Mathematics / advanced
Affine algebraic geometry
Connect polynomial equations to algebraic sets, coordinate rings, morphisms, tangent spaces, and local geometry.
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Module 01 advanced
Affine algebraic geometry
Connect polynomial equations to algebraic sets, coordinate rings, morphisms, tangent spaces, and local geometry.
- 01
Affine varieties and vanishing ideals
Translate polynomial solution sets into ideals while distinguishing equations, point sets, and their intrinsic polynomial functions.
29 min - 02
Hilbert's Nullstellensatz and radical ideals
State the field hypotheses of the algebra–geometry correspondence and distinguish radical membership from ordinary ideal membership.
31 min - 03
Coordinate rings and regular maps
Recover polynomial maps from homomorphisms in the opposite direction and check isomorphisms through explicit inverse formulas.
29 min - 04
Irreducibility and Noetherian decomposition
Find the finite geometric components of a polynomial system and connect irreducibility to prime ideals and domains.
30 min - 05
Dimension and Noether normalization
Identify algebraic dimension through prime chains and finite maps while testing whether a chosen projection is actually finite.
32 min - 06
Tangent spaces and Jacobian criteria
Linearize a complete equation ideal, compare tangent and local dimensions, and keep reducedness and perfect-field assumptions explicit.
32 min - 07
Singular points and local rings
Study functions near one point through localization and use the maximal ideal's cotangent quotient to recognize a singularity intrinsically.
31 min - 08
Rational maps and function fields
Work on dense open subsets, recognize birational inverses, and distinguish a rational function from a globally regular one.
31 min - 09
Gröbner bases and elimination
Choose an elimination order, compute polynomial consequences, and distinguish the closed elimination variety from the actual constructible projection.
34 min - 10
Normalization and birational maps
Enlarge a domain to its integral closure in the same function field and analyze finite normalization maps at cusps and nodes.
34 min