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

A. E. Druzhinin

Sbornik. Mathematics, Tome 214 (2023) no. 2, pp. 210-225

Résumé

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 $(X-D)\times(\mathbb{A}^1_k-Z_0)\times\dots\times(\mathbb{A}^1_k-Z_l)$, where $Z_0,\dots,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 titles.

Détail

Keywords: Gersten conjecture, Cousin complex, motivic cohomologies.

Bibliographie

[1] S. M. Gersten, “Some exact sequences in the higher K-theory of rings”, Algebraic K-theory (Battelle Memorial Inst., Seattle, WA 1972), v. I, Lecture Notes in Math., 341, Higher K-theories, Springer, Berlin, 1973, 211–243 | DOI | MR | Zbl

[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 ; Cohomology of groups and algebraic K-theory (Hangzhou 2007), Adv. Lect. Math. (ALM), 12, Int. Press, Somerville, MA; Higher Education Press, Beijing, 2010, 413–478 | DOI | MR | Zbl | MR | Zbl

[3] I. A. Panin, “The equicharacteristic case of the Gersten conjecture”, Number theory, algebra and algebraic geometry, Tr. Mat. Inst. Steklova, 241, Nauka, MAIK “Nauka/Interperiodika”, Moscow, 2003, 169–178 ; English transl. in Proc. Steklov Inst. Math., 241 (2003), 154–163 | MR | Zbl

[4] S. Bloch and A. Ogus, “Gersten's conjecture and the homology of schemes”, Ann. Sci. École Norm. Sup. (4), 7 (1974), 181–201 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[6] I. A. Panin, “Nice triples and moving lemmas for motivic spaces”, Izv. Ross. Akad. Nauk Ser. Mat., 83:4 (2019), 158–193 ; English transl. in Izv. Math., 83:4 (2019), 796–829 | DOI | MR | Zbl | DOI

[7] I. Panin and A. Smirnov, Push-forwards in oriented cohomology theories of algebraic varieties https://conf.math.illinois.edu/K-theory/0459/

[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 | MR | Zbl

[9] F. Morel, “The stable $\mathbb A^1$-connectivity theorems”, K-Theory, 35:1–2 (2005), 1–68 | DOI | MR | Zbl

[10] G. Garkusha and I. Panin, “Homotopy invariant presheaves with framed transfers”, Camb. J. Math., 8:1 (2020), 1–94 | DOI | MR | Zbl

[11] A. Druzhinin and I. Panin, “Surjectivity of the étale excision map for homotopy invariant framed presheaves”, Tr. Mat. Inst. Steklova, 320 (2023), 103–127 ; English transl. in Proc. Steklov Inst. Math., 320 (2023), 91–114 | DOI | MR | Zbl | DOI

[12] A. Druzhinin and J. I. Kylling, Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic, arXiv: 1809.03238

[13] F. Morel, $\mathbb A^1$-algebraic topology over a field, Lecture Notes in Math., 2052, Springer, Heidelberg, 2012, x+259 pp. | DOI | MR | Zbl

[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 | MR | Zbl

[15] F. Morel and A. Sawant, Cellular $\mathbb A^1$-homology and the motivic version of Matsumoto's theorem, arXiv: 2007.14770

[16] A. Druzhinin, Strict $\mathbb A^1$-homotopy invariance theorem with integral coefficients over fields, arXiv: 2108.01006

[17] R. W. Thomason and 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 | DOI | MR | Zbl

[18] M. Schlichting, “The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes”, Invent. Math., 179:2 (2010), 349–433 | DOI | MR | Zbl

[19] I. Panin and C. Walter, “On the motivic commutative ring spectrum $\mathbf{BO}$”, Algebra i Analiz, 30:6 (2018), 43–96 ; St. Petersburg Math. J., 30:6 (2019), 933–972 | MR | Zbl | DOI

[20] J. P. Serre, “Les espaces fibrés algébriques”, Anneaux de Chow et applications, Seminaire C. Chevalley, 3, Secrétariat mathématique, Paris, 1958, Exp. No. 1, 37 pp. | MR | Zbl

[21] A. Grothendieck, “Le groupe de Brauer. II. Théorie cohomologique”, Dix exposés sur la cohomologie de schémas, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968, 67–87 ; Séminaire N. Bourbaki, v. 1965/66, W. A. Benjamin, Inc., New York, 1966, Exp. No. 297, 21 pp. | MR | Zbl | MR

[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 Sér. I Math., 309:10 (1989), 651–655 | MR | Zbl

[23] R. Fedorov and I. Panin, “A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields”, Publ. Math. Inst. Hautes Études Sci., 122:1 (2015), 169–193 | DOI | MR | Zbl

[24] I. A. Panin, “Proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing a field”, Izv. Ross. Akad. Nauk Ser. Mat., 84:4 (2020), 169–186 ; English transl. in Izv. Math., 84:4 (2020), 780–795 | DOI | MR | Zbl | DOI

[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 | DOI | MR | Zbl

[26] R. Fedorov, On the purity conjecture of Nisnevich for torsors under reductive group schemes, arXiv: 2109.10332v3

[27] A. Grothendieck, “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de sch'emas”, Quatrième partie, Inst. Hautes Études Sci. Publ. Math., 32 (1967), 5–361 | MR | Zbl

[28] I. Panin, “Oriented cohomology theories of algebraic varieties”, K-Theory, 30:3 (2003), 265–314 | DOI | MR | Zbl

[29] I. Panin, “Oriented cohomology theories of algebraic varieties. II”, Homology Homotopy Appl., 11:1 (2009), 349–405 | DOI | MR | Zbl

[30] A. Druzhinin, H. Kolderup and P. A. Østvær, Strict $\mathbb A^1$-invariance over the integers, arXiv: 2012.07365v1

[31] L. Gruson, “Une propriété des couples henséliens”, Colloque d'algèbre commutative (Rennes 1972), Publ. Sém. Math. Univ. Rennes, 1972, no. 4, Univ. Rennes, Rennes, 1972, Exp. No. 10, 13 | MR | Zbl

[32] R. Elkik, “Solutions d'équations à coefficients dans un anneau hensélien”, Ann. Sci. École Norm. Sup. (4), 6:4 (1973), 553–603 | DOI | MR | Zbl