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Model category structures and spectral sequences

Part of: Homological algebra

Published online by Cambridge University Press:  01 August 2019

Joana Cirici ,
Daniela Egas Santander ,
Muriel Livernet  and
Sarah Whitehouse
Joana Cirici
Affiliation:
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585 08007 Barcelona, Spain ([email protected])
Daniela Egas Santander
Affiliation:
Ecole polytechnique fédérale de Lausanne, SV BMI UPHESS, MA B3 425 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland ([email protected])
Muriel Livernet
Affiliation:
Univ Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS, Sorbonne Université, 8 place Aurélie Nemours, F-75013, Paris, France ([email protected])
Sarah Whitehouse
Affiliation:
School of Mathematics and Statistics, University of Sheffield S3 7RH, Sheffield, England ([email protected])
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Abstract

Let R be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of R-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.

Keywords

filtered complexbicomplexspectral sequencemodel category

MSC classification

Primary: 18G55: Homotopical algebra
Secondary: 18G40: Spectral sequences, hypercohomology
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2815 - 2848
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Cartan, H. and Eilenberg, S.. Homological Algebra (Princeton, N. J.: Princeton University Press, 1956).Google Scholar
2Cirici, J., Egas Santander, D., Livernet, M. and Whitehouse, S.. Derived A-infinity algebras and their homotopies. Topol. Appl. 235 (2018), 214–268.Google Scholar
3Cirici, J. and Guillén, F.. E 1-formality of complex algebraic varieties. Algebr. Geom. Topol. 14 (2014), 3049–3079.CrossRefGoogle Scholar
4Cirici, J. and Guillén, F.. Homotopy theory of mixed Hodge complexes. Tohoku Math. J. 68 (2016), 349–375.CrossRefGoogle Scholar
5Cordero, L. A., Fernández, M., Ugarte, L. and Gray, A.. A general description of the terms in the Frölicher spectral sequence. Differ. Geom. Appl. 7 (1997), 75–84.CrossRefGoogle Scholar
6Deligne, P.. Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.CrossRefGoogle Scholar
7Di Natale, C.. Derived moduli of complexes and derived Grassmannians. Appl. Categ. Struct. 25 (2017), 809–861.CrossRefGoogle Scholar
8Fausk, H.. t-model structures on chain complexes of presheaves. arXiv:math/0612414.Google Scholar
9Félix, Y., Oprea, J. and Tanré, D.. Algebraic models in geometry, volume 17 of Oxford Graduate Texts in Mathematics (Oxford: Oxford University Press, 2008).Google Scholar
10Halperin, S. and Tanré, D.. Homotopie filtrée et fibrés C ∞. Illinois J. Math. 34 (1990), 284–324.CrossRefGoogle Scholar
11Hirschhorn, P. S.. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs (Providence, RI: American Mathematical Society, 2003).Google Scholar
12Hovey, M.. Model categories, volume 63 of Mathematical Surveys and Monographs (Providence, RI: American Mathematical Society, 1999).Google Scholar
13Illusie, L.. Complexe cotangent et déformations. I, volume 239 of Lecture Notes in Mathematics (Berlin: Springer-Verlag, 1971).CrossRefGoogle Scholar
14Laumon, G.. Sur la catégorie dérivée des 𝒟-modules filtrés. In Raynaud, Michel and Shioda, Tetsuji (eds.) Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pp. 151–237 (Berlin: Springer, 1983).CrossRefGoogle Scholar
15McCleary, J.. A user's guide to spectral sequences, volume 58 of Cambridge Studies in Advanced Mathematics, 2nd edn (Cambridge: Cambridge University Press, 2001).Google Scholar
16Morgan, J. W.. The algebraic topology of smooth algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204.CrossRefGoogle Scholar
17Muro, F. and Roitzheim, C.. Homotopy theory of bicomplexes. J. Pure Appl. Algebra 223 (2019), 1913–1939.CrossRefGoogle Scholar
18Paranjape, K. H.. Some spectral sequences for filtered complexes and applications. J. Algebra 186 (1996), 793–806.CrossRefGoogle Scholar
19Riehl, E.. Category theory in context (Mineola, NY: Dover Publications, 2016).Google Scholar