arXiv subjects

Pavel Sechin

Publications and source records attributed to Pavel Sechin.

7 recordsLinked to original sources

arXiv2025

Invertible Morava motives in quadrics

We associate to any element in the Milnor K-theory of a field $k$ modulo 2 an invertible Morava K-theory motive over $k$. Specifically, for $α$ in $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ we construct an invertible $\mathrm{K}(n)$-motive $L_α$ in a way that is natural in the base field and additive in $α$. This can be seen as categorification of $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ in motives. The motives $L_α$ are constructed as direct summands of the $\mathrm{K}(n)$-motives of quadrics, and we develop the necessary framework for the study of the latter. We show that passing to the field of functions of quadrics of dimension greater than or equal to $2^{n+1}-1$ does not lose any information about the structure of $\mathrm{K}(n)$-motives. This is based on the study of "decomposition of the diagonal" in Morava K-theory of quadrics. For quadrics of dimension less than $2^{n+1}-1$, we show that their Chow motives can be "reconstructed" from their $\mathrm{K}(n)$-motives, although the latter appear structurally simpler. Our proof of this result relies on the use of the unstable symmetric operations of Vishik on algebraic cobordism. The occurrence of the motive $L_α$ as a direct summand of the $\mathrm{K}(n)$-motive of $X$ can be seen as evidence that $α$ is a cohomological invariant of $X$. We study this occurrence for quadrics and relate it to Kahn's Descent conjecture.

math.AG↗

arXiv2024

Morava $J$-invariant

We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the $J$-invariant in terms of the ordinary (Chow) $J$-invariant.

math.KT↗

arXiv2024

Morava K-theory of orthogonal groups and motives of projective quadrics

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.

math.KT↗

arXiv2020

Applications of the Morava $K$-theory to algebraic groups

In the present article we discuss an approach to cohomological invariants of algebraic groups over fields of characteristic zero based on the Morava $K$-theories, which are generalized oriented cohomology theories in the sense of Levine--Morel. We show that the second Morava $K$-theory detects the triviality of the Rost invariant and, more generally, relate the triviality of cohomological invariants and the splitting of Morava motives. We describe the Morava $K$-theory of generalized Rost motives, compute the Morava $K$-theory of some affine varieties, and characterize the powers of the fundamental ideal of the Witt ring with the help of the Morava $K$-theory. Besides, we obtain new estimates on torsion in Chow groups of codimensions up to $2^n$ of quadrics from the $(n+2)$-nd power of the fundamental ideal of the Witt ring. We compute torsion in Chow groups of $K(n)$-split varieties with respect to a prime $p$ in all codimensions up to $\frac{p^n-1}{p-1}$ and provide a combinatorial tool to estimate torsion up to codimension $p^n$. An important role in the proof is played by the gamma filtration on Morava $K$-theories, which gives a conceptual explanation of the nature of the torsion. Furthermore, we show that under some conditions the $K(n)$-motive of a smooth projective variety splits if and only if its $K(m)$-motive splits for all $m\le n$.

math.AG↗

arXiv2018

On the Structure of Algebraic Cobordism

In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration $Ω^*_{(r)}(X)$ are unions of finitely presented $\mathbb{L}$-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules $\mathbb{L}/I(p,n)x$ where $\mathrm{deg\ } x\ge \frac{p^n-1}{p-1}$. As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free $\mathbb{L}$-resolutions of $Ω^*(X)$, and show that algebraic cobordism of a smooth surface can be described in terms of $K_0$ together with a topological filtration.

math.AG↗

arXiv2018

Chern classes from Morava K-theories to $p^n$-typical oriented theories

We study non-additive operations from algebraic Morava K-theories to oriented cohomology theories in algebraic geometry. For oriented cohomology theory $A$ that has a {$p^n$}-typical formal group law over a $\mathbb{Z}_{(p)}$-algebra we construct `Chern classes' from the algebraic $n$-th Morava K-theory with $p$-local coefficients to $A$. If the coefficient ring of $A$ is a free $\mathbb{Z}_{(p)}$-module we also prove that these Chern classes freely generate all operations from $\mathrm{K}(n)$ to $A$. Examples of such theories are algebraic Morava K-theories $\mathrm{K}(nm)^*$ for all $m\in\mathbb{N}$ and Chow groups with $p$-local coefficients. The universal $p^n$-typical oriented theory is $BP\{n\}^*$ whose coefficient ring is also a free $\mathbb{Z}_{(p)}$-module. Chern classes from the $n$-th algebraic Morava K-theory $\mathrm{K}(n)$ to itself allow us to introduce the gamma filtration on $\mathrm{K}(n)$. This is the best approximation to the topological filtration obtained by values of operations and it satisfies properties similar to that of the classical gamma filtration on $\mathrm{K}_0$. The major difference from the classical case is that Chern classes from the graded factors $gr^i_\gamma \mathrm{K}(n)^*$ to Chow groups with p-local coefficients are surjective for $i\le p^n$, which allows to estimate $p$-torsion in Chow groups of codimension up to $p^n$ of some varieties.

math.AG↗

arXiv2017

Chern classes from Morava K-theories to Chow groups

In this paper we calculate the ring of unstable (possibly non-additive) operations from algebraic Morava K-theory K(n) to Chow groups with $\mathbb{Z}_{(p)}$-coefficients. More precisely, we prove that it is a formal power series ring on generators $c_i:K(n)\rightarrow CH^i \otimes \mathbb{Z}_{(p)}$, which satisfy a Cartan-type formula.

math.AG↗