相关论文: The Fundamental Group of an Algebraic Stack

We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. Then use this additional structure to analyze geometric/ topological properties of stacks.…

We define the algebraic fundamental group functor of a reductive group scheme over an arbitrary (non-empty) base scheme and prove that this functor is exact.

We prove that if a group scheme of multiplicative type acts on an algebraic stack with affine, finitely presented diagonal then the stack of fixed points is algebraic. For this, we extend two theorems of [SGA3.2] on functors of subgroups of…

We give the basic definitions of group actions on (algebraic) stacks, and prove the existence of fixed points and quotients as (algebraic) stacks.

In this note, we investigate how different fundamental groups of presentations of a fixed algebra $A$ can be. For finitely many finitely presented groups $G_i$, we construct an algebra $A$ such that all $G_i$ appear as fundamental groups of…

Passing from arithmetic schemes to algebraic schemes, in a similar manner we will have the computation of the \'etale fundamental group of an algebraic scheme and then will define and discuss the qc fundamental group of an algebraic scheme…

We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf…

The goal of this note is to spell out the (apparently well-known and intuitively clear) notion of abelian category over an algebraic stack. In the future we will discuss the (much less evident) notion, when instead of an abelian category…

We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded…

We show that for a stratum of projectivized abelian differentials with sufficiently many simple zeroes, the inclusion into the appropriate moduli space of pointed curves induces an injection at the level of orbifold fundamental group,…

In this paper, we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications.

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their \'etale coverings and their fundamental groups are also defined. These fundamental group approximates…

Given an algebraic stack, we compare its Nori fundamental group with that of its coarse moduli space. We also study conditions under which the stack can be uniformized by an algebraic space.

We extend the notion of algebraic stack to an arbitrary subcanonical site C. If the topology on C is local on the target and satisfies descent for morphisms, we show that algebraic stacks are precisely those which are weakly equivalent to…

We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which…

We define a category whose objects are finite etale coverings of an algebraic stack and prove that it is a Galois category and that it allows one to compute the fundamental group of the stack. We then prove a Van Kampen theorem for…

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in…

We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes,…

In this note I introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group $G$ as a sheaf on some geometric object (a stack). This point of…

We extend the definition of fundamental group scheme to non reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.