The Euler characteristic of an ideal sheaf #
Let X be a scheme over a field k and I ⊆ 𝒪_X an ideal sheaf whose closed subscheme
D = V(I) is finite over k, of rank dim_k Γ(D, 𝒪_D). The exact sequence
0 ⟶ I ⟶ 𝒪_X ⟶ ι_* 𝒪_D ⟶ 0 and the vanishing of higher cohomology on the affine Noetherian
scheme D give
χ(I) = χ(𝒪_X) - dim_k Γ(D, 𝒪_D)
(Scheme.IdealSheafData.eulerCharBelow_sheaf), where χ is the degree-2 truncation
dim H⁰ - dim H¹ and H⁰(X, 𝒪_X), H¹(X, 𝒪_X) are assumed finite-dimensional.
More generally, for a vector bundle E of constant rank r on X, tensoring the sequence with
E gives 0 ⟶ E ⊗ I ⟶ E ⟶ E ⊗ ι_* 𝒪_D ⟶ 0. By the projection formula the last term is the
direct image of the pullback of E to D, a vector bundle of rank r on a scheme finite over k
and hence free (FiniteLocallyFreeSheaf.FixedRank.nonempty_iso_free_of_isFinite). So E ⊗ I has
finite-dimensional H⁰ and H¹ exactly when E does
(Scheme.IdealSheafData.finiteDimensional_cohomology_tensor_sheaf_iff), and then
χ(E ⊗ I) = χ(E) - r · dim_k Γ(D, 𝒪_D)
(Scheme.IdealSheafData.eulerCharBelow_tensor_sheaf). No hypothesis on X is needed.
The rank is Mathlib's Scheme.Hom.finrank of the finite morphism D ⟶ Spec k, evaluated at a
point of Spec k; it is dim_k Γ(D, 𝒪_D) by
AlgebraicGeometry.Scheme.Hom.finrank_eq_finrank_of_field.
References #
- R. Hartshorne, Algebraic Geometry: Lemma III.2.10 and Theorem III.3.5 (the cohomology of
𝒪_D), and the proof of Theorem IV.1.3 (Riemann–Roch), where degrees are computed from the same kind of exact sequence. - The Stacks Project, Section 33.44: degrees on proper curves, and the change of Euler characteristic under twisting by an effective Cartier divisor.
Twisting by an ideal sheaf preserves finiteness of cohomology. If I ⊆ 𝒪_X cuts out a
closed subscheme finite over the field k and E is a vector bundle of constant rank on X,
then H⁰ and H¹ of E ⊗ I are finite-dimensional exactly when those of E are.
The Euler characteristic of a twisted ideal sheaf. If I ⊆ 𝒪_X cuts out a closed
subscheme D finite over the field k and E is a vector bundle of constant rank r on X
with finite-dimensional H⁰(X, E) and H¹(X, E), then χ(E ⊗ I) = χ(E) - r · dim_k Γ(D, 𝒪_D),
with χ = dim H⁰ - dim H¹ and the dimension of Γ(D, 𝒪_D) written as the rank of D ⟶ Spec k at
the point s. For the trivial line bundle this is eulerCharBelow_sheaf.
The Euler characteristic of an ideal sheaf. If I ⊆ 𝒪_X cuts out a closed subscheme
D finite over the field k, and H⁰(X, 𝒪_X) and H¹(X, 𝒪_X) are finite-dimensional, then
χ(I) = χ(𝒪_X) - dim_k Γ(D, 𝒪_D), with χ = dim H⁰ - dim H¹ and the dimension of Γ(D, 𝒪_D)
written as the rank of D ⟶ Spec k at the point s. No dimension or properness hypothesis on
X is needed.