The structure sheaf on PrimeSpectrum R
. #
We define the structure sheaf on TopCat.of (PrimeSpectrum R)
, for a commutative ring R
and prove
basic properties about it. We define this as a subsheaf of the sheaf of dependent functions into the
localizations, cut out by the condition that the function must be locally equal to a ratio of
elements of R
.
Because the condition "is equal to a fraction" passes to smaller open subsets, the subset of functions satisfying this condition is automatically a subpresheaf. Because the condition "is locally equal to a fraction" is local, it is also a subsheaf.
(It may be helpful to refer back to Mathlib/Topology/Sheaves/SheafOfFunctions.lean
,
where we show that dependent functions into any type family form a sheaf,
and also Mathlib/Topology/Sheaves/LocalPredicate.lean
, where we characterise the predicates
which pick out sub-presheaves and sub-sheaves of these sheaves.)
We also set up the ring structure, obtaining
structureSheaf : Sheaf CommRingCat (PrimeSpectrum.Top R)
.
We then construct two basic isomorphisms, relating the structure sheaf to the underlying ring R
.
First, StructureSheaf.stalkIso
gives an isomorphism between the stalk of the structure sheaf
at a point p
and the localization of R
at the prime ideal p
. Second,
StructureSheaf.basicOpenIso
gives an isomorphism between the structure sheaf on basicOpen f
and the localization of R
at the submonoid of powers of f
.
References #
- [Robin Hartshorne, Algebraic Geometry][Har77]
The prime spectrum, just as a topological space.
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The type family over PrimeSpectrum R
consisting of the localization over each point.
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- ⋯ = ⋯
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- AlgebraicGeometry.StructureSheaf.instInhabitedLocalizations R P = { default := 1 }
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- ⋯ = ⋯
The predicate saying that a dependent function on an open U
is realised as a fixed fraction
r / s
in each of the stalks (which are localizations at various prime ideals).
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- One or more equations did not get rendered due to their size.
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The predicate IsFraction
is "prelocal",
in the sense that if it holds on U
it holds on any open subset V
of U
.
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- One or more equations did not get rendered due to their size.
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We will define the structure sheaf as
the subsheaf of all dependent functions in Π x : U, Localizations R x
consisting of those functions which can locally be expressed as a ratio of
(the images in the localization of) elements of R
.
Quoting Hartshorne:
For an open set $U ⊆ Spec A$, we define $𝒪(U)$ to be the set of functions $s : U → ⨆_{𝔭 ∈ U} A_𝔭$, such that $s(𝔭) ∈ A_𝔭$ for each $𝔭$, and such that $s$ is locally a quotient of elements of $A$: to be precise, we require that for each $𝔭 ∈ U$, there is a neighborhood $V$ of $𝔭$, contained in $U$, and elements $a, f ∈ A$, such that for each $𝔮 ∈ V, f ∉ 𝔮$, and $s(𝔮) = a/f$ in $A_𝔮$.
Now Hartshorne had the disadvantage of not knowing about dependent functions,
so we replace his circumlocution about functions into a disjoint union with
Π x : U, Localizations x
.
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The functions satisfying isLocallyFraction
form a subring.
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- One or more equations did not get rendered due to their size.
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The structure sheaf (valued in Type
, not yet CommRingCat
) is the subsheaf consisting of
functions satisfying isLocallyFraction
.
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The structure presheaf, valued in CommRingCat
, constructed by dressing up the Type
valued
structure presheaf.
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- One or more equations did not get rendered due to their size.
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Some glue, verifying that the structure presheaf valued in CommRingCat
agrees
with the Type
valued structure presheaf.
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- One or more equations did not get rendered due to their size.
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The structure sheaf on $Spec R$, valued in CommRingCat
.
This is provided as a bundled SheafedSpace
as Spec.SheafedSpace R
later.
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- AlgebraicGeometry.Spec.structureSheaf R = { val := AlgebraicGeometry.structurePresheafInCommRing R, cond := ⋯ }
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The section of structureSheaf R
on an open U
sending each x ∈ U
to the element
f/g
in the localization of R
at x
.
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- AlgebraicGeometry.StructureSheaf.const R f g U hu = ⟨fun (x : ↥(Opposite.unop (Opposite.op U))) => IsLocalization.mk' (AlgebraicGeometry.StructureSheaf.Localizations R ↑x) f ⟨g, ⋯⟩, ⋯⟩
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The canonical ring homomorphism interpreting an element of R
as
a section of the structure sheaf.
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- One or more equations did not get rendered due to their size.
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The canonical ring homomorphism interpreting an element of R
as an element of
the stalk of structureSheaf R
at x
.
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- One or more equations did not get rendered due to their size.
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The canonical ring homomorphism from the localization of R
at p
to the stalk
of the structure sheaf at the point p
.
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The ring homomorphism that takes a section of the structure sheaf of R
on the open set U
,
implemented as a subtype of dependent functions to localizations at prime ideals, and evaluates
the section on the point corresponding to a given prime ideal.
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- One or more equations did not get rendered due to their size.
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The ring homomorphism from the stalk of the structure sheaf of R
at a point corresponding to
a prime ideal p
to the localization of R
at p
,
formed by gluing the openToLocalization
maps.
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- One or more equations did not get rendered due to their size.
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The ring isomorphism between the stalk of the structure sheaf of R
at a point p
corresponding to a prime ideal in R
and the localization of R
at p
.
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- One or more equations did not get rendered due to their size.
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- ⋯ = ⋯
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- ⋯ = ⋯
The canonical ring homomorphism interpreting s ∈ R_f
as a section of the structure sheaf
on the basic open defined by f ∈ R
.
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- ⋯ = ⋯
The ring isomorphism between the structure sheaf on basicOpen f
and the localization of R
at the submonoid of powers of f
.
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- AlgebraicGeometry.StructureSheaf.basicOpenIso R f = (CategoryTheory.asIso (let_fun this := AlgebraicGeometry.StructureSheaf.toBasicOpen R f; this)).symm
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Stalk of the structure sheaf at a prime p as localization of R
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- ⋯ = ⋯
Sections of the structure sheaf of Spec R on a basic open as localization of R
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- ⋯ = ⋯
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- ⋯ = ⋯
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- ⋯ = ⋯
The ring isomorphism between the ring R
and the global sections Γ(X, 𝒪ₓ)
.
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Given a ring homomorphism f : R →+* S
, an open set U
of the prime spectrum of R
and an open
set V
of the prime spectrum of S
, such that V ⊆ (comap f) ⁻¹' U
, we can push a section s
on U
to a section on V
, by composing with Localization.localRingHom _ _ f
from the left and
comap f
from the right. Explicitly, if s
evaluates on comap f p
to a / b
, its image on V
evaluates on p
to f(a) / f(b)
.
At the moment, we work with arbitrary dependent functions s : Π x : U, Localizations R x
. Below,
we prove the predicate isLocallyFraction
is preserved by this map, hence it can be extended to
a morphism between the structure sheaves of R
and S
.
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- AlgebraicGeometry.StructureSheaf.comapFun f U V hUV s y = (Localization.localRingHom ((PrimeSpectrum.comap f) ↑y).asIdeal (↑y).asIdeal f ⋯) (s ⟨(PrimeSpectrum.comap f) ↑y, ⋯⟩)
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For a ring homomorphism f : R →+* S
and open sets U
and V
of the prime spectra of R
and
S
such that V ⊆ (comap f) ⁻¹ U
, the induced ring homomorphism from the structure sheaf of R
at U
to the structure sheaf of S
at V
.
Explicitly, this map is given as follows: For a point p : V
, if the section s
evaluates on p
to the fraction a / b
, its image on V
evaluates on p
to the fraction f(a) / f(b)
.
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- One or more equations did not get rendered due to their size.
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For an inclusion i : V ⟶ U
between open sets of the prime spectrum of R
, the comap of the
identity from OO_X(U) to OO_X(V) equals as the restriction map of the structure sheaf.
This is a generalization of the fact that, for fixed U
, the comap of the identity from OO_X(U)
to OO_X(U) is the identity.
The comap of the identity is the identity. In this variant of the lemma, two open subsets U
and
V
are given as arguments, together with a proof that U = V
. This is useful when U
and V
are not definitionally equal.