arXiv subjects

Grigory Garkusha

Publications and source records attributed to Grigory Garkusha.

At least 19 recordsLinked to original sources

arXiv2026

Computing framed motives

We develop methods for computing framed motives associated with motivic Thom spectra. Our main tool is a motivic Atiyah--Hirzebruch spectral sequence relating framed motives to framed motivic cohomology. As a consequence, after inverting a finite set of primes, the bigraded homotopy sheaves of motivic Thom spectra are computed in terms of framed motivic cohomology. We further analyze the symmetric-group actions inherent in framed correspondences and introduce a theory of torsion framed motivic cohomology that yields new computational descriptions of framed motivic cohomology groups. These constructions lead to a category of permutation-free framed correspondences from which we reconstruct rational stable motivic homotopy theory.

math.AG↗

arXiv2025

K-theory and matrix transfers

We introduce and study matrix transfers to achieve elementary models for bivariant $K$-theory. They share lots of common properties with Voevodsky's framed correspondences and lead to symmetric matrix motives of algebraic varieties introduced in this paper. Symmetric matrix motives recover $K$-motives and fit in a closed symmetric monoidal triangulated category of symmetric matrix motives constructed in this paper by using methods of enriched motivic homotopy theory.

math.KT↗

arXiv2025

Symmetric quasi-coherent sheaves

Using methods of stable homotopy theory, the category of symmetric quasi-coherent sheaves associated with non-commutative graded algebras with extra symmetries is introduced and studied in this paper. It is shown to be a closed symmetric monoidal Grothendieck category with invertible generators. It is proven that the category of quasi-coherent sheaves on a projective scheme is recovered out of symmetric quasi-coherent sheaves. As an application, symmetric projective schemes associated to such algebras are introduced and studied. It is shown that classical projective schemes are recovered from symmetric ones.

math.AG↗

arXiv2024

The Ziegler spectrum for enriched ringoids and schemes

The Ziegler spectrum for categories enriched in closed symmetric monoidal Grothendieck categories is defined and studied in this paper. It recovers the classical Ziegler spectrum of a ring. As an application, the Ziegler spectrum as well as the category of generalised quasi-coherent sheaves of a reasonable scheme is introduced and studied. It is shown that there is a closed embedding of the injective spectrum of a coherent scheme endowed with the tensor fl-topology (respectively of a noetherian scheme endowed with the dual Zariski topology) into its Ziegler spectrum. It is also shown that quasi-coherent sheaves and generalised quasi-coherent sheaves are related to each other by a recollement.

math.AG↗

arXiv2023

Correspondences and stable homotopy theory

A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra $SH$ is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory $SH(k)$ from spectral modules over associated spectral categories.

math.AT↗

arXiv2023

Fibrant resolutions for motivic Thom spectra

Using the theory of framed correspondences developed by Voevodsky [24] and the machinery of framed motives introduced and developed in [6], various explicit fibrant resolutions for a motivic Thom spectrum $E$ are constructed in this paper. It is shown that the bispectrum $$M_E^{\mathbb G}(X)=(M_{E}(X),M_{E}(X)(1),M_{E}(X)(2),\ldots),$$ each term of which is a twisted $E$-framed motive of $X$, introduced in the paper, represents $X_+\wedge E$ in the category of bispectra. As a topological application, it is proved that the $E$-framed motive with finite coefficients $M_E(pt)(pt)/N$, $N>0$, of the point $pt=Spec (k)$ evaluated at $pt$ is a quasi-fibrant model of the topological $S^2$-spectrum $Re^ε(E)/N$ whenever the base field $k$ is algebraically closed of characteristic zero with an embedding $ε:k\hookrightarrow\mathbb C$. Furthermore, the algebraic cobordism spectrum $MGL$ is computed in terms of $Ω$-correspondences in the sense of [15]. It is also proved that $MGL$ is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasi-projective varieties.

math.AG↗

arXiv2022

Semilocal Milnor K-theory

In this paper, semilocal Milnor $K$-theory of fields is introduced and studied. A strongly convergent spectral sequence relating semilocal Milnor $K$-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups $H^p_{Zar}(k,\mathbb Z(2))$, $p\leq 1$, are computed as semilocal Milnor $K$-theory groups $\widehat{K}^M_{2,3-p}(k)$. The following applications are given: (i) several criteria for the Beilinson-Soulé Vanishing Conjecture; (ii) computation of $K_4$ of a field; (iii) the Beilinson conjecture for rational $K$-theory of fields of prime characteristic is shown to be equivalent to vanishing of rational semilocal Milnor $K$-theory.

math.KT↗

arXiv2022

The triangulated categories of framed bispectra and framed motives

An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{nis}^{fr}(k)$ and $SH_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{eff}(k)$ respectively. We also recover $SH(k)$ and $SH^{eff}(k)$ as the triangulated category of framed motivic spectral functors $SH_{S^1}^{fr}[\mathcal Fr_0(k)]$ and the triangulated category of framed motives $\mathcal {SH}^{fr}(k)$ respectively constructed in the paper.

math.KT↗

arXiv2022

Framed motivic $Γ$-spaces

We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $Γ$-spaces and framed correspondences into the concept of framed motivic $Γ$-spaces; these are continuous or enriched functors of two variables that take values in motivic spaces and are equipped with a framing. We craft proofs of our main results by imposing further axioms on framed motivic $Γ$-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, ${\mathbb A}^{1}$- and $σ$-invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the ${\mathbb A}^{1}$-homotopy theory of algebraic varieties.

math.AG↗

arXiv2022

Group schemes and motivic spectra

By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of classical $S^1$-spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra are introduced and studied. It is shown that the stable homotopy theory of motivic spectra is recovered from each of these types of spectra. An application is given for the localization functor $C_*\mathcal Fr:SH_{nis}(k)\to SH_{nis}(k)$ in the sense of [15] that converts the Morel-Voevodsky stable motivic homotopy theory $SH(k)$ into the equivalent local theory of framed bispectra [15].

math.AG↗

arXiv2021

Cancellation theorem for framed motives of algebraic varieties

The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $$M_{fr}(X)(n)\to\underline{\textrm{Hom}}(\mathbb G,M_{fr}(X)(n+1)),\quad n\geq 0,$$ is a schemewise stable equivalence, where $M_{fr}(X)(n)$ is the $n$th twisted framed motive of $X$. This result is also necessary for the proof of the main theorem of [GP1] computing fibrant resolutions of suspension $\mathbb P^1$-spectra $Σ^\infty_{\mathbb P^1}X_+$ with $X$ a smooth algebraic variety. The Cancellation Theorem for framed motives is reduced to the Cancellation Theorem for linear framed motives stating that the natural map of complexes of abelian groups \[ \mathbb ZF(Δ^\bullet \times X,Y) \to \mathbb ZF((Δ^\bullet \times X)\wedge (\mathbb G_m,1),Y\wedge (\mathbb G_m,1)),\quad X,Y\in Sm/k, \] is a quasi-isomorphism, where $\mathbb ZF(X,Y)$ is the group of stable linear framed correspondences in the sense of [GP1].

math.KT↗

arXiv2021

Framed motives of relative motivic spheres

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra $$M_{fr}(X\times (\mathbb{A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\times T^n).$$ Moreover, it is proven that the sequence of $S^1$-spectra $$M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times\mathbb A^1) \to M_{fr}(X \times T^{n+1})$$ is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of [GP1]. This computation is crucial for the whole machinery of framed motives [GP1].

math.KT↗

arXiv2019

Recollements for derived categories of enriched functors and triangulated categories of motives

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit into recollements of derived categories of associated Grothendieck categories of Nisnevich sheaves with specific transfers.

math.KT↗

arXiv2019

Reconstructing rational stable motivic homotopy theory

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy-Levine-Panin, a theorem of Cisinski-Deglise and a version of the Roendigs-Ostvaer theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor-Witt correspondences in the sense of Calmes-Fasel.

math.AT↗

arXiv2018

Comparing motives of smooth algebraic varieties

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad q\geq 0,$$ are quasi-isomorphisms, it is proved that for every $k$-smooth algebraic variety $X$ the morphisms of twisted motives of $X$ with $\mathbb Z[1/e]$-coefficients $$M_{\mathcal A}(X)(q)\otimes\mathbb Z[1/e]\to M_{\mathcal B}(X)(q)\otimes\mathbb Z[1/e]$$ are quasi-isomorphisms. Furthermore, it is shown that the induced functors between triangulated categories of motives $$DM_{\mathcal A}^{eff}(k)[1/e]\to DM_{\mathcal B}^{eff}(k)[1/e],\quad DM_{\mathcal A}(k)[1/e]\to DM_{\mathcal B}(k)[1/e]$$ are equivalences. As an application, the Cor-, $K_0^\oplus$-, $K_0$- and $\mathbb K_0$-motives of smooth algebraic varieties with $\mathbb Z[1/e]$-coefficients are locally quasi-isomorphic to each other. Moreover, their triangulated categories of motives with $\mathbb Z[1/e]$-coefficients are shown to be equivalent. Another application is given for the bivariant motivic spectral sequence.

math.AG↗

arXiv2018

Derived categories for Grothendieck categories of enriched functors

The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.

math.CT↗

arXiv2018

Framed motives of algebraic varieties (after V. Voevodsky)

Using the theory of framed correspondences developed by Voevodsky, we introduce and study framed motives of algebraic varieties. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension $\mathbb P^1$-spectrum of any smooth scheme $X\in Sm/k$. Moreover, it is shown that the bispectrum $$(M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots),$$ each term of which is a twisted framed motive of $X$, has motivic homotopy type of the suspension bispectrum of $X$. Furthermore, an explicit computation of infinite $\mathbb P^1$-loop motivic spaces is given in terms of spaces with framed correspondences. We also introduce big framed motives of bispectra and show that they convert the classical Morel--Voevodsky motivic stable homotopy theory into an equivalent local theory of framed bispectra. As a topological application, it is proved that the framed motive $M_{fr}(pt)(pt)$ of the point $pt=Spec(k)$ evaluated at $pt$ is a quasi-fibrant model of the classical sphere spectrum whenever the base field $k$ is algebraically closed of characteristic zero.

math.KT↗

arXiv2018

Homotopy invariant presheaves with framed transfers

The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$, the associated Nisnevich sheaf $\mathcal F_{nis}$ is $\mathbb A^1$-invariant whenever the base field $k$ is infinite of characteristic different from 2. Moreover, if the base field $k$ is infinite perfect of characteristic different from 2, then every $\mathbb A^1$-invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly $\mathbb A^1$-invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$ is a presheaf of $\mathbb Z[1/2]$-modules. This result and the paper are inspired by Voevodsky's paper [13].

math.AG↗