The differential line bundle and canonical divisors #
On an integral curve over an exact constant field, with invertible differential sheaf and
discrete valuation rings at its codimension-one points, the sheaf of relative differentials is the
sheaf of a canonical divisor. A separating rational function t gives the rational basis
dt; its Weil divisor K identifies a regular differential f dt with a section f of
𝒪_X(K). The isomorphism is characterized by this rational-coordinate formula. The
codimension-one points need not be rational, and the constant field need not be perfect: the
local comparison only uses that the differentials of each codimension-one local ring are free of
rank one, which follows from invertibility of the differential sheaf. In particular this applies
to smooth curves over imperfect fields such as 𝔽_p(t).
For a proper curve this identifies the differential line bundle with the canonical bundle
constructed from Weil differentials. A separating rational function exists because invertibility
of the differential sheaf makes the differentials of the function field one-dimensional.
Consequently, for every line bundle L, the dual of H¹(X, L) is linearly equivalent to
H⁰(X, Ω_{X/k} ⊗ L⁻¹). This is only the existence of a noncanonical equivalence: the comparison
does not provide a natural trace pairing.
The numerical invariants of Ω_{X/k} follow: it has g independent global sections,
one-dimensional H¹, and degree 2g - 2, and every line bundle L satisfies the Riemann–Roch
theorem dim H⁰(X, L) - dim H⁰(X, Ω_{X/k} ⊗ L⁻¹) = deg L + 1 - g.
References #
- R. Hartshorne, Algebraic Geometry, II, Section 8, and IV, Section 1.
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., Sections I.5 and IV.3.
The local comparison is mem_range_rationalDifferential_iff_weilDifferentialOrder_nonneg, the
separating function comes from exists_transcendental_and_isSeparable_adjoin_of_isInvertible, and
the rational basis and Weil divisor are those of kaehlerBasisOfSeparating and
weilDifferentialDivisor.
Reading a differential in the rational basis dt identifies Ω_{X/k} with 𝒪_X(K),
where K has the Weil orders of dt as its coefficients. The constant field is exact but need
not be perfect, and the differential sheaf is assumed invertible; the codimension-one places may
have any degree.
Instances For
The divisor-sheaf isomorphism sends a differential to its rational coefficient in dt.
The inverse divisor-sheaf isomorphism sends a rational coefficient f to f dt.
On a curve over an exact constant field with all the places of its function field, the invertible sheaf of relative differentials is isomorphic to the Weil-differential canonical bundle. The constant field need not be perfect.
Over an exact constant field, the dual of H¹(X, L) is linearly equivalent to
H⁰(X, Ω_{X/k} ⊗ L⁻¹), by the identification of the differential line bundle with the canonical
bundle. This asserts only the existence of a noncanonical linear equivalence, not a natural
duality pairing.
The global differentials form a space of dimension the genus. Over an exact constant
field, dim H⁰(X, Ω_{X/k}) = g, where g = dim H¹(X, 𝒪_X).
Over an exact constant field, H¹(X, Ω_{X/k}) is one-dimensional.
The Riemann–Roch theorem with the sheaf of differentials. Over an exact constant field,
every line bundle L satisfies
dim H⁰(X, L) - dim H⁰(X, Ω_{X/k} ⊗ L⁻¹) = deg L + 1 - g,
where deg L = χ(L) - χ(𝒪_X) is the Euler-characteristic degree and g = dim H¹(X, 𝒪_X).
The sheaf of differentials of a proper curve has degree 2g - 2. Over an exact constant
field, the Euler-characteristic degree of Ω_{X/k} is 2g - 2, where
g = dim H¹(X, 𝒪_X).