Cousin complex on the complement to the strict normal-crossing divisor in a local essentially smooth scheme over a field

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

For any $\mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)\times(\mathbb{A}^1_k-Z_0)\times…\times(\mathbb{A}^1_k-Z_l)$, where $Z_0,…,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 titles.

About the authors

Andrei Eduardovich Druzhinin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Email: andrei.druzh@gmail.com
Candidate of physico-mathematical sciences

References

  1. S. M. Gersten, “Some exact sequences in the higher K-theory of rings”, Algebraic K-theory (Battelle Memorial Inst., Seattle, Wash., 1972), v. I, Lecture Notes in Math., 341, Higher K-theories, Springer, Berlin, 1973, 211–243
  2. D. Quillen, “Higher algebraic K-theory. I”, Algebraic K-theory (Battelle Memorial Inst., Seattle, WA, 1972), v. I, Lecture Notes in Math., 341, Higher K-theories, Springer, Berlin, 1973, 85–147
  3. I. A. Panin, “The equicharacteristic case of the Gersten conjecture”, Теория чисел, алгебра и алгебраическая геометрия, Сборник статей. К 80-летию со дня рождения академика Игоря Ростиславовича Шафаревича, Труды МИАН, 241, Наука, МАИК «Наука/Интерпериодика», М., 2003, 169–178
  4. S. Bloch, A. Ogus, “Gersten's conjecture and the homology of schemes”, Ann. Sci. Ecole Norm. Sup. (4), 7 (1974), 181–201
  5. V. Voevodsky, “Cohomological theory of presheaves with transfers”, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000, 87–137
  6. И. А. Панин, “Совершенные тройки и гомотопии отображений мотивных пространств”, Изв. РАН. Сер. матем., 83:4 (2019), 158–193
  7. I. Panin, A. Smirnov, Push-forwards in oriented cohomology theories of algebraic varieties
  8. F. Morel, “An introduction to $mathbb A^1$-homotopy theory”, Contemporary developments in algebraic K-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 357–441
  9. F. Morel, “The stable $mathbb A^1$-connectivity theorems”, K-Theory, 35:1-2 (2005), 1–68
  10. G. Garkusha, I. Panin, “Homotopy invariant presheaves with framed transfers”, Camb. J. Math., 8:1 (2020), 1–94
  11. А. Дружинин, И. Панин, “Сюръективность этального вырезания для гомотопически инвариантных предпучков с оснащенными трансферами”, Труды МИАН, 320 (2023) (в печати)
  12. A. Druzhinin, J. I. Kylling, Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic
  13. F. Morel, $mathbb A^1$-algebraic topology over a field, Lecture Notes in Math., 2052, Springer, Heidelberg, 2012, x+259 pp.
  14. V. Voevodsky, “$mathbb A^1$-homotopy theory”, Proceedings of the international congress of mathematicians (Berlin, 1998), v. I, Doc. Math., Extra Vol. 1, 1998, 579–604
  15. F. Morel, A. Sawant, Cellular $mathbb A^1$-homology and the motivic version of Matsumoto's theorem
  16. A. Druzhinin, Strict $mathbb A^1$-homotopy invariance theorem with integral coefficients over fields
  17. R. W. Thomason, T. Trobaugh, “Higher algebraic K-theory of schemes and of derived categories”, The Grothendieck Festschrift, v. III, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990, 247–435
  18. M. Schlichting, “The Mayer–Vietoris principle for Grothendieck–Witt groups of schemes”, Invent. Math., 179:2 (2010), 349–433
  19. I. Panin, C. Walter, “On the motivic commutative ring spectrum $mathbf{BO}$”, Алгебра и анализ, 30:6 (2018), 43–96
  20. J. P. Serre, “Les espaces fibres algebriques”, Anneaux de Chow et applications, Seminaire C. Chevalley, 3, Secretariat mathematique, Paris, 1958, Exp. No. 1, 37 pp.
  21. A. Grothendieck, “Le groupe de Brauer. II. Theorie cohomologique”, Dix exposes sur la cohomologie de schemas, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968, 67–87
  22. Y. Nisnevich, “Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional regular local rings”, C. R. Acad. Sci. Paris Ser. I Math., 309:10 (1989), 651–655
  23. R. Fedorov, I. Panin, “A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields”, Publ. Math. Inst. Hautes Etudes Sci., 122:1 (2015), 169–193
  24. И. А. Панин, “Доказательство гипотезы Гротендика–Серра о главных расслоениях над регулярным локальным кольцом, содержащим поле”, Изв. РАН. Сер. матем., 84:4 (2020), 169–186
  25. I. Panin, “On Grothendieck–Serre conjecture concerning principal bundles”, Proceedings of the international congress of mathematicians (ICM 2018) (Rio de Janeiro, 2018), v. 2, World Sci. Publ., Hackensack, NJ, 2018, 201–221
  26. R. Fedorov, On the purity conjecture of Nisnevich for torsors under reductive group schemes
  27. A. Grothendieck, “Elements de geometrie algebrique. IV. Etude locale des schemas et des morphismes de sch'emas”, Quatrième partie, Inst. Hautes Etudes Sci. Publ. Math., 32 (1967), 5–361
  28. I. Panin, “Oriented cohomology theories of algebraic varieties”, K-Theory, 30:3 (2003), 265–314
  29. I. Panin, “Oriented cohomology theories of algebraic varieties. II”, Homology Homotopy Appl., 11:1 (2009), 349–405
  30. A. Druzhinin, H. Kolderup, P. A. Ostvaer, Strict $mathbb A^1$-invariance over the integers
  31. L. Gruson, “Une propriete des couples henseliens”, Colloque d'algèbre commutative (Rennes, 1972), Publ. Sem. Math. Univ. Rennes, 1972, no. 4, Univ. Rennes, Rennes, 1972, Exp. No. 10, 13 pp.
  32. R. Elkik, “Solutions d'equations à coefficients dans un anneau henselien”, Ann. Sci. Ecole Norm. Sup. (4), 6:4 (1973), 553–603

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2023 Druzhinin A.E.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).