The Riemann existence theorem: statements and assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Riemann existence theorem — Theorem 2.1. Every scheme X locally of finite type over C: Y -> Y(C) is an equivalence from finite étale covers of X to finite covering spaces of X(C). No normality or reducedness hypothesis.
• Analytic form — Corollary 2.3. Varieties X: finite étale covers of X are equivalent to analytic spaces over the analytification that are finite coverings and local isomorphisms.
• Fibre functors — Theorem 3.1. Connected X and a complex point x: the étale fundamental group is the automorphism group of the fibre functor at x on finite coverings of X(C).
• Comparison of fundamental groups — Theorem 3.2. Connected X locally of finite type over C: the étale fundamental group is the profinite completion of the topological fundamental group. Proposition 3.3 supplies the semilocal simple connectedness of X(C), including singular, nonreduced and nonseparated X, from the stated triangulation dependencies.
• Connected components — Theorem 2.1. Schemes locally of finite type over C: open and closed subsets of X and of X(C) correspond.
• Extension across a normal crossings boundary — Theorem 3.3. A finite covering of degree d of a product of punctured discs and discs: its locally bounded holomorphic functions form a locally free algebra of rank d on the polydisc.
• The smooth quasi-projective case — Theorem 2.1. Smooth quasi-projective varieties over C: every finite covering of X(C) comes from a finite étale cover.
• Descent along proper surjective morphisms — Theorem 3.4. Noetherian schemes: every proper surjective morphism is an effective descent morphism for finite étale covers.
• Contractible neighbourhoods at singular points — Proposition 3.3. Every scheme X locally of finite type over C: X(C) has a basis at each point of open sets strongly deformation retracting onto that point. No smoothness, reducedness or separatedness hypothesis.

Reading order

The first two lessons set up the comparison and prove full faithfulness. The third and fourth prove the theorem for smooth varieties, through finite covers of punctured polydiscs, smooth compactification and GAGA. The fifth proves descent along proper surjective morphisms, and the sixth proves the theorem in general and compares fundamental groups.