Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T02:18:24.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Milnor descent for cohesive dg-categories

Published online by Cambridge University Press:  20 August 2013

Oren Ben-Bassat  and
Jonathan Block
Oren Ben-Bassat
Affiliation:
Mathematical Institute, University of Oxford, Oxford, UK, oren.benbassat@gmail.com
Jonathan Block
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA, blockj@math.upenn.edu
Get access

Abstract

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

Keywords

descentprojective modulesdg categoriesMayer-Vietoris46L8758B34
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 3 , December 2013 , pp. 433 - 459
Copyright
Copyright © ISOPP 2013 

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.)

References

B.Block, J., Duality and Equivalence of Module Categories in Noncommutative Geometry, CRM Proceedings and Lecture Notes 50 (2010), 311–339.Google Scholar
BB.Ben-Bassat, O., Block, J., Cohesive dg-categories I: Milnor descent, arXiv:1201.6118.Google Scholar
BBP.Ben-Bassat, O., Block, J., Pantev, T., Non-commutative tori and Fourier-Mukai duality, Compos. Math. 143(2) (2007), 423–475.Google Scholar
BK.Bondal, A. I., Kapranov, M. M., Enhanced triangulated categories, Math. USSR Sbornik 70(1) (1991), 93–107.Google Scholar
BT.Ben-Bassat, O., Temkin, M., Berkovich Spaces and Tubular Descent, Adv. in Math. 234 (2013), 217–238.Google Scholar
Dr.Drinfeld, V., DG quotients of dg-categories, J. Algebra 272(2) (2004), 643–691.Google Scholar
Ke.Keller, B., Deriving dg-categories, Ann. Scient. Éc. Norm. Sup., 4e-série 27 (1994), 63–102.Google Scholar
Ke2.Keller, B., On differential graded categories, International Congress of Mathematicians 2, 151–190, Eur. Math. Soc., Zürich, 2006.Google Scholar
L.Landsburg, S., K-Theory and Patching for Categories of Complexes, Duke Math. Journal 12(2) (1991), 359–384.Google Scholar
LV.Lowen, W., Van den Bergh, M., Hochschild cohomology of Abelian categories and ringed spaces, Adv. in Math. 198(1) (2005), 172–221.Google Scholar
M.Milnor, J., Introduction to Algebraic K-Theory, Ann. of Math Studies 72, Princeton University Press, 1971.Google Scholar
Or.Orlov, D. O., Triangulated categories of singularities and D-branes in Landau-Ginzburg models (Russian), Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 240–262; translation in Proc. Steklov Inst. Math. 246(3) (2004), 227–248.Google Scholar
PP.Polishchuk, A.Positselski, L., Quadratic algebras, University Lecture Series 37. American Mathematical Society, Providence, RI, 2005. xii+159 pp.Google Scholar
S.Schwarz, A., Noncommutative supergeometry, duality and deformations, Nuclear Physics B 650(3) (2003), 475–496.CrossRefGoogle Scholar
Ta.Tabuada, G., Homotopy Theory of dg-categories via Localizing Pairs and Drinfeld's DG Quotient, Homology, Homotopy, and Applications 12(1) (2010), 187–219.Google Scholar
T.Toën, B., Lectures on DG-categories, available from http://ens.math.univ-montp2.fr/~toen/.Google Scholar
To.Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167(3) (2007), 615–667.Google Scholar
Yu.Yu, S., Dolbeault dga of a formal neighborhood, preprint, arXiv:1206.5155.Google Scholar