On standard norm varieties

Nikita A. Karpenko; Alexander S. Merkurjev

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 1, page 177-216
  • ISSN: 0012-9593

Abstract

top
Let  p be a prime integer and F a field of characteristic 0 . Let  X be thenorm varietyof a symbol in the Galois cohomology group H n + 1 ( F , μ p ⊗ n ) (for some n ≥ 1 ), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field F ( X ) has the following property: for any equidimensional variety Y , the change of field homomorphism CH ( Y ) → CH ( Y F ( X ) ) of Chow groups with coefficients in integers localized at  p is surjective in codimensions < ( dim X ) / ( p - 1 ) . One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in the appendix). Another important ingredient is A -trivialityof  X , the property saying that the degree homomorphism on  CH 0 ( X L ) is injective for any field extension L / F with X ( L ) ≠ ∅ . The proof involves the theory of rational correspondences reviewed in the appendix.

How to cite

top

Karpenko, Nikita A., and Merkurjev, Alexander S.. "On standard norm varieties." Annales scientifiques de l'École Normale Supérieure 46.1 (2013): 177-216. <http://eudml.org/doc/272131>.

@article{Karpenko2013,
abstract = {Let $p$ be a prime integer and $F$ a field of characteristic $0$. Let $X$ be thenorm varietyof a symbol in the Galois cohomology group $H^\{n+1\}(F,\mu _p^\{\otimes n\})$ (for some $n\ge 1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F(X)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\mathop \{\mathrm \{CH\}\}\nolimits (Y)\rightarrow \mathop \{\mathrm \{CH\}\}\nolimits (Y_\{F(X)\})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $&lt; (\dim X)/(p-1)$. One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in the appendix). Another important ingredient is$A$-trivialityof $X$, the property saying that the degree homomorphism on $\mathop \{\mathrm \{CH\}\}\nolimits _0(X_L)$ is injective for any field extension $L/F$ with $X(L)\ne \emptyset $. The proof involves the theory of rational correspondences reviewed in the appendix.},
author = {Karpenko, Nikita A., Merkurjev, Alexander S.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {norm varieties; Chow groups and motives; Steenrod operations},
language = {eng},
number = {1},
pages = {177-216},
publisher = {Société mathématique de France},
title = {On standard norm varieties},
url = {http://eudml.org/doc/272131},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Karpenko, Nikita A.
AU - Merkurjev, Alexander S.
TI - On standard norm varieties
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 1
SP - 177
EP - 216
AB - Let $p$ be a prime integer and $F$ a field of characteristic $0$. Let $X$ be thenorm varietyof a symbol in the Galois cohomology group $H^{n+1}(F,\mu _p^{\otimes n})$ (for some $n\ge 1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F(X)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\mathop {\mathrm {CH}}\nolimits (Y)\rightarrow \mathop {\mathrm {CH}}\nolimits (Y_{F(X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $&lt; (\dim X)/(p-1)$. One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in the appendix). Another important ingredient is$A$-trivialityof $X$, the property saying that the degree homomorphism on $\mathop {\mathrm {CH}}\nolimits _0(X_L)$ is injective for any field extension $L/F$ with $X(L)\ne \emptyset $. The proof involves the theory of rational correspondences reviewed in the appendix.
LA - eng
KW - norm varieties; Chow groups and motives; Steenrod operations
UR - http://eudml.org/doc/272131
ER -

References

top
  1. [1] A. R. Boisvert, A new definition of the Steenrod operations in algebraic geometry, ProQuest LLC, Ann Arbor, MI, 2007, Thesis (Ph.D.)–University of California, Los Angeles. MR2710781
  2. [2] A. R. Boisvert, A new definition of the Steenrod operations in algebraic geometry, preprint arXiv:0805.1414. 
  3. [3] P. Brosnan, A short proof of Rost nilpotence via refined correspondences, Doc. Math.8 (2003), 69–78. Zbl1044.11017MR2029161
  4. [4] P. Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc.355 (2003), 1869–1903. Zbl1045.55005MR1953530
  5. [5] V. Chernousov & A. S. Merkurjev, Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem, Transform. Groups11 (2006), 371–386. Zbl1111.14009MR2264459
  6. [6] P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series 81, Cambridge Univ. Press, 1983. MR696937
  7. [7] R. Elman, N. Karpenko & A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, Amer. Math. Soc., 2008. Zbl1165.11042MR2427530
  8. [8] R. Fino, Around rationality of integral cycles, J. Pure Appl. Algebra (2012), doi://10.1016/j.jpaa.2012.12.003. Zbl1295.14008MR3042631
  9. [9] R. Fino, Around rationality of cycles, to appear in Cent. Eur. J. Math. Zbl1300.14006MR3036018
  10. [10] W. Fulton, Intersection theory, second éd., Ergeb. Math. Grenzg. 2, Springer, 1998. Zbl0885.14002MR1644323
  11. [11] S. Garibaldi, Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc. 200 (2009), 81. Zbl1191.11009MR2528487
  12. [12] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, in Séminaire Bourbaki, vol. 6, exp. no 221, Soc. Math. France, 1995, 249–276. MR1611822
  13. [13] B. Kahn & R. Sujatha, Birational motives, I, K -theory Preprint Archives (preprint server) 596, 2002. 
  14. [14] N. Karpenko & A. Merkurjev, Rost projectors and Steenrod operations, Doc. Math.7 (2002), 481–493. Zbl1030.11013MR2015051
  15. [15] N. A. Karpenko, Criteria of motivic equivalence for quadratic forms and central simple algebras, Math. Ann.317 (2000), 585–611. Zbl0965.11015MR1776119
  16. [16] N. A. Karpenko, Weil transfer of algebraic cycles, Indag. Math. (N.S.) 11 (2000), 73–86. Zbl1047.14004MR1809664
  17. [17] N. A. Karpenko, Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties, J. reine angew. Math. (2012), doi://10.1515/crelle.2012.011. Zbl1267.14009MR3039776
  18. [18] N. A. Karpenko & A. S. Merkurjev, Canonical p -dimension of algebraic groups, Adv. Math.205 (2006), 410–433. Zbl1119.14041MR2258262
  19. [19] J. I. Manin, Correspondences, motifs and monoidal transformations, Mat. Sb. (N.S.) 77 (119) (1968), 475–507. Zbl0199.24803MR258836
  20. [20] A. S. Merkurʼev & A. Suslin, K -cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136. MR675529
  21. [21] A. Merkurjev, Rost invariants of simply connected algebraic groups, in Cohomological invariants in Galois cohomology, Univ. Lecture Ser. 28, Amer. Math. Soc., 2003, 101–158. MR1999385
  22. [22] A. Merkurjev, Unramified elements in cycle modules, J. Lond. Math. Soc.78 (2008), 51–64. Zbl1155.14017MR2427051
  23. [23] A. Merkurjev & A. Suslin, Motivic cohomology of the simplicial motive of a Rost variety, J. Pure Appl. Algebra214 (2010), 2017–2026. Zbl1200.14041MR2645334
  24. [24] A. S. Merkurjev, Essential dimension, in Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math. 493, Amer. Math. Soc., 2009, 299–325. Zbl1188.14006MR2537108
  25. [25] D. H. Nguyen, On p-generic splitting varieties for Milnor K-symbols mod p, ProQuest LLC, Ann Arbor, MI, 2009, Thesis (Ph.D.)–University of California, Los Angeles. MR2714019
  26. [26] D. H. Nguyen, On p -generic splitting varieties for Milnor K -symbols mod p , preprint arXiv:1003.3971. MR2714019
  27. [27] I. Panin, Application of K -theory in algebraic geometry, Thèse, LOMI, Leningrad, 1984. 
  28. [28] M. Rost, Chow groups with coefficients, Doc. Math.1 (1996), No. 16, 319–393. Zbl0864.14002MR1418952
  29. [29] M. Rost, On the basic correspondence of a splitting variety, preprint http://www.math.uni-bielefeld.de/~rost/data/bkc-c.pdf. 
  30. [30] A. Suslin & S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra206 (2006), 245–276. MR2220090
  31. [31] M. L. Thakur, Isotopy and invariants of Albert algebras, Comment. Math. Helv.74 (1999), 297–305. Zbl0931.17021MR1691951
  32. [32] A. Vishik, Generic points of quadrics and Chow groups, Manuscripta Math.122 (2007), 365–374. Zbl1154.14003MR2305424
  33. [33] A. Vishik, Rationality of integral cycles, Doc. Math. (2010), 661–670, Extra volume: Andrei A. Suslin sixtieth birthday. Zbl1257.14003MR2804267
  34. [34] A. Vishik & K. Zainoulline, Motivic splitting lemma, Doc. Math.13 (2008), 81–96. Zbl1132.14303MR2393083
  35. [35] V. Voevodsky, On motivic cohomology with 𝐙 / l -coefficients, Ann. of Math.174 (2011), 401–438. Zbl1236.14026MR2811603
  36. [36] K. Zainoulline, Special correspondences and Chow traces of Landweber-Novikov operations, J. reine angew. Math. 628 (2009), 195–204. Zbl1171.14004MR2503240

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.