53 Fundamental Groups of Schemes

Section 53.1: Introduction
Section 53.2: Schemes étale over a point
- Definition 53.2.1
- Lemma 53.2.2
- Remark 53.2.3
Section 53.3: Galois categories
- Equation 53.3.0.1
- Lemma 53.3.1
- Example 53.3.2
  - Equation 53.3.2.1
- Lemma 53.3.3
- Lemma 53.3.4
- Example 53.3.5
  - Equation 53.3.5.1
- Definition 53.3.6 reference
- Lemma 53.3.7
- Lemma 53.3.8
- Lemma 53.3.9 reference
- Proposition 53.3.10 reference
- Lemma 53.3.11
Section 53.4: Functors and homomorphisms
- Equation 53.4.0.1
- Lemma 53.4.1
- Lemma 53.4.2
- Lemma 53.4.3
- Lemma 53.4.4
- Lemma 53.4.5
Section 53.5: Finite étale morphisms
- Example 53.5.1
- Lemma 53.5.2
- Remark 53.5.3
- Lemma 53.5.4
- Lemma 53.5.5
Section 53.6: Fundamental groups
- Definition 53.6.1
- Theorem 53.6.2
- Lemma 53.6.3
- Remark 53.6.4
Section 53.7: Galois covers of connected schemes
Section 53.8: Topological invariance of the fundamental group
- Lemma 53.8.1
- Lemma 53.8.2
- Lemma 53.8.3
- Proposition 53.8.4
Section 53.9: Finite étale covers of proper schemes
- Lemma 53.9.1
- Lemma 53.9.2
Section 53.10: Local connectedness
- Lemma 53.10.1
- Lemma 53.10.2
- Lemma 53.10.3 slogan reference
- Lemma 53.10.4
- Lemma 53.10.5
- Lemma 53.10.6
- Lemma 53.10.7
  - Equation 53.10.7.1
- Lemma 53.10.8
- Lemma 53.10.9
Section 53.11: Fundamental groups of normal schemes
- Lemma 53.11.1
- Lemma 53.11.2
- Proposition 53.11.3
- Lemma 53.11.4
Section 53.12: Group actions and integral closure
- Lemma 53.12.1
- Lemma 53.12.2
- Lemma 53.12.3
- Lemma 53.12.4
- Lemma 53.12.5
Section 53.13: Ramification theory
- Equation 53.13.0.1
- Lemma 53.13.1
- Lemma 53.13.2
- Example 53.13.3
- Lemma 53.13.4
- Lemma 53.13.5
- Lemma 53.13.6
Section 53.14: Geometric and arithmetic fundamental groups
- Lemma 53.14.1
- Lemma 53.14.2
- Lemma 53.14.3
Section 53.15: Homotopy exact sequence
- Lemma 53.15.1 reference
- Proposition 53.15.2
Section 53.16: Specialization maps
- Lemma 53.16.1
- Lemma 53.16.2
- Lemma 53.16.3
- Lemma 53.16.4
Section 53.17: Restriction to a closed subscheme
- Lemma 53.17.1
- Lemma 53.17.2
- Lemma 53.17.3
- Lemma 53.17.4
- Lemma 53.17.5
- Lemma 53.17.6
- Lemma 53.17.7
Section 53.18: Pushouts and fundamental groups
- Lemma 53.18.1
Section 53.19: Finite étale covers of punctured spectra, I
- Situation 53.19.1
- Lemma 53.19.2
- Lemma 53.19.3
- Lemma 53.19.4
- Lemma 53.19.5
- Lemma 53.19.6 reference
- Lemma 53.19.7
Section 53.20: Purity in local case, I
- Lemma 53.20.1
- Lemma 53.20.2
- Lemma 53.20.3
- Lemma 53.20.4
- Lemma 53.20.5
- Lemma 53.20.6
- Lemma 53.20.7
Section 53.21: Purity of branch locus
- Lemma 53.21.1
- Lemma 53.21.2
- Lemma 53.21.3
- Lemma 53.21.4: Purity of branch locus historical remark reference
- Lemma 53.21.5
- Lemma 53.21.6
Section 53.22: Finite étale covers of punctured spectra, II
- Lemma 53.22.1
- Lemma 53.22.2
- Lemma 53.22.3
Section 53.23: Finite étale covers of punctured spectra, III
- Lemma 53.23.1
- Lemma 53.23.2
- Remark 53.23.3
Section 53.24: Finite étale covers of punctured spectra, IV
- Lemma 53.24.1
- Lemma 53.24.2
Section 53.25: Purity in local case, II
- Lemma 53.25.1
- Lemma 53.25.2
- Proposition 53.25.3
Section 53.26: Purity in local case, III
- Lemma 53.26.1
- Lemma 53.26.2 slogan
- Lemma 53.26.3
- Proposition 53.26.4
Section 53.27: Lefschetz for the fundamental group
- Proposition 53.27.1
- Proposition 53.27.2
- Proposition 53.27.3
Section 53.28: Purity of ramification locus
- Lemma 53.28.1
- Lemma 53.28.2
- Lemma 53.28.3: Purity of ramification locus reference
Section 53.29: Affineness of complement of ramification locus
- Lemma 53.29.1
- Lemma 53.29.2
- Theorem 53.29.3
Section 53.30: Specialization maps in the smooth proper case
- Lemma 53.30.1
- Proposition 53.30.2
- Theorem 53.30.3
Section 53.31: Tame ramification
- Lemma 53.31.1
- Lemma 53.31.2
- Example 53.31.3: Standard tamely ramified morphism
- Lemma 53.31.4: Abhyankar's lemma for regular divisor
- Lemma 53.31.5
- Lemma 53.31.6
- Lemma 53.31.7
- Lemma 53.31.8