Atom-molecule correspondence in Grothendieck categories with - - PowerPoint PPT Presentation

Atom-molecule correspondence in 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 and two-sided primes. Refine the definition of integral noncommutative space introduced by Paul Smith. Introduce an operation called artinianization. Give a categorical proof of Goldie’s theorem for right noetherian rings.

Λ : right noetherian ring Theorem (Gabriel 1962) We have maps { indecomposable injectives in Mod Λ } ∼ =

ϕ

⇄

ψ

Spec Λ such that Ass I = {ϕ(I)} E(Λ/P) = ψ(P) ⊕ · · · ⊕ ψ(P) ϕψ = id (ϕ is surjective, ψ is injective) Mod Λ := { right Λ-modules } Spec Λ := { two-sided prime ideals of Λ }

Λ : right noetherian ring Overview

• ne-sided prime

two-sided prime { indec injs in Mod Λ } ∼ =

ϕ

Spec Λ

ψ

Λ : right noetherian ring Overview

• ne-sided prime

(atom) two-sided prime (molecule) { indec injs in Mod Λ } ∼ =

ϕ

• 1−1
• Spec Λ

ψ

• 1−1
• ASpec(Mod Λ)

ϕ

MSpec(Mod Λ)

ψ

Λ : right noetherian ring Overview

• ne-sided prime

(atom) two-sided prime (molecule) { indec injs in Mod Λ } ∼ =

ϕ

• 1−1
• Spec Λ

ψ

• 1−1
• ASpec(Mod Λ)

ϕ

• ∪

MSpec(Mod Λ)

ψ

• ∪

AMin(Mod Λ) MMin(Mod Λ)

Λ : right noetherian ring Overview

• ne-sided prime

(atom) two-sided prime (molecule) { indec injs in Mod Λ } ∼ =

ϕ

• 1−1
• Spec Λ

ψ

• 1−1
• ASpec(Mod Λ)

ϕ

• ∪

MSpec(Mod Λ)

ψ

• ∪

AMin(Mod Λ)

1−1

MMin(Mod Λ)

Atoms (=one-sided primes)

A : Grothendieck category (e.g. Mod Λ for a ring Λ) Definition H ∈ A is called monoform if H = 0 For every 0 = L ⊂ H, { subobjects of H } ∼ = ∩ { subobjects of H/L } ∼ = = {0} Proposition H ∈ A is monoform, 0 = L ⊂ H = ⇒ L is monoform.

Definition H1 is called atom-equivalent to H2 if { subobjects of H1 } ∼ = ∩ { subobjects of H2 } ∼ = = {0} Definition (Storrer 1972, K 2012) The atom spectrum of A is ASpec A := { monoform objects in A } atom-equivalence . H denotes the equivalence class. An atom is an element of ASpec A.

Proposition (Storrer 1972) Let R be a commutative ring. ASpec(Mod R) 1−1 ← → Spec R R/p ← p Proposition Let Λ be a right artinian ring. ASpec(Mod Λ) 1−1 ← → { simple Λ-modules } ∼ = S ← S

A : locally noetherian Grothendieck category (e.g. Mod Λ for a right noetherian ring Λ) Theorem (Matlis 1958, K 2012) ASpec A 1−1 ← → { indecomposable injectives in A } ∼ = H → E(H)

A : locally noetherian Grothendieck category (e.g. Mod Λ for a right noetherian ring Λ) Theorem (Gabriel 1962, Herzog 1997, Krause 1997, K 2012) { localizing subcats of A } 1−1 ← → { localizing subsets of ASpec A } X → ASpec X Moreover, ASpec A X = ASpec A \ ASpec X. localizing subcat := full subcat closed under sub, quot, ext,

A : Grothendieck category (e.g. Mod Λ for a ring Λ) Definition (K 2015) Define a partial order ≤ on ASpec A 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(Mod R), ≤) ∼ = (Spec R, ⊂) R/p ← p

A : Grothendieck category having a noetherian generator (e.g. Mod Λ for a right noetherian ring Λ) Theorem (K) For each α ∈ ASpec A, there exists β ∈ AMin A satisfying β ≤ α. # AMin A < ∞. There exists the smallest weakly closed subcategory Aa-red satisfying ASpec Aa-red = ASpec A. (atomically reduced part of A) (Aa-red)a-red = Aa-red. AMin A := { minimal elements of ASpec A } weakly closed subcat := full subcat closed under sub, quot,

A : Grothendieck category having a noetherian generator (e.g. Mod Λ for a right noetherian ring Λ) Definition (K) A is called a-reduced if A = Aa-red a-irreducible if # AMin A = 1 a-integral if A is a-reduced and a-irreducible a- := atomically Question (We will see the answer soon!) When is Mod Λ a-reduced/a-irreducible/a-integral?

A : Grothendieck category having a noetherian generator (e.g. Mod Λ for a right noetherian ring Λ) Definition (K) In fact, ASpec A \ AMin A 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 (N˘ ast˘ asescu 1981) Let A be a Grothendieck category having an artinian generator. Then there exists a right artinian ring Λ such that A ∼ = Mod Λ.

Λ : right noetherian ring Overview

• ne-sided prime

(atom) two-sided prime (molecule) { indec injs in Mod Λ } ∼ =

ϕ

• 1−1
• Spec Λ

ψ

• 1−1
• ASpec(Mod Λ)

ϕ

• ∪

MSpec(Mod Λ)

ψ

• ∪

AMin(Mod Λ)

1−1

MMin(Mod Λ)

Molecules (=two-sided primes)

Theorem (Rosenberg 1995) Let Λ be a ring. { closed subcats of Mod Λ } 1−1 ← → { two-sided ideals of Λ } Mod Λ I ← I Mod Λ J ∗ Mod Λ I ← IJ Mclosed ← AnnΛ(M) closed subcat := full subcat closed under sub, quot, , C1 ∗ C2 := { M ∈ A | ∃ 0 → M1 → M → M2 → 0, Mi ∈ Ci }

A : Grothendieck category (e.g. Mod Λ for a ring Λ) Definition H ∈ A is called prime if H = 0 For every 0 = L ⊂ H, Lclosed = Hclosed Proposition H ∈ A is prime, 0 = L ⊂ H = ⇒ L is prime.

Definition H1 is called molecule-equivalent to H2 if H1closed = H2closed Definition (K) The molecule spectrum of A is MSpec A := { prime objects in A } molecule-equivalence .

• H denotes the equivalence class. A molecule is an element of

MSpec A.

Definition Define a partial order ≤ on MSpec A by ρ ≤ σ ⇐ ⇒ ρclosed ⊃ σclosed If ρ = H, then ρclosed := Hclosed. Proposition Let Λ be a ring. (MSpec(Mod Λ), ≤) ∼ = (Spec Λ, ⊂)

• Λ/P

← P Spec Λ := { two-sided prime ideals of Λ }

A : Grothendieck category having a noetherian generator (e.g. Mod Λ for a right noetherian ring Λ) Proposition For each ρ ∈ MSpec A, there exists σ ∈ MMin A satisfying σ ≤ ρ. # MMin A < ∞. There exists the smallest closed subcategory Am-red satisfying MSpec Am-red = MSpec A. (molecularly reduced part of A) (Am-red)m-red = Am-red. MMin A := { minimal elements of MSpec A } closed subcat := full subcat closed under sub, quot, ,

A : Grothendieck category having a noetherian generator (e.g. Mod Λ for a right noetherian ring Λ) Definition (K) A is called m-reduced if A = Am-red m-irreducible if # MMin A = 1 m-integral if A is m-reduced and m-irreducible Proposition Let Λ be a right noetherian ring. Then Mod Λ is m-reduced ⇐ ⇒ Λ is a semiprime ring m-irreducible ⇐ ⇒ the prime radical √ 0 belongs to Spec Λ m-integral ⇐ ⇒ Λ is a prime ring

Λ : right noetherian ring Overview

• ne-sided prime

(atom) two-sided prime (molecule) { indec injs in Mod Λ } ∼ =

ϕ

• 1−1
• Spec Λ

ψ

• 1−1
• ASpec(Mod Λ)

ϕ

• ∪

MSpec(Mod Λ)

ψ

• ∪

AMin(Mod Λ)

1−1

MMin(Mod Λ)

Atom-molecule correspondence

A : Grothendieck cat having a noetherian generator, Ab4* (e.g. Mod Λ for a right noetherian ring Λ) Theorem (K) ϕ: ASpec A → MSpec A given by H → H is a surjective poset homomorphism. ψ: MSpec A → ASpec A given by ψ(ρ) = min{ α ∈ ASpec A | ϕ(α) = ρ } induces a poset isomorphism MSpec A ∼ − → Im ψ. ϕψ = id.

Atom-molecule correspondence

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

ϕ

⇄

ψ

MSpec A induces AMin A ∼ = MMin A. 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 quotient ring Λ′, which is semisimple.

Application

Theorem (Goldie 1960) Every semiprime right noetherian ring Λ has a right quotient ring Λ′, which is semisimple. Sketch Mod Λ is m-reduced = ⇒ Mod Λ is a-reduced = ⇒ (Mod Λ)artin is semisimple = ⇒ Mod Λ → (Mod Λ)artin sends Λ to a projective generator P = ⇒ Λ = EndΛ(Λ) → End(P) =: Λ′

Λ : right noetherian ring Overview

• ne-sided prime

(atom) two-sided prime (molecule) { indec injs in Mod Λ } ∼ =

ϕ

• 1−1
• Spec Λ

ψ

• 1−1
• ASpec(Mod Λ)

ϕ

• ∪

MSpec(Mod Λ)

ψ

• ∪

AMin(Mod Λ)

1−1

MMin(Mod Λ)

Appendix

A : Grothendieck category (e.g. Mod Λ for a ring Λ) Definition For each M ∈ A, AAss M := { H ∈ ASpec A | H ⊂ M } ASupp M := { H ∈ ASpec A | H is a subquot of M } Proposition Let 0 → L → M → N → 0 be an exact sequence. AAss L ⊂ AAss M ⊂ AAss L ∪ AAss N ASupp M = ASupp L ∪ ASupp N Proposition AAss

• i∈I

Mi =

• i∈I

AAss Mi, ASupp

• i∈I

Mi =

• i∈I

ASupp Mi

A : locally noetherian Grothendieck category (e.g. Mod Λ for a right noetherian ring Λ) Theorem (Gabriel 1962, Herzog 1997, Krause 1997, K 2012) { localizing subcats of A } 1−1 ← → { localizing subsets of ASpec A } X →

• M∈X

ASupp M { M ∈ A | ASupp M ⊂ Φ } ← Φ localizing subcat := full subcat closed under sub, quot, ext, localizing subset := union of subsets of the form ASupp M = subset of the form ASupp M

A : Grothendieck cat having a noetherian generator, Ab4* (e.g. Mod Λ for a right noetherian ring Λ) Key Lemma Let M ∈ A satisfying AAss M ⊂ AMin A. Then Mweakly closed = Mclosed weakly closed subcat · · · sub, quot, closed subcat · · · sub, quot, ,