The Euler characteristic of 𝒪_X(D) and the degree of a line bundle #
Let X be a Noetherian integral scheme over a field k whose codimension-one local rings are
discrete valuation rings, and whose codimension-one points are closed with finite residue fields
over k. Adding a point y to a Weil divisor D changes the Euler characteristic by the residue
degree [κ(y) : k] (SchemeWeilDivisor.eulerCharBelow_sheaf_add_ofPoint). Inducting over the
divisor gives
χ(𝒪_X(D)) = deg D + χ(𝒪_X),
where deg D = Σ_y D(y) [κ(y) : k] is SchemeWeilDivisor.relativeDegree (X ↘ Spec k) and
χ(M) = dim H⁰(X, M) - dim H¹(X, M). The same induction shows that whether Hⁱ(X, 𝒪_X(D)) is
finite-dimensional does not depend on D, so the only finiteness input needed is that of the
cohomology of 𝒪_X itself.
On a proper curve over k, H⁰ is always finite-dimensional
(SchemeWeilDivisor.finiteDimensional_cohomology_zero_sheaf) and every line bundle is some
𝒪_X(D) (SchemeWeilDivisor.exists_nonempty_iso_sheaf). As soon as H¹(X, 𝒪_X) is
finite-dimensional, the Euler-characteristic degree χ(L) - χ(𝒪_X) of a line bundle therefore
agrees with the degree of any divisor of L, and is additive under tensor product.
Main declarations #
SchemeWeilDivisor.finiteDimensional_cohomology_sheaf_iff:Hⁱ(X, 𝒪_X(D))is finite-dimensional exactly whenHⁱ(X, 𝒪_X(E))is;SchemeWeilDivisor.eulerCharBelow_sheaf_eq_relativeDegree_add:χ(𝒪_X(D)) = deg D + χ(𝒪_X);SchemeWeilDivisor.relativeDegree_eq_of_linearlyEquivalentandSchemeWeilDivisor.relativeDegree_principalDivisor: the degree is a linear-equivalence invariant, and a principal divisor has degree zero;SchemeWeilDivisor.finiteDimensional_cohomology_one_sheafandInvertibleSheaf.finiteDimensional_cohomology_one: on a proper curve withH¹(X, 𝒪_X)finite-dimensional,H¹of every𝒪_X(D)and of every line bundle is finite-dimensional;InvertibleSheaf.eulerDegree_eq_relativeDegreeandLineBundleClass.eulerDegree_toLineBundleClass: the Euler-characteristic degree of𝒪_X(D)isdeg D;LineBundleClass.eulerDegree_mul: the Euler-characteristic degree is additive under tensor product.
References #
- R. Hartshorne, Algebraic Geometry, IV, Theorem 1.3 and its proof.
- J.-P. Serre, Algebraic Groups and Class Fields, Chapter II, §3.
Finite-dimensionality of Hⁱ(X, 𝒪_X(D)) does not depend on D. If every
codimension-one point of X is closed with finite residue field over k, then Hⁱ(X, 𝒪_X(D))
is finite-dimensional exactly when Hⁱ(X, 𝒪_X(E)) is.
χ(𝒪_X(D)) = deg D + χ(𝒪_X(0)). Let X be a Noetherian integral scheme over a field k
whose codimension-one local rings are discrete valuation rings and whose codimension-one points
are closed with finite residue fields over k. If H⁰(X, 𝒪_X(0)) and H¹(X, 𝒪_X(0)) are
finite-dimensional, then for every Weil divisor D
χ(𝒪_X(D)) = Σ_y D(y) [κ(y) : k] + χ(𝒪_X(0)),
where χ(M) = dim H⁰(X, M) - dim H¹(X, M).
On a curve, 𝒪_X(0) is the trivial line bundle 𝒪_X.
Equations
- One or more equations did not get rendered due to their size.
Instances For
H¹(X, 𝒪_X(D)) is finite-dimensional once H¹(X, 𝒪_X) is. On a proper integral curve
over a field k whose codimension-one local rings are discrete valuation rings, if
H¹(X, 𝒪_X) is finite-dimensional over k, then so is H¹(X, 𝒪_X(D)) for every Weil
divisor D.
χ(𝒪_X(D)) = deg D + χ(𝒪_X). On a proper integral curve over a field k whose
codimension-one local rings are discrete valuation rings, if H¹(X, 𝒪_X) is finite-dimensional
over k, then for every Weil divisor D
χ(𝒪_X(D)) = Σ_y D(y) [κ(y) : k] + χ(𝒪_X),
where χ(M) = dim H⁰(X, M) - dim H¹(X, M).
The degree is a linear-equivalence invariant. On a proper integral curve over k whose
codimension-one local rings are discrete valuation rings, with H¹(X, 𝒪_X) finite-dimensional,
linearly equivalent Weil divisors have the same degree: their sheaves are isomorphic, so their
Euler characteristics agree.
A principal divisor has degree zero. On a proper integral curve over k whose
codimension-one local rings are discrete valuation rings, with H¹(X, 𝒪_X) finite-dimensional,
the divisor of a nonzero rational function has degree zero.
The residue-degree weights kill principal divisors. This is
SchemeWeilDivisor.relativeDegree_principalDivisor in the form consumed by the abstract
degree-zero divisor class group WeilDivisor.OrderSystem.picZero.
H¹ of a line bundle on a proper curve is finite-dimensional once H¹(X, 𝒪_X) is. On a
proper integral curve over a field k whose codimension-one local rings are discrete valuation
rings, if H¹(X, 𝒪_X) is finite-dimensional over k, then so is H¹(X, L) for every line
bundle L.
The Euler-characteristic degree of 𝒪_X(D) is deg D. On a proper integral curve over
a field k whose codimension-one local rings are discrete valuation rings, with H¹(X, 𝒪_X)
finite-dimensional over k, a line bundle L ≅ 𝒪_X(D) has
χ(L) - χ(𝒪_X) = Σ_y D(y) [κ(y) : k].
The Euler-characteristic degree of the line-bundle class of 𝒪_X(D) is deg D, on a
proper integral curve over k whose codimension-one local rings are discrete valuation rings
and whose H¹(X, 𝒪_X) is finite-dimensional.
The Euler-characteristic degree is additive under tensor product, on a proper integral
curve over k whose codimension-one local rings are discrete valuation rings and whose
H¹(X, 𝒪_X) is finite-dimensional.