117 (2011)
Lieblich, M.: Moduli of complexes on a proper morphism. Journal of Algebraic Geometry 15, 175–206 (2006)
Lurie, J.: Derived Algebraic Geometry, PhD thesis, available at www.math.harvard.edu/lurie/papers/DAG.pdf (2004)
Lurie, J.: Higher topos theory annals of mathematics studies (2009)
Lurie, J.: Derived Algebraic Geometry XII – Proper Morphisms, Completions and the Grothendieck Existence Theorem, available at http://www.math.harvard.edu/lurie/papers/DAG-XII.pdf (2011)
Manetti, M.: Lectures on deformations of complex manifolds. Rendiconti di Matematica VII(4), 1–183 (2004)
Pandit, P.: Moduli Problems in Derived Noncommutative Geometry, PhD thesis. ProQuest LLC, Ann Arbor (2011)
Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted symplectic structures. Publications Mathématiques de l’I.H.É.S. 117(1), 271–328 (2013)
Pridham, J.P.: Presenting higher stacks as simplicial schemes. Adv. Math. 238, 184–245 (2013)
Pridham, J.P.: Representability of derived stacks. Journal of K-Theory (10) 2, 413–453 (2012)
Pridham, J.P.: Derived Moduli of Schemes and Sheaves. Journal of K-Theory (10) 1, 41–85 (2012). arXiv:1011.2742v6
Pridham, J.P.: Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting, arXiv:1104.1409 (2011)
Quillen, D.G.: Homotopical Algebra Lecture Notes in Mathematics, vol. 43. Springer, Berlin (1967)
Schapira, P., Schneiders, J.-P.: Derived Categories of Filtered Objects, arXiv:1306.1359 (2013)
Schürg, T., Toën, B., Vezzosi, G.: Derived Algebraic Geometry, Determinants of Perfect Complexes and Applications to Obstruction Theories for Maps and Complexes. Journal Für die Reine und Angewandte Mathematik (Crelle), 1–42 (2013). doi:10.1515/crelle-2013-0037
Sernesi, E.: Deformations of algebraic schemes. Springer (2006)
Simpson, C: Algebraic (Geometric) n-Stacks arXiv:math.AG/9609014 (1996)
Simpson, C.: Geometricity of the Hodge Filtration on the ∞-Stack of Perfect Complexes over x DR. Moscow Mathematical Journal 9, 665–721 (2009)
Tabuada, G.: Une Structure de catégorie de modèles de Quillen sur la Catégorie des DG-Catégories, Les Comptes Rendus de l’Académie des Sciences. Séries I Mathématiques 340(1), 15–19 (2005)
Tabuada, G.: Differential graded versus simplicial categories. Topology and its Applications 157(3), 563–593 (2010)
Toën, B.: The Homotopy Theory of DG-categories and Derived Morita Theory. Invent. Math. 167, 615–667 (2007)
Toën, B: Lectures on DG-categories. In: Topics in Algebraic and Topological K-Theory, pp. 243–302. Springer (2008)
Toën, B.: Higher and Derived Stacks: a Global Overview. In: Proceedings of Symposia in Pure Mathematics Algebraic Geometry - Seattle 2005, 80, Part 1. American Mathematical Society, Providence (2009)
Toën, B., Vaquié, M.: Moduli of Objects in DG-categories. Annales Scientifiques de l’École Normale Supérieure (4) 40, 387–444 (2007)
Toën, B., Vezzosi, G.: Homotopical Algebraic Geometry, II: Geometric stacks and applications, vol. 193 (2008)
Weibel, C.: An introduction to homological algebra. Cambridge University Press (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/1].
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Di Natale, C. Derived Moduli of Complexes and Derived Grassmannians. Appl Categor Struct 25, 809–861 (2017). https://doi.org/10.1007/s10485-016-9439-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9439-x
Keywords
- Model category
- Derived geometric stack
- Lurie-Pridham representability
- Filtered complex
- Grassmannian
- Rees functor