arXiv subjects

Alexander Neshitov

Publications and source records attributed to Alexander Neshitov.

10 recordsLinked to original sources

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.

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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The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2].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].

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arXiv2019

Relative equivariant motives and modules

We introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization techniques of Kostant-Kumar in the context of generalized oriented cohomology as well as the Rost nilpotence principle for algebraic cobordism and its generic version. As an application, we obtain new proofs and examples of indecomposable Chow motives of versal flag varieties.

math.AG↗

arXiv2018

Framed and MW-transfers for homotopy modules

In the paper we use the theory of framed correpondences to construct Milnor-Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $\mathbb{A}^1$-homotopy sheaves of smooth varieties with the zeroth homology of corresponding MW-motivic complexes and prove that the hearts of homotopy $t$-structures on the stable $\mathbb{A}^1$-derived category and the category of Milnor-Witt motives are equivalent.

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arXiv2016

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Let G be a split semisimple linear algebraic group over a field k0. Let E be a G-torsor over a field extension k of k0. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k. Following the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the h-motivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecke-type algebra H which depends on the root datum of G, on the torsor E and on the formal group law of the theory h. In particular, taking h to be the Chow groups with finite coefficients Fp and E to be a generic G-torsor we prove that all indecomposable submodules of an affine nil-Hecke algebra H of G with coefficients in Fp are isomorphic to each other and correspond to the (non-graded) generalized Rost-Voevodsky motive for (G,p).

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arXiv2015

Framed correspondences and the Milnor-Witt K-theory

The article is to construct a graded ring isomorphism between $H_0(ZF(Δ^{\bullet}_k,\mathbb{G}_m^{\wedge *}))$ and the Milnor-Witt K-theory ring $K^{MW}_{*\geqslant 0}(k)$, where $k$ is a field of characteristic zero and $ZF_*(k)$ is the category of linear framed correspondences of algebraic k-varieties, introduced by Garkusha and Panin. As it was shown by Garkusha and Panin, this partially recovers the computation of the motivic cohomotopy groups $π^{n,n}(Σ^{\infty}_{S^1}Σ^{\infty}_{\mathbb{G}_m}S^0)(k)$, which is originally due to Morel.

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arXiv2014

Invariants of degree 3 and torsion in the Chow group of a versal flag

We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.

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arXiv2013

Motives and oriented cohomology of a linear algebraic group

For a cellular variety $X$ over a field $k$ of characteristic 0 and an algebraic oriented cohomology theory $\hh$ of Levine-Morel we construct a filtration on the cohomology ring $\hh(X)$ such that the associated graded ring is isomorphic to the Chow ring of $X$. Taking $X$ to be the variety of Borel subgroups of a split semisimple linear algebraic group $G$ over $k$ we apply this filtration to relate the oriented cohomology of $G$ to its Chow ring. As an immediate application we compute the algebraic cobordism ring of a group of type $G_2$, of groups $SO_n$ and $Spin_m$ for $n=3,4$ and $m=3,4,5,6$ and $PGL_k$ for $k\geqslant 2$. Using this filtration we also establish the following comparison result between Chow motives and $\hh$-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically split variety $Y$ gives rise to a $\hh$-motivic decomposition of $Y$ with the same generating function. Moreover, under some conditions on the coefficient ring of $\hh$ the obtained $\hh$-motivic decomposition will be irreducible. We also prove that if Chow motives of two twisted forms of $Y$ coincide, then their $\hh$-motives coincide as well.

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arXiv2012

Rigidity theorem for presheaves with Ω-transfers

In 1983 A. Suslin proved the Quillen-Lichtenbaum conjecture about algebraic K-theory of algebraically closed fields. The proof was based on a theorem called the Suslin rigidity theorem. In the present paper we prove the rigidity theorem for homotopy invariant presheaves with Ω-transfers, introduced by I. Panin. This type of presheaves includes the K-functor and algebraic cobordism of M. Levine and F. Morel. Keywords: rigidity theorem, presheaves with transfers, cohomology theories. MSC2000: 14F43

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arXiv2011

Algebraic analogue of Atiyah's theorem

In topology there is a theorem of Atiyah, concerning K-theory of classifying space of connected compact Lie group. We consider an algebraic analogue of this theorem. We prove that for a split reductive algebraic group G over a field there is an isomorphism between K-theory of etale classifying space of group G and a completion of the G-equivariant K-theory of the base field.

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