Loops on schemes and the algebraic fundamental group

TL;DR

This paper reinterprets the algebraic fundamental group of proper schemes using interval schemes, creating an algebraic analogue of continuous loops and homotopy, with results based on Macaulayfication and Lefschetz theorems.

Contribution

It introduces a novel topological perspective for algebraic fundamental groups by employing interval schemes and homotopy concepts.

Findings

Reinterpretation of algebraic fundamental group using interval schemes

Establishment of algebraic loops and homotopy via monodromy action

Results depend on Macaulayfication and Lefschetz type theorems

Abstract

In this note we give a re-interpretation of the algebraic fundamental group for proper schemes that is rather close to the original definition of the fundamental group for topological spaces. The idea is to replace the standard interval from topology by what we call interval schemes. This leads to an algebraic version of continuous loops, and the homotopy relation is defined in terms of the monodromy action. Our main results hinge on Macaulayfication for proper schemes and Lefschetz type results.