Sheaf cohomology, Riemann–Roch, Serre duality, and differentials on the two-chart model #
Let F / k be an algebraic function field, x ∈ F transcendental over k, and
X_F = TauCeti.AlgebraicGeometry.twoChartModel hF hx its regular proper curve model over k.
Using the divisor equivalence twoChartModel.equivDivisor (Div(X_F) ≃+ Div(F)), the
function-field isomorphism twoChartModel.functionFieldEquiv (k(X_F) ≃ₐ[k] F), and the
transport of Riemann–Roch spaces, repartitions, and Weil differentials, this file identifies the
sheaf cohomology and canonical invariants of X_F with their function-field counterparts on F:
- Over an arbitrary base field
k:Γ(X_F, 𝒪(D)) ≃ₗ[k] L(D)(twoChartModel.globalSectionsEquivRiemannRochSpace),H⁰(X_F, 𝒪(D)) ≃ₗ[k] L(D)(twoChartModel.cohomologyZeroEquivRiemannRochSpace),dim_k H⁰(X_F, 𝒪(D)) = ℓ(D)(twoChartModel.finrank_cohomology_zero_sheaf_eq_dim),A_F ⧸ (A_F(D) + F) ≃ₗ[k] H¹(X_F, 𝒪(D))(twoChartModel.repartitionQuotientEquivCohomologyOne), andH¹(X_F, 𝒪(D))^∨ ≃ₗ[k] Ω_F(D)(twoChartModel.cohomologyOneDualEquivWeilDifferentialFiltration). - When
kis integrally closed inF(IsIntegrallyClosedIn k F):dim_k H¹(X_F, 𝒪(D)) = i(D) = ℓ(W - D)(twoChartModel.finrank_cohomology_one_sheaf_eq_indexOfSpecialtyandtwoChartModel.finrank_cohomology_one_sheaf_eq_dim_sub), the Euler-characteristic and dimension forms of Riemann–Roch onX_F(twoChartModel.eulerCharBelow_sheaf_eq_degree_add_one_sub_genusandtwoChartModel.finrank_cohomology_zero_sheaf_sub_finrank_cohomology_one_sheaf), and Serre dualityH¹(X_F, 𝒪(D))^∨ ≃ₗ[k] H⁰(X_F, 𝒪(K - D))(twoChartModel.cohomologyOneDualEquivCohomologyZero) with explicit pairing(f, [a]) ↦ ω(f · a)onF(twoChartModel.cohomologyOneDualEquivCohomologyZero_symm_apply_repartitionToCohomologyOne). - Canonical bundle and relative differentials (still for
IsIntegrallyClosedIn k F):twoChartModel.canonicalBundlemaps undertwoChartModel.picardEquivClassGroup(Pic(X_F) ≃+ Cl(F)) toTauCeti.canonicalClassand has Euler-characteristic degree2g - 2,gglobal sections, and one-dimensionalH¹. WhenΩ_{X_F/k}is invertible — in particular wheneverX_Fis smooth overk(twoChartModel.smoothOfRelativeDimension_one,twoChartModel.isInvertible_relativeDifferentials), as it always is over a perfect field (twoChartModel.smoothOfRelativeDimension_one_of_perfectField) —[Ω_{X_F/k}]also maps toTauCeti.canonicalClassand satisfiesdeg Ω_{X_F/k} = 2g - 2,dim_k H⁰(X_F, Ω_{X_F/k}) = g, anddim_k H¹(X_F, Ω_{X_F/k}) = 1.
References #
- J.-P. Serre, Algebraic Groups and Class Fields, Chapter II.
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Section I.5 and Appendix B.
- R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977, Chapter IV, Section 1.
The top open of twoChartModel hF hx is nonempty.
Global sections, H⁰, repartition quotients, and H¹^∨ over an arbitrary field #
Global sections of 𝒪_{X_F}(D) form the Riemann–Roch space L(D) in F.
Instances For
globalSectionsEquivRiemannRochSpace sends a global section of 𝒪_{X_F}(D) to its underlying
element of F via functionFieldEquiv.
(globalSectionsEquivRiemannRochSpace hxt D).symm sends f ∈ L(D) to the global section whose
underlying rational function is (functionFieldEquiv hxt).symm f.
Zeroth cohomology of 𝒪_{X_F}(D) is the Riemann–Roch space L(D) in F.
Instances For
cohomologyZeroEquivRiemannRochSpace reads a degree-zero cohomology class as an element of F
via cohomologyZeroEquiv and functionFieldEquiv.
(cohomologyZeroEquivRiemannRochSpace hxt D).symm sends f ∈ L(D) to the degree-zero
cohomology class whose global section has rational function (functionFieldEquiv hxt).symm f.
dim_k H⁰(X_F, 𝒪_{X_F}(D)) = ℓ(D) over any base field k.
The first cohomology class in H¹(X_F, 𝒪_{X_F}(D)) attached to a repartition of F.
Instances For
repartitionToCohomologyOne transports a ∈ A_F along (functionFieldEquiv hxt).symm and
applies SchemeWeilDivisor.repartitionToCohomologyOne.
Every class in H¹(X_F, 𝒪_{X_F}(D)) is represented by a repartition of F.
First cohomology of 𝒪_{X_F}(D) as a repartition quotient of F:
A_F ⧸ (A_F(D) + F) ≃ₗ[k] H¹(X_F, 𝒪_{X_F}(D)).
Instances For
repartitionQuotientEquivCohomologyOne sends the class of a ∈ A_F to
repartitionToCohomologyOne hxt D a.
The kernel of repartitionToCohomologyOne hxt D is (A_F(equivDivisor hxt D) + F) ∩ A_F.
The dual of H¹(X_F, 𝒪_{X_F}(D)) is the space Ω_F(D) of Weil differentials of F bounded
by D. No exact-constant assumption is needed.
Instances For
The Weil differential on F attached to a linear form φ on H¹(X_F, 𝒪_{X_F}(D)) evaluates a
repartition a ∈ A_F by applying φ to repartitionToCohomologyOne hxt D a.
Exact constant field: h¹ = i(D) = ℓ(W - D), Riemann–Roch, and Serre duality #
dim_k H¹(X_F, 𝒪_{X_F}(D)) = i(D): when k is integrally closed in F, the first
cohomology dimension of 𝒪_{X_F}(D) equals the index of specialty of equivDivisor hxt D.
dim_k H¹(X_F, 𝒪_{X_F}(D)) = ℓ(W - D) for every canonical divisor W of F.
Serre duality on X_F, in dimensions: for every divisor K on X_F whose image in F
represents canonicalClass hF hk, dim_k H¹(X_F, 𝒪(D)) = dim_k H⁰(X_F, 𝒪(K - D)).
Serre duality on the two-chart model. Let ω be a nonzero Weil differential of F and K
the Weil divisor on X_F corresponding to (ω). For every Weil divisor D on X_F, the dual of
H¹(X_F, 𝒪(D)) is k-linearly isomorphic to H⁰(X_F, 𝒪(K - D)); a class in H⁰(X_F, 𝒪(K - D))
with underlying function f ∈ F pairs with the class of a ∈ A_F to ω(f · a)
(twoChartModel.cohomologyOneDualEquivCohomologyZero_symm_apply_repartitionToCohomologyOne).
Instances For
The Serre duality pairing on X_F in terms of F: a class s ∈ H⁰(X_F, 𝒪(K - D)) pairs
with the cohomology class of a repartition a ∈ A_F to ω(f · a), where f ∈ F is the function
attached to s by cohomologyZeroEquivRiemannRochSpace.
The Riemann–Roch theorem on X_F in Euler-characteristic form: when k is integrally
closed in F, χ(𝒪_{X_F}(D)) = deg D + 1 - g(F).
The Riemann–Roch theorem on X_F in dimension form: when k is integrally closed in F,
dim_k H⁰(X_F, 𝒪(D)) - dim_k H¹(X_F, 𝒪(D)) = deg D + 1 - g(F).
Canonical bundle and relative differentials [Ω_{X_F/k}] #
A canonical line bundle on X_F, obtained from InvertibleSheaf.canonicalBundle when k
is integrally closed in F. Its class in Pic(X_F) ≃+ Cl(F) is canonicalClass hF hk.
Instances For
A Weil divisor K on X_F represents the class of canonicalBundle if and only if
equivDivisor hxt K represents canonicalClass hF hk in Cl(F).
Under Pic(X_F) ≃+ Cl(F), the canonical bundle maps to canonicalClass hF hk.
The canonical bundle on X_F has Euler-characteristic degree 2g - 2.
dim_k H⁰(X_F, ω_{X_F}) = g(F).
dim_k H¹(X_F, ω_{X_F}) = 1.
When Ω_{X_F/k} is invertible and k is integrally closed in F, Ω_{X_F/k} is isomorphic
to twoChartModel.canonicalBundle.
When Ω_{X_F/k} is invertible, its class in Pic(X_F) ≃+ Cl(F) is canonicalClass hF hk.
When Ω_{X_F/k} is invertible, its Euler-characteristic degree is 2g - 2.
When Ω_{X_F/k} is invertible, dim_k H⁰(X_F, Ω_{X_F/k}) = g(F).
When Ω_{X_F/k} is invertible, dim_k H¹(X_F, Ω_{X_F/k}) = 1.
Line-bundle Riemann–Roch on X_F with Ω_{X_F/k}: when Ω_{X_F/k} is invertible and k
is integrally closed in F, every line bundle L on X_F satisfies
dim_k H⁰(X_F, L) - dim_k H⁰(X_F, Ω_{X_F/k} ⊗ L⁻¹) = deg L + 1 - g(F).
If twoChartModel hF hx is smooth over Spec k, it is smooth of relative dimension one.
If twoChartModel hF hx is smooth over Spec k, its sheaf of relative differentials
Ω_{X_F/k} is invertible.
On a smooth two-chart model, [Ω_{X_F/k}] maps to canonicalClass hF hk under
Pic(X_F) ≃+ Cl(F).
On a smooth two-chart model, Ω_{X_F/k} has Euler-characteristic degree 2g - 2.