Algebraic dimension

Chains of prime ideals define Krull dimension and connect algebraic structure to geometric dimension.

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Algebraic singularities
Singular points fail suitable smoothness conditions measured by local geometry or tangent behavior.
Localization
Localization formally inverts a chosen multiplicatively closed set of ring elements.
Rings
A ring supports addition and multiplication linked by distributive laws.
Sheaves
A sheaf organizes local data that restrict consistently and glue uniquely when compatible.
Affine varieties
Common polynomial zero sets define affine algebraic sets over a chosen field.