Scope and references

AG-DFG-02

This further source reference is outside the selected collection. The complete supporting lesson retains its stated prerequisites; no new proof of this result is asserted here.

Complex cover comparison

Comparison with the topological fundamental group over the complex numbers treats schemes locally of finite type over C, including nilpotents, nonseparated schemes and non-quasi-compact schemes. Its cover equivalences include every morphism. The fundamental-group comparison assumes connectedness and a complex base point.

The analytic construction, local rings and coherent modules are proved in Complex analytic spaces and analytification, Theorem 3.2, Theorem 4.1 and Proposition 5.1. Proper GAGA, including nilpotents, is proved in Serre's comparison theorems and Chow's theorem, Theorem 6.11, with morphisms in Proposition 6.9 and cohomology in Theorem 6.7.

The Riemann existence theorem, Theorem 2.1, proves the general cover equivalence. Its Proposition 3.3 proves contractible open neighbourhoods at every complex point; Theorem 3.2 then gives the profinite comparison. Its analytic-category form in the full scheme scope uses Proposition 2.1 of the comparison lesson.

The neighbourhood proof uses Subanalytic triangulations on analytic manifolds, Relative polyhedron theorem. That provider assumes lower subanalytic uniformization, set calculus and analytic facts; its linked analytic-finiteness component retains its separate licence. These remain external prerequisites of this lesson. Inclusion here does not assert recursive prerequisite closure or completion of the parent course.