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A. Druzhinin

Publications and source records attributed to A. Druzhinin.

5 recordsLinked to original sources
arXiv2026

The trivial fiber topology and framed motives over the integers

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.

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arXiv2024

Cousin complexes in motivic homotopy theory

We investigate Cousin (bi-)complexes in the setting of motives. Over essentially smooth local schemes, the columns of the Cousin bicomplex with coefficients in any stable motivic homotopy type are shown to be acyclic. On the other hand, we also construct a family of non-acyclic Cousin complexes over any positive dimensional base scheme. Our method of proof employs the notion of extended compactified framed correspondences. Three major motivations for this study are to further our understanding of strict homotopy invariance, motivic infinite loop spaces, and connectivity in stable motivic homotopy theory. As applications of our main results on motivic Cousin complexes, we generalize several fundamental results in these topics to finite dimensional base schemes.

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arXiv2020

Rigidity for smooth affine pairs over a field

Let $Z\to X$ be a closed immersion of smooth affine schemes over an arbitrary field $k$, and $X^h_Z$ denote the henselization of $X$ along $Z$. For each presheaf $E\colon \mathbf{SH}(k)\to \mathrm{Ab}^\mathrm{op}$ on the stable motivic homotopy category over $k$ and the induced continuous presheaf $E\colon \mathrm{EssSm}_k\to \mathrm{Ab}^\mathrm{op}$ on the category of essentially smooth schemes there is a homomorphism \[E(X^h_Z)\to E(Z).\] We prove that this is an isomorphism for any $l_\varepsilon$-torsion presheaf $E$, for $l\in \mathbb Z$, $(l,\mathrm{chark}\,k)=1$, and $l_\varepsilon=\sum_{i=1}^n \langle (-1)^i \rangle$. More generally, the isomorphism holds for any homotopy invariant $l_\varepsilon$-torsion linear $σ$-stable framed additive presheaf $F$ over $k$. The case of $l$-torsion presheaves follows as well. The result generalises known Gabber's rigidity theorems for local henselian schemes to the case of smooth affine henselian pairs. The above isomorphism is proven by constructing of (stable) $\mathbb{A}^1$-homotopies of motivic spaces via algebro-geometric techniques. To achieve this in our setting we replace often used Quillen's trick by an alternative construction that provides required smooth relative curves over smooth affine schemes for an arbitrary base field.

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arXiv2020

Stable connectivity over a base

Morel's stable connectivity theorems state that for any connective $S^1$-spectrum $F$ of motivic spaces (Nisnevich simplicial sheaves) over an arbitrary field, the spectrum $L_{\mathbb A^1}(F)$ is connective, and the same property for $\mathbb P^1$-spectra of motivic spaces. Here $L_{\mathbb A^1}$ denotes the $\mathbb A^1$-localisation in the category of motivic spectra over a field $k$. Originally the same property was conjectured for the case of motivic $S^1$-spectra over a base scheme $S$.In view of Ayoub's conterexamples the modified version of conjecture states that $L_{\mathbb A^1}(F)$ is $(-d)$-connective for any connective $F$, where $d=\mathrm{dim} S$ is the Krull dimension. The conjecture is proven under the infiniteness assumption on the residue fields for the cases of Dedekind schemes by J.~Schmidt and F.~Strunk and noetherian domains of arbitrary dimension by N.~Deshmukh, A.~Hogadi, G.~Kulkarni and S.~Yadavand. In the article we prove the result or general base with out the assumption on the residue fields. So by the result for any smooth scheme $X$ over a base scheme $S$ of Krull dimension $d$ the Nisnevich sheaves of $S^1$-stable motivic homotopy groups $π_i^{S^1}(X)$ and $\mathbb P^1$-stable motivic homotopy groups $π_{i+j,j}^{\mathbb P^1}(X)$ vanishes for all $i<-d$.

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arXiv2018

The homomorphism of presheaves ${\mathrm{K}}^\mathrm{MW}_*\to π^{*,*}_s$ over a base

We construct the homomorphism of presheaves ${\mathrm{K}}^\mathrm{MW}_* \to π^{*,*}$ over an arbitrary base scheme $S$, where $\mathrm{K}^\mathrm{MW}$ is the (naive) Milnor-Witt K-theory presheave. Also we discuss some partly alternative proof (or proofs) of the isomorphism of sheaves $\unKMW_n\simeq \underlineπ^{n,n}_s$, $n\in \mathbb Z$, over a filed $k$ originally proved in \cite{M02} and \cite{M-A1Top}.

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