Mathematics / research
Schemes and projective geometry
Extend algebraic geometry through locally ringed spaces, projective constructions, sheaves, divisors, and cohomology.
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Module 01 research
Schemes and projective geometry
Extend algebraic geometry through locally ringed spaces, projective constructions, sheaves, divisors, and cohomology.
- 01
Sheaves and locally ringed spaces
Organize compatible local functions and require geometric morphisms to preserve the maximal ideals of their stalks.
30 min - 02
Affine schemes and structure sheaves
Recover a ring from its affine scheme while retaining generic points, residue fields, and nilpotent structure.
29 min - 03
Gluing schemes and fiber products
Assemble compatible affine charts and compute geometric base changes with tensor products rather than set-theoretic pairs.
31 min - 04
Projective space and the Proj construction
Build projective geometry from homogeneous prime ideals and degree-zero localizations while tracking the grading assumptions.
31 min - 05
Separatedness, properness, and valuative criteria
Use diagonals and valuation-ring extension problems to distinguish unique limits, existing limits, and proper algebraic families.
32 min - 06
Quasi-coherent sheaves and vector bundles
Translate sheaves into localized modules and distinguish arbitrary module data from finite locally free bundles.
31 min - 07
Divisors, line bundles, and linear systems
Relate local equations and invertible sheaves while keeping Cartier divisors distinct from codimension-one Weil data.
33 min - 08
Čech cohomology and sheaf cohomology
Compute global gluing obstructions with an acyclic affine cover and justify when the Čech complex computes derived cohomology.
34 min - 09
Projective curves and Riemann–Roch
Count sections on a smooth projective curve using degree, genus, the canonical bundle, and the special correction term.
33 min - 10
Flat families and Hilbert polynomials
Track projective families through flatness and a chosen polarization while allowing smooth fibers to degenerate into reducible ones.
35 min