Serre duality for divisor sheaves on a curve #
Let X be an integral separated Noetherian curve over a field k whose codimension-one local
rings are discrete valuation rings and whose structure morphism satisfies the existence part of
the valuative criterion (a proper curve, for instance), so that its codimension-one points are the
places of the function field k(X). The cohomology of the sheaves πͺ_X(D) is then identified
with the function-field objects of Weil's theory of repartitions and differentials, and this file
deduces, when k is integrally closed in k(X) (IsIntegrallyClosedIn k X.functionField),
Serre duality in the form
HΒΉ(X, πͺ_X(D))^β¨ β Hβ°(X, πͺ_X(K - D)),
for K a canonical divisor, that is, a divisor whose class in the function field is the
canonical class of Weil differentials.
The two halves are:
Hβ°(X, πͺ_X(D)) = L(D)(SchemeWeilDivisor.globalSectionsEquivRiemannRochSpace): a global section ofπͺ_X(D)is a rational function whose order at every codimension-one pointxis at least-D(x), which is exactly membership in the RiemannβRoch space of the corresponding function-field divisor;HΒΉ(X, πͺ_X(D))^β¨ = Ξ©(D)(SchemeWeilDivisor.cohomologyOneDualEquivWeilDifferentialFiltration): first cohomology is the repartition quotientA_{k(X)} / (A_{k(X)}(D) + k(X)), and the Weil differentials bounded byDare by definition the linear forms on the repartitions that vanish onA_{k(X)}(D) + k(X).
Combined with the function-field duality L(W - D) β Ξ©(D), x β¦ x Β· Ο, for W the divisor of a
nonzero Weil differential Ο, this is Serre duality. The pairing is explicit: a section f of
πͺ_X(K - D) pairs with the class of a repartition a to Ο(f Β· a).
Under the same constant-field hypothesis, taking D = 0 computes the genus:
dim_k HΒΉ(X, πͺ_X) is the genus of the function field, and a canonical divisor has degree
2g - 2.
Main declarations #
SchemeWeilDivisor.cohomologyOneDualEquivCohomologyZero: Serre duality,HΒΉ(X, πͺ_X(D))^β¨ β Hβ°(X, πͺ_X(K - D)), with the pairingSchemeWeilDivisor.cohomologyOneDualEquivCohomologyZero_symm_apply_repartitionToCohomologyOne;SchemeWeilDivisor.finrank_cohomology_one_sheaf_eq_finrank_cohomology_zero_sheaf_sub:dim_k HΒΉ(X, πͺ_X(D)) = dim_k Hβ°(X, πͺ_X(K - D))for every canonical divisorK;SchemeWeilDivisor.genus_eq_genus_functionField: the genusdim_k HΒΉ(X, πͺ_X)ofXis the genus of its function field;SchemeWeilDivisor.relativeDegree_eq_two_mul_genus_sub_two: a canonical divisor has degree2g - 2.
References #
- J.-P. Serre, Algebraic Groups and Class Fields, Chapter II (repartitions and the duality theorem).
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Section I.5, in particular Theorem 1.5.14.
- R. Hartshorne, Algebraic Geometry, Chapter III, Corollary 7.7, and Chapter IV, Section 1.
The top open of an integral scheme is nonempty.
Serre duality #
Serre duality for divisor sheaves on a curve with k integrally closed in k(X).
Let Ο be a nonzero Weil differential of the function field and K the divisor on X
corresponding to its divisor (Ο). For every Weil divisor D, the dual of HΒΉ(X, πͺ_X(D)) is
Hβ°(X, πͺ_X(K - D)); a section f pairs with the
class of a repartition a to Ο(f Β· a)
(SchemeWeilDivisor.cohomologyOneDualEquivCohomologyZero_symm_apply_repartitionToCohomologyOne).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Serre duality pairing. Under SchemeWeilDivisor.cohomologyOneDualEquivCohomologyZero,
a section s of πͺ_X(K - D), with underlying rational function f, is the linear form on
HΒΉ(X, πͺ_X(D)) sending the class of a repartition a to Ο(f Β· a).
Not @[simp]: SchemeWeilDivisor.repartitionToCohomologyOne_apply already rewrites the class of a
repartition on the left-hand side, so this rule would not be in simp-normal form.
Dimensions, the genus and the canonical degree #
Serre duality, in dimensions. Assume k is integrally closed in k(X). For every
canonical divisor K on the curve X (a divisor whose function-field divisor represents the
canonical class) and every Weil divisor D,
dim_k HΒΉ(X, πͺ_X(D)) = dim_k Hβ°(X, πͺ_X(K - D)).
When k is integrally closed in k(X), the first cohomology of πͺ_X(0) has dimension the
genus of the function field: its dual is the space of regular Weil differentials.
The genus of a curve is the genus of its function field when k is integrally closed in
k(X). The genus dim_k HΒΉ(X, πͺ_X) of X agrees with the genus of k(X) defined through
Riemann's theorem.
A canonical divisor has degree 2g - 2 when k is integrally closed in k(X). Every
divisor K on X whose function-field divisor represents the canonical class has degree
2g - 2, where g = dim_k HΒΉ(X, πͺ_X) is the genus of X.