injective hulls of simple modules over noetherian rings

Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras

Indecomposable injective modules over Down-Upalgebras

Christian Lompjointly with Paula Carvalho and Dilek Pusat-Yilmaz

Universidade do Porto

21. May 2010

Christian Lomp Indecomposable injective modules over Down-Up algebras

Prufer Groups

Prime number p: the Prufer group Zp∞ is the union of the chain

Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂∞⋃

Zpn = Zp∞ .

Every proper subgroup of Zp∞ is finite.

The Prufer groups are the injective hulls of Zp.

Definition (Injective Hull)

M ⊆ E (M) :

E (M) is injective;

M ⊆ E (M) is essential, i.e.

∀U ⊆ E (M) : U ∩M = 0⇒ U = 0.

Prufer Groups

Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂∞⋃

Zpn = Zp∞ .

M ⊆ E (M) :

E (M) is injective;

∀U ⊆ E (M) : U ∩M = 0⇒ U = 0.

Prufer Groups

Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂∞⋃

Zpn = Zp∞ .

M ⊆ E (M) :

E (M) is injective;

∀U ⊆ E (M) : U ∩M = 0⇒ U = 0.

Prufer Groups

Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂∞⋃

Zpn = Zp∞ .

M ⊆ E (M) :

E (M) is injective;

∀U ⊆ E (M) : U ∩M = 0⇒ U = 0.

Matlis Theory

Theorem (Matlis, 1960)

Injective hulls of simples over Noetherian commutative rings areArtinian.

Theorem (Vamos, 1968)

For a commutative ring R the following are equivalent:

Injective hulls of simples are Artinian;

Rm is Noetherian, ∀m ∈ MaxSpec(R).

Matlis Theory

Theorem (Matlis, 1960)

Injective hulls of simples over Noetherian commutative rings areArtinian.

Theorem (Vamos, 1968)

For a commutative ring R the following are equivalent:

Injective hulls of simples are Artinian;

Rm is Noetherian, ∀m ∈ MaxSpec(R).

Weyl algebras

Theorem (Hirano, 2002)

Injective hulls of simples over A1(Z) = Z[x , y | yx = xy + 1]are Artinian.

The injective hull of Q[x ] as A1(Q)-module is not Artinian.

Observation

Injective hulls of simples over A1(Q) are locally Artinian.

Theorem (Stafford, 1984)

There exist simple modules over An(C) (n ≥ 2) whose injectivehull is not locally Artinian.

Weyl algebras

Observation

Weyl algebras

Observation

Weyl algebras

Observation

Quantum Plane

Injective hulls of simple modules over K [x , y ] are Artinian.

Theorem (Carvalho,Musson 2010)

Injective hulls of simples over Kq[x , y ] = K [x , y | yx = qxy ] arelocally Artinian if and only if q is a root of unity.

Question

Over which non-commutative Noetherian rings are injective hullsof simples locally Artinian?

Quantum Plane

Question

Quantum Plane

Question

Theorem (Carvalho, Pusat-Yilmaz,L.)

Let A be a countably generated Noetherian C-algebra withNoetherian centre.

1 Injective hulls of simples over A are locally Artinian

2 Injective hulls of simples over A/mA are locally Artinian,∀m ∈ MaxSpec(Z (A)).

(3-dimensional complex Heisenberg Lie algebra)

h generated by x , y , z subject to [x , y ] = z , [x , z ] = 0 = [y , z ].Let A = U(h), then Z (A) = C[z ] and

A/mA ' C[x , y ] or A/mA ' A1(C).

Hence injective hulls of simples over U(h) are locally Artinian.

h generated by x , y , z subject to [x , y ] = z , [x , z ] = 0 = [y , z ].

Let A = U(h), then Z (A) = C[z ] and

A/mA ' C[x , y ] or A/mA ' A1(C).

(2n + 1-dimensional complex Heisenberg Lie algebra)

h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for1 ≤ i ≤ n and zero for all other combinations of generators.

Then A = U(h) admits a non-locally Artinian injective hull of asimple if n ≥ 2, because A/〈z − 1〉 ' An(C).

Theorem (Musson, 1982)

∀ non-nilpotent soluble finite dimensional complex Lie algebras g

∃ a non-locally Artinian injective hulls of a simple U(g)-module.Any such algebra has C[x , y | yx = xy + x ] as a factor algebra.

h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for1 ≤ i ≤ n and zero for all other combinations of generators.Then A = U(h) admits a non-locally Artinian injective hull of asimple if n ≥ 2, because A/〈z − 1〉 ' An(C).

∃ a non-locally Artinian injective hulls of a simple U(g)-module.

Any such algebra has C[x , y | yx = xy + x ] as a factor algebra.

Down-Up algebras

Theorem (Dahlberg, 1988)

Injective hulls of simples over U(sl2) are locally Artinian.

U(sl2) = A(2,−1, 1) and U(h) = A(2,−1, 0) are NoetherianDown-up Algebras

Theorem (Benkart, Roby 1999)

For (α, β, γ) ∈ C3, the Down-Up algebra A(α, β, γ) is generatedby two elements u and d subject to the relations

d2u = αdud + βud2 + γd

du2 = αudu + βu2d + γu

Down-Up algebras

Fully bounded Noetherian

Question (Smith - X.Antalya Algebra Days 2008)

For which Noetherian Down-Up algebras are all injective hulls ofsimple modules locally Artinian?

Theorem (Jategaonkar, 1974)

Injective hulls of simples over an FBN ring are locally Artinian.

Definition (Fully Bounded Noetherian)

R is Fully bounded Noetherian if it is Noetherian and any essentialleft/right ideal of a prime factor of R contains a non-zero ideal.

(Jacobson’s conjecture)

If J is the Jacobson radical of a Noetherian ring, then⋂∞

n=1 Jn = 0.

FBN Down-Up algebras

Question

Which Noetherian Down-Up algebras are FBN ?

The following statements are equivalent for a Noetherian Down-upalgebra A = A(α, β, γ):

1 A is module-finite over a central subalgebra;

2 A satisfies a polynomial identity;

3 A is fully bounded Noetherian;

4 The roots of the polynomial X 2−αX − β are distinct roots ofunity such that both are also different from 1 if γ 6= 0.

FBN Down-Up algebras

Question

Which Noetherian Down-Up algebras are FBN ?

The following statements are equivalent for a Noetherian Down-upalgebra A = A(α, β, γ):

1 A is module-finite over a central subalgebra;

2 A satisfies a polynomial identity;

3 A is fully bounded Noetherian;

4 The roots of the polynomial X 2−αX − β are distinct roots ofunity such that both are also different from 1 if γ 6= 0.

Generalized Weyl algebras

Definition (Bavula 1992, Rosenberg 1995)

The generalized Weyl algebra R(σ, a) is the R-algebra generatedby X + and X− subject to

X +r = σ(r)X + and X−r = σ−1(r)X− ∀r ∈ R;

X +X− = a and X−X + = σ−1(a).

Observation (Kirkman-Musson-Passman, 2000)

A is isomorphic to a generalized Weyl algebra, where R = C[x , y ]and σ(x) = β−1(y −αx −γ) and σ(y) = x and ϕ : A→ R(σ, x) by

u 7→ X +; d 7→ X−; ud 7→ x ; du 7→ y

Observation (Kulkarni, 2001)

R(σ, x) is f.g. over its centre if and only if σ has finite order.

X +X− = a and X−X + = σ−1(a).

u 7→ X +; d 7→ X−; ud 7→ x ; du 7→ y

X +X− = a and X−X + = σ−1(a).

u 7→ X +; d 7→ X−; ud 7→ x ; du 7→ y

Conclusion

Injective hulls of simples over A(1− η, η, 1) are locally Artinian, ifη is a root of unity.

(Bavula+Lenagan, 2001): Krull dimension ofAη := A(1− η, η, 1) is 2.

(Praton 2004): Z (Aη) = C[w ]

⇒ Aη/mAη has Krull dimension 1 for m 6= 〈w〉.

Corollary

Injective hulls of simples over A(α, β, γ) are locally Artinian, if theroots of X 2 − αX − β are distinct roots of unity or both 1.

Conclusion

(Praton 2004): Z (Aη) = C[w ]

Corollary

Conclusion

(Praton 2004): Z (Aη) = C[w ]

Corollary

Conclusion

(Praton 2004): Z (Aη) = C[w ]

Corollary

Conclusion

(Praton 2004): Z (Aη) = C[w ]

Corollary

Progress ...

Injective hulls of simples over A(α, β, γ) are locally Artinian, if(and only if ?) the roots of X 2 − αX − β are roots of unity.

P.Carvalho, C.L., D.Pusat-Yilmaz, ”Injective Modules over

Down-Up algebras”, arxiv:0906.2930 to appear in Glasgow Math. J.

P.Carvalho, I.Musson, ”Monolithic Modules over Noetherian Rings”,

arxiv:1001.1466

Tesekkur Ederim !

Progress ...

arxiv:1001.1466

Tesekkur Ederim !

Progress ...

arxiv:1001.1466

Tesekkur Ederim !

injective hulls of simple modules over noetherian rings

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algebras christian lomp

injective hulls of zp

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injective hull of qx

denition injective hull

artinian rm

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