Injective hulls of simple modules over Noetherian rings

Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Indecomposable injective modules over Down-Up
algebras
Christian Lomp
jointly with Paula Carvalho and Dilek Pusat-Yilmaz
Universidade do Porto
21. May 2010
Christian Lomp Indecomposable injective modules over Down-Up algebras

Pr¨ufer Groups
Prime number p: the Pr¨ufer group Zp∞ is the union of the chain
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .

Pr¨ufer Groups
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
Every proper subgroup of Zp∞ is ﬁnite.

Pr¨ufer Groups
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
The Pr¨ufer groups are the injective hulls of Zp.

Prüfer Groups
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
The Prüfer 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.

Matlis Theory
Theorem (Matlis, 1960)
Injective hulls of simples over Noetherian commutative rings are
Artinian.

Matlis Theory
Theorem (Matlis, 1960)
Injective hulls of simples over Noetherian commutative rings are
Artinian.
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.

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

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

Weyl algebras
are 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 injective
hull is not locally Artinian.

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

Quantum Plane
Theorem (Carvalho,Musson 2010)
Injective hulls of simples over Kq[x, y] = K[x, y | yx = qxy] are
locally Artinian if and only if q is a root of unity.

Quantum Plane
Injective hulls of simples over Kq[x, y] = K[x, y | yx = qxy] are
locally Artinian if and only if q is a root of unity.
Question
Over which non-commutative Noetherian rings are injective hulls
of simples locally Artinian?

Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be a countably generated Noetherian C-algebra with
Noetherian 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)).

Noetherian centre.
(3-dimensional complex Heisenberg Lie algebra)
h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z].

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

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

Noetherian centre.
A/mA C[x, y] or A/mA A1(C).
Hence injective hulls of simples over U(h) are locally Artinian.

(2n + 1-dimensional complex Heisenberg Lie algebra)
h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for
1 ≤ i ≤ n and zero for all other combinations of generators.

Then A = U(h) admits a non-locally Artinian injective hull of a
simple 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.

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.

Down-Up algebras
Theorem (Dahlberg, 1988)
Injective hulls of simples over U(sl2) are locally Artinian.

Down-Up algebras
U(sl2) = A(2, −1, 1) and U(h) = A(2, −1, 0) are Noetherian
Down-up Algebras

Down-Up algebras
U(sl2) = A(2, −1, 1) and U(h) = A(2, −1, 0) are Noetherian
Down-up Algebras
Theorem (Benkart, Roby 1999)
For (α, β, γ) ∈ C3, the Down-Up algebra A(α, β, γ) is generated
by two elements u and d subject to the relations
d2
u = αdud + βud2
+ γd
du2
= αudu + βu2
d + γu

Fully bounded Noetherian
Question (Smith - X.Antalya Algebra Days 2008)
For which Noetherian Down-Up algebras are all injective hulls of
simple 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 essential
left/right ideal of a prime factor of R contains a non-zero ideal.

Deﬁnition (Fully Bounded Noetherian)
R is Fully bounded Noetherian if it is Noetherian and any essential
left/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 ?

FBN Down-Up algebras
Question
Which Noetherian Down-Up algebras are FBN ?
The following statements are equivalent for a Noetherian Down-up
algebra 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 X2 − αX − β are distinct roots of
unity such that both are also different from 1 if γ = 0.

Generalized Weyl algebras
Deﬁnition (Bavula 1992, Rosenberg 1995)
The generalized Weyl algebra R(σ, a) is the R-algebra generated
by 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).

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 → X+
; d → X−
; ud → x; du → y

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 → X+
; d → X−
; ud → x; du → y
Observation (Kulkarni, 2001)
R(σ, x) is f.g. over its centre if and only if σ has ﬁnite order.

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

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

Conclusion
Aη := A(1 − η, η, 1) is 2.
(Praton 2004): Z(Aη) = C[w]

Conclusion
Aη := A(1 − η, η, 1) is 2.
⇒ Aη/mAη has Krull dimension 1 for m = w .

Conclusion
Aη := A(1 − η, η, 1) is 2.
⇒ Aη/mAη has Krull dimension 1 for m = w .
Corollary
Injective hulls of simples over A(α, β, γ) are locally Artinian, if the
roots of X2 − αX − β are distinct roots of unity or both 1.

Progress ...
Injective hulls of simples over A(α, β, γ) are locally Artinian, if
(and only if ?) the roots of X2 − αX − β are roots of unity.

Progress ...
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

Progress ...
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
Te¸sekk¨ur Ederim !

Injective hulls of simple modules over Noetherian rings

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