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Invariance of the tame fundamental group under base change between algebraically closed fields

Aaron Landesman

Abstract

We show that the tame étale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic $p \geq 0$.

Invariance of the tame fundamental group under base change between algebraically closed fields

Abstract

We show that the tame étale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic .

Paper Structure

This paper contains 11 sections, 16 theorems, 4 equations.

Table of Contents

  1. Statement of theorem
  2. Acknowledgements
  3. Proof of theorem
  4. Idea of proof of \ref{['theorem:isomorphism-on-algebraically-closed-base-change']}
  5. Surjectivity
  6. Proof of injectivity in the curve case
  7. Dominating compactifications
  8. Proof of injectivity in the smooth and quasi-projective case
  9. Proof of injectivity in the smooth case
  10. Proof of injectivity in the general case
  11. Collected Lemmas

Key Result

Theorem 1.1

Suppose $k$ is an algebraically closed field of characteristic $p\geq 0$ and $U$ is a connected normal separated finite type scheme over $k$. Let $L$ be any algebraically closed field containing $k$ and $\overline U$ any normal compactification of $U$. Then, the natural map $\pi_1^{\operatorname{tam

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Example 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.3
  • ...and 29 more