On the motivic description of truncated fundamental group rings

2024-03-06

Eduard Looijenga

A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable … A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable topology and denote by $I$ the augmentation ideal of its fundamental group ring. Then for every field F and positive integer n, the space of F-valued linear forms on $ I/I^{n+1}$ is naturally isomorphic to $H^n(X^n,X(n,*); F)$, where $X(n,*)$ is an explicitly defined subspace of $X^n$. We here construct a simple isomorphism between $I/I^{n+1}$ and $H_n(X^n,X(n,*); \mathbf{Z})$ and express the maps that define the Hopf algebra structure on the $I$-adic completion of the fundamental group ring of $(X,*)$ in these terms.

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Rational motivic path spaces and Kim’s relative unipotent section conjecture

2022-07-13

Ishai Dan‐Cohen , Tomer M. Schlank

We develop the foundations of commutative algebra objects in the category of motives, which we call “motivic dga’s.” Works of White and Cisinski and Déglise provide us with a suitable … We develop the foundations of commutative algebra objects in the category of motives, which we call “motivic dga’s.” Works of White and Cisinski and Déglise provide us with a suitable model structure. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga and provides us with a factorization of Kim’s relative unipotent section conjecture into several smaller conjectures with a homotopical flavor.

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On the proalgebraic fundamental group of topological spaces and amalgamated products of affine group schemes

2023-01-01

Christopher Deninger , Michael Wibmer

The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces … The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on $X$. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert-van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk-Scholze spaces and compare it to the motivic Galois group.

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Motivic Atiyah-Segal completion theorem

2020-01-01

Gonçalo Tabuada , Michel Van den Bergh

Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, … Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic K-theory, of Thomason's completion theorem in \'etale K-theory with coefficients, and also of Atiyah-Segal's completion theorem in topological K-theory. These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological spaces no longer needs to be compact and the equivariant topological K-theory groups no longer need to be finitely generated over the representation ring. As a second application, we obtain new completion theorems in l-adic \'etale K-theory, in (real) semi-topological K-theory and also in periodic cyclic homology. As a third application, we obtain a purely algebraic description of the different equivariant cohomology groups in the literature (motivic, l-adic, (real) morphic, Betti, de Rham, etc). Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy K-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of (real) semi-topological K-theory.

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Framed motivic $\Gamma$-spaces

2023-01-01

Grigory Garkusha , Ivan Panin , Paul Arne Østvær

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 $\Gamma$-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 $\Gamma$-spaces and Voevodsky's framed correspondences into the concept of framed motivic $\Gamma$-spaces; these are continuous or enriched functors of two variables that take values in framed motivic spaces. We craft proofs of our main results by imposing further axioms on framed motivic $\Gamma$-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, $\mathbb{A}^1$- and $\sigma$-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.

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Rational motivic path spaces and Kim's relative unipotent section conjecture

2017-01-01

Ishai Dan‐Cohen , Tomer M. Schlank

We initiate a study of path spaces in the nascent context of "motivic dga's", under development in doctoral work by Gabriella Guzman. This enables us to reconstruct the unipotent fundamental … We initiate a study of path spaces in the nascent context of "motivic dga's", under development in doctoral work by Gabriella Guzman. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga, and provides us with a factorization of Kim's relative unipotent section conjecture into several smaller conjectures with a homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that the path spaces of the punctured projective line over a number field are concentrated in degree zero with respect to Levine's t-structure for mixed Tate motives. This constitutes a step in the direction of Kim's conjecture.

A category of kernels for equivariant factorizations, II: Further implications

2014-02-14

Matthew R. Ballard , David Favero , Ludmil Katzarkov

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Integral Models for Spaces via the Higher Frobenius

2019-01-01

Allen Yuan

We give a fully faithful integral model for spaces in terms of $\mathbb{E}_{\infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius … We give a fully faithful integral model for spaces in terms of $\mathbb{E}_{\infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete $\mathbb{E}_{\infty}$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$ is the data of its Spanier-Whitehead dual as an $\mathbb{E}_{\infty}$-ring together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen's $Q$-construction acts on the $\infty$-category of $\mathbb{E}_{\infty}$-rings with "genuine equivariant multiplication," which we call global algebras. The second is a "pre-group-completed" variant of algebraic $K$-theory which we call partial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of $\mathbb{F}_p$ up to $p$-completion.

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A Pro-algebraic Fundamental Group for Topological Spaces

2023-03-01

Christopher Deninger

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Motivic homotopy theory with ramification filtrations

2025-04-02

Junnosuke Koizumi , Hiroyasu Miyazaki , Shuji Saito

The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme … The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme $S$, in which cohomology theories with ramification filtrations are representable. Every such cohomology theory enjoys basic properties such as the Nisnevich descent, the cube-invariance, the blow-up invariance, the smooth blow-up excision, the Gysin sequence, the projective bundle formula and the Thom isomorphism. In case $S$ is the spectrum of a perfect field, the cohomology of every reciprocity sheaf is upgraded to a cohomology theory with a ramification filtration represented in our categories. We also address relations of our theory with other non-$\mathbb{A}^1$-invariant motivic homotopy theories such as the logarithmic motivic homotopy theory of Binda, Park, and {\O}stv{\ae}r and the theory of motivic spectra of Annala-Iwasa.

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Rational enriched motivic spaces

2023-01-01

Peter Bonart

Rational enriched motivic spaces are introduced and studied in this thesis to provide new models for connective and very effective motivic spectra with rational coefficients.We first study homological algebra for … Rational enriched motivic spaces are introduced and studied in this thesis to provide new models for connective and very effective motivic spectra with rational coefficients.We first study homological algebra for Grothendieck categories of functors enriched in Nisnevich sheaves with specific transfers A. Following constructions of Voevodsky for triangulated categories of motives and framed motivic Γ-spaces, we introduce and study motivic structures on unbounded chain complexes of enriched functors yielding two new models of the triangulated category of big motives with A-tranfers DM A .We next define enriched motivic spaces as certain enriched functors of simplicial A-sheaves.We then use the properties of enriched motivic spaces and the above reconstruction results to recover SH(k) 0,Q and SH veff (k) Q . DeclarationsThis work has not previously

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Framed motivic $Γ$-spaces

2019-01-01

Grigory Garkusha , Ivan Panin , Paul Arne Østvær

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 $\Gamma$-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 $\Gamma$-spaces and framed correspondences into the concept of framed motivic $\Gamma$-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 $\Gamma$-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, ${\mathbb A}^{1}$- and $\sigma$-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.

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Motivic Steenrod problem away from the characteristic

2024-07-09

Toni Annala , Tobias Shin

In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in … In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in algebraic geometry: is every element on the Chow line of the motivic cohomology of $X$ the pushforward of a fundamental class along a projective derived-lci morphism? If $X$ is a smooth variety over a field of characteristic $p \geq 0$, then a positive answer to this question follows up to $p$-torsion from resolution of singularities by alterations. However, if $X$ is singular, then this is no longer necessarily so: we give examples of motivic cohomology classes of a singular scheme $X$ that are not $p$-torsion and are not expressible as such pushforwards. A consequence of our result is that the Chow ring of a singular variety cannot be expressed as a quotient of its algebraic cobordism ring, as suggested by the first-named-author in his thesis.

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A Proof of the Integral Identity via Braden's Theorem

2024-10-15

Florian Ivorra

The purpose of this paper is to provide a very short proof of a generalized categorified version, within the motivic stable homotopy category of Morel and Voevodsky, of the integral … The purpose of this paper is to provide a very short proof of a generalized categorified version, within the motivic stable homotopy category of Morel and Voevodsky, of the integral identity for virtual motives conjectured by Kontsevich and Soibelman. Our proof is an application of an important result in geometric representation theory due to Braden and known as the hyperbolic localization/restriction theorem. Though originally proved in the context of etale sheaves (or sheaves on the associated complex analytic space in the case of complex algebraic varieties) Braden's theorem turns out to hold also in the context of motivic sheaves, at least in the special case of vector bundles with a linear G_m-action.

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A pro-algebraic fundamental group for topological spaces

2020-01-01

Christopher Deninger

Consider a connected topological space $X$ with a point $x \in X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite dimensional … Consider a connected topological space $X$ with a point $x \in X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite dimensional (flat) vector bundles on $X$ and its Tannakian dual $π_K (X,x)$ with respect to the fibre functor in $x$. The maximal pro-étale quotient of $π_K (X,x)$ is the étale fundamental group of $X$ studied by Kucharczyk and Scholze. For well behaved topological spaces, $π_K (X,x)$ is the pro-algebraic completion of the ordinary fundamental group $π_1 (X,x)$. We obtain some structural results on $π_K (X,x)$ by studying (pseudo-)torsors attached to its quotients. This approach uses ideas of Nori in algebraic geometry and a result of Deligne on Tannakian categories. We also calculate $π_K (X,x)$ for some generalized solenoids.

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Dimensional homotopy t-structures in motivic homotopy theory

2017-03-02

Mikhail V. Bondarko , Frédéric Déglise

Motivic Complexes and Relative Cycles

2019-01-01

Denis-Charles Cisinski , Frédéric Déglise

Thomason's completion for K-theory and cyclic homology of quotient stacks

2025-02-13

Amalendu Krishna , Rajat Kanti Nath

We prove several completion theorems for equivariant K-theory and cyclic homology of schemes with group action over a field. One of these shows that for an algebraic space over a … We prove several completion theorems for equivariant K-theory and cyclic homology of schemes with group action over a field. One of these shows that for an algebraic space over a field acted upon by a linear algebraic group, the derived completion of equivariant K'-theory at the augmentation ideal of the representation ring of the group coincides with the ordinary K'-theory of the bar construction associated to the group action. This provides a solution to Thomason's completion problem. For action with finite stabilizers, we show that the equivariant K-theory and cyclic homology have non-equivariant descriptions even without passing to their completions. As an application, we describe all equivariant Hochschild and other homology groups for such actions.

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The categorical form of Fargues' conjecture for tori

2022-01-01

Konrad Zou

We prove the main conjecture of arXiv:2102.13459 for integral coefficients in the case of tori. Along the way we prove that the spectral action as constructed in that manuscript is … We prove the main conjecture of arXiv:2102.13459 for integral coefficients in the case of tori. Along the way we prove that the spectral action as constructed in that manuscript is compatible with the action of the excursion algebra and preserves the grading by $\pi_1(G)_Q$ on both sides. We additionally develop a (non-solidified) version of condensed group (co)homology and show that many constructions from classical group (co)homology extend to that case.

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Derived categories of symmetric products and moduli spaces of vector bundles on a curve

2023-01-01

Kyoung-Seog Lee , Han‐Bom Moon

We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the … We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.

On the motivic description of truncated fundamental group rings

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