Related papers: On the S-fundamental group scheme II

We introduce the notion of a neutral representation of a finite group, or finite group scheme, $G$; a representation $V$ with the property that if a gerbe $\mathcal{G}$ over a field $k$ that is a form of the classifying stack $\mathcal{B}…

Let $k$ be an algebraically closed field. Chambert-Loir proved that the \'etale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain…

This paper is a summary of author's results on finite flat commutative group schemes. The properties of the generic fibre functor are discussed. A complete classification of finite local flat commutative group schemes over mixed…

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…

We show that there exist flat surface bundles with closed leaves having non-trivial normal bundles. This leads us to compute the Abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that…

We study the Fourier-Mukai functor D(Y) -> D(X) induced by the universal family on a fine moduli space Y for simple semihomogeneous vector bundles on an abelian variety X. The main result is that the Fourier-Mukai transform of a very…

Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete…

Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the category of smooth manifolds. We prove that a…

Let $G$ be a finite group and $H$ a core-free subgroup of $G$. We will show that if there exists a solvable, generating transversal of $H$ in $G$, then $G$ is a solvable group. Further, if $S$ is a generating transversal of $H$ in $G$ and…

In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that…

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.

Let $X$ be an abelian variety defined over an algebraically closed field $k$. We consider theta groups associated to \emph{simple semi-homogenous vector bundles of separable type} on $X$. We determine the structure and representation theory…

A decorated vector bundle is a vector bundle equipped with a reduction of structure group to a complex reductive subgroup $G \subseteq \mathbf{GL}(r,\mathbb{C})$. Examples include symplectic and special-orthogonal vector bundles, as well as…

To paraphrase, part I constructs a bundle of $A _{\infty}$ categories given the input of a Hamiltonian fibration over a smooth manifold. Here we show that this bundle is generally non-trivial by a sample computation. One principal…

We study the partially wrapped Fukaya category of a surface with boundary with an action of a group of order two. Inspired by skew-group algebras and categories, we define the notion of a skew-group $A_\infty$-category and let it play the…

Let $\pi:\CA\ra S$ be an abelian scheme over a scheme $S$ which is quasi-projective over an affine noetherian scheme and let $\CL$ be a symmetric, rigidified, relatively ample line bundle on $\CA$. We show that there is an isomorphism…

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm…

A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of…

Let $R$ be an associative ring with unit. Given an $R$-module $M$, we can associate the following covariant functor from the category of $R$-algebras to the category of abelian groups: $S\mapsto M\otimes_R S$. With the corresponding notion…

The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth…