@[reducible, inline]

abbrev Formalization.ProjectiveSpace (k : Type u_1) [Field k] (n : ℕ) : Type u_1

Projective n-space over k, identified with the projectivization of k^{n+1}.

@[reducible, inline]

abbrev Formalization.PuncturedAffineSpace (k : Type u_1) [Field k] (n : ℕ) : Type u_1

The punctured affine space A^{n+1} \ {0} over k, used as the domain of the quotient map defining P^n.

def Formalization.ProjectiveSpace.π (k : Type u_1) [Field k] (n : ℕ) : PuncturedAffineSpace k n → ProjectiveSpace k n

The quotient map A^{n+1} \ {0} → P^n sending a nonzero vector to its projective class (Lecture 2, Definition 4).

theorem Formalization.ProjectiveSpace.π_eq_iff {k : Type u_1} [Field k] {n : ℕ} (v w : PuncturedAffineSpace k n) : π k n v = π k n w ↔ ∃ (a : kˣ), a • ↑w = ↑v

Two nonzero vectors define the same point of P^n iff they differ by a nonzero scalar.

theorem Formalization.ProjectiveSpace.π_surjective {k : Type u_1} [Field k] {n : ℕ} : Function.Surjective (π k n)

The quotient map A^{n+1} \ {0} → P^n is surjective.

def Formalization.AffineSpace.zeroLocus (k : Type u_2) [Field k] (n : ℕ) (S : Set (MvPolynomial (Fin (n + 1)) k)) : Set (Fin (n + 1) → k)

The vanishing locus in A^{n+1} of a set of polynomials: points where every polynomial in S evaluates to zero.

def Formalization.AffineSpace.mulPairs (k : Type u_2) [Field k] (n : ℕ) (S T : Set (MvPolynomial (Fin (n + 1)) k)) : Set (MvPolynomial (Fin (n + 1)) k)

The set of all products f * g with f ∈ S and g ∈ T, used in showing the union of two zero loci is itself a zero locus.

@[implicit_reducible]

instance Formalization.zariskiTopologyAffine (k : Type u_2) [Field k] (n : ℕ) : TopologicalSpace (Fin (n + 1) → k)

The Zariski topology on A^{n+1}, whose closed sets are the vanishing loci of sets of polynomials.

@[implicit_reducible]

instance Formalization.projectiveSpaceTopology (k : Type u_2) [Field k] (n : ℕ) : TopologicalSpace (ProjectiveSpace k n)

The Zariski topology on P^n, defined as the quotient topology coinduced from the Zariski topology on A^{n+1} via the projection π.

def Formalization.ProjectiveSpace.IsRegular (k : Type u_2) [Field k] (n : ℕ) (U : Set (ProjectiveSpace k n)) (f : ↑U → k) : Prop

A function f : U → k on an open subset U ⊆ P^n is regular (Lecture 2, Def 4) if locally on the affine cone it can be written as a ratio p/q of polynomials with q non-vanishing.