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 ϕ { indecomposable injectives in Mod Λ } Spec Λ ⇄ ∼ = ψ 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 one-sided prime two-sided prime ϕ { indec injs in Mod Λ } � Spec Λ ∼ = ψ

� � � � � � � Λ : right noetherian ring Overview one-sided prime two-sided prime ( atom ) ( molecule ) ϕ { indec injs in Mod Λ } Spec Λ ∼ = ψ 1 − 1 1 − 1 ϕ � MSpec(Mod Λ ) ASpec(Mod Λ ) ψ

� � � � � � � � Λ : right noetherian ring Overview one-sided prime two-sided prime ( atom ) ( molecule ) ϕ { indec injs in Mod Λ } Spec Λ ∼ = ψ 1 − 1 1 − 1 ϕ ASpec(Mod Λ ) MSpec(Mod Λ ) ψ ∪ ∪ AMin(Mod Λ ) MMin(Mod Λ )

� � � � � � � � Λ : right noetherian ring Overview one-sided prime two-sided prime ( atom ) ( molecule ) ϕ { indec injs in Mod Λ } Spec Λ ∼ = ψ 1 − 1 1 − 1 ϕ ASpec(Mod Λ ) MSpec(Mod Λ ) ψ ∪ ∪ 1 − 1 � MMin(Mod Λ ) AMin(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 H 1 is called atom-equivalent to H 2 if { subobjects of H 1 } ∩ { subobjects of H 2 } � = { 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. → { simple Λ -modules } ASpec(Mod Λ ) 1 − 1 ← ∼ = S ← � S

A : locally noetherian Grothendieck category (e.g. Mod Λ for a right noetherian ring Λ ) Theorem (Matlis 1958, K 2012) → { indecomposable injectives in A } ASpec A 1 − 1 ← ∼ = 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 A a-red satisfying ASpec A a-red = ASpec A . ( atomically reduced part of A ) ( A a-red ) a-red = A a-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 = A a-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 . A artin := A / X is called the artinianization of A . Proposition A ∼ − → A artin ⇐ ⇒ 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 one-sided prime two-sided prime ( atom ) ( molecule ) ϕ { indec injs in Mod Λ } Spec Λ ∼ = ψ 1 − 1 1 − 1 ϕ ASpec(Mod Λ ) MSpec(Mod Λ ) ψ ∪ ∪ 1 − 1 � MMin(Mod Λ ) AMin(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 ← � IJ I � M � closed ← � Ann Λ ( M ) closed subcat := full subcat closed under sub, quot, � , � C 1 ∗ C 2 := { M ∈ A | ∃ 0 → M 1 → M → M 2 → 0 , M i ∈ C i }

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 , � L � closed = � H � closed Proposition H ∈ A is prime, 0 � = L ⊂ H = ⇒ L is prime.

Definition H 1 is called molecule-equivalent to H 2 if � H 1 � closed = � H 2 � closed 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 ⇐ ⇒ � ρ � closed ⊃ � σ � closed If ρ = � H , then � ρ � closed := � H � closed . 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 A m-red satisfying MSpec A m-red = MSpec A . ( molecularly reduced part of A ) ( A m-red ) m-red = A m-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 = A m-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 one-sided prime two-sided prime ( atom ) ( molecule ) ϕ { indec injs in Mod Λ } Spec Λ ∼ = ψ 1 − 1 1 − 1 ϕ ASpec(Mod Λ ) MSpec(Mod Λ ) ψ ∪ ∪ 1 − 1 � MMin(Mod Λ ) AMin(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) ϕ MSpec A induces AMin A ∼ ASpec A = MMin A . ⇄ ψ A a-red = A m-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 = End Λ ( Λ ) → End( P ) =: Λ ′ =

� � � � � � � � Λ : right noetherian ring Overview one-sided prime two-sided prime ( atom ) ( molecule ) ϕ { indec injs in Mod Λ } Spec Λ ∼ = ψ 1 − 1 1 − 1 ϕ ASpec(Mod Λ ) MSpec(Mod Λ ) ψ ∪ ∪ 1 − 1 � MMin(Mod Λ ) AMin(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 M i = AAss M i , ASupp M i = ASupp M i i ∈ I i ∈ I i ∈ I i ∈ I