Sheaves and schemes

The language of schemes: sheaves on topological spaces, ringed spaces and their modules, affine schemes and the equivalence between modules and quasi-coherent sheaves, schemes glued from affine pieces, the functor of points and representability, fibre products, local properties, quasi-coherent sheaves and closed subschemes, and Proj with projective space and projective bundles. The course cites the site's lesson on sheaves of modules for the homological facts it proves and supplies the sheaf-theoretic background that lesson assumes. Lessons prove what they teach; the Stacks project (read in its AI Integrated Stacks Project edition) is cited by tag.

1. Sheaves on topological spaces
2. Ringed spaces and sheaves of modules
3. Quasi-coherent, coherent and locally free modules
4. Affine schemes
5. Schemes, gluing and immersions
6. The functor of points and representability
7. Fibre products and base change
8. Properties of schemes
9. Quasi-coherent sheaves on schemes
10. Closed subschemes and scheme-theoretic images
11. Proj of a graded ring
12. Projective space, relative Proj and maps to projective space

Original text CC0 1.0 unless the lesson states other terms.