Equivariant connective 𝐾-theory

2021 ◽  
Vol 31 (1) ◽  
pp. 181-204
Author(s):  
Nikita Karpenko ◽  
Alexander Merkurjev
Keyword(s):  
Spectral Sequence ◽  
Finite Type ◽  
Group Scheme ◽  
Affine Group ◽  
Homotopy Invariance ◽  
Content Type ◽  
Basic Properties ◽  
Equivariant Version

For separated schemes of finite type over a field with an action of an affine group scheme of finite type, we construct the bi-graded equivariant connective K K -theory mapping to the equivariant K K -homology of Guillot and the equivariant algebraic K K -theory of Thomason. It has all the standard basic properties as the homotopy invariance and localization. We also get the equivariant version of the Brown-Gersten-Quillen spectral sequence and study its convergence.

scholarly journals On Path Homology of Vertex Colored (Di)Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 965
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Homology Theory ◽  
Homotopy Invariance ◽  
Homology Groups ◽  
Basic Properties

In this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from the notion of the path complex. Any graph naturally gives rise to a path complex in which for a given set of vertices, paths go along the edges of the graph. We define path complexes of vertex colored (di)graphs using the natural restrictions that are given by coloring. Thus, we obtain a new collection of colored-path homology theories. We introduce the notion of colored homotopy and prove functoriality as well as homotopy invariance of homology groups. For any colored digraph, we construct the spectral sequence of colored-path homology groups which gives the effective method of computations in the general case since any (di)graph can be equipped with various colorings. We provide a lot of examples to illustrate our results as well as methods of computations. We introduce the notion of homotopy and prove functoriality and homotopy invariance of introduced vertexed colored-path homology groups. For any colored digraph, we construct the spectral sequence of path homology groups which gives the effective method of computations in the constructed theory. We provide a lot of examples to illustrate obtained results as well as methods of computations.

scholarly journals Uniformizing the Moduli Stacks of Global G-Shtukas

2019 ◽  
Author(s):  
E Arasteh Rad ◽  
Urs Hartl
Keyword(s):  
Finite Type ◽  
Moduli Spaces ◽  
Function Field ◽  
Group Scheme ◽  
Base Change ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Moduli Stacks ◽  
Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

scholarly journals Tannakization in derived algebraic geometry

2014 ◽  
Vol 14 (3) ◽  
pp. 642-700 ◽  
Author(s):  
Isamu Iwanari
Keyword(s):  
Algebraic Geometry ◽  
Galois Group ◽  
Monoidal Category ◽  
Group Scheme ◽  
Affine Group ◽  
Stable Category ◽  
Group Schemes ◽  
Symmetric Monoidal Category ◽  
Derived Algebraic Geometry ◽  
Mixed Motives

AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

The Trace Formula for BunG(X)

2019 ◽  
pp. 239-308
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Trace Formula ◽  
Algebraic Curve ◽  
Group Scheme ◽  
Affine Group ◽  
Generic Fiber ◽  
Prove Theorem ◽  
Lefschetz Trace Formula ◽  
Moduli Stack ◽  
Smooth Affine

This chapter aims to prove Theorem 1.4.4.1, which is formulated as follows: Theorem 5.0.0.3, let X be an algebraic curve over F q and let G be a smooth affine group scheme over X. Suppose that the fibers of G are connected and that the generic fiber of G is semisimple. Then the moduli stack BunG(X) satisfies the Grothendieck–Lefschetz trace formula. However, Theorem 5.0.0.3 cannot be deduced directly from the Grothendieck–Lefschetz trace formula for global quotient stacks because the moduli stack BunG(X) is usually not quasi-compact. The strategy instead will be to decompose BunG (X) into locally closed substacks BunG(X)[P,ν‎] which are more directly amenable to analysis.

scholarly journals Sur l'existence du sch\'ema en groupes fondametal

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Marco Antei ◽  
Michel Emsalem ◽  
Carlo Gasbarri
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Group Scheme ◽  
Final Version ◽  
Finite Fundamental Group ◽  
Galois Covers ◽  
The One

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group. Comment: in French. Final version (finally!)

scholarly journals Smooth affine group schemes over the dual numbers

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Matthieu ROMAGNY ◽  
Dajano Tossici
Keyword(s):  
Commutative Ring ◽  
Group Algebra ◽  
Group Scheme ◽  
Affine Group ◽  
Unipotent Group ◽  
Dual Numbers ◽  
Group Schemes ◽  
Weil Restriction ◽  
International Audience ◽  
Smooth Affine

International audience We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \to \text{Lie}(G, I) \to E \to G \to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$. Nous construisons une équivalence entre la catégorie des schémas en groupes affines et lisses sur l'anneau des nombres duaux généralisés k[I], et la catégorie des extensions de la forme 1 → Lie(G, I) → E → G → 1 où G est un schéma en groupes affine, lisse sur k. Ici k est un anneau commutatif arbitraire et k[I] = k ⊕ I avec I 2 = 0. L'équivalence est donnée par la restriction de Weil, et nous construisons un foncteur quasi-inverse explicite que nous appelons extension de Weil. Ces foncteurs sont compatibles avec les structures exactes et avec les structures de champs en O k-modules des deux catégories. Nos constructions s'appuient sur le schéma en algèbres de groupe d'un schéma en groupes affines, que nous introduisons et dont nous donnons les propriétés principales. En application, nous donnons une classification de Dieudonné pour les schémas en groupes commutatifs, lisses, unipotents sur k[I] lorsque k est un corps parfait.

scholarly journals Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rafik Medjati ◽  
Hanifi Zoubir ◽  
Brahim Medjahdi
Keyword(s):  
Heisenberg Group ◽  
Finite Type ◽  
Fundamental Form ◽  
Natural Extension ◽  
Laplacian Operator ◽  
Translation Surfaces ◽  
Content Type ◽  
Flat Metric ◽  
First Fundamental Form

PurposeIn the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ1(x)*γ2(y), where γ1 and γ2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.Design/methodology/approachIn this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔri = λiri. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.FindingsClassification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.Originality/valueThe subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

Universal smooth k-tame central extension

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Finite Type ◽  
Central Extension ◽  
Group Scheme ◽  
Reductive Groups ◽  
Central Extensions ◽  
Weil Restriction ◽  
Group A ◽  
Subgroup Scheme ◽  
Minimal Type ◽  
Simply Connected

This chapter describes the construction of canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. The chapter establishes good properties of the universal smooth k-tame central extension, noting that the property “locally of minimal type” is inherited by pseudo-reductive central quotients of pseudo-reductive groups. Although inseparable Weil restriction does not generally preserve perfectness, the chapter shows that the formation of the universal smooth k-tame central extension interacts with derived groups of Weil restrictions.

scholarly journals Coding of substitution dynamical systems as shifts of finite type

2014 ◽  
Vol 36 (3) ◽  
pp. 944-972 ◽  
Author(s):  
PAUL SURER
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Periodic Points ◽  
Basic Properties ◽  
Substitution Dynamical Systems ◽  
Shift Map ◽  
Special Case

We develop a theory that allows us to code dynamical systems induced by primitive substitutions continuously as shifts of finite type in many different ways. The well-known prefix–suffix coding turns out to correspond to one special case. We precisely analyse the basic properties of these codings (injectivity, coding of the periodic points, properties of the presentation graph, interaction with the shift map). A lot of examples illustrate the theory and show that, depending on the particular coding, several amazing effects may occur. The results give new insights into the theory of substitution dynamical systems and might serve as a powerful tool for further researches.

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