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Loops on schemes and the algebraic fundamental group

Kay Rülling, Stefan Schröer

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.

Loops on schemes and the algebraic fundamental group

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.

Paper Structure

This paper contains 7 sections, 17 theorems, 40 equations.

Table of Contents

  1. Monodromy
  2. Galois categories
  3. Interval schemes
  4. The algebraic loop group
  5. A Lefschetz type theorem
  6. The non-proper case
  7. Proof of the main theorem

Key Result

Theorem 1

Let $X$ be a connected scheme that is separated and of finite type over a ground field $k$, endowed with a geometric point $x_0:\operatorname{Spec} (k^{\operatorname{sep}})\to X$. Then the injection monodromy transformation has dense image. It is actually bijective, provided that $X$ is proper.

Theorems & Definitions (32)

  • Theorem
  • Proposition 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 3.1
  • Proposition 3.2
  • Example 3.3
  • Proposition 3.4
  • ...and 22 more