# Coherent Sheaves

- [Introduction](introduction.md)

- [Creation Functions](creation.md)

  - [`Sheaf(M, X): ModMPol, Sch → ShfCoh`](creation.md#function-sheaf-modmpol-sch)

  - [`StructureSheaf(X): Sch → ShfCoh`](creation.md#function-structuresheaf-sch)

  - [`StructureSheaf(X, n): Sch, RngIntElt → ShfCoh`](creation.md#function-structuresheaf-sch-rngintelt)

  - [`CanonicalSheaf(X): Sch → ShfCoh`](creation.md#function-canonicalsheaf-sch)

  - [`CanonicalSheaf(X, n): Sch, RngIntElt → ShfCoh`](creation.md#function-canonicalsheaf-sch-rngintelt)

  - [`Twist(S, n): ShfCoh, RngIntElt → ShfCoh`](creation.md#function-twist-shfcoh-rngintelt)

  - [`Example: Creat Exs`](creation.md#example-ex-ac24b9)

  - [`SheafOfDifferentials(X): Sch → ShfCoh`](creation.md#function-sheafofdifferentials-sch)

  - [`TangentSheaf(X): Sch → ShfCoh`](creation.md#function-tangentsheaf-sch)

  - [`HorrocksMumfordBundle(P): Prj → ShfCoh`](creation.md#function-horrocksmumfordbundle-prj)

- [Accessor Functions](accessors.md)

  - [`Module(S): ShfCoh → ModMPol`](accessors.md#function-module-shfcoh)

  - [`Scheme(S): ShfCoh → Sch`](accessors.md#function-scheme-shfcoh)

  - [`FullModule(S): ShfCoh → ModMPol`](accessors.md#function-fullmodule-shfcoh)

  - [`GlobalSectionSubmodule(S): ShfCoh → ModMPol`](accessors.md#function-globalsectionsubmodule-shfcoh)

  - [`SaturateSheaf(~S): ShfCoh`](accessors.md#function-saturatesheaf-shfcoh-ref)

  - [`Example: Access Exs`](accessors.md#example-ex-0b6d9f)

- [Basic Constructions](more-cons.md)

  - [`TensorProduct(S, T): ShfCoh, ShfCoh → ShfCoh`](more-cons.md#function-tensorproduct-shfcoh-shfcoh)

  - [`TensorPower(S, n): ShfCoh, RngIntElt → ShfCoh`](more-cons.md#function-tensorpower-shfcoh-rngintelt)

  - [`Dual(S): ShfCoh → ShfCoh`](more-cons.md#function-dual-shfcoh)

  - [`SheafHoms(S, T): ShfCoh, ShfCoh → ShfCoh, Map`](more-cons.md#function-shf-shfhoms)

  - [`DirectSum(S, T): ShfCoh, ShfCoh → ShfCoh`](more-cons.md#function-directsum-shfcoh-shfcoh)

  - [`Restriction(S, Y): ShfCoh, Sch → ShfCoh`](more-cons.md#function-restriction-shfcoh-sch)

  - [`Example: Cons Exs`](more-cons.md#example-ex-22d504)

- [Sheaf Homomorphisms](sheaf-homs.md)

  - [`SheafHomomorphism(S, T, h): ShfCoh, ShfCoh, ModMPolHom → ShfHom`](sheaf-homs.md#function-sheafhomomorphism-shfcoh-shfcoh-modmpolhom)

  - [`Domain(f): ShfHom → ShfCoh`](sheaf-homs.md#function-domain-shfhom)

  - [`Codomain(f): ShfHom → ShfCoh`](sheaf-homs.md#function-codomain-shfhom)

  - [`Degree(f): ShfHom → RngIntElt`](sheaf-homs.md#function-degree-shfhom)

  - [`ModuleHomomorphism(f): ShfHom → ModMPolHom`](sheaf-homs.md#function-modulehomomorphism-shfhom)

  - [`Kernel(f): ShfHom → ShfCoh, ShfHom`](sheaf-homs.md#function-kernel-shfhom)

  - [`Image(f): ShfHom → ShfCoh, ShfHom, ShfHom`](sheaf-homs.md#function-image-shfhom)

  - [`Cokernel(f): ShfHom → ShfCoh, ShfHom`](sheaf-homs.md#function-cokernel-shfhom)

  - [`Expand(hms): SeqEnum[ShfHom] → ShfHom`](sheaf-homs.md#function-expand-seqenum-shfhom)

- [Divisor Maps and Riemann-Roch Spaces](div_maps.md)

  - [`DivisorMap(S): ShfCoh → Map, Sch`](div_maps.md#function-divisormap-shfcoh)

  - [`DivisorToSheaf(X, I): Sch, RngMPol → ShfCoh`](div_maps.md#function-shf-div-to-shf)

  - [`RiemannRochBasis(X, I): Sch, RngMPol → SeqEnum, RngMPolElt, ShfCoh`](div_maps.md#function-riemannrochbasis-sch-rngmpol)

  - [`IneffectiveDivisorToSheaf(X, I, J): Sch, RngMPol, RngMPol → ShfCoh`](div_maps.md#function-shf-ineff-div-to-shf)

  - [`IneffectiveRiemannRochBasis(X, I, J): Sch, RngMPol, RngMPol → SeqEnum, RngMPolElt, ShfCoh`](div_maps.md#function-ineffectiveriemannrochbasis-sch-rngmpol-rngmpol)

  - [`Example: Shf:rr Scroll`](div_maps.md#example-ex-72931f)

  - [`Example: shf:dp3to6`](div_maps.md#example-ex-774e58)

- [Predicates](shf:preds.md)

  - [`IsLocallyFree(S): ShfCoh → BoolElt, RngIntElt`](shf:preds.md#function-shf-loc-free)

  - [`IsIsomorphic(S, T): ShfCoh, ShfCoh → BoolElt, ShfHom`](shf:preds.md#function-isisomorphic-shfcoh-shfcoh)

  - [`IsIsomorphicWithTwist(S, T): ShfCoh, ShfCoh → BoolElt, RngIntElt, ShfHom`](shf:preds.md#function-isisomorphicwithtwist-shfcoh-shfcoh)

  - [`IsArithmeticallyCohenMacaulay(S): ShfCoh → BoolElt`](shf:preds.md#function-shf-arith-cm)

- [Miscellaneous](shf:misc.md)

  - [`CohomologyDimension(S, r, n): ShfCoh, RngIntElt, RngIntElt → RngIntElt`](shf:misc.md#function-cohomologydimension-shfcoh-rngintelt-rngintelt)

  - [`DimensionOfGlobalSections(S): ShfCoh → RngIntElt`](shf:misc.md#function-dimensionofglobalsections-shfcoh)

  - [`IntersectionPairing(S, T): ShfCoh, ShfCoh → RngIntElt`](shf:misc.md#function-intersectionpairing-shfcoh-shfcoh)

  - [`ZeroSubscheme(S, s): ShfCoh, ModMPolElt → Sch`](shf:misc.md#function-zerosubscheme-shfcoh-modmpolelt)

- [Examples](shf:exs.md)

  - [`Example: Big Ex`](shf:exs.md#example-ex-1d7a31)

  - [`Example: Ell Crv Norm`](shf:exs.md#example-ex-362695)