Documentation

Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf

The structure sheaf on ProjectiveSpectrum ๐’œ. #

In Mathlib/AlgebraicGeometry/Topology.lean, we have given a topology on ProjectiveSpectrum ๐’œ; in this file we will construct a sheaf on ProjectiveSpectrum ๐’œ.

Notation #

Main definitions and results #

We define the structure sheaf as the subsheaf of all dependent function f : ฮ  x : U, HomogeneousLocalization ๐’œ x such that f is locally expressible as ratio of two elements of the same grading, i.e. โˆ€ y โˆˆ U, โˆƒ (V โŠ† U) (i : โ„•) (a b โˆˆ ๐’œ i), โˆ€ z โˆˆ V, f z = a / b.

Then we establish that Proj ๐’œ is a LocallyRingedSpace:

References #

source
def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.IsFraction {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] {U : TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ)} (f : (x : โ†ฅU) โ†’ HomogeneousLocalization.AtPrime ๐’œ (โ†‘x).asHomogeneousIdeal.toIdeal) :
Prop

The predicate saying that a dependent function on an open U is realised as a fixed fraction r / s of same grading in each of the stalks (which are localizations at various prime ideals).

Equations
    Instances For
      source
      def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isFractionPrelocal {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
      TopCat.PrelocalPredicate fun (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) => HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal

      The predicate IsFraction is "prelocal", in the sense that if it holds on U it holds on any open subset V of U.

      Equations
        Instances For
          source
          def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
          TopCat.LocalPredicate fun (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) => HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal

          We will define the structure sheaf as the subsheaf of all dependent functions in ฮ  x : U, HomogeneousLocalization ๐’œ x consisting of those functions which can locally be expressed as a ratio of A of same grading.

          Equations
            Instances For
              source
              theorem AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.zero_mem' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) :
              (isLocallyFraction ๐’œ).pred 0
              source
              theorem AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.one_mem' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) :
              (isLocallyFraction ๐’œ).pred 1
              source
              theorem AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.add_mem' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) (a b : (x : โ†ฅ(Opposite.unop U)) โ†’ HomogeneousLocalization.AtPrime ๐’œ (โ†‘x).asHomogeneousIdeal.toIdeal) (ha : (isLocallyFraction ๐’œ).pred a) (hb : (isLocallyFraction ๐’œ).pred b) :
              (isLocallyFraction ๐’œ).pred (a + b)
              source
              theorem AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.neg_mem' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) (a : (x : โ†ฅ(Opposite.unop U)) โ†’ HomogeneousLocalization.AtPrime ๐’œ (โ†‘x).asHomogeneousIdeal.toIdeal) (ha : (isLocallyFraction ๐’œ).pred a) :
              (isLocallyFraction ๐’œ).pred (-a)
              source
              theorem AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.mul_mem' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) (a b : (x : โ†ฅ(Opposite.unop U)) โ†’ HomogeneousLocalization.AtPrime ๐’œ (โ†‘x).asHomogeneousIdeal.toIdeal) (ha : (isLocallyFraction ๐’œ).pred a) (hb : (isLocallyFraction ๐’œ).pred b) :
              (isLocallyFraction ๐’œ).pred (a * b)
              source
              def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.sectionsSubring {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) :
              Subring ((x : โ†ฅ(Opposite.unop U)) โ†’ HomogeneousLocalization.AtPrime ๐’œ (โ†‘x).asHomogeneousIdeal.toIdeal)

              The functions satisfying isLocallyFraction form a subring of all dependent functions ฮ  x : U, HomogeneousLocalization ๐’œ x.

              Equations
                Instances For
                  source
                  def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structureSheafInType {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
                  TopCat.Sheaf (Type u_2) (ProjectiveSpectrum.top ๐’œ)

                  The structure sheaf (valued in Type, not yet CommRing) is the subsheaf consisting of functions satisfying isLocallyFraction.

                  Equations
                    Instances For
                      source
                      instance AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.commRingStructureSheafInTypeObj {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) :
                      CommRing ((structureSheafInType ๐’œ).val.obj U)
                      Equations
                        source
                        def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
                        TopCat.Presheaf CommRingCat (ProjectiveSpectrum.top ๐’œ)

                        The structure presheaf, valued in CommRing, constructed by dressing up the Type valued structure presheaf.

                        Equations
                          Instances For
                            source
                            @[simp]
                            theorem AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj_carrier {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (U : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–) :
                            โ†‘((structurePresheafInCommRing ๐’œ).obj U) = (structureSheafInType ๐’œ).val.obj U
                            source
                            def AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafCompForget {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
                            CategoryTheory.Functor.comp (structurePresheafInCommRing ๐’œ) (CategoryTheory.forget CommRingCat) โ‰… (structureSheafInType ๐’œ).val

                            Some glue, verifying that the structure presheaf valued in CommRing agrees with the Type valued structure presheaf.

                            Equations
                              Instances For
                                source
                                def AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
                                TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top ๐’œ)

                                The structure sheaf on Proj ๐’œ, valued in CommRing.

                                Equations
                                  Instances For
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.res_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {U V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (i : V โŸถ U) (s : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)) (x : โ†ฅ(Opposite.unop U)) :
                                    โ†‘((CategoryTheory.ConcreteCategory.hom ((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.map i)) s) x = โ†‘s (i.unop x)
                                    source
                                    theorem AlgebraicGeometry.Proj.ext {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (s t : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)) (h : โ†‘s = โ†‘t) :
                                    s = t
                                    source
                                    theorem AlgebraicGeometry.Proj.ext_iff {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {๐’œ : โ„• โ†’ Submodule R A} [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} {s t : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)} :
                                    s = t โ†” โ†‘s = โ†‘t
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.add_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (s t : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)) (x : โ†ฅ(Opposite.unop V)) :
                                    โ†‘(s + t) x = โ†‘s x + โ†‘t x
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.mul_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (s t : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)) (x : โ†ฅ(Opposite.unop V)) :
                                    โ†‘(s * t) x = โ†‘s x * โ†‘t x
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.sub_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (s t : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)) (x : โ†ฅ(Opposite.unop V)) :
                                    โ†‘(s - t) x = โ†‘s x - โ†‘t x
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.pow_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (s : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj V)) (x : โ†ฅ(Opposite.unop V)) (n : โ„•) :
                                    โ†‘(s ^ n) x = โ†‘s x ^ n
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.zero_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (x : โ†ฅ(Opposite.unop V)) :
                                    โ†‘0 x = 0
                                    source
                                    @[simp]
                                    theorem AlgebraicGeometry.Proj.one_apply {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] {V : (TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ))แต’แต–} (x : โ†ฅ(Opposite.unop V)) :
                                    โ†‘1 x = 1
                                    source
                                    def AlgebraicGeometry.Proj.toSheafedSpace {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
                                    SheafedSpace CommRingCat

                                    Proj of a graded ring as a SheafedSpace

                                    Equations
                                      Instances For
                                        source
                                        def AlgebraicGeometry.openToLocalization {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (U : TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ)) (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (hx : x โˆˆ U) :
                                        (ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj (Opposite.op U) โŸถ CommRingCat.of (HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal)

                                        The ring homomorphism that takes a section of the structure sheaf of Proj on the open set U, implemented as a subtype of dependent functions to localizations at homogeneous prime ideals, and evaluates the section on the point corresponding to a given homogeneous prime ideal.

                                        Equations
                                          Instances For
                                            source
                                            def AlgebraicGeometry.stalkToFiberRingHom {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) :
                                            (ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.stalk x โŸถ CommRingCat.of (HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal)

                                            The ring homomorphism from the stalk of the structure sheaf of Proj at a point corresponding to a homogeneous prime ideal x to the homogeneous localization at x, formed by gluing the openToLocalization maps.

                                            Equations
                                              Instances For
                                                source
                                                @[simp]
                                                theorem AlgebraicGeometry.germ_comp_stalkToFiberRingHom {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (U : TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ)) (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (hx : x โˆˆ U) :
                                                CategoryTheory.CategoryStruct.comp ((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.germ U x hx) (stalkToFiberRingHom ๐’œ x) = openToLocalization ๐’œ U x hx
                                                source
                                                @[simp]
                                                theorem AlgebraicGeometry.stalkToFiberRingHom_germ {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (U : TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ)) (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (hx : x โˆˆ U) (s : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj (Opposite.op U))) :
                                                (CategoryTheory.ConcreteCategory.hom (stalkToFiberRingHom ๐’œ x)) ((CategoryTheory.ConcreteCategory.hom ((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.germ U x hx)) s) = โ†‘s โŸจx, hxโŸฉ
                                                source
                                                theorem AlgebraicGeometry.mem_basicOpen_den {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (f : HomogeneousLocalization.NumDenSameDeg ๐’œ x.asHomogeneousIdeal.toIdeal.primeCompl) :
                                                x โˆˆ ProjectiveSpectrum.basicOpen ๐’œ โ†‘f.den
                                                source
                                                def AlgebraicGeometry.sectionInBasicOpen {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (f : HomogeneousLocalization.NumDenSameDeg ๐’œ x.asHomogeneousIdeal.toIdeal.primeCompl) :
                                                โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj (Opposite.op (ProjectiveSpectrum.basicOpen ๐’œ โ†‘f.den)))

                                                Given a point x corresponding to a homogeneous prime ideal, there is a (dependent) function such that, for any f in the homogeneous localization at x, it returns the obvious section in the basic open set D(f.den).

                                                Equations
                                                  Instances For
                                                    source
                                                    def AlgebraicGeometry.homogeneousLocalizationToStalk {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (y : HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal) :
                                                    โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.stalk x)

                                                    Given any point x and f in the homogeneous localization at x, there is an element in the stalk at x obtained by sectionInBasicOpen. This is the inverse of stalkToFiberRingHom.

                                                    Equations
                                                      Instances For
                                                        source
                                                        theorem AlgebraicGeometry.homogeneousLocalizationToStalk_stalkToFiberRingHom {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (z : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.stalk x)) :
                                                        homogeneousLocalizationToStalk ๐’œ x ((CategoryTheory.ConcreteCategory.hom (stalkToFiberRingHom ๐’œ x)) z) = z
                                                        source
                                                        theorem AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (z : HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal) :
                                                        (CategoryTheory.ConcreteCategory.hom (stalkToFiberRingHom ๐’œ x)) (homogeneousLocalizationToStalk ๐’œ x z) = z
                                                        source
                                                        def AlgebraicGeometry.Proj.stalkIso' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) :
                                                        โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.stalk x) โ‰ƒ+* HomogeneousLocalization.AtPrime ๐’œ x.asHomogeneousIdeal.toIdeal

                                                        Using homogeneousLocalizationToStalk, we construct a ring isomorphism between stalk at x and homogeneous localization at x for any point x in Proj.

                                                        Equations
                                                          Instances For
                                                            source
                                                            @[simp]
                                                            theorem AlgebraicGeometry.Proj.stalkIso'_germ {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (U : TopologicalSpace.Opens โ†‘(ProjectiveSpectrum.top ๐’œ)) (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (hx : x โˆˆ U) (s : โ†‘((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).val.obj (Opposite.op U))) :
                                                            (stalkIso' ๐’œ x) ((CategoryTheory.ConcreteCategory.hom ((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.germ U x hx)) s) = โ†‘s โŸจx, hxโŸฉ
                                                            source
                                                            @[simp]
                                                            theorem AlgebraicGeometry.Proj.stalkIso'_symm_mk {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] (x : โ†‘(ProjectiveSpectrum.top ๐’œ)) (f : HomogeneousLocalization.NumDenSameDeg ๐’œ x.asHomogeneousIdeal.toIdeal.primeCompl) :
                                                            (stalkIso' ๐’œ x).symm (HomogeneousLocalization.mk f) = (CategoryTheory.ConcreteCategory.hom ((ProjectiveSpectrum.Proj.structureSheaf ๐’œ).presheaf.germ (ProjectiveSpectrum.basicOpen ๐’œ โ†‘f.den) x โ‹ฏ)) (sectionInBasicOpen ๐’œ x f)
                                                            source
                                                            def AlgebraicGeometry.Proj.toLocallyRingedSpace {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (๐’œ : โ„• โ†’ Submodule R A) [GradedAlgebra ๐’œ] :
                                                            LocallyRingedSpace

                                                            Proj of a graded ring as a LocallyRingedSpace

                                                            Equations
                                                              Instances For