Irreducible components under extension of the base field

Let A be an algebra over a field K and let L / K be a field extension. The irreducible components of Spec (L ⊗[K] A) correspond to the minimal primes Q of L ⊗[K] A. This file shows that each such Q contracts to a minimal prime P of A, and that when A is finitely generated the component V(Q) has the dimension of V(P): dim ((L ⊗[K] A) ⧸ Q) = dim (A ⧸ P). Consequently Spec A is pure-dimensional of dimension d exactly when Spec (L ⊗[K] A) is. This is the affine case of the invariance of pure-dimensionality of a scheme locally of finite type over a field under extension of the base field, which makes pure relative dimension stable under base change.

The contraction is minimal because L ⊗[K] A is flat over A, so it satisfies going down. For the dimension, choose a transcendence basis of L / K, generating an intermediate field E that is a rational function field over K, with L / E algebraic.

• Over the algebraic extension L / E, the algebra L ⊗[K] A = L ⊗[E] (E ⊗[K] A) is integral and flat over E ⊗[K] A. So Q contracts to a minimal prime Q' of E ⊗[K] A and the integral injective extension (E ⊗[K] A) ⧸ Q' → (L ⊗[K] A) ⧸ Q preserves the dimension.
• Over the rational function field E, the ring E ⊗[K] (A ⧸ P) is a domain (TauCeti.isDomain_fractionRing_mvPolynomial_tensorProduct), so the extension of P to E ⊗[K] A is prime. It is contained in Q', hence equals it by minimality, and (E ⊗[K] A) ⧸ Q' ≅ E ⊗[K] (A ⧸ P) has the dimension of A ⧸ P (TauCeti.ringKrullDim_tensorProduct_field_of_finiteType).

Main results

• Ideal.ringKrullDim_quotient_tensorProduct_of_mem_minimalPrimes: for finitely generated A, a minimal prime Q of L ⊗[K] A satisfies dim ((L ⊗[K] A) ⧸ Q) = dim (A ⧸ Q ∩ A).
• TauCeti.isPureDimensional_primeSpectrum_tensorProduct_iff: pure-dimensionality of the spectrum of a finitely generated algebra is invariant under extension of the base field.

References

• Stacks Project, Tag 00P4, the pointwise form of the invariance of dimension under extension of the base field

theorem Ideal.comap_mem_minimalPrimes_and_ringKrullDim_eq_of_isIntegral {K : Type u_1} {A : Type u_2} [CommSemiring K] [CommSemiring A] [Algebra K A] {L : Type u_3} [CommRing L] [Algebra K L] (Q : Ideal (TensorProduct K L A)) (E : Type u_4) [CommRing E] [Algebra K E] [Algebra E L] [IsScalarTower K E L] [Module.Flat E L] [Algebra.IsIntegral E L] (hQ : Q ∈ minimalPrimes (TensorProduct K L A)) : comap (Algebra.TensorProduct.map (IsScalarTower.toAlgHom K E L) (AlgHom.id K A)) Q ∈ minimalPrimes (TensorProduct K E A) ∧ ringKrullDim (TensorProduct K L A ⧸ Q) = ringKrullDim (TensorProduct K E A ⧸ comap (Algebra.TensorProduct.map (IsScalarTower.toAlgHom K E L) (AlgHom.id K A)) Q)

For a flat, integral E-algebra L with compatible K-algebra structures, the ring L ⊗[K] A is integral and flat over E ⊗[K] A. So a minimal prime of L ⊗[K] A contracts to a minimal prime of E ⊗[K] A, and the two quotients have the same Krull dimension.

theorem Ideal.ringKrullDim_quotient_tensorProduct_of_mem_minimalPrimes {K : Type u_1} [Field K] {A : Type u_2} [CommRing A] [Algebra K A] {L : Type u_3} [Field L] [Algebra K L] [Algebra.FiniteType K A] (Q : Ideal (TensorProduct K L A)) (hQ : Q ∈ minimalPrimes (TensorProduct K L A)) : ringKrullDim (TensorProduct K L A ⧸ Q) = ringKrullDim (A ⧸ comap Algebra.TensorProduct.includeRight Q)

Let A be a finitely generated algebra over a field K and L / K a field extension. For a minimal prime Q of L ⊗[K] A, the irreducible component V(Q) of Spec (L ⊗[K] A) has the dimension of the irreducible component V(Q ∩ A) of Spec A.

theorem TauCeti.isPureDimensional_primeSpectrum_tensorProduct_iff {K : Type u_1} [Field K] {A : Type u_2} [CommRing A] [Algebra K A] (L : Type u_3) [Field L] [Algebra K L] [Algebra.FiniteType K A] {d : ℕ} : IsPureDimensional d (PrimeSpectrum (TensorProduct K L A)) ↔ IsPureDimensional d (PrimeSpectrum A)

The spectrum of a finitely generated algebra A over a field K is pure-dimensional of dimension d if and only if the spectrum of L ⊗[K] A is, for any field extension L / K.