Equivariant Euler characteristics and sheaf resolvents

Ph. Cassou-Noguès[1]; M.J. Taylor[2]

  • [1] Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération 33405 Talence Cedex France
  • [2] The University of Manchester School of Mathematics Alan Turing Building Oxford Road Manchester, M13 9PL UK

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2315-2345
  • ISSN: 0373-0956

Abstract

top
For certain tame abelian covers of arithmetic surfaces we obtain formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf and also its square root. These formulas allow us to carry out explicit calculations; in particular, we are able to exhibit examples where these two Euler characteristics and that of the structure sheaf are all different and non-trivial. Our results are obtained by using resolvent techniques together with the local Riemann-Roch Theorem.

How to cite

top

Cassou-Noguès, Ph., and Taylor, M.J.. "Equivariant Euler characteristics and sheaf resolvents." Annales de l’institut Fourier 62.6 (2012): 2315-2345. <http://eudml.org/doc/251109>.

@article{Cassou2012,
abstract = {For certain tame abelian covers of arithmetic surfaces we obtain formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf and also its square root. These formulas allow us to carry out explicit calculations; in particular, we are able to exhibit examples where these two Euler characteristics and that of the structure sheaf are all different and non-trivial. Our results are obtained by using resolvent techniques together with the local Riemann-Roch Theorem.},
affiliation = {Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération 33405 Talence Cedex France; The University of Manchester School of Mathematics Alan Turing Building Oxford Road Manchester, M13 9PL UK},
author = {Cassou-Noguès, Ph., Taylor, M.J.},
journal = {Annales de l’institut Fourier},
keywords = {Euler characteristic; resolvent; intersection numbers},
language = {eng},
number = {6},
pages = {2315-2345},
publisher = {Association des Annales de l’institut Fourier},
title = {Equivariant Euler characteristics and sheaf resolvents},
url = {http://eudml.org/doc/251109},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Cassou-Noguès, Ph.
AU - Taylor, M.J.
TI - Equivariant Euler characteristics and sheaf resolvents
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2315
EP - 2345
AB - For certain tame abelian covers of arithmetic surfaces we obtain formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf and also its square root. These formulas allow us to carry out explicit calculations; in particular, we are able to exhibit examples where these two Euler characteristics and that of the structure sheaf are all different and non-trivial. Our results are obtained by using resolvent techniques together with the local Riemann-Roch Theorem.
LA - eng
KW - Euler characteristic; resolvent; intersection numbers
UR - http://eudml.org/doc/251109
ER -

References

top
  1. T. Chinburg, Galois module structure of de Rham cohomology, J. de Théorie des Nombres de Bordeaux 4 (1991), 1-18 Zbl0768.14008MR1183915
  2. T. Chinburg, Galois structure of the de Rham cohomology of tame covers of schemes, Ann. of Math. 139 (1994), 443-490 Zbl0828.14007MR1274097
  3. T. Chinburg, B. Erez, Equivariant Euler-Poincaré characteristics and tameness, Astérisque 209 (1992), 179-194 Zbl0796.11051MR1211011
  4. T. Chinburg, B. Erez, G. Pappas, M. J. Taylor, Tame actions of group schemes: integrals amd slices, Duke Math. J. 82 (1996), 269-308 Zbl0907.14021MR1387229
  5. T. Chinburg, B. Erez, G. Pappas, M. J. Taylor, Riemann-Roch type theorems for arithmetic schemes with a finite group action, J. Reine Angew. Math. 489 (1997), 151-187 Zbl0903.19001MR1461208
  6. T. Chinburg, B. Erez, G. Pappas, M. J. Taylor, ε -constants and the Galois structure of de Rham cohomology, Ann. of Math. 146 (1997), 411-473 Zbl0939.14009MR1477762
  7. T. Chinburg, G. Pappas, M. J. Taylor, Cubic structures, equivariant Euler characteristics and lattices of modular forms, Ann. of Math. 170 (2009), 561-608 Zbl1255.14010MR2552102
  8. B. Erez, M. J. Taylor, Hermitian modules in Galois extensions of number fields and Adams operations, Ann. of Math. 135 (1992), 271-296 Zbl0756.11035MR1154594
  9. A. Fröhlich, Arithmetic and Galois module structure for tame extensions, J. Crelle 286/287 (1976), 380-440 Zbl0385.12004MR432595
  10. A. Fröhlich, Galois module structure of algebraic integers, (1983), Springer-Verlag Zbl0501.12012MR717033
  11. W. Fulton, Intersection theory, 2nd edition, (1998), Folge, Springer-Verlag Zbl0885.14002MR1644323
  12. R. Hartshorne, Residues and Duality, 20 (1966), Springer-Verlag MR222093
  13. R. Hartshorne, Algebraic Geometry, 52 (1977), Springer-Verlag Zbl0531.14001MR463157
  14. S. Lang, Introduction to Arakelov theory, (1988), Springer-Verlag Zbl0667.14001MR969124
  15. G. Pappas, Galois modules and the Theorem of the Cube, Invent. Math. 133 (1998), 193-225 Zbl0923.14030MR1626489
  16. G. Pappas, Galois module structure and the γ -filtration, Compositio Mathematica 121 (2000), 79-104 Zbl0968.14026MR1753111
  17. J.-P. Serre, Revêtements à ramification impaire et thêta-caractéristiques, C. R. Acad. Sci. Paris 311, Série 1 (1990), 547-552 Zbl0742.14030MR1078120
  18. M. J. Taylor, On the self duality of a ring of integers as a Galois module, Invent. Math. 46 (1978), 173-177 Zbl0381.12007MR472770
  19. M. J. Taylor, On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math. 63 (1981), 41-79 Zbl0469.12003MR608528
  20. M. J. Taylor, Class groups of group rings, (1983), C. U. P. Zbl0597.13002
  21. L. Washington, Introduction to cyclotomic fields, (1980), Springer Verlag Zbl0484.12001MR1421575

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.