• Corpus ID: 237605086

Algebraic loop groups

  title={Algebraic loop groups},
  author={Kay Rulling and Stefan Schroer},
In this note we introduce algebraic loops, starting from the notion of an interval scheme, and define the algebraic loop group of a connected scheme with a geometric base point x0 as the set of homotopy classes of algebraic loops based at x0. The group structure is induced by concatenating algebraic loops. The main result is an isomorphism of this algebraic loop group to Grothendieck’s algebraic fundamental group for proper connected schemes over a field. 


Universal covering spaces and fundamental groups in algebraic geometry as schemes
In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a
A General Seifert–Van Kampen Theorem for Algebraic Fundamental Groups
A Seifert–Van Kampen theorem describes the fundamental group of a space in terms of the fundamental groups of the constituents of a covering and the configuration of connected components of the
Hypersurfaces in projective schemes and a moving lemma
Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We
Ample subvarieties of algebraic varieties
Ample divisors.- Affine open subsets.- Generalization to higher codimensions.- The grothendieck-lefschetz theorems.- Formal-rational functions along a subvariety.- Algebraic geometry and analytic
Galois theory for schemes
Introduction 1–5 Coverings of topological spaces. The fundamental group. Finité etale coverings of a scheme. An example. Contents of the sections. Prerequisites and conventions. 1. Statement of the
Survey on some aspects of Lefschetz theorems in algebraic geometry
We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne’s program in Weil II.
On Macaulayfication of Noetherian schemes
The Macaulayfication of a Noetherian scheme X is a birational proper morphism from a Cohen-Macaulay scheme to X. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its
Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2)
New updated edition by Yves Laszlo of the book ``Cohomologie locale des faisceaux coh\'erents et th\'eor\`emes de Lefschetz locaux et globaux (SGA 2)'', Advanced Studies in Pure Mathematics 2,
Revêtements étales et groupe fondamental (SGA 1)
Le texte pr\'esente les fondements d'une th\'eorie du groupe fondamental en G\'eom\'etrie Alg\'ebrique, dans le point de vue ``kroneckerien'' permettant de traiter sur le m\^eme pied le cas d'une
Conducteur, descente et pincement
Une somme amalgamee de schemas est decrite localement par un produit fibre d'anneaux. Ce texte donne un resultat global d'existence ( 5.4) de schemas definis comme certaines sommes amalgamees et un