noncomputable def fiberDim {R : Type u_1} [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (𝔭 : Ideal R) [𝔭.IsPrime] : ℕ

Fiber dimension of an R-module M at a prime 𝔭: the dimension of the residue field tensor product κ(𝔭) ⊗_R M over κ(𝔭).

theorem fiberDim_eq_rankAtStalk {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Flat R M] [Module.Finite R M] (p : PrimeSpectrum R) : fiberDim M p.asIdeal = Module.rankAtStalk M p

For a flat finite module, the fiber dimension agrees with the rank at the stalk.

theorem lemma25_fiber_finite {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Finite R M] (𝔭 : Ideal R) [𝔭.IsPrime] : Module.Finite 𝔭.ResidueField (TensorProduct R 𝔭.ResidueField M)

Lemma 25 (Lec 12), part 1: the fiber κ(𝔭) ⊗_R M of a finitely generated module is finite-dimensional over the residue field.

theorem lemma25_fiber_zero_iff_not_mem_support {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Finite R M] (p : PrimeSpectrum R) : fiberDim M p.asIdeal = 0 ↔ p ∉ Module.support R M

Lemma 25 (Lec 12): the fiber M ⊗_R κ(p) vanishes at p iff p is not in the support of M.

theorem lemma25_support_isClosed {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Finite R M] : IsClosed (Module.support R M)

Lemma 25 (Lec 12): the support of a finitely generated module is a closed subset of Spec R (cut out by the annihilator).

noncomputable def topEquivTensor (A R M : Type u) [CommRing R] [CommRing A] [Algebra R A] [AddCommGroup M] [Module R M] : TensorProduct R A ↥⊤ ≃ₗ[A] TensorProduct R A M

The base change A ⊗_R (⊤ : Submodule R M) ≃ₗ[A] A ⊗_R M: tensoring with the inclusion of the top submodule is an isomorphism.

theorem spanFinrank_top_eq_of_linearEquiv {R : Type u_1} {M N : Type u} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (e : M ≃ₗ[R] N) : ⊤.spanFinrank = ⊤.spanFinrank

The minimal number of generators of M (spanFinrank of the top submodule) is invariant under linear equivalences.

theorem finrank_residueField_tensor_eq_spanFinrank_top {R : Type u} [CommRing R] [IsLocalRing R] {M : Type u} [AddCommGroup M] [Module R M] [Module.Finite R M] : Module.finrank (IsLocalRing.ResidueField R) (TensorProduct R (IsLocalRing.ResidueField R) M) = ⊤.spanFinrank

Over a local ring, the dimension of the fiber κ ⊗_R M equals the minimal number of generators of M (Nakayama).

theorem fiberDim_eq_spanFinrank_stalk {R : Type u} [CommRing R] {M : Type u} [AddCommGroup M] [Module R M] [Module.Finite R M] (𝔭 : Ideal R) [𝔭.IsPrime] : fiberDim M 𝔭 = ⊤.spanFinrank

Bridge between fiber dimension and the localized module: fiberDim M 𝔭 equals the minimal number of generators of the stalk M_𝔭.

theorem lemma25_fiber_upper_semicontinuity {R : Type u} [CommRing R] [IsNoetherianRing R] {M : Type u} [AddCommGroup M] [Module R M] [Module.Finite R M] (𝔭 𝔮 : Ideal R) [𝔭.IsPrime] [𝔮.IsPrime] (h : 𝔭 ≤ 𝔮) : fiberDim M 𝔭 ≤ fiberDim M 𝔮

Lemma 25 (Lec 12) — Upper semi-continuity of fiber dimension: if 𝔭 ⊆ 𝔮 are primes, then fiberDim M 𝔭 ≤ fiberDim M 𝔮.

theorem lemma25_locally_free_criterion_mp {R : Type u} [CommRing R] [IsNoetherianRing R] [IsLocalRing R] [IsDomain R] {M : Type u} [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Free R M] : fiberDim M (IsLocalRing.maximalIdeal R) = Module.finrank (FractionRing R) (TensorProduct R (FractionRing R) M)

Forward direction of the local-free criterion: if M is free over a local domain, then fiber dimension at the closed point equals fiber dimension at the generic point.

theorem lemma25_locally_free_criterion_mpr {R : Type u} [CommRing R] [IsNoetherianRing R] [IsLocalRing R] [IsDomain R] {M : Type u} [AddCommGroup M] [Module R M] [Module.Finite R M] : fiberDim M (IsLocalRing.maximalIdeal R) = Module.finrank (FractionRing R) (TensorProduct R (FractionRing R) M) → Module.Free R M

Reverse direction of the local-free criterion: over a local Noetherian domain, if the fiber dimensions at the closed and generic points coincide, then M is free.

theorem lemma25_locally_free_criterion {R : Type u} [CommRing R] [IsNoetherianRing R] [IsLocalRing R] [IsDomain R] {M : Type u} [AddCommGroup M] [Module R M] [Module.Finite R M] : Module.Free R M ↔ fiberDim M (IsLocalRing.maximalIdeal R) = Module.finrank (FractionRing R) (TensorProduct R (FractionRing R) M)

Lemma 25 (Lec 12) — Local-free criterion: over a local Noetherian domain, a finite module is free iff its fiber dimensions at the closed and generic points agree.