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All Outputs (13)

On the Balmer spectrum of the Morel-Voevodsky category (2025)
Journal Article
Du, P., & Vishik, A. (2025). On the Balmer spectrum of the Morel-Voevodsky category. Duke Mathematical Journal, 174(6), 1013-1044. https://doi.org/10.1215/00127094-2024-0051

We introduce the Morava-isotropic stable homotopy category and, more generally, the stable homotopy category of an extension E∕k. These “local” versions of the Morel–Voevodsky stable A1-homotopy category SH(k) are analogues of local motivic categorie... Read More about On the Balmer spectrum of the Morel-Voevodsky category.

Isotropic and numerical equivalence for Chow groups and Morava K-theories (2024)
Journal Article
Vishik, A. (2024). Isotropic and numerical equivalence for Chow groups and Morava K-theories. Inventiones Mathematicae, 237(2), 779-808. https://doi.org/10.1007/s00222-024-01267-z

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with Fp-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular,... Read More about Isotropic and numerical equivalence for Chow groups and Morava K-theories.

Operations in connective K-theory (2023)
Journal Article
Vishik, A., & Merkurjev, A. (2023). Operations in connective K-theory. Algebra and Number Theory, 17(9), 1595–1636. https://doi.org/10.2140/ant.2023.17.1595

We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the ˆZ case. Moreover, although integral additive operations are topologically... Read More about Operations in connective K-theory.

Torsion Motives (2023)
Journal Article
Vishik, A. (2023). Torsion Motives. International Mathematics Research Notices, 2023(23), 20252–20295. https://doi.org/10.1093/imrn/rnad056

In this paper we study Chow motives whose identity map is killed by a natural number. Examples of such objects were constructed by Gorchinskiy-Orlov [10]. We introduce various invariants of torsion motives, in particular, the p-level. We show that th... Read More about Torsion Motives.

On isotropic and numerical equivalence of cycles (2022)
Journal Article
Vishik, A. (2022). On isotropic and numerical equivalence of cycles. Selecta Mathematica (New Series), 29(1), Article 8. https://doi.org/10.1007/s00029-022-00812-z

We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with Fp-coefficients). This conjecture is essential for understanding the structure of the isotropic motivic category and that of... Read More about On isotropic and numerical equivalence of cycles.

ISOTROPIC MOTIVES (2020)
Journal Article
Vishik, A. (2020). ISOTROPIC MOTIVES. Journal of the Institute of Mathematics of Jussieu, 21(4), 1271-1330. https://doi.org/10.1017/S1474748020000560

In this article we introduce the local versions of the Voevodsky category of motives with Fp coefficients over a field k, parameterized by finitely-generated extensions of k. We introduce the, so-called, flexible fields, passage to which is conservat... Read More about ISOTROPIC MOTIVES.

Operations and poly-operations in algebraic cobordism (2020)
Journal Article
Vishik, A. (2020). Operations and poly-operations in algebraic cobordism. Advances in Mathematics, 366, Article 107066. https://doi.org/10.1016/j.aim.2020.107066

© 2020 Elsevier Inc. In the case of a field of characteristic zero, we describe all operations (including non-additive ones) from a theory A⁎ obtained from Algebraic Cobordism Ω⁎ of M. Levine-F. Morel by change of coefficients to any oriented cohomol... Read More about Operations and poly-operations in algebraic cobordism.

Affine quadrics and the Picard group of the motivic category (2019)
Journal Article
Vishik, A. (2019). Affine quadrics and the Picard group of the motivic category. Compositio Mathematica, 155(8), 1500-1520. https://doi.org/10.1112/S0010437X19007401

In this paper we study the subgroup of the Picard group of Voevodsky's category of geometric motives DMgm(k; Z/2) generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives o... Read More about Affine quadrics and the Picard group of the motivic category.

Stable and unstable operations in algebraic cobordism (2019)
Journal Article
Vishik, A. (2019). Stable and unstable operations in algebraic cobordism. Annales Scientifiques de l'École Normale Supérieure, 1(52), 561-630. https://doi.org/10.24033/asens.2393

© 2019 Société Mathématique de France. Tous droits réservés - We describe additive (unstable) operations from a theory A* obtained from the Levine-Morel algebraic cobordism by change of coefficients to any oriented cohomology theory B* (over a field... Read More about Stable and unstable operations in algebraic cobordism.

Rost nilpotence and free theories (2018)
Journal Article
Gille, S., & Vishik, A. (2018). Rost nilpotence and free theories. Documenta Mathematica, 23, 1635-1657

We introduce coherent cohomology theories h_* and prove that if such a theory is moreover generically constant then the Rost nilpotence principle holds for projective homogeneous varieties in the category of h_*-motives. Examples of such theories are... Read More about Rost nilpotence and free theories.

Algebraic Cobordism as a module over the Lazard ring (2015)
Journal Article
Vishik, A. (2015). Algebraic Cobordism as a module over the Lazard ring. Mathematische Annalen, 363(3-4), 973-983. https://doi.org/10.1007/s00208-015-1190-3

In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of Lev... Read More about Algebraic Cobordism as a module over the Lazard ring.