Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case

Rémi Shankar Lodh[1]

  • [1] The University of Utah Department of Mathematics 155 S 1400 E Salt Lake City UT 84112 (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 1875-1942
  • ISSN: 0373-0956

Abstract

top
Let K be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic p > 0 , and let K + be the valuation ring of K . We relate the log-crystalline cohomology of the special fibre of certain affine K + -schemes X = Spec ( R ) with good or semi-stable reduction to the Galois cohomology of the fundamental group π 1 ( X K ¯ ) of the geometric generic fibre with coefficients in a Fontaine ring constructed from R . This is based on Faltings’ theory of almost étale extensions.

How to cite

top

Lodh, Rémi Shankar. "Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case." Annales de l’institut Fourier 61.5 (2011): 1875-1942. <http://eudml.org/doc/219736>.

@article{Lodh2011,
abstract = {Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p&gt;0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\operatorname\{Spec\}(R)$ with good or semi-stable reduction to the Galois cohomology of the fundamental group $\pi _1(X_\{\bar\{K\}\})$ of the geometric generic fibre with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings’ theory of almost étale extensions.},
affiliation = {The University of Utah Department of Mathematics 155 S 1400 E Salt Lake City UT 84112 (USA)},
author = {Lodh, Rémi Shankar},
journal = {Annales de l’institut Fourier},
keywords = {$p$-adic Hodge theory; almost étale extensions; crystalline cohomology; log-structures; -adic Hodge theory},
language = {eng},
number = {5},
pages = {1875-1942},
publisher = {Association des Annales de l’institut Fourier},
title = {Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case},
url = {http://eudml.org/doc/219736},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Lodh, Rémi Shankar
TI - Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 1875
EP - 1942
AB - Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p&gt;0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\operatorname{Spec}(R)$ with good or semi-stable reduction to the Galois cohomology of the fundamental group $\pi _1(X_{\bar{K}})$ of the geometric generic fibre with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings’ theory of almost étale extensions.
LA - eng
KW - $p$-adic Hodge theory; almost étale extensions; crystalline cohomology; log-structures; -adic Hodge theory
UR - http://eudml.org/doc/219736
ER -

References

top
  1. F. Andreatta, O. Brinon, Acyclicité géométrique d’un B cris relatif, (2007) 
  2. P. Berthelot, Cohomologie cristalline des schémas de caractéristique p &gt; 0 , 407 (1974), Springer Zbl0298.14012MR384804
  3. P. Berthelot, A. Ogus, Notes on crystalline cohomology, (1978), Princeton University Press Zbl0383.14010MR491705
  4. N. Bourbaki, Algèbre commutative, (1985), Masson Zbl0547.13002
  5. C. Breuil, Topologie log-syntomique, cohomologie log-cristalline, et cohomologie de Čech, Bull. S.M.F. 124 (1996), 587-647 Zbl0865.19004MR1432059
  6. G. Faltings, Crystalline cohomology and p -adic Galois representations, Algebraic Analysis, Geometry and Number Theory (1989), 25-80, Johns Hopkins University Press Zbl0805.14008MR1463696
  7. G. Faltings, Almost étale extensions, Astérisque 279 (2002), 185-270 Zbl1027.14011MR1922831
  8. J.-M. Fontaine, Le corps des périodes p -adiques, Astérisque 223 (1994), 59-111 Zbl0940.14012MR1293971
  9. O. Gabber, L. Ramero, Almost ring theory, 1800 (2003), Springer Zbl1045.13002MR2004652
  10. A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique, Publ. math. I.H.E.S. 4,8,11,17,20,24,28,32 (1960-1967) Zbl0153.22301
  11. L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. E.N.S. (4ème série) 12 (1979), 501-661 Zbl0436.14007MR565469
  12. K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, Geometry and Number Theory (1989), 191-224, Johns Hopkins University Press Zbl0776.14004MR1463703
  13. K. Kato, Semi-stable reduction and p -adic étale cohomology, Astérisque 223 (1994), 269-293 Zbl0847.14009MR1293975
  14. J.-P. Serre, Corps locaux, (1968), Hermann Zbl0137.02501MR354618
  15. J.-P. Serre, Local Algebra, (2000), Springer Zbl0959.13010MR1771925

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.