Pullback of Fitting ideal sheaves

Fitting ideal sheaves of quasicoherent modules of finite type commute with pullback. Their affine computation is the algebraic identity Fitt_k(S ⊗_R M) = Fitt_k(M) S. This compatibility is an ingredient for base change of relative singular subschemes. That application also requires a pullback comparison for relative differentials, which is not established here.

References

• The Stacks Project, Tag 0C3C (Fitting ideals of quasicoherent modules).
• The Stacks Project, Tag 07ZA (base change of Fitting ideals).

@[simp]

theorem AlgebraicGeometry.Scheme.Modules.fittingIdeal_pullback {X Y : Scheme} (f : X ⟶ Y) (M : Y.Modules) [SheafOfModules.IsQuasicoherent M] [SheafOfModules.IsFiniteType M] (k : ℕ) : ((pullback f).obj M).fittingIdeal ⋯ k = (M.fittingIdeal ⋯ k).comap f

Fitting ideal sheaves of quasicoherent modules of finite type commute with pullback along arbitrary morphisms of schemes, without a flatness hypothesis.