Searcharxiv⌕ Search

arXiv subjects

Alexander Vishik

Publications and source records attributed to Alexander Vishik.

At least 19 recordsLinked to original sources
arXiv2026

On the $p$-primary and $p$-adic cases of the Isotropy Conjecture

The purpose of this note is to show that, in contrast to the ${\Bbb F}_p$-case (proven in [7]), the $p$-primary and $p$-adic cases of the Isotropy Conjecture, claiming that the isotropic Chow groups with ${\Bbb Z}/p^r$, $r>1$, respectively, with ${\Bbb Z}_p$-coefficients over a flexible field coincide with the numerical ones, don't hold. We show that the $BP$-theory with $I(\infty)$-primary, respectively, $I(\infty)$-adic coefficients may serve as a regular substitute for $p$-primary, respectively, $p$-adic Chow groups, which permits to extend the results of [6] to arbitrary primes.

math.AG↗
arXiv2026

On the numerical triviality of $BP$-cycles

We show that, in the case of a prime $2$, the numerical triviality of $BP$-cycles modulo various powers of the (augmentation) invariant ideal $I(\infty)$ is controlled by pure symbols in $K^M_*/2$ over the flexible closure of the base field.

math.AG↗
arXiv2025

The Balmer spectrum of Voevodsky motives and pure symbols

In this article we introduce invariants of points of the Balmer spectrum of the Voevodsky motivic category whose values are "light Rost cycle submodules" of the module of pure symbols in Milnor's K-theory (mod 2). As an application, we show that isotropic points of the Balmer spectrum are closed. We also introduce the notion of points of a boundary type and show that this class contains isotropic points, but not the etale one.

math.AG↗
arXiv2025

Varieties and pure symbols

In this article, we prove that the 2-isotropy of any projective variety is controlled by a pure symbol in the Milnor's K-theory (mod 2) of the flexible closure of the base field. We also show that such pure symbols control the 2-equivalence of field extensions as well as the numerical equivalence of algebraic cycles (with mod 2 coefficients).

math.AG↗
arXiv2023

On the Balmer spectrum of the Morel-Voevodsky category

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 ${\Bbb{A}}^1$-homotopy category $SH(k)$ are analogues of local motivic categories introduced in [22], but with a substantially more general notion of "isotropy". This permits to construct the, so-called, isotropic Morava points of the Balmer spectrum $\operatorname{Spc}(SH^c(k))$ of (the compact part of) the Morel-Voevodsky category. These analogues of topological Morava points are parametrized by the choice of Morava K-theory and a $K(p,m)$-equivalence class of extensions $E/k$. This provides a large supply of new points, and substantially improves our understanding of the spectrum. An interesting new feature is that the specialization among isotropic points behaves differently than in topology.

math.AG↗
arXiv2023

On torsion spaces

The purpose of this note is to construct examples of compact torsion objects of ${\cal SH}(F)$ of every $p$-level over an arbitrary field of characteristic different from $p$. We adapt the approach of Mitchell to the algebraic situation. We show that the respective level is determined by the action of the $v_m$-elements on the $MGL$-motive, and also prove the refinement which permits to distinguish isotropic Morava points of the Balmer spectrum of ${\cal SH}(F)$.

math.AG↗
arXiv2023

Isotropic and numerical equivalence for Chow groups and Morava K-theories

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and $\otimes$ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky ${\Bbb{A}}^1$-stable homotopic category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.

math.AG↗
arXiv2022

On isotropic and numerical equivalence of cycles

We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This conjecture is essential for understanding the structure of the isotropic motivic category and that of the tensor triangulated spectrum of Voevodsky category of motives. We prove the conjecture for the new range of cases. In particular, we show that, for a given variety $X$, it holds for sufficiently large primes $p$. We also prove the $p$-adic analogue. This permits to interpret integral numerically trivial classes in $CH(X)$ as $p^{\infty}$-anisotropic ones.

math.AG↗
arXiv2021

Subtle Characteristic Classes

We construct new subtle Stiefel--Whitney classes of quadratic forms. These classes are much more informative than the ones introduced by Milnor. In particular, they see all the powers of the fundamental ideal of the Witt ring, contain the Arason invariant and it's higher analogues. Moreover, the new classes allow to treat the J-invariant of quadrics. This invariant, introduced in 2005, has been so far completely isolated from characteristic classes. In addition, our classes allow to describe explicitly the structure of some motives associated with quadratic forms.

math.AG↗
arXiv2021

Torsion motives

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. We introduce various invariants of torsion motives, in particular, the $p$-level. We show that this invariant bounds from below the dimension of the variety a torsion motive $M$ is a direct summand of and imposes restrictions on motivic and singular cohomology of $M$. We study in more details the $p$-torsion motives of surfaces, in particular, the Godeaux torsion motive. We show that such motives are in 1-to-1 correspondence with certain Rost cycle submodules of free modules over $H^*_{et}$. This description is parallel to that of mod-$p$ reduced motives of curves.

math.AG↗
arXiv2020

Isotropic motives

In this article we introduce the local versions of the Voevodsky category of motives with Z/p-coefficients over a field k, parameterized by finitely-generated extensions of k. We introduce the, so-called, flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy "local" invariants from which one can read motivic information. We compute the local motivic cohomology of a point, for p=2, and study the local Chow motivic category. We introduce local Chow groups and conjecture that, over flexible fields, these should coincide with Chow groups modulo numerical equivalence with Z/p-coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this Conjecture in various cases.

math.AG↗
arXiv2020

Operations in connective K-theory

In this article 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 $\hat{{\mathbb{Z}}}$ one. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stable operations with $\hat{{\mathbb{Z}}}$-coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.

math.KT↗
arXiv2020

Operations and poly-operations in Algebraic Cobordism

We describe all operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^* (in the case of a field of characteristic zero). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done earlier, as well as the topological one obtained by T.Kashiwabara. The key new ingredients which permit us to treat the non-additive operations are: the use of "poly-operations" and the "Discrete Taylor expansion". As an application we construct the only missing, the 0-th (non-additive) Symmetric operation, for arbitrary p, which permits to sharpen results on the structure of Algebraic Cobordism. We also prove the general Riemann-Roch theorem for arbitrary (even non-additive) operations (over an arbitrary field). This extends the multiplicative case proved by I.Panin.

math.AG↗
arXiv2019

Affine quadrics and the Picard group of the motivic category

In this article we study the subgroup of the Picard group of Voevodsky's category of geometric motives generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann, but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the Criterion of motivic equivalence of projective quadrics.

math.AG↗
arXiv2018

Rost nilpotence and free theories

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 algebraic cobordism and its descendants the free theories.

math.AG↗
arXiv2017

Stable and Unstable operations in Algebraic Cobordism

We describe additive (unstable) operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence between the set of operations, and the set of transformations: A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying certain simple properties. This provides an effective tool of constructing such operations. As an application, we prove that (unstable) additive operations in Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear combinations of Landweber-Novikov operations which take integral values on the products of projective spaces. On our way we obtain that stable operations there are exactly L-linear combinations of Landweber-Novikov operations. We also show that multiplicative operations A^* ---> B^* are in 1-to-1 correspondence with the morphisms of the respective formal group laws. We construct Integral (!) Adams Operations in Algebraic Cobordism, and all theories obtained from it by change of coefficients, giving classical Adams operations in the case of K_0. Finally, we construct Symmetric Operations for all primes p (these operations in Algebraic Cobordism, previously known only for p=2, are more subtle than the Landweber-Novikov operations, and have applications to rationality questions), as well as the T.tom Dieck - style Steenrod operations in Algebraic Cobordism. As a bi-product of the proof of our main theorem we get the Riemann-Roch Theorem for additive (unstable) operations.

math.AG↗
arXiv2015

Symmetric operations for all primes and Steenrod operations in Algebraic Cobordism

In this article we construct Symmetric operations for all primes (previously known only for p=2). These unstable operations are more subtle than the Landweber-Novikov operations, and encode all p-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map L ---> Z[b_1,b_2,...], providing an important structure on Algebraic Cobordism. Applications include: questions of rationality of Chow group elements - see [11], and the structure of the Graded Algebraic Cobordism. We also construct Steenrod operations of T.tom Dieck-style in Algebraic Cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.

math.AG↗