58 Fundamental Groups of Schemes

Section 58.1: Introduction
Section 58.2: Schemes étale over a point
- Definition 58.2.1
- Lemma 58.2.2
- Remark 58.2.3
Section 58.3: Galois categories
- Equation 58.3.0.1
- Lemma 58.3.1
- Example 58.3.2
  - Equation 58.3.2.1
- Lemma 58.3.3
- Lemma 58.3.4
- Example 58.3.5
  - Equation 58.3.5.1
- Definition 58.3.6 reference
- Lemma 58.3.7
- Lemma 58.3.8
- Lemma 58.3.9 reference
- Proposition 58.3.10 reference
- Lemma 58.3.11
Section 58.4: Functors and homomorphisms
- Equation 58.4.0.1
- Lemma 58.4.1
- Lemma 58.4.2
- Lemma 58.4.3
- Lemma 58.4.4
- Lemma 58.4.5
Section 58.5: Finite étale morphisms
- Example 58.5.1
- Lemma 58.5.2
- Remark 58.5.3
- Lemma 58.5.4
- Lemma 58.5.5
Section 58.6: Fundamental groups
- Definition 58.6.1
- Theorem 58.6.2
- Lemma 58.6.3
- Remark 58.6.4
Section 58.7: Galois covers of connected schemes
Section 58.8: Topological invariance of the fundamental group
- Lemma 58.8.1
- Lemma 58.8.2
- Lemma 58.8.3
- Proposition 58.8.4
Section 58.9: Finite étale covers of proper schemes
- Lemma 58.9.1
- Lemma 58.9.2
- Lemma 58.9.3
Section 58.10: Local connectedness
- Lemma 58.10.1
- Lemma 58.10.2
- Lemma 58.10.3
- Lemma 58.10.4
- Lemma 58.10.5
- Lemma 58.10.6
  - Equation 58.10.6.1
- Lemma 58.10.7
- Lemma 58.10.8
Section 58.11: Fundamental groups of normal schemes
- Lemma 58.11.1
- Lemma 58.11.2
- Proposition 58.11.3
- Lemma 58.11.4
Section 58.12: Group actions and integral closure
- Lemma 58.12.1
- Lemma 58.12.2
- Lemma 58.12.3
- Lemma 58.12.4
- Lemma 58.12.5
Section 58.13: Ramification theory
- Equation 58.13.0.1
- Lemma 58.13.1
- Lemma 58.13.2
- Example 58.13.3
- Lemma 58.13.4
- Lemma 58.13.5
- Lemma 58.13.6
Section 58.14: Geometric and arithmetic fundamental groups
- Lemma 58.14.1
- Lemma 58.14.2
- Lemma 58.14.3
Section 58.15: Homotopy exact sequence
- Lemma 58.15.1 reference
- Proposition 58.15.2
Section 58.16: Specialization maps
- Lemma 58.16.1
- Lemma 58.16.2
- Lemma 58.16.3
- Lemma 58.16.4
Section 58.17: Restriction to a closed subscheme
- Lemma 58.17.1
- Lemma 58.17.2
- Lemma 58.17.3
- Lemma 58.17.4
- Lemma 58.17.5
- Lemma 58.17.6
- Lemma 58.17.7
Section 58.18: Pushouts and fundamental groups
- Lemma 58.18.1
Section 58.19: Finite étale covers of punctured spectra, I
- Situation 58.19.1
- Lemma 58.19.2
- Lemma 58.19.3
- Lemma 58.19.4
- Lemma 58.19.5
- Lemma 58.19.6 reference
- Lemma 58.19.7
Section 58.20: Purity in local case, I
- Lemma 58.20.1
- Lemma 58.20.2
- Lemma 58.20.3
- Lemma 58.20.4
- Lemma 58.20.5
- Lemma 58.20.6
- Lemma 58.20.7
Section 58.21: Purity of branch locus
- Lemma 58.21.1
- Lemma 58.21.2
- Lemma 58.21.3
- Lemma 58.21.4: Purity of branch locus historical remark reference
- Lemma 58.21.5
- Lemma 58.21.6
Section 58.22: Finite étale covers of punctured spectra, II
- Lemma 58.22.1
- Lemma 58.22.2
- Lemma 58.22.3
Section 58.23: Finite étale covers of punctured spectra, III
- Lemma 58.23.1
- Lemma 58.23.2
- Remark 58.23.3
Section 58.24: Finite étale covers of punctured spectra, IV
- Lemma 58.24.1
- Lemma 58.24.2
Section 58.25: Purity in local case, II
- Lemma 58.25.1
- Lemma 58.25.2
- Proposition 58.25.3
Section 58.26: Purity in local case, III
- Lemma 58.26.1
- Lemma 58.26.2 slogan
- Lemma 58.26.3
- Proposition 58.26.4
Section 58.27: Lefschetz for the fundamental group
- Proposition 58.27.1
- Proposition 58.27.2
- Proposition 58.27.3
Section 58.28: Purity of ramification locus
- Lemma 58.28.1
- Lemma 58.28.2
- Lemma 58.28.3: Purity of ramification locus reference
Section 58.29: Affineness of complement of ramification locus
- Lemma 58.29.1
- Lemma 58.29.2
- Theorem 58.29.3
Section 58.30: Specialization maps in the smooth proper case
- Lemma 58.30.1
- Proposition 58.30.2
- Theorem 58.30.3
Section 58.31: Tame ramification
- Lemma 58.31.1
- Lemma 58.31.2
- Lemma 58.31.3
- Example 58.31.4: Standard tamely ramified morphism
- Lemma 58.31.5: Abhyankar's lemma for regular divisor
- Lemma 58.31.6
- Lemma 58.31.7
- Lemma 58.31.8
- Lemma 58.31.9
Section 58.32: Tricks in positive characteristic
- Lemma 58.32.1
- Lemma 58.32.2
- Lemma 58.32.3
- Proposition 58.32.4