statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_closed_range_comap_of_surjective (hf : surjective f) : is_closed (set.range (comap f)) | begin
rw range_comap_of_surjective _ f hf,
exact is_closed_zero_locus ↑(ker f),
end | lemma | prime_spectrum.is_closed_range_comap_of_surjective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_closed",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_comap_of_surjective (hf : surjective f) : closed_embedding (comap f) | { induced := (comap_inducing_of_surjective S f hf).induced,
inj := comap_injective_of_surjective f hf,
closed_range := is_closed_range_comap_of_surjective S f hf } | lemma | prime_spectrum.closed_embedding_comap_of_surjective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"closed_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open (r : R) : topological_space.opens (prime_spectrum R) | { carrier := { x | r ∉ x.as_ideal },
is_open' := ⟨{r}, set.ext $ λ x, set.singleton_subset_iff.trans $ not_not.symm⟩ } | def | prime_spectrum.basic_open | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set.ext",
"topological_space.opens"
] | `basic_open r` is the open subset containing all prime ideals not containing `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_basic_open (f : R) (x : prime_spectrum R) :
x ∈ basic_open f ↔ f ∉ x.as_ideal | iff.rfl | lemma | prime_spectrum.mem_basic_open | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_basic_open {a : R} : is_open ((basic_open a) : set (prime_spectrum R)) | (basic_open a).is_open | lemma | prime_spectrum.is_open_basic_open | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_open",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_zero_locus_compl (r : R) :
(basic_open r : set (prime_spectrum R)) = (zero_locus {r})ᶜ | set.ext $ λ x, by simpa only [set.mem_compl_iff, mem_zero_locus, set.singleton_subset_iff] | lemma | prime_spectrum.basic_open_eq_zero_locus_compl | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set.ext",
"set.mem_compl_iff",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_one : basic_open (1 : R) = ⊤ | topological_space.opens.ext $ by simp | lemma | prime_spectrum.basic_open_one | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_zero : basic_open (0 : R) = ⊥ | topological_space.opens.ext $ by simp | lemma | prime_spectrum.basic_open_zero | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_le_basic_open_iff (f g : R) :
basic_open f ≤ basic_open g ↔ f ∈ (ideal.span ({g} : set R)).radical | by rw [← set_like.coe_subset_coe, basic_open_eq_zero_locus_compl,
basic_open_eq_zero_locus_compl, set.compl_subset_compl,
zero_locus_subset_zero_locus_singleton_iff] | lemma | prime_spectrum.basic_open_le_basic_open_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.span",
"set.compl_subset_compl",
"set_like.coe_subset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul (f g : R) : basic_open (f * g) = basic_open f ⊓ basic_open g | topological_space.opens.ext $ by {simp [zero_locus_singleton_mul]} | lemma | prime_spectrum.basic_open_mul | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul_le_left (f g : R) : basic_open (f * g) ≤ basic_open f | by { rw basic_open_mul f g, exact inf_le_left } | lemma | prime_spectrum.basic_open_mul_le_left | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul_le_right (f g : R) : basic_open (f * g) ≤ basic_open g | by { rw basic_open_mul f g, exact inf_le_right } | lemma | prime_spectrum.basic_open_mul_le_right | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_pow (f : R) (n : ℕ) (hn : 0 < n) : basic_open (f ^ n) = basic_open f | topological_space.opens.ext $ by simpa using zero_locus_singleton_pow f n hn | lemma | prime_spectrum.basic_open_pow | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"topological_space.opens.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_basic_opens : topological_space.is_topological_basis
(set.range (λ (r : R), (basic_open r : set (prime_spectrum R)))) | begin
apply topological_space.is_topological_basis_of_open_of_nhds,
{ rintros _ ⟨r, rfl⟩,
exact is_open_basic_open },
{ rintros p U hp ⟨s, hs⟩,
rw [← compl_compl U, set.mem_compl_iff, ← hs, mem_zero_locus, set.not_subset] at hp,
obtain ⟨f, hfs, hfp⟩ := hp,
refine ⟨basic_open f, ⟨f, rfl⟩, hfp, _⟩,
... | lemma | prime_spectrum.is_topological_basis_basic_opens | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"compl_compl",
"is_topological_basis_basic_opens",
"prime_spectrum",
"set.compl_subset_compl",
"set.mem_compl_iff",
"set.not_subset",
"set.range",
"topological_space.is_topological_basis",
"topological_space.is_topological_basis_of_open_of_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_basis_basic_opens :
topological_space.opens.is_basis (set.range (@basic_open R _)) | begin
unfold topological_space.opens.is_basis,
convert is_topological_basis_basic_opens,
rw ← set.range_comp,
end | lemma | prime_spectrum.is_basis_basic_opens | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_topological_basis_basic_opens",
"set.range",
"set.range_comp",
"topological_space.opens.is_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_basic_open (f : R) : is_compact (basic_open f : set (prime_spectrum R)) | is_compact_of_finite_subfamily_closed $ λ ι Z hZc hZ,
begin
let I : ι → ideal R := λ i, vanishing_ideal (Z i),
have hI : ∀ i, Z i = zero_locus (I i) := λ i,
by simpa only [zero_locus_vanishing_ideal_eq_closure] using (hZc i).closure_eq.symm,
rw [basic_open_eq_zero_locus_compl f, set.inter_comm, ← set.diff_eq,... | lemma | prime_spectrum.is_compact_basic_open | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"is_compact",
"is_compact_of_finite_subfamily_closed",
"prime_spectrum",
"set.diff_eq",
"set.diff_eq_empty",
"set.inter_comm",
"set.mem_singleton",
"set.singleton_subset_iff",
"submodule.exists_finset_of_mem_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_bot_iff (f : R) :
basic_open f = ⊥ ↔ is_nilpotent f | begin
rw [← topological_space.opens.coe_inj, basic_open_eq_zero_locus_compl],
simp only [set.eq_univ_iff_forall, set.singleton_subset_iff,
topological_space.opens.coe_bot, nilpotent_iff_mem_prime, set.compl_empty_iff, mem_zero_locus,
set_like.mem_coe],
exact ⟨λ h I hI, h ⟨I, hI⟩, λ h ⟨I, hI⟩, h I hI⟩
end | lemma | prime_spectrum.basic_open_eq_bot_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_nilpotent",
"nilpotent_iff_mem_prime",
"set.compl_empty_iff",
"set.eq_univ_iff_forall",
"set.singleton_subset_iff",
"set_like.mem_coe",
"topological_space.opens.coe_bot",
"topological_space.opens.coe_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_away_comap_range (S : Type v) [comm_ring S] [algebra R S] (r : R)
[is_localization.away r S] : set.range (comap (algebra_map R S)) = basic_open r | begin
rw localization_comap_range S (submonoid.powers r),
ext,
simp only [mem_zero_locus, basic_open_eq_zero_locus_compl, set_like.mem_coe, set.mem_set_of_eq,
set.singleton_subset_iff, set.mem_compl_iff, disjoint_iff_inf_le],
split,
{ intros h₁ h₂,
exact h₁ ⟨submonoid.mem_powers r, h₂⟩ },
{ rintros ... | lemma | prime_spectrum.localization_away_comap_range | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra",
"algebra_map",
"comm_ring",
"disjoint_iff_inf_le",
"is_localization.away",
"set.mem_compl_iff",
"set.range",
"set.singleton_subset_iff",
"set_like.mem_coe",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_away_open_embedding (S : Type v) [comm_ring S] [algebra R S] (r : R)
[is_localization.away r S] : open_embedding (comap (algebra_map R S)) | { to_embedding := localization_comap_embedding S (submonoid.powers r),
open_range := by { rw localization_away_comap_range S r, exact is_open_basic_open } } | lemma | prime_spectrum.localization_away_open_embedding | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_localization.away",
"open_embedding",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_ideal_le_as_ideal (x y : prime_spectrum R) : x.as_ideal ≤ y.as_ideal ↔ x ≤ y | iff.rfl | lemma | prime_spectrum.as_ideal_le_as_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"as_ideal_le_as_ideal",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_ideal_lt_as_ideal (x y : prime_spectrum R) : x.as_ideal < y.as_ideal ↔ x < y | iff.rfl | lemma | prime_spectrum.as_ideal_lt_as_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"as_ideal_lt_as_ideal",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_mem_closure (x y : prime_spectrum R) :
x ≤ y ↔ y ∈ closure ({x} : set (prime_spectrum R)) | by rw [← as_ideal_le_as_ideal, ← zero_locus_vanishing_ideal_eq_closure,
mem_zero_locus, vanishing_ideal_singleton, set_like.coe_subset_coe] | lemma | prime_spectrum.le_iff_mem_closure | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"as_ideal_le_as_ideal",
"closure",
"le_iff_mem_closure",
"prime_spectrum",
"set_like.coe_subset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_specializes (x y : prime_spectrum R) :
x ≤ y ↔ x ⤳ y | (le_iff_mem_closure x y).trans specializes_iff_mem_closure.symm | lemma | prime_spectrum.le_iff_specializes | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"le_iff_mem_closure",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_order_embedding : prime_spectrum R ↪o filter (prime_spectrum R) | order_embedding.of_map_le_iff nhds $ λ a b, (le_iff_specializes a b).symm | def | prime_spectrum.nhds_order_embedding | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"filter",
"nhds",
"order_embedding.of_map_le_iff",
"prime_spectrum"
] | `nhds` as an order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_map_of_specializes {x y : prime_spectrum R} (h : x ⤳ y) :
localization.at_prime y.as_ideal →+* localization.at_prime x.as_ideal | @is_localization.lift _ _ _ _ _ _ _ _
localization.is_localization (algebra_map R (localization.at_prime x.as_ideal))
begin
rintro ⟨a, ha⟩,
rw [← prime_spectrum.le_iff_specializes, ← as_ideal_le_as_ideal, ← set_like.coe_subset_coe,
← set.compl_subset_compl] at h,
exact (is_localization.map_units _... | def | prime_spectrum.localization_map_of_specializes | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra_map",
"as_ideal_le_as_ideal",
"is_localization.lift",
"localization.at_prime",
"prime_spectrum",
"prime_spectrum.le_iff_specializes",
"set.compl_subset_compl",
"set_like.coe_subset_coe"
] | If `x` specializes to `y`, then there is a natural map from the localization of `y` to the
localization of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_point : prime_spectrum R | ⟨maximal_ideal R, (maximal_ideal.is_maximal R).is_prime⟩ | def | local_ring.closed_point | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | The closed point in the prime spectrum of a local ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_ring_hom_iff_comap_closed_point {S : Type v} [comm_ring S] [local_ring S]
(f : R →+* S) : is_local_ring_hom f ↔ prime_spectrum.comap f (closed_point S) = closed_point R | by { rw [(local_hom_tfae f).out 0 4, prime_spectrum.ext_iff], refl } | lemma | local_ring.is_local_ring_hom_iff_comap_closed_point | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"comm_ring",
"is_local_ring_hom",
"local_ring",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_closed_point {S : Type v} [comm_ring S] [local_ring S] (f : R →+* S)
[is_local_ring_hom f] : prime_spectrum.comap f (closed_point S) = closed_point R | (is_local_ring_hom_iff_comap_closed_point f).mp infer_instance | lemma | local_ring.comap_closed_point | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"comm_ring",
"is_local_ring_hom",
"local_ring",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_closed_point (x : prime_spectrum R) :
x ⤳ closed_point R | (prime_spectrum.le_iff_specializes _ _).mp (local_ring.le_maximal_ideal x.2.1) | lemma | local_ring.specializes_closed_point | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"local_ring.le_maximal_ideal",
"prime_spectrum",
"prime_spectrum.le_iff_specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_point_mem_iff (U : topological_space.opens $ prime_spectrum R) :
closed_point R ∈ U ↔ U = ⊤ | begin
split,
{ rw eq_top_iff, exact λ h x _, (specializes_closed_point x).mem_open U.2 h },
{ rintro rfl, trivial }
end | lemma | local_ring.closed_point_mem_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"eq_top_iff",
"prime_spectrum",
"topological_space.opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.prime_spectrum.comap_residue (x : prime_spectrum (residue_field R)) :
prime_spectrum.comap (residue R) x = closed_point R | begin
rw subsingleton.elim x ⊥,
ext1,
exact ideal.mk_ker,
end | lemma | prime_spectrum.comap_residue | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.mk_ker",
"prime_spectrum",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_of_Df (f) : set (prime_spectrum R) | {p : prime_spectrum R | ∃ i : ℕ , (coeff f i) ∉ p.as_ideal} | def | algebraic_geometry.polynomial.image_of_Df | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/is_open_comap_C.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.polynomial.basic"
] | [
"prime_spectrum"
] | Given a polynomial `f ∈ R[x]`, `image_of_Df` is the subset of `Spec R` where at least one
of the coefficients of `f` does not vanish. Lemma `image_of_Df_eq_comap_C_compl_zero_locus`
proves that `image_of_Df` is the image of `(zero_locus {f})ᶜ` under the morphism
`comap C : Spec R[x] → Spec R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_image_of_Df : is_open (image_of_Df f) | begin
rw [image_of_Df, set_of_exists (λ i (x : prime_spectrum R), coeff f i ∉ x.as_ideal)],
exact is_open_Union (λ i, is_open_basic_open),
end | lemma | algebraic_geometry.polynomial.is_open_image_of_Df | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/is_open_comap_C.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.polynomial.basic"
] | [
"is_open",
"is_open_Union",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_C_mem_image_of_Df {I : prime_spectrum R[X]}
(H : I ∈ (zero_locus {f} : set (prime_spectrum R[X]))ᶜ ) :
prime_spectrum.comap (polynomial.C : R →+* R[X]) I ∈ image_of_Df f | exists_C_coeff_not_mem (mem_compl_zero_locus_iff_not_mem.mp H) | lemma | algebraic_geometry.polynomial.comap_C_mem_image_of_Df | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/is_open_comap_C.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.polynomial.basic"
] | [
"polynomial.C",
"prime_spectrum",
"prime_spectrum.comap"
] | If a point of `Spec R[x]` is not contained in the vanishing set of `f`, then its image in
`Spec R` is contained in the open set where at least one of the coefficients of `f` is non-zero.
This lemma is a reformulation of `exists_C_coeff_not_mem`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_of_Df_eq_comap_C_compl_zero_locus :
image_of_Df f = prime_spectrum.comap (C : R →+* R[X]) '' (zero_locus {f})ᶜ | begin
ext x,
refine ⟨λ hx, ⟨⟨map C x.as_ideal, (is_prime_map_C_of_is_prime x.is_prime)⟩, ⟨_, _⟩⟩, _⟩,
{ rw [mem_compl_iff, mem_zero_locus, singleton_subset_iff],
cases hx with i hi,
exact λ a, hi (mem_map_C_iff.mp a i) },
{ ext x,
refine ⟨λ h, _, λ h, subset_span (mem_image_of_mem C.1 h)⟩,
rw ← ... | lemma | algebraic_geometry.polynomial.image_of_Df_eq_comap_C_compl_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/is_open_comap_C.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.polynomial.basic"
] | [
"prime_spectrum.comap"
] | The open set `image_of_Df f` coincides with the image of `basic_open f` under the
morphism `C⁺ : Spec R[x] → Spec R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_comap_C :
is_open_map (prime_spectrum.comap (C : R →+* R[X])) | begin
rintros U ⟨s, z⟩,
rw [← compl_compl U, ← z, ← Union_of_singleton_coe s, zero_locus_Union, compl_Inter, image_Union],
simp_rw [← image_of_Df_eq_comap_C_compl_zero_locus],
exact is_open_Union (λ f, is_open_image_of_Df),
end | theorem | algebraic_geometry.polynomial.is_open_map_comap_C | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/is_open_comap_C.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.polynomial.basic"
] | [
"compl_compl",
"is_open_Union",
"is_open_map",
"prime_spectrum.comap"
] | The morphism `C⁺ : Spec R[x] → Spec R` is open.
Stacks Project "Lemma 00FB", first part.
https://stacks.math.columbia.edu/tag/00FB | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_spectrum | (as_ideal : ideal R)
(is_maximal : as_ideal.is_maximal) | structure | maximal_spectrum | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"ideal"
] | The maximal spectrum of a commutative ring `R` is the type of all maximal ideals of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prime_spectrum (x : maximal_spectrum R) : prime_spectrum R | ⟨x.as_ideal, x.is_maximal.is_prime⟩ | def | maximal_spectrum.to_prime_spectrum | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"maximal_spectrum",
"prime_spectrum"
] | The natural inclusion from the maximal spectrum to the prime spectrum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prime_spectrum_injective : (@to_prime_spectrum R _).injective | λ ⟨_, _⟩ ⟨_, _⟩ h, by simpa only [mk.inj_eq] using (prime_spectrum.ext_iff _ _).mp h | lemma | maximal_spectrum.to_prime_spectrum_injective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prime_spectrum_range :
set.range (@to_prime_spectrum R _) = {x | is_closed ({x} : set $ prime_spectrum R)} | begin
simp only [is_closed_singleton_iff_is_maximal],
ext ⟨x, _⟩,
exact ⟨λ ⟨y, hy⟩, hy ▸ y.is_maximal, λ hx, ⟨⟨x, hx⟩, rfl⟩⟩
end | lemma | maximal_spectrum.to_prime_spectrum_range | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"is_closed",
"prime_spectrum",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zariski_topology : topological_space $ maximal_spectrum R | prime_spectrum.zariski_topology.induced to_prime_spectrum | instance | maximal_spectrum.zariski_topology | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"maximal_spectrum",
"topological_space"
] | The Zariski topology on the maximal spectrum of a commutative ring is defined as the subspace
topology induced by the natural inclusion into the prime spectrum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prime_spectrum_continuous : continuous $ @to_prime_spectrum R _ | continuous_induced_dom | lemma | maximal_spectrum.to_prime_spectrum_continuous | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"continuous",
"continuous_induced_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_localization_eq_bot :
(⨅ v : maximal_spectrum R,
localization.subalgebra.of_field K _ v.as_ideal.prime_compl_le_non_zero_divisors) = ⊥ | begin
ext x,
rw [algebra.mem_bot, algebra.mem_infi],
split,
{ apply imp_of_not_imp_not,
intros hrange hlocal,
let denom : ideal R := (submodule.span R {1} : submodule R K).colon (submodule.span R {x}),
have hdenom : (1 : R) ∉ denom :=
begin
intro hdenom,
rcases submodule.mem_span_sin... | theorem | maximal_spectrum.infi_localization_eq_bot | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"algebra.mem_bot",
"algebra.mem_infi",
"algebra.smul_def",
"algebra_map",
"ideal",
"imp_of_not_imp_not",
"inv_mul_cancel_right₀",
"inv_one",
"localization.subalgebra.of_field",
"map_mul",
"map_one",
"maximal_spectrum",
"mul_comm",
"mul_one",
"no_zero_smul_divisors.algebra_map_injective",... | An integral domain is equal to the intersection of its localizations at all its maximal ideals
viewed as subalgebras of its field of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_localization_eq_bot :
(⨅ v : prime_spectrum R,
localization.subalgebra.of_field K _ $ v.as_ideal.prime_compl_le_non_zero_divisors) = ⊥ | begin
ext x,
rw [algebra.mem_infi],
split,
{ rw [← maximal_spectrum.infi_localization_eq_bot, algebra.mem_infi],
exact λ hx ⟨v, hv⟩, hx ⟨v, hv.is_prime⟩ },
{ rw [algebra.mem_bot],
rintro ⟨y, rfl⟩ ⟨v, hv⟩,
exact ⟨y, 1, v.ne_top_iff_one.mp hv.ne_top, by rw [map_one, inv_one, mul_one]⟩ }
end | theorem | prime_spectrum.infi_localization_eq_bot | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/maximal.lean | [
"algebraic_geometry.prime_spectrum.basic",
"ring_theory.localization.as_subring"
] | [
"algebra.mem_bot",
"algebra.mem_infi",
"inv_one",
"localization.subalgebra.of_field",
"map_one",
"maximal_spectrum.infi_localization_eq_bot",
"mul_one",
"prime_spectrum"
] | An integral domain is equal to the intersection of its localizations at all its prime ideals
viewed as subalgebras of its field of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_prime_spectrum_prod_le (I : ideal R) :
∃ (Z : multiset (prime_spectrum R)), multiset.prod (Z.map as_ideal) ≤ I | begin
refine is_noetherian.induction (λ (M : ideal R) hgt, _) I,
by_cases h_prM : M.is_prime,
{ use {⟨M, h_prM⟩},
rw [multiset.map_singleton, multiset.prod_singleton],
exact le_rfl },
by_cases htop : M = ⊤,
{ rw htop,
exact ⟨0, le_top⟩ },
have lt_add : ∀ z ∉ M, M < M + span R {z},
{ intros z h... | lemma | prime_spectrum.exists_prime_spectrum_prod_le | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/noetherian.lean | [
"algebraic_geometry.prime_spectrum.basic",
"topology.noetherian_space"
] | [
"ideal",
"ideal.mem_sup_right",
"ideal.mul_le_left",
"ideal.mul_le_right",
"is_noetherian.induction",
"le_rfl",
"le_sup_left",
"multiset",
"multiset.map_add",
"multiset.map_singleton",
"multiset.prod",
"multiset.prod_add",
"multiset.prod_singleton",
"prime_spectrum",
"set.singleton_mul_s... | In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3]) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_prime_spectrum_prod_le_and_ne_bot_of_domain
(h_fA : ¬ is_field A) {I : ideal A} (h_nzI: I ≠ ⊥) :
∃ (Z : multiset (prime_spectrum A)), multiset.prod (Z.map as_ideal) ≤ I ∧
multiset.prod (Z.map as_ideal) ≠ ⊥ | begin
revert h_nzI,
refine is_noetherian.induction (λ (M : ideal A) hgt, _) I,
intro h_nzM,
have hA_nont : nontrivial A,
apply is_domain.to_nontrivial A,
by_cases h_topM : M = ⊤,
{ rcases h_topM with rfl,
obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ (p : ideal A), p ≠ ⊥ ∧ p.is_prime,
{ apply ring.not_is_field_i... | lemma | prime_spectrum.exists_prime_spectrum_prod_le_and_ne_bot_of_domain | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/noetherian.lean | [
"algebraic_geometry.prime_spectrum.basic",
"topology.noetherian_space"
] | [
"ideal",
"ideal.mul_le_left",
"ideal.mul_le_right",
"is_field",
"is_noetherian.induction",
"le_sup_left",
"le_top",
"multiset",
"multiset.map_add",
"multiset.map_singleton",
"multiset.prod",
"multiset.prod_add",
"multiset.prod_singleton",
"ne_bot_of_gt",
"nontrivial",
"prime_spectrum",... | In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
product or prime ideals ([samuel, § 3.3, Lemma 3]) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
carrier : ideal (A⁰_ f) | ideal.comap (algebra_map (A⁰_ f) (away f))
(ideal.span $ algebra_map A (away f) '' x.val.as_homogeneous_ideal) | def | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra_map",
"ideal",
"ideal.comap",
"ideal.span"
] | For any `x` in `Proj| (pbo f)`, the corresponding ideal in `Spec A⁰_f`. This fact that this ideal
is prime is proven in `Top_component.forward.to_fun` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier_iff (z : A⁰_ f) :
z ∈ carrier 𝒜 x ↔
z.val ∈ ideal.span (algebra_map A (away f) '' x.1.as_homogeneous_ideal) | iff.rfl | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra_map",
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier.clear_denominator' [decidable_eq (away f)]
{z : localization.away f}
(hz : z ∈ span ((algebra_map A (away f)) '' x.val.as_homogeneous_ideal)) :
∃ (c : algebra_map A (away f) '' x.1.as_homogeneous_ideal →₀ away f)
(N : ℕ) (acd : Π y ∈ c.support.image c, A),
f ^ N • z = algebra_map A (away f)
... | begin
rw [←submodule_span_eq, finsupp.span_eq_range_total, linear_map.mem_range] at hz,
rcases hz with ⟨c, eq1⟩,
rw [finsupp.total_apply, finsupp.sum] at eq1,
obtain ⟨⟨_, N, rfl⟩, hN⟩ := is_localization.exist_integer_multiples_of_finset (submonoid.powers f)
(c.support.image c),
choose acd hacd using hN,
... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra_map",
"finsupp.span_eq_range_total",
"finsupp.total_apply",
"is_localization.exist_integer_multiples_of_finset",
"linear_map.mem_range",
"localization.away",
"smul_eq_mul",
"smul_mul_assoc",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier.clear_denominator [decidable_eq (away f)]
{z : A⁰_ f} (hz : z ∈ carrier 𝒜 x) :
∃ (c : algebra_map A (away f) '' x.1.as_homogeneous_ideal →₀ away f)
(N : ℕ) (acd : Π y ∈ c.support.image c, A),
f ^ N • z.val = algebra_map A (away f)
(∑ i in c.support.attach, acd (c i) (finset.mem_image.mpr ... | mem_carrier.clear_denominator' x $ (mem_carrier_iff 𝒜 x z).mpr hz | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint :
(disjoint (x.1.as_homogeneous_ideal.to_ideal : set A) (submonoid.powers f : set A)) | begin
by_contra rid,
rw [set.not_disjoint_iff] at rid,
choose g hg using rid,
obtain ⟨hg1, ⟨k, rfl⟩⟩ := hg,
by_cases k_ineq : 0 < k,
{ erw x.1.is_prime.pow_mem_iff_mem _ k_ineq at hg1,
exact x.2 hg1 },
{ erw [show k = 0, by linarith, pow_zero, ←ideal.eq_top_iff_one] at hg1,
apply x.1.is_prime.1,
... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.disjoint | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"by_contra",
"disjoint",
"pow_zero",
"set.not_disjoint_iff",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_ne_top :
carrier 𝒜 x ≠ ⊤ | begin
have eq_top := disjoint x,
classical,
contrapose! eq_top,
obtain ⟨c, N, acd, eq1⟩ := mem_carrier.clear_denominator _ x ((ideal.eq_top_iff_one _).mp eq_top),
rw [algebra.smul_def, homogeneous_localization.one_val, mul_one] at eq1,
change localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1,
simp only [mk... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier_ne_top | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra.smul_def",
"disjoint",
"homogeneous_localization.one_val",
"ideal.eq_top_iff_one",
"is_localization.eq",
"localization.mk",
"mk_eq_mk'",
"mul_one",
"one_mul",
"set.not_disjoint_iff_nonempty_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun (x : Proj.T| (pbo f)) : (Spec.T (A⁰_ f)) | ⟨carrier 𝒜 x, carrier_ne_top x, λ x1 x2 hx12, begin
classical, simp only [mem_carrier_iff] at hx12 ⊢,
let J := span (⇑(algebra_map A (away f)) '' x.val.as_homogeneous_ideal),
suffices h : ∀ (x y : localization.away f), x * y ∈ J → x ∈ J ∨ y ∈ J,
{ rw [homogeneous_localization.mul_val] at hx12, exact h x1.val x... | def | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.to_fun | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra.smul_def",
"algebra_map",
"homogeneous_localization.mul_val",
"ideal.mul_mem_right",
"ideal.subset_span",
"is_localization.eq",
"localization.away",
"localization.induction_on",
"localization.mk",
"localization.mk_mul",
"mk_eq_mk'",
"mul_one",
"one_mul",
"submonoid.coe_one"
] | The function between the basic open set `D(f)` in `Proj` to the corresponding basic open set in
`Spec A⁰_f`. This is bundled into a continuous map in `Top_component.forward`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 𝒜 k)
(b_mem2 : b ∈ submonoid.powers f) : to_fun 𝒜 f ⁻¹'
((@prime_spectrum.basic_open (A⁰_ f) _
(quotient.mk' ⟨k, ⟨a, a_mem⟩, ⟨b, b_mem1⟩, b_mem2⟩)) :
set (prime_spectrum (homogeneous_localization.away 𝒜 f)))
= {x | x.1 ∈ (pbo f) ⊓ (... | begin
classical,
ext1 y, split; intros hy,
{ refine ⟨y.2, _⟩,
rw [set.mem_preimage, set_like.mem_coe, prime_spectrum.mem_basic_open] at hy,
rw projective_spectrum.mem_coe_basic_open,
intro a_mem_y,
apply hy,
rw [to_fun, mem_carrier_iff, homogeneous_localization.val_mk', subtype.coe_mk],
ds... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra.smul_def",
"homogeneous_localization.away",
"homogeneous_localization.val_mk'",
"ideal.mul_mem_left",
"ideal.subset_span",
"is_localization.eq",
"localization.mk",
"mk_eq_mk'",
"mul_comm",
"mul_one",
"one_mul",
"prime_spectrum",
"prime_spectrum.basic_open",
"prime_spectrum.mem_bas... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Spec {f : A} : (Proj.T| (pbo f)) ⟶ (Spec.T (A⁰_ f)) | { to_fun := to_Spec.to_fun 𝒜 f,
continuous_to_fun := begin
apply is_topological_basis.continuous (prime_spectrum.is_topological_basis_basic_opens),
rintros _ ⟨⟨k, ⟨a, ha⟩, ⟨b, hb1⟩, ⟨k', hb2⟩⟩, rfl⟩, dsimp,
erw to_Spec.preimage_eq f a b k ha hb1 ⟨k', hb2⟩,
refine is_open_induced_iff.mpr ⟨(pbo f).1 ⊓ ... | def | algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"is_open.inter",
"prime_spectrum.is_topological_basis_basic_opens",
"set.mem_preimage"
] | The continuous function between the basic open set `D(f)` in `Proj` to the corresponding basic
open set in `Spec A⁰_f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_tac : tactic unit | let b : tactic unit :=
`[exact pow_mem_graded _ (submodule.coe_mem _) <|> exact nat_cast_mem_graded _ _ <|>
exact pow_mem_graded _ f_deg] in
b <|> `[by repeat { all_goals { apply graded_monoid.mul_mem } }; b] | def | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_tac | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"submodule.coe_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier (q : Spec.T (A⁰_ f)) : set A | {a | ∀ i, (quotient.mk' ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
⟨f^i, by rw mul_comm; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.1} | def | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"mul_comm",
"quotient.mk'"
] | The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
`q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
of the `i`-th projection of `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier_iff (q : Spec.T (A⁰_ f)) (a : A) :
a ∈ carrier f_deg q ↔
∀ i, (quotient.mk' ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩, ⟨f^i, by rw mul_comm; mem_tac⟩, ⟨_, rfl⟩⟩
: A⁰_ f) ∈ q.1 | iff.rfl | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"mul_comm",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier_iff' (q : Spec.T (A⁰_ f)) (a : A) :
a ∈ carrier f_deg q ↔
∀ i, (localization.mk (proj 𝒜 i a ^ m) ⟨f^i, ⟨i, rfl⟩⟩ : localization.away f) ∈
(algebra_map (homogeneous_localization.away 𝒜 f) (localization.away f)) '' q.1.1 | (mem_carrier_iff f_deg q a).trans begin
split; intros h i; specialize h i,
{ rw set.mem_image, refine ⟨_, h, rfl⟩, },
{ rw set.mem_image at h, rcases h with ⟨x, h, hx⟩,
convert h, rw [ext_iff_val, val_mk'], dsimp only [subtype.coe_mk], rw ←hx, refl, },
end | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"algebra_map",
"homogeneous_localization.away",
"localization.away",
"localization.mk",
"set.mem_image",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.add_mem (q : Spec.T (A⁰_ f)) {a b : A} (ha : a ∈ carrier f_deg q)
(hb : b ∈ carrier f_deg q) :
a + b ∈ carrier f_deg q | begin
refine λ i, (q.2.mem_or_mem _).elim id id,
change (quotient.mk' ⟨_, _, _, _⟩ : A⁰_ f) ∈ q.1, dsimp only [subtype.coe_mk],
simp_rw [←pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul],
let g : ℕ → A⁰_ f := λ j, (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m
then quotient.mk' ⟨m * i, ⟨... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.add_mem | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"add_pow",
"add_smul",
"algebra_map",
"homogeneous_localization.away",
"homogeneous_localization.smul_val",
"homogeneous_localization.val",
"localization.away",
"mul_assoc",
"mul_comm",
"nsmul_eq_mul",
"pow_add",
"quotient.mk'",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.zero_mem : (0 : A) ∈ carrier f_deg q | λ i, begin
convert submodule.zero_mem q.1 using 1,
rw [ext_iff_val, val_mk', zero_val], simp_rw [map_zero, zero_pow hm],
convert localization.mk_zero _ using 1,
end | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"localization.mk_zero",
"submodule.zero_mem",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q | begin
revert c,
refine direct_sum.decomposition.induction_on 𝒜 _ _ _,
{ rw zero_smul, exact carrier.zero_mem f_deg hm _ },
{ rintros n ⟨a, ha⟩ i,
simp_rw [subtype.coe_mk, proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha],
split_ifs,
{ convert_to (quotient.mk' ⟨_, ⟨a^m, pow_mem_graded ... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.smul_mem | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"add_smul",
"direct_sum.decomposition.induction_on",
"ideal.mul_mem_left",
"localization.mk_mul",
"mul_comm",
"mul_pow",
"pow_add",
"quotient.mk'",
"smul_eq_mul",
"subtype.coe_mk",
"zero_pow",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.as_ideal : ideal A | { carrier := carrier f_deg q,
zero_mem' := carrier.zero_mem f_deg hm q,
add_mem' := λ a b, carrier.add_mem f_deg q,
smul_mem' := carrier.smul_mem f_deg hm q } | def | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"ideal"
] | For a prime ideal `q` in `A⁰_f`, the set `{a | aᵢᵐ/fⁱ ∈ q}` as an ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
carrier.as_ideal.homogeneous : (carrier.as_ideal f_deg hm q).is_homogeneous 𝒜 | λ i a ha j, (em (i = j)).elim
(λ h, h ▸ by simpa only [proj_apply, decompose_coe, of_eq_same] using ha _)
(λ h, begin
simp only [proj_apply, decompose_of_mem_ne 𝒜 (submodule.coe_mem (decompose 𝒜 a i)) h,
zero_pow hm], convert carrier.zero_mem f_deg hm q j, rw [map_zero, zero_pow hm],
end) | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.homogeneous | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"em",
"submodule.coe_mem",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.as_homogeneous_ideal : homogeneous_ideal 𝒜 | ⟨carrier.as_ideal f_deg hm q, carrier.as_ideal.homogeneous f_deg hm q⟩ | def | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_homogeneous_ideal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"homogeneous_ideal"
] | For a prime ideal `q` in `A⁰_f`, the set `{a | aᵢᵐ/fⁱ ∈ q}` as a homogeneous ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
carrier.denom_not_mem : f ∉ carrier.as_ideal f_deg hm q | λ rid, q.is_prime.ne_top $ (ideal.eq_top_iff_one _).mpr
begin
convert rid m,
simpa only [ext_iff_val, one_val, proj_apply, decompose_of_mem_same _ f_deg, val_mk'] using
(mk_self (⟨_, m, rfl⟩ : submonoid.powers f)).symm,
end | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.denom_not_mem | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"ideal.eq_top_iff_one",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.relevant :
¬homogeneous_ideal.irrelevant 𝒜 ≤ carrier.as_homogeneous_ideal f_deg hm q | λ rid, carrier.denom_not_mem f_deg hm q $ rid $ direct_sum.decompose_of_mem_ne 𝒜 f_deg hm.ne' | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.relevant | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"direct_sum.decompose_of_mem_ne",
"homogeneous_ideal.irrelevant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.as_ideal.ne_top : (carrier.as_ideal f_deg hm q) ≠ ⊤ | λ rid, carrier.denom_not_mem f_deg hm q (rid.symm ▸ submodule.mem_top) | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.ne_top | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier.as_ideal.prime : (carrier.as_ideal f_deg hm q).is_prime | (carrier.as_ideal.homogeneous f_deg hm q).is_prime_of_homogeneous_mem_or_mem
(carrier.as_ideal.ne_top f_deg hm q) $ λ x y ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy,
show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _, begin
rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left],
{ apply q.2.mem_or_mem, convert hxy (nx + ny) using 1,
... | lemma | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.prime | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"and_forall_ne",
"mul_pow",
"pow_add",
"subtype.coe_mk",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun : (Spec.T (A⁰_ f)) → (Proj.T| (pbo f)) | λ q, ⟨⟨carrier.as_homogeneous_ideal f_deg hm q, carrier.as_ideal.prime f_deg hm q,
carrier.relevant f_deg hm q⟩,
(projective_spectrum.mem_basic_open _ f _).mp $ carrier.denom_not_mem f_deg hm q⟩ | def | algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.to_fun | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/scheme.lean | [
"algebraic_geometry.projective_spectrum.structure_sheaf",
"algebraic_geometry.Spec",
"ring_theory.graded_algebra.radical"
] | [
"projective_spectrum.mem_basic_open"
] | The function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction {U : opens (projective_spectrum.Top 𝒜)} (f : Π x : U, at x.1) : Prop | ∃ (i : ℕ) (r s : 𝒜 i),
∀ x : U, ∃ (s_nin : s.1 ∉ x.1.as_homogeneous_ideal),
(f x) = quotient.mk' ⟨i, r, s, s_nin⟩ | def | algebraic_geometry.projective_spectrum.structure_sheaf.is_fraction | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top",
"quotient.mk'"
] | 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). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction_prelocal : prelocal_predicate (λ (x : projective_spectrum.Top 𝒜), at x) | { pred := λ U f, is_fraction f,
res := by rintros V U i f ⟨j, r, s, w⟩; exact ⟨j, r, s, λ y, w (i y)⟩ } | def | algebraic_geometry.projective_spectrum.structure_sheaf.is_fraction_prelocal | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | The predicate `is_fraction` is "prelocal", in the sense that if it holds on `U` it holds on any open
subset `V` of `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_locally_fraction : local_predicate (λ (x : projective_spectrum.Top 𝒜), at x) | (is_fraction_prelocal 𝒜).sheafify | def | algebraic_geometry.projective_spectrum.structure_sheaf.is_locally_fraction | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | We will define the structure sheaf as the subsheaf of all dependent functions in
`Π x : U, homogeneous_localization 𝒜 x` consisting of those functions which can locally be expressed
as a ratio of `A` of same grading. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mem' (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ) :
(is_locally_fraction 𝒜).pred (0 : Π x : unop U, at x.1) | λ x, ⟨unop U, x.2, 𝟙 (unop U), ⟨0, ⟨0, zero_mem _⟩, ⟨1, one_mem⟩, λ y, ⟨_, rfl⟩⟩⟩ | lemma | algebraic_geometry.projective_spectrum.structure_sheaf.section_subring.zero_mem' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem' (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ) :
(is_locally_fraction 𝒜).pred (1 : Π x : unop U, at x.1) | λ x, ⟨unop U, x.2, 𝟙 (unop U), ⟨0, ⟨1, one_mem⟩, ⟨1, one_mem⟩, λ y, ⟨_, rfl⟩⟩⟩ | lemma | algebraic_geometry.projective_spectrum.structure_sheaf.section_subring.one_mem' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem' (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ)
(a b : Π x : unop U, at x.1)
(ha : (is_locally_fraction 𝒜).pred a) (hb : (is_locally_fraction 𝒜).pred b) :
(is_locally_fraction 𝒜).pred (a + b) | λ x,
begin
rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩,
rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩,
refine ⟨Va ⊓ Vb, ⟨ma, mb⟩, opens.inf_le_left _ _ ≫ ia, ja + jb,
⟨sb * ra + sa * rb, add_mem (add_comm jb ja ▸ mul_mem sb_mem ra_mem : sb * ra ∈ 𝒜 (ja + jb))
(... | lemma | algebraic_geometry.projective_spectrum.structure_sheaf.section_subring.add_mem' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"mul_comm",
"projective_spectrum.Top",
"ring_hom.map_mul",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mem' (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ)
(a : Π x : unop U, at x.1)
(ha : (is_locally_fraction 𝒜).pred a) :
(is_locally_fraction 𝒜).pred (-a) | λ x,
begin
rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, w⟩,
choose nin hy using w,
refine ⟨V, m, i, j, ⟨-r, submodule.neg_mem _ r_mem⟩, ⟨s, s_mem⟩, λ y, ⟨nin y, _⟩⟩,
simp only [ext_iff_val, val_mk', ←subtype.val_eq_coe] at hy,
simp only [pi.neg_apply, ext_iff_val, neg_val, hy, val_mk', ←subtype.val_e... | lemma | algebraic_geometry.projective_spectrum.structure_sheaf.section_subring.neg_mem' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top",
"submodule.neg_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem' (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ)
(a b : Π x : unop U, at x.1)
(ha : (is_locally_fraction 𝒜).pred a) (hb : (is_locally_fraction 𝒜).pred b) :
(is_locally_fraction 𝒜).pred (a * b) | λ x,
begin
rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩,
rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩,
refine ⟨Va ⊓ Vb, ⟨ma, mb⟩, opens.inf_le_left _ _ ≫ ia, ja + jb,
⟨ra * rb, set_like.mul_mem_graded ra_mem rb_mem⟩,
⟨sa * sb, set_like.mul_mem_graded sa_mem sb_mem... | lemma | algebraic_geometry.projective_spectrum.structure_sheaf.section_subring.mul_mem' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"pi.mul_apply",
"projective_spectrum.Top",
"ring_hom.map_mul",
"set_like.mul_mem_graded",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_subring (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ) : subring (Π x : unop U, at x.1) | { carrier := { f | (is_locally_fraction 𝒜).pred f },
zero_mem' := zero_mem' U,
one_mem' := one_mem' U,
add_mem' := add_mem' U,
neg_mem' := neg_mem' U,
mul_mem' := mul_mem' U } | def | algebraic_geometry.projective_spectrum.structure_sheaf.sections_subring | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top",
"subring"
] | The functions satisfying `is_locally_fraction` form a subring of all dependent functions
`Π x : U, homogeneous_localization 𝒜 x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
structure_sheaf_in_Type : sheaf Type* (projective_spectrum.Top 𝒜) | subsheaf_to_Types (is_locally_fraction 𝒜) | def | algebraic_geometry.projective_spectrum.structure_sheaf.structure_sheaf_in_Type | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | The structure sheaf (valued in `Type`, not yet `CommRing`) is the subsheaf consisting of
functions satisfying `is_locally_fraction`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring_structure_sheaf_in_Type_obj (U : (opens (projective_spectrum.Top 𝒜))ᵒᵖ) :
comm_ring ((structure_sheaf_in_Type 𝒜).1.obj U) | (sections_subring U).to_comm_ring | instance | algebraic_geometry.projective_spectrum.structure_sheaf.comm_ring_structure_sheaf_in_Type_obj | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"comm_ring",
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
structure_presheaf_in_CommRing : presheaf CommRing (projective_spectrum.Top 𝒜) | { obj := λ U, CommRing.of ((structure_sheaf_in_Type 𝒜).1.obj U),
map := λ U V i,
{ to_fun := ((structure_sheaf_in_Type 𝒜).1.map i),
map_zero' := rfl,
map_add' := λ x y, rfl,
map_one' := rfl,
map_mul' := λ x y, rfl, }, } | def | algebraic_geometry.projective_spectrum.structure_sheaf.structure_presheaf_in_CommRing | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing",
"CommRing.of",
"projective_spectrum.Top"
] | The structure presheaf, valued in `CommRing`, constructed by dressing up the `Type` valued
structure presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
structure_presheaf_comp_forget :
structure_presheaf_in_CommRing 𝒜 ⋙ (forget CommRing) ≅ (structure_sheaf_in_Type 𝒜).1 | nat_iso.of_components (λ U, iso.refl _) (by tidy) | def | algebraic_geometry.projective_spectrum.structure_sheaf.structure_presheaf_comp_forget | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing"
] | Some glue, verifying that that structure presheaf valued in `CommRing` agrees with the `Type`
valued structure presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Proj.structure_sheaf : sheaf CommRing (projective_spectrum.Top 𝒜) | ⟨structure_presheaf_in_CommRing 𝒜,
-- We check the sheaf condition under `forget CommRing`.
(is_sheaf_iff_is_sheaf_comp _ _).mpr
(is_sheaf_of_iso (structure_presheaf_comp_forget 𝒜).symm
(structure_sheaf_in_Type 𝒜).cond)⟩ | def | algebraic_geometry.projective_spectrum.Proj.structure_sheaf | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing",
"projective_spectrum.Top"
] | The structure sheaf on `Proj` 𝒜, valued in `CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res_apply (U V : opens (projective_spectrum.Top 𝒜)) (i : V ⟶ U)
(s : (Proj.structure_sheaf 𝒜).1.obj (op U)) (x : V) :
((Proj.structure_sheaf 𝒜).1.map i.op s).1 x = (s.1 (i x) : _) | rfl | lemma | algebraic_geometry.res_apply | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Proj.to_SheafedSpace : SheafedSpace CommRing | { carrier := Top.of (projective_spectrum 𝒜),
presheaf := (Proj.structure_sheaf 𝒜).1,
is_sheaf := (Proj.structure_sheaf 𝒜).2 } | def | algebraic_geometry.Proj.to_SheafedSpace | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing",
"Top.of",
"projective_spectrum"
] | `Proj` of a graded ring as a `SheafedSpace` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_to_localization (U : opens (projective_spectrum.Top 𝒜)) (x : projective_spectrum.Top 𝒜)
(hx : x ∈ U) :
(Proj.structure_sheaf 𝒜).1.obj (op U) ⟶ CommRing.of (at x) | { to_fun := λ s, (s.1 ⟨x, hx⟩ : _),
map_one' := rfl,
map_mul' := λ _ _, rfl,
map_zero' := rfl,
map_add' := λ _ _, rfl } | def | algebraic_geometry.open_to_localization | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing.of",
"projective_spectrum.Top"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber_ring_hom (x : projective_spectrum.Top 𝒜) :
(Proj.structure_sheaf 𝒜).presheaf.stalk x ⟶ CommRing.of (at x) | limits.colimit.desc (((open_nhds.inclusion x).op) ⋙ (Proj.structure_sheaf 𝒜).1)
{ X := _,
ι :=
{ app := λ U, open_to_localization 𝒜 ((open_nhds.inclusion _).obj (unop U)) x (unop U).2, } } | def | algebraic_geometry.stalk_to_fiber_ring_hom | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing.of",
"projective_spectrum.Top"
] | 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 `open_to_localization` maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_comp_stalk_to_fiber_ring_hom (U : opens (projective_spectrum.Top 𝒜)) (x : U) :
(Proj.structure_sheaf 𝒜).presheaf.germ x ≫ stalk_to_fiber_ring_hom 𝒜 x =
open_to_localization 𝒜 U x x.2 | limits.colimit.ι_desc _ _ | lemma | algebraic_geometry.germ_comp_stalk_to_fiber_ring_hom | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_ring_hom_germ' (U : opens (projective_spectrum.Top 𝒜))
(x : projective_spectrum.Top 𝒜) (hx : x ∈ U) (s : (Proj.structure_sheaf 𝒜).1.obj (op U)) :
stalk_to_fiber_ring_hom 𝒜 x
((Proj.structure_sheaf 𝒜).presheaf.germ ⟨x, hx⟩ s) = (s.1 ⟨x, hx⟩ : _) | ring_hom.ext_iff.1 (germ_comp_stalk_to_fiber_ring_hom 𝒜 U ⟨x, hx⟩ : _) s | lemma | algebraic_geometry.stalk_to_fiber_ring_hom_germ' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_ring_hom_germ (U : opens (projective_spectrum.Top 𝒜)) (x : U)
(s : (Proj.structure_sheaf 𝒜).1.obj (op U)) :
stalk_to_fiber_ring_hom 𝒜 x ((Proj.structure_sheaf 𝒜).presheaf.germ x s) = s.1 x | by { cases x, exact stalk_to_fiber_ring_hom_germ' 𝒜 U _ _ _ } | lemma | algebraic_geometry.stalk_to_fiber_ring_hom_germ | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_localization.mem_basic_open (x : projective_spectrum.Top 𝒜) (f : at x) :
x ∈ projective_spectrum.basic_open 𝒜 f.denom | by { rw projective_spectrum.mem_basic_open, exact f.denom_mem } | lemma | algebraic_geometry.homogeneous_localization.mem_basic_open | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top",
"projective_spectrum.basic_open",
"projective_spectrum.mem_basic_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
section_in_basic_open (x : projective_spectrum.Top 𝒜) :
Π (f : at x),
(Proj.structure_sheaf 𝒜).1.obj (op (projective_spectrum.basic_open 𝒜 f.denom)) | λ f, ⟨λ y, quotient.mk' ⟨f.deg, ⟨f.num, f.num_mem_deg⟩, ⟨f.denom, f.denom_mem_deg⟩, y.2⟩,
λ y, ⟨projective_spectrum.basic_open 𝒜 f.denom, y.2,
⟨𝟙 _, ⟨f.deg, ⟨⟨f.num, f.num_mem_deg⟩, ⟨f.denom, f.denom_mem_deg⟩,
λ z, ⟨z.2, rfl⟩⟩⟩⟩⟩⟩ | def | algebraic_geometry.section_in_basic_open | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top",
"projective_spectrum.basic_open",
"quotient.mk'"
] | 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.denom)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_localization_to_stalk (x : projective_spectrum.Top 𝒜) :
(at x) → (Proj.structure_sheaf 𝒜).presheaf.stalk x | λ f, (Proj.structure_sheaf 𝒜).presheaf.germ
(⟨x, homogeneous_localization.mem_basic_open _ x f⟩ : projective_spectrum.basic_open _ f.denom)
(section_in_basic_open _ x f) | def | algebraic_geometry.homogeneous_localization_to_stalk | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"projective_spectrum.Top",
"projective_spectrum.basic_open"
] | Given any point `x` and `f` in the homogeneous localization at `x`, there is an element in the
stalk at `x` obtained by `section_in_basic_open`. This is the inverse of `stalk_to_fiber_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Proj.stalk_iso' (x : projective_spectrum.Top 𝒜) :
(Proj.structure_sheaf 𝒜).presheaf.stalk x ≃+* CommRing.of (at x) | ring_equiv.of_bijective (stalk_to_fiber_ring_hom _ x)
⟨λ z1 z2 eq1, begin
obtain ⟨u1, memu1, s1, rfl⟩ := (Proj.structure_sheaf 𝒜).presheaf.germ_exist x z1,
obtain ⟨u2, memu2, s2, rfl⟩ := (Proj.structure_sheaf 𝒜).presheaf.germ_exist x z2,
obtain ⟨v1, memv1, i1, ⟨j1, ⟨a1, a1_mem⟩, ⟨b1, b1_mem⟩, hs1⟩⟩ := s1.2 ⟨x, ... | def | algebraic_geometry.Proj.stalk_iso' | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"CommRing.of",
"homogeneous_localization.ext_iff_val",
"homogeneous_localization.val_mk'",
"is_localization.eq",
"localization.at_prime",
"localization.mk",
"localization.mk_eq_mk'",
"projective_spectrum.Top",
"projective_spectrum.basic_open",
"quotient.eq",
"ring_equiv.of_bijective",
"subtype... | Using `homogeneous_localization_to_stalk`, we construct a ring isomorphism between stalk at `x`
and homogeneous localization at `x` for any point `x` in `Proj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Proj.to_LocallyRingedSpace : LocallyRingedSpace | { local_ring := λ x, @@ring_equiv.local_ring _
(show local_ring (at x), from infer_instance) _
(Proj.stalk_iso' 𝒜 x).symm,
..(Proj.to_SheafedSpace 𝒜) } | def | algebraic_geometry.Proj.to_LocallyRingedSpace | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/structure_sheaf.lean | [
"algebraic_geometry.projective_spectrum.topology",
"topology.sheaves.local_predicate",
"ring_theory.graded_algebra.homogeneous_localization",
"algebraic_geometry.locally_ringed_space"
] | [
"local_ring",
"ring_equiv.local_ring"
] | `Proj` of a graded ring as a `LocallyRingedSpace` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_spectrum | (as_homogeneous_ideal : homogeneous_ideal 𝒜)
(is_prime : as_homogeneous_ideal.to_ideal.is_prime)
(not_irrelevant_le : ¬(homogeneous_ideal.irrelevant 𝒜 ≤ as_homogeneous_ideal)) | structure | projective_spectrum | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"homogeneous_ideal",
"homogeneous_ideal.irrelevant"
] | The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals
that are prime and do not contain the irrelevant ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_locus (s : set A) : set (projective_spectrum 𝒜) | {x | s ⊆ x.as_homogeneous_ideal} | def | projective_spectrum.zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | The zero locus of a set `s` of elements of a commutative ring `A` is the set of all relevant
homogeneous prime ideals of the ring that contain the set `s`.
An element `f` of `A` can be thought of as a dependent function on the projective spectrum of `𝒜`.
At a point `x` (a homogeneous prime ideal) the function (i.e., ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_zero_locus (x : projective_spectrum 𝒜) (s : set A) :
x ∈ zero_locus 𝒜 s ↔ s ⊆ x.as_homogeneous_ideal | iff.rfl | lemma | projective_spectrum.mem_zero_locus | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"projective_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_span (s : set A) :
zero_locus 𝒜 (ideal.span s) = zero_locus 𝒜 s | by { ext x, exact (submodule.gi _ _).gc s x.as_homogeneous_ideal.to_ideal } | lemma | projective_spectrum.zero_locus_span | algebraic_geometry.projective_spectrum | src/algebraic_geometry/projective_spectrum/topology.lean | [
"ring_theory.graded_algebra.homogeneous_ideal",
"topology.category.Top.basic",
"topology.sets.opens"
] | [
"ideal.span",
"submodule.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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