atom-molecule correspondence in grothendieck categories ... · overview one-sided prime (atom)...

Atom-molecule correspondencein Grothendieck categories

with applications to noetherian rings

Ryo Kanda

Nagoya University

July 6, 2015

Aims of this talk

Investigate the relationship between one-sided primes andtwo-sided primes.

Refine the definition of integral noncommutative spaceintroduced by Paul Smith.

Introduce an operation called artinianization.

Give a categorical proof of Goldie’s theorem for rightnoetherian rings.

Λ : right noetherian ring

Theorem (Gabriel 1962)

We have maps

{ indecomposable injectives in ModΛ }∼=

ϕ

�ψ

SpecΛ

such that

Ass I = {ϕ(I)}E(Λ/P ) = ψ(P )⊕ · · · ⊕ ψ(P )

ϕψ = id(ϕ is surjective, ψ is injective)

ModΛ := { right Λ-modules }SpecΛ := { two-sided prime ideals of Λ }


Overview

one-sided prime two-sided prime

{ indec injs in ModΛ }∼=

ϕ //SpecΛ

ψoo


Overview

one-sided prime

(atom)

two-sided prime

(molecule)


ϕ //

OO

1−1��

SpecΛψ

oo OO

1−1��

ASpec(ModΛ)ϕ //

MSpec(ModΛ)ψ

oo


Overview

one-sided prime

(atom)

two-sided prime

(molecule)


ϕ //

OO

1−1��

SpecΛψ

oo OO

1−1��

ASpec(ModΛ)ϕ //

∪MSpec(ModΛ)

ψoo

∪AMin(ModΛ) MMin(ModΛ)


Overview

one-sided prime

(atom)

two-sided prime

(molecule)


ϕ //

OO

1−1��

SpecΛψ

oo OO

1−1��

ASpec(ModΛ)ϕ //

∪MSpec(ModΛ)

ψoo

∪AMin(ModΛ) oo

1−1 //MMin(ModΛ)

Atoms (=one-sided primes)

A : Grothendieck category (e.g. ModΛ for a ring Λ)

Definition

H ∈ A is called monoform if

H 6= 0

For every 0 6= L ⊂ H,

{ subobjects of H }∼=

∩ { subobjects of H/L }∼=

= {0}

Proposition

H ∈ A is monoform, 0 6= L ⊂ H =⇒ L is monoform.

Definition

H1 is called atom-equivalent to H2 if

{ subobjects of H1 }∼=

∩ { subobjects of H2 }∼=

6= {0}

Definition (Storrer 1972, K 2012)

The atom spectrum of A is

ASpecA :={monoform objects in A}

atom-equivalence.

H denotes the equivalence class. An atom is an element ofASpecA.

Proposition (Storrer 1972)

Let R be a commutative ring.

ASpec(ModR)1−1←→ SpecR

R/p ←[ p

Proposition

Let Λ be a right artinian ring.

ASpec(ModΛ)1−1←→ { simple Λ-modules }

∼=S ←[ S

A : locally noetherian Grothendieck category(e.g. ModΛ for a right noetherian ring Λ)

Theorem (Matlis 1958, K 2012)

ASpecA 1−1←→ { indecomposable injectives in A}∼=

H 7→ E(H)


Theorem (Gabriel 1962, Herzog 1997, Krause 1997, K 2012)

{ localizing subcats of A} 1−1←→ { localizing subsets of ASpecA}X 7→ ASpecX

Moreover,

ASpecAX

= ASpecA \ASpecX .

localizing subcat := full subcat closed under sub, quot, ext,⊕


Definition (K 2015)

Define a partial order ≤ on ASpecA by

α ≤ β ⇐⇒ ∀H = α, ∃L = β

such that L is a subquotient of H.

subquotient := subobj of a quot obj = quot obj of a subobj

Proposition

Let R be a commutative ring.

(ASpec(ModR),≤) ∼= (SpecR,⊂)

R/p ←[ p

A : Grothendieck category having a noetherian generator(e.g. ModΛ for a right noetherian ring Λ)

Theorem (K)

For each α ∈ ASpecA, there exists β ∈ AMinA satisfyingβ ≤ α.

# AMinA <∞.

There exists the smallest weakly closed subcategory Aa-red

satisfying ASpecAa-red = ASpecA.(atomically reduced part of A)

(Aa-red)a-red = Aa-red.

AMinA := {minimal elements of ASpecA}weakly closed subcat := full subcat closed under sub, quot,

⊕


Definition (K)

A is called

a-reduced if A = Aa-red

a-irreducible if # AMinA = 1

a-integral if A is a-reduced and a-irreducible

a- := atomically

Question (We will see the answer soon!)

When is ModΛ a-reduced/a-irreducible/a-integral?


Definition (K)

In fact, ASpecA \AMinA is a localizing subset.

Let X be the corresponding localizing subcat of A.

Aartin := A/X is called the artinianization of A.

Proposition

A ∼−→ Aartin ⇐⇒ A has a generator of finite length.

Theorem (Nastasescu 1981)

Let A be a Grothendieck category having an artinian generator.Then there exists a right artinian ring Λ such that A ∼= ModΛ.


Overview

one-sided prime

(atom)

two-sided prime

(molecule)


ϕ //

OO

1−1��

SpecΛψ

oo OO

1−1��

ASpec(ModΛ)ϕ //

∪MSpec(ModΛ)

ψoo

∪AMin(ModΛ) oo

1−1 //MMin(ModΛ)

Molecules (=two-sided primes)

Theorem (Rosenberg 1995)

Let Λ be a ring.

{ closed subcats of ModΛ } 1−1←→ { two-sided ideals of Λ }

ModΛ

I←[ I

ModΛ

J∗Mod

Λ

I←[ IJ

〈M〉closed ←[ AnnΛ(M)

closed subcat := full subcat closed under sub, quot,⊕

,∏

C1 ∗ C2 := {M ∈ A | ∃ 0→M1 →M →M2 → 0, Mi ∈ Ci }


Definition

H ∈ A is called prime if

H 6= 0

For every 0 6= L ⊂ H,

〈L〉closed = 〈H〉closed

Proposition

H ∈ A is prime, 0 6= L ⊂ H =⇒ L is prime.

Definition

H1 is called molecule-equivalent to H2 if

〈H1〉closed = 〈H2〉closed

Definition (K)

The molecule spectrum of A is

MSpecA :={ prime objects in A}molecule-equivalence

.

H denotes the equivalence class. A molecule is an element ofMSpecA.

Definition

Define a partial order ≤ on MSpecA by

ρ ≤ σ ⇐⇒ 〈ρ〉closed ⊃ 〈σ〉closed

If ρ = H, then 〈ρ〉closed := 〈H〉closed.

Proposition

Let Λ be a ring.

(MSpec(ModΛ),≤) ∼= (SpecΛ,⊂)

Λ/P ←[ P

SpecΛ := { two-sided prime ideals of Λ }


Proposition

For each ρ ∈ MSpecA, there exists σ ∈ MMinA satisfyingσ ≤ ρ.

# MMinA <∞.

There exists the smallest closed subcategory Am-red

satisfying MSpecAm-red = MSpecA.(molecularly reduced part of A)

(Am-red)m-red = Am-red.

MMinA := {minimal elements of MSpecA}closed subcat := full subcat closed under sub, quot,

⊕,∏


Definition (K)

A is called

m-reduced if A = Am-red

m-irreducible if # MMinA = 1

m-integral if A is m-reduced and m-irreducible

Proposition

Let Λ be a right noetherian ring. Then ModΛ is

m-reduced ⇐⇒ Λ is a semiprime ring

m-irreducible⇐⇒ the prime radical√

0 belongs to SpecΛ

m-integral ⇐⇒ Λ is a prime ring


Overview

one-sided prime

(atom)

two-sided prime

(molecule)


ϕ //

OO

1−1��

SpecΛψ

oo OO

1−1��

ASpec(ModΛ)ϕ //

∪MSpec(ModΛ)

ψoo

∪AMin(ModΛ) oo

1−1 //MMin(ModΛ)

Atom-molecule correspondence

A : Grothendieck cat having a noetherian generator, Ab4*(e.g. ModΛ for a right noetherian ring Λ)

Theorem (K)

ϕ : ASpecA → MSpecA given by H 7→ H is a surjectiveposet homomorphism.

ψ : MSpecA → ASpecA given by

ψ(ρ) = min{α ∈ ASpecA | ϕ(α) = ρ }

induces a poset isomorphism MSpecA ∼−→ Imψ.

ϕψ = id.

Atom-molecule correspondence


Theorem (K)

ASpecAϕ

�ψ

MSpecA induces AMinA ∼= MMinA.

Aa-red = Am-red.

Corollary (K)

A is a-reduced/a-irreducible/a-integral⇐⇒ A is m-reduced/m-irreducible/m-integral

Application

Theorem (Goldie 1960)

Every semiprime right noetherian ring Λ has a right quotientring Λ′, which is semisimple.

Sketch

ModΛ is m-reduced=⇒ ModΛ is a-reduced=⇒ (ModΛ)artin is semisimple=⇒ ModΛ→ (ModΛ)artin sends Λ to a projective generator P=⇒ Λ = EndΛ(Λ)→ End(P ) =: Λ′


Overview

one-sided prime

(atom)

two-sided prime

(molecule)


ϕ //

OO

1−1��

SpecΛψ

oo OO

1−1��

ASpec(ModΛ)ϕ //

∪MSpec(ModΛ)

ψoo

∪AMin(ModΛ) oo

1−1 //MMin(ModΛ)

Appendix


Definition

For each M ∈ A,

AAssM := {H ∈ ASpecA | H ⊂M }ASuppM := {H ∈ ASpecA | H is a subquot of M }

Proposition

Let 0→ L→M → N → 0 be an exact sequence.

AAssL ⊂ AAssM ⊂ AAssL ∪AAssN

ASuppM = ASuppL ∪ASuppN

Proposition

AAss⊕i∈I

Mi =⋃i∈I

AAssMi, ASupp⊕i∈I

Mi =⋃i∈I

ASuppMi


Theorem (Gabriel 1962, Herzog 1997, Krause 1997, K 2012)

{ localizing subcats of A} 1−1←→ { localizing subsets of ASpecA}

X 7→⋃M∈X

ASuppM

{M ∈ A | ASuppM ⊂ Φ } ←[ Φ

localizing subcat := full subcat closed under sub, quot, ext,⊕

localizing subset := union of subsets of the form ASuppM= subset of the form ASuppM


Key Lemma

Let M ∈ A satisfying AAssM ⊂ AMinA. Then

〈M〉weakly closed = 〈M〉closed

weakly closed subcat · · · sub, quot,⊕

closed subcat · · · sub, quot,⊕

,∏

atom-molecule correspondence in grothendieck categories ... · overview one-sided prime (atom)...

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