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Andrei Druzhinin

Publications and source records attributed to Andrei Druzhinin.

14 recordsLinked to original sources
arXiv2025

Strict homotopy invariance via compactified homotopies and correspondences, and fibres of essentially smooth schemes over one-dimensional base schemes

We develop the technique of compactified correspondences and homotopies over one-dimensional base schemes, and illuminate the perfectness and the inverting of characteristic assumptions from the celebrating Voevodsky's strict homotopy invariance theorem and its framed correspondences generalisation over an arbitrary base field. The assumption in this crucial theorem for Voevodsky's motives theory %over a field was kept from the origins of the study, and came later into more modern theory of framed motives by Garkusha-Panin. Applying the technique, we obtain also analogs of Gersten and Nisnevich conjectures for Cousin complexes of generalised motivic cohomotopies over a field, and acyclicity of Cousin complexes on generic fibres of essentially smooth local schemes over one-dimensional base schemes.

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arXiv2024

Zariski-local framed $\mathbb{A}^1$-homotopy theory

For any (not necessarily perfect) field $k$ we obtain equivalences of $\infty$-categories \[\mathbf{H}^{\mathrm{fr},\mathrm{gp}}(k)\simeq \mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(k) \text{ and } \mathbf{DM}(k)\simeq\mathbf{DM}_{\mathrm{zar}}(k).\] We also construct an equivalence of $\infty$-categories \[ \mathbf{H}^{\mathrm{fr},\mathrm{gp}}(S) \simeq \mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(S) \] of group-like framed motivic spaces over a separated noetherian scheme $S$ of finite Krull dimension with respect to the Nisnevich topology at one side and the Zariski fibre topology $\mathrm{zf}$ generated by the Zariski one and the trivial fibre topology (introduced by Druzhinin, Kolderup and Østvær) on the other side. Over a field, the Zariski fibre topology equals the Zariski topology and the result follows from the previous one. To prove it in the case of a general base scheme, we prove a localisation theorem for $\mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(-)$ employing the ideas from the proof of the {\it affine localisation theorem} for the trivial fibre topology by the first author, Kolderup and Østvær.

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arXiv2024

Hypercomplete étale framed motives and comparison of stable homotopy groups of motivic spectra and étale realizations over a field

For any base field and integer $l$ invertible in $k$, we prove that $Ω^\infty_{\mathbb{G}_m}$ and $Ω^\infty_{\mathbb{P}^1}$ commute with hyper étale sheafification $L_{\acute{e}t}$ and Betti realization through infinite loop space theory in motivic homotopy theory. The central subject of this article is an $l$-complete hypercomplete étale analog of the framed motives theory developed by Garkusha and Panin. Using Bachman's hypercomplete étale \RigidityTheorem and the $\infty$-categorical approach of framed motivic spaces by Elmanto, Hoyois, Khan, Sosnilo, Yakerson, we prove the recognition principle and the framed motives formula for the composite functor \[Δ^\mathrm{op}\mathrm{Sm}_k\to \mathrm{Spt}^{\mathbb{G}_m^{-1}}_{\mathbb{A}^1,\acute{e}t}(\mathrm{Sm}_k)\xrightarrow{Ω^\infty_{\mathbb{G}_m}} \mathrm{Spt}_{\acute{e}t,\hat{n}}(\mathrm{Sm}_k).\] The first applications include the hypercomplete étale stable motivic connectivity theorem and an étale local isomorphism \[π^{\mathbb{A}^1,\mathrm{Nis}}_{i,j}(E)\simeqπ^{\mathbb{A}^1,\acute{e}t}_{i,j}(E)\] for any $l$-complete effective motivic spectra $E$, and $j\geq 0$. Furthermore, we obtain a new proof for Levine's comparison isomorphism over $\mathbb C$, $π_{i,0}^{\mathbb{A}^1,\mathrm{Nis}}(E)(\mathbb{C})\cong π_i(Be(E))$, and Zargar's generalization for algebraically closed fields, that applies to an arbitrary base field.

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arXiv2020

Cohomological correspondence categories

We prove that homotopy invariance and cancellation properties are satisfied by any linear category of correspondences that is defined, via Calmès and Fasel's construction, by an underlying cohomology theory. In particular, this includes any category of correspondences arising from the cohomology theory defined by an MSL-algebra.

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arXiv2018

Geometric models for fibrant resolutions of motivic suspension spectra

We construct geometric models for the $\mathbb P^1$-spectrum $M_{\mathbb P^1}(Y)$, which computes in Garkusha-Panin's theory of framed motives \cite{GP14} a positively motivically fibrant $Ω_{\mathbb P^1}$ replacement of $Σ_{\mathbb P^1}^\infty Y$ for a smooth scheme $Y\in \Sm_k$ over a perfect field $k$. Namely, we get the $T$-spectrum in the category of pairs of smooth ind-schemes that defines $\mathbb P^1$-spectrum of pointed sheaves termwise motivically equivalent to $M_{\mathbb P^1}(Y)$.

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arXiv2018

Surjectivity of the etale excision map for homotopy invariant framed presheaves

The category of framed correspondences Fr_*(k), framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [17]. Based on the notes [17] a new approach to the classical Morel--Voevodsky motivic stable homotopy theory was developed by G.Garkusha and I.Panin in [8]. The purpose of this paper is to prove Theorem 1.1 stating that if the ground field k is infinite, then the surjectivity of the etale excision property is true for any A1-invariant stable radditive framed presheaf of Abelian groups F. The injectivity of the etale excision was proved in [9]. The surjectivity of the etale excision was proved in [9] if the ground field is infinite of characteristic not 2. In this preprint the surjectivity of the etale excision is proved in the case of any infinite ground field. As explained in the introduction to [8] all the results of [9], [1], [10] and [8] are true now automatically without any restrictions on the characteristic of the ground field.

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arXiv2018

Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences

The cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences between smooth varieties over an infinite prefect field $k$, $char k \neq 2$, is proved, the isomorphism $$Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet,B^\bullet) \simeq Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet(1),B^\bullet(1)),$$ for $A^\bullet,B^\bullet\in \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ in the category of effective Grothendieck-Witt-motives constructed in \cite{AD_DMGWeff} is obtained (and similarly for Witt-motives). This implies that the canonical functor $Σ_{\mathbb G_m^{\wedge 1}}^\infty\colon \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)\to \mathbf{DM}^\mathrm{GW}(k)$ is fully faithful, where $\mathbf{DM}^\mathrm{GW}(k)$ is the category of non-effective GW-motives (defined by stabilization of $\mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ along $\mathbb G_m^{\wedge 1}$) and yields the main property of motives of smooth varieties in the category $\mathbf{DM}^\mathrm{GW}(k)$: $$ Hom_{\mathbf{DM}^\mathrm{GW}(k)}(M^{GW}(X), Σ_{\mathbb G_m^{\wedge 1}}^\infty\mathcal F[i]) \simeq H^i_{Nis}(X,\mathcal F) ,$$ for any smooth variety $X$ and homotopy invariant sheave with GW-transfers $\mathcal F$ (and similarly for $\mathbf{DM}^\mathrm{W}(k)$).

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arXiv2018

Rigidity for linear framed presheaves and generalized motivic cohomology theories

A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a $ϕ$-torsion spectrum with $ϕ\in\mathrm{GW}(k)$ of rank coprime to the (exponential) characteristic of the base field $k$. It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by $n$-torsion spectra as well as to the ones representable by $η$-periodic spectra and spectra related to Witt groups.

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arXiv2018

Effective Grothendieck-Witt motives of smooth varieties

The category of effective Grothendieck-Witt-motives $\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ (and Witt-motives $\mathbf{DM}^W_{\mathrm{eff},-}(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $char\,k\neq 2$, is defined. The functor $M^{GW}_{eff}\colon Sm_k\to \mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$ Hom_{\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)}(M^{GW}_{eff}(X), F[i]) \simeq H^i_{nis}(X, F) $$ naturally in $X$ and $F$.

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arXiv2018

Strictly homotopy invariance of Nisnevich sheaves with GW-transfers

The strictly homotopy invariance of the associated Nisnevish sheave $\widetilde{\mathcal F}_{Nis}$ of a homotopy invariant presheave $\mathcal F$ with GW-transfers (or Witt-transfers) on the category of smooth varieties over a prefect field $k$, $char\,k \neq 2$, is proved, i.e. the isomorphism $$H^i_{Nis}(\mathbb A^1\times X,\widetilde{\mathcal F}_{Nis})\simeq H^i_{Nis}(X,\widetilde{\mathcal F}_{Nis})$$ for any $X\in Sm_k$ is obtained. This theorem is necessary for the construction of the triangulated category of GW-motives $\mathbf{DM}^{GW}(k)$ and Witt-motives $\mathbf{DM}^W(k)$ by the Voevodsky-Suslin method originally used for the construction of the category of motives $\mathbf{DM}(k)$. In particular, the result of the article gives the direct prove of the strictly homotopy invariance of the Nisnevich sheaves associated to hermitian K-theory and Witt-groups (without using of the representability of these cohomology theories in the motivic homotopy category $\mathbf H_{\mathbb A^1}(k)$ proved by Hornbostle [Horn_ReprKOWitt]); and on other side the strictly homotopy invariance theorem proved here and the representability criteria proved in [Horn_ReprKOWitt] implies that cohomologies $H^i_{nis}(-,\widetilde{\mathcal F}_{nis})$ of the associated sheaf of a homotopy invariant presheave with GW-(Witt-)transfers $\mathcal F$ are representable in $\mathbf H_{\mathbb A^1}(k)$.

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arXiv2017

Rigidity theorem for presheaves with Witt-transfers

The rigidity theorem for homotopy invariant presheaves with Witt-transfers on the category of smooth affine varieties over a field $k$ with characteristic not equal to 2 is proved. Namely for such a presheaf $\mathcal F$ the isomorphism $\mathcal F(U)\simeq \mathcal F(x)$ where $U$ is henseliation of a variety at smooth closed point with separable residue field (over $k$) is proved. The rigidity for presheaves $W^i(X\times -)$ where $X$ is smooth variety and $W^i(-)$ are derived Witt-groups ($i\in \mathbb Z/4\mathbb Z$) follows as corollary.

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arXiv2016

Triangulated category of effective $Witt$-motives $DWM^-_{eff}(k)$

The category of effective $Witt$-motives $DWM^-(k)$ with functor $WM\colon Sm_k\to DWM^-(k)$ defining motives of smooth affine varieties for perfect field $k$, $char k\neq 2$ is constructed. In the construction Voevodsky-Suslin method is applyed to a category of $Witt$-correspondence between affine smooth varieties $WCor_k$ that morphisms are defined by class in $Witt$-group of quadratic space $(P,q_P)$ with $P$ being $k[X\times Y]$-module finitely generated projective over $k[X]$ and $q_P\colon P\to Hom_{k[X]}(P,k[X])$ being $k[X\times Y]$-liner isomorphism. And the natural isomorphism $$Hom_{DWM^-_{eff}(k)}(WM(X),\mathcal F[i]) \simeq H^i_{Nis}(X,\mathcal F) $$ for any smooth affine $X$ and homotopy invariant Nisnevich sheave $\mathcal F$ with $Witt$-transfers (that is presheave on the category $WCor_k$ such that its restriction on the category $Sm_k$ is a sheave) is proved.

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