A Riemann-Roch theorem for dg algebras

François Petit

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 2, page 197-223
  • ISSN: 0037-9484

Abstract

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Given a smooth proper dg algebra A , a perfect dg A -module M and an endomorphism f of M , we define the Hochschild class of the pair ( M , f ) with values in the Hochschild homology of the algebra A . Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

How to cite

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Petit, François. "A Riemann-Roch theorem for dg algebras." Bulletin de la Société Mathématique de France 141.2 (2013): 197-223. <http://eudml.org/doc/272742>.

@article{Petit2013,
abstract = {Given a smooth proper dg algebra $A$, a perfect dg $A$-module $M$ and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of the algebra $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.},
author = {Petit, François},
journal = {Bulletin de la Société Mathématique de France},
keywords = {differential graded algebra; perfect module; Serre duality; Hochschild homology; Hochschild class; Riemann-Roch theorem},
language = {eng},
number = {2},
pages = {197-223},
publisher = {Société mathématique de France},
title = {A Riemann-Roch theorem for dg algebras},
url = {http://eudml.org/doc/272742},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Petit, François
TI - A Riemann-Roch theorem for dg algebras
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 2
SP - 197
EP - 223
AB - Given a smooth proper dg algebra $A$, a perfect dg $A$-module $M$ and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of the algebra $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.
LA - eng
KW - differential graded algebra; perfect module; Serre duality; Hochschild homology; Hochschild class; Riemann-Roch theorem
UR - http://eudml.org/doc/272742
ER -

References

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  1. [1] M. van den Bergh – « Existence theorems for dualizing complexes over non-commutative graded and filtered rings », J. Algebra195 (1997), p. 662–679. Zbl0894.16020MR1469646
  2. [2] A. I. Bondal & M. Kapranov – « Representable functors, Serre functors, and reconstructions », Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), p. 1183–1205, 1337. MR1039961
  3. [3] A. Căldăraru – « The Mukai pairing, I: the Hochschild structure, arXiv:math/0308079 », ArXiv Mathematics e-prints (2003). 
  4. [4] —, « The Mukai pairing. II. The Hochschild-Kostant-Rosenberg isomorphism », Adv. Math.194 (2005), p. 34–66. Zbl1098.14011MR2141853
  5. [5] A. Căldăraru & S. Willerton – « The Mukai pairing. I. A categorical approach », New York J. Math.16 (2010), p. 61–98. MR2657369
  6. [6] M. Engeli & G. Felder – « A Riemann-Roch-Hirzebruch formula for traces of differential operators », Ann. Sci. Éc. Norm. Supér. 41 (2008), p. 621–653. Zbl1163.32009MR2489635
  7. [7] B. Fresse – Modules over operads and functors, Lecture Notes in Math., vol. 1967, Springer, 2009. MR2494775
  8. [8] V. Ginzburg – « Lectures on Noncommutative Geometry, arXiv:math/0506603 », ArXiv Mathematics e-prints (2005). 
  9. [9] V. Hinich – « Homological algebra of homotopy algebras », Comm. Algebra25 (1997), p. 3291–3323. Zbl0894.18008MR1465117
  10. [10] M. Hovey – Model categories, Mathematical Surveys and Monographs, vol. 63, Amer. Math. Soc., 1999. MR1650134
  11. [11] P. Jørgensen – « Auslander-Reiten theory over topological spaces », Comment. Math. Helv.79 (2004), p. 160–182. Zbl1053.55010MR2031704
  12. [12] M. Kashiwara & P. Schapira – Deformation quantization modules, Astérisque, vol. 345, Soc. Math. France, 2012. MR3012169
  13. [13] B. Keller – « Invariance and localization for cyclic homology of DG algebras », J. Pure Appl. Algebra123 (1998), p. 223–273. Zbl0890.18007MR1492902
  14. [14] —, « On differential graded categories », in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, p. 151–190. MR2275593
  15. [15] M. Kontsevich & Y. Soibelman – « Notes on A ∞ -algebras, A ∞ -categories and non-commutative geometry », in Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, 2009, p. 153–219. Zbl1202.81120MR2596638
  16. [16] J. L. Loday – Cyclic homology, second éd., Grund. Math. Wiss., vol. 301, Springer, 1998, Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR1600246
  17. [17] N. Markarian – « The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, arXiv:math/0610553 », ArXiv Mathematics e-prints (2006). MR2472137
  18. [18] D. Murfet – « Residues and duality for singularity categories of isolated Gorenstein singularities, arXiv:0912.1629v2 », ArXiv e-prints (2009). MR3143706
  19. [19] A. Neeman – « The connection between the K -theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel », Ann. Sci. École Norm. Sup.25 (1992), p. 547–566. Zbl0868.19001MR1191736
  20. [20] —, Triangulated categories, Annals of Math. Studies, vol. 148, Princeton Univ. Press, 2001. MR1812507
  21. [21] A. Polishchuk & A. Vaintrob – « Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations, arXiv:1002.2116v2 », ArXiv e-prints (2010). MR2954619
  22. [22] K. Ponto & M. Shulman – « Traces in symmetric monoidal categories, arXiv:1107.6032 », ArXiv e-prints (2011). 
  23. [23] A. C. Ramadoss – « The Mukai pairing and integral transforms in Hochschild homology », Mosc. Math. J. 10 (2010), p. 629–645, 662–663. Zbl1208.14013MR2732577
  24. [24] D. C. Ravenel – « Localization with respect to certain periodic homology theories », Amer. J. Math.106 (1984), p. 351–414. Zbl0586.55003MR737778
  25. [25] D. Shklyarov – « On Serre duality for compact homologically smooth DG algebras, arXiv:math/0702590 », ArXiv Mathematics e-prints (2007). 
  26. [26] D. Shklyarov – « Hirzebruch-Riemann-Roch-type formula for DG algebras », Proceedings of the London Mathematical Society (2012). MR3020737
  27. [27] B. Toën & M. Vaquié – « Moduli of objects in dg-categories », Ann. Sci. École Norm. Sup.40 (2007), p. 387–444. Zbl1140.18005MR2493386

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