Krull's Height Theorem Algebraic Geometry Nakayama's Lemma (Special Case) Algebraic Geometry Second Inequality Algebraic Geometry Complement of a Hypersurface is Affine Algebraic Geometry Smooth Points Form an Open Dense Subset Algebraic Geometry Basic Dimension Bounds for $\mathcal{L}(D)$ Algebraic Geometry Element Outside $\mathfrak{m}_p^2$ Generates $\mathfrak{m}_p$ Algebraic Geometry Projective Varieties Are Covered by Affine Pieces Algebraic Geometry