arXiv subjects

Egor Zolotarev

Publications and source records attributed to Egor Zolotarev.

4 recordsLinked to original sources

arXiv2026

On the metalinear algebraic cobordism spectrum

In this paper, we study the metalinear algebraic cobordism spectrum $\mathrm{MML}$ (also sometimes denoted $\mathrm{MSL}^c$), which is built from the structure groups of oriented vector bundles. We establish an interpolation between $\mathrm{MSL}$ and $\mathrm{MML}$ and deduce that the canonical morphism $\mathrm{MSL}\to \mathrm{MML}$ admits a retraction. We parametrize all such retractions in the category of $\mathrm{MSL}$-modules and, after fixing one of them, obtain an equivalence $\mathrm{MML}\cong\mathrm{MSL}\oplus \Sigma^{2,1}\mathrm{MGL}$. As an application of these results, we determine various invariants of the metalinear algebraic cobordism spectrum over a field (after inverting the exponential characteristic). More precisely, we determine the first few Milnor-Witt stems of $\mathrm{MML}$ in terms of the very effective algebraic and hermitian K-theory spectra, and the geometric diagonal of $\mathrm{MML}$ in terms of Stong's complex-spin cobordism ring. We also compute the slices and use them to describe the category of 2-inverted modules over the $\mathbb{E}_\infty$-ring spectrum $\mathrm{MML}$.

math.AT↗

arXiv2026

Slices of the special linear algebraic cobordism spectrum

Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.

math.AT↗

arXiv2026

Counter-examples to a conjecture of Karpenko via truncated Brown-Peterson cohomology

Let $G$ be a split semisimple linear algebraic group and let $X$ denote the generically twisted variety of Borel subgroups in $G$. Nikita Karpenko conjectured that the map from the Chow ring of $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. After having been verified for many $G$, the conjecture was disproved by Nobuaki Yagita for some spinor groups. Later, other counter-examples were constructed by Baek-Karpenko and Baek-Devyatov. We present a new method for constructing counter-examples that is based on the connection of the truncated Brown-Peterson cohomology with the connective K-theory. Using this method, we disprove the conjecture for new groups, including $\mathrm{Spin}_{15}$, which is now the smallest known spinor group for which the conjecture fails.

math.AG↗

arXiv2024

The geometric diagonal of the special linear algebraic cobordism

The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\mathbb{P}^1$-diagonal of the homotopy groups of the special linear algebraic cobordism $\pi_{2*,*}(\mathrm{MSL})$ over a local Dedekind domain $k$ with $1/2\in k$ after inverting the exponential characteristic of the residue field of $k$. We discuss the action of the motivic Hopf element $\eta$ on this ring, obtain a description of the localization away from $2$ and compute the $2$-primary torsion subgroup. The complete answer is given in terms of the special unitary cobordism ring. An important component of the computation is the construction of Pontryagin characteristic numbers with values in the hermitian K-theory. We also construct Chern numbers in this setting, prove the motivic version of the Anderson-Brown-Peterson theorem and briefly discuss classes of Calabi-Yau varieties in the $\mathrm{SL}$-cobordism ring.

math.AT↗