Prime spectrum of a commutative (semi)ring #
For the Zariski topology, see Mathlib.RingTheory.Spectrum.Prime.Topology
.
(It is also naturally endowed with a sheaf of rings,
which is constructed in AlgebraicGeometry.StructureSheaf
.)
Main definitions #
zeroLocus s
: The zero locus of a subsets
ofR
is the subset ofPrimeSpectrum R
consisting of all prime ideals that contains
.vanishingIdeal t
: The vanishing ideal of a subsett
ofPrimeSpectrum R
is the intersection of points int
(viewed as prime ideals).
Conventions #
We denote subsets of (semi)rings with s
, s'
, etc...
whereas we denote subsets of prime spectra with t
, t'
, etc...
Inspiration/contributors #
The contents of this file draw inspiration from https://github.com/ramonfmir/lean-scheme which has contributions from Ramon Fernandez Mir, Kevin Buzzard, Kenny Lau, and Chris Hughes (on an earlier repository).
References #
The prime spectrum of the zero ring is empty.
The prime spectrum is in bijection with the set of prime ideals.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The map from the direct sum of prime spectra to the prime spectrum of a direct product.
Equations
- PrimeSpectrum.primeSpectrumProdOfSum R S (Sum.inl { asIdeal := I, isPrime := isPrime }) = { asIdeal := I.prod ⊤, isPrime := ⋯ }
- PrimeSpectrum.primeSpectrumProdOfSum R S (Sum.inr { asIdeal := J, isPrime := isPrime }) = { asIdeal := ⊤.prod J, isPrime := ⋯ }
Instances For
The prime spectrum of R × S
is in bijection with the disjoint unions of the prime spectrum of
R
and the prime spectrum of S
.
Equations
Instances For
The zero locus of a set s
of elements of a commutative (semi)ring R
is the set of all
prime ideals of the ring that contain the set s
.
An element f
of R
can be thought of as a dependent function on the prime spectrum of R
.
At a point x
(a prime ideal) the function (i.e., element) f
takes values in the quotient ring
R
modulo the prime ideal x
. In this manner, zeroLocus s
is exactly the subset of
PrimeSpectrum R
where all "functions" in s
vanish simultaneously.
Equations
- PrimeSpectrum.zeroLocus s = {x : PrimeSpectrum R | s ⊆ ↑x.asIdeal}
Instances For
The vanishing ideal of a set t
of points of the prime spectrum of a commutative ring R
is
the intersection of all the prime ideals in the set t
.
An element f
of R
can be thought of as a dependent function on the prime spectrum of R
.
At a point x
(a prime ideal) the function (i.e., element) f
takes values in the quotient ring
R
modulo the prime ideal x
. In this manner, vanishingIdeal t
is exactly the ideal of R
consisting of all "functions" that vanish on all of t
.
Equations
- PrimeSpectrum.vanishingIdeal t = ⨅ x ∈ t, x.asIdeal
Instances For
zeroLocus
and vanishingIdeal
form a galois connection.
zeroLocus
and vanishingIdeal
form a galois connection.
The specialization order #
We endow PrimeSpectrum R
with a partial order induced from the ideal lattice.
This is exactly the specialization order.
See the corresponding section at Mathlib.RingTheory.Spectrum.Prime.Topology
.
Equations
Equations
- PrimeSpectrum.instUnique = { default := ⊥, uniq := ⋯ }
Also see PrimeSpectrum.isClosed_singleton_iff_isMaximal
In a noetherian ring, every ideal contains a product of prime ideals ([samuel1967, § 3.3, Lemma 3]).
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 ([samuel1967, § 3.3, Lemma 3])