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# Injective hulls of simple modules over Noetherian rings

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### Injective hulls of simple modules over Noetherian rings

1. 1. 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
2. 2. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra 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∞ . Christian Lomp Indecomposable injective modules over Down-Up algebras
3. 3. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra 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∞ . Every proper subgroup of Zp∞ is ﬁnite. Christian Lomp Indecomposable injective modules over Down-Up algebras
4. 4. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra 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∞ . Every proper subgroup of Zp∞ is ﬁnite. The Pr¨ufer groups are the injective hulls of Zp. Christian Lomp Indecomposable injective modules over Down-Up algebras
5. 5. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra 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∞ . Every proper subgroup of Zp∞ is ﬁnite. The Pr¨ufer 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
6. 6. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Matlis Theory Theorem (Matlis, 1960) Injective hulls of simples over Noetherian commutative rings are Artinian. Christian Lomp Indecomposable injective modules over Down-Up algebras
7. 7. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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). Christian Lomp Indecomposable injective modules over Down-Up algebras
8. 8. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Weyl algebras Theorem (Hirano, 2002) Injective hulls of simples over A1(Z) = Z[x, y | yx = xy + 1] are Artinian. Christian Lomp Indecomposable injective modules over Down-Up algebras
9. 9. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
10. 10. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
11. 11. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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 injective hull is not locally Artinian. Christian Lomp Indecomposable injective modules over Down-Up algebras
12. 12. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Quantum Plane Injective hulls of simple modules over K[x, y] are Artinian. Christian Lomp Indecomposable injective modules over Down-Up algebras
13. 13. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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] are locally Artinian if and only if q is a root of unity. Christian Lomp Indecomposable injective modules over Down-Up algebras
14. 14. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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] 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? Christian Lomp Indecomposable injective modules over Down-Up algebras
15. 15. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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)). Christian Lomp Indecomposable injective modules over Down-Up algebras
16. 16. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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)). (3-dimensional complex Heisenberg Lie algebra) h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z]. Christian Lomp Indecomposable injective modules over Down-Up algebras
17. 17. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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)). (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 Christian Lomp Indecomposable injective modules over Down-Up algebras
18. 18. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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)). (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). Christian Lomp Indecomposable injective modules over Down-Up algebras
19. 19. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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)). (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. Christian Lomp Indecomposable injective modules over Down-Up algebras
20. 20. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras (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. Christian Lomp Indecomposable injective modules over Down-Up algebras
21. 21. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras (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). Christian Lomp Indecomposable injective modules over Down-Up algebras
22. 22. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras (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. Christian Lomp Indecomposable injective modules over Down-Up algebras
23. 23. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras (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. Any such algebra has C[x, y | yx = xy + x] as a factor algebra. Christian Lomp Indecomposable injective modules over Down-Up algebras
24. 24. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Down-Up algebras Theorem (Dahlberg, 1988) Injective hulls of simples over U(sl2) are locally Artinian. Christian Lomp Indecomposable injective modules over Down-Up algebras
25. 25. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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 Noetherian Down-up Algebras Christian Lomp Indecomposable injective modules over Down-Up algebras
26. 26. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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 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 Christian Lomp Indecomposable injective modules over Down-Up algebras
27. 27. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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? Christian Lomp Indecomposable injective modules over Down-Up algebras
28. 28. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
29. 29. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
30. 30. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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. (Jacobson’s conjecture) If J is the Jacobson radical of a Noetherian ring, then ∞ n=1 Jn = 0. Christian Lomp Indecomposable injective modules over Down-Up algebras
31. 31. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras FBN Down-Up algebras Question Which Noetherian Down-Up algebras are FBN ? Christian Lomp Indecomposable injective modules over Down-Up algebras
32. 32. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras FBN Down-Up algebras Question Which Noetherian Down-Up algebras are FBN ? Theorem (Carvalho, Pusat-Yilmaz,L.) 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
33. 33. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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). Christian Lomp Indecomposable injective modules over Down-Up algebras
34. 34. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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). 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 Christian Lomp Indecomposable injective modules over Down-Up algebras
35. 35. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras 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). 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
36. 36. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Conclusion Theorem (Carvalho, Pusat-Yilmaz,L.) Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if η is a root of unity. Christian Lomp Indecomposable injective modules over Down-Up algebras
37. 37. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Conclusion Theorem (Carvalho, Pusat-Yilmaz,L.) Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if η is a root of unity. (Bavula+Lenagan, 2001): Krull dimension of Aη := A(1 − η, η, 1) is 2. Christian Lomp Indecomposable injective modules over Down-Up algebras
38. 38. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Conclusion Theorem (Carvalho, Pusat-Yilmaz,L.) Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if η is a root of unity. (Bavula+Lenagan, 2001): Krull dimension of Aη := A(1 − η, η, 1) is 2. (Praton 2004): Z(Aη) = C[w] Christian Lomp Indecomposable injective modules over Down-Up algebras
39. 39. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Conclusion Theorem (Carvalho, Pusat-Yilmaz,L.) Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if η is a root of unity. (Bavula+Lenagan, 2001): Krull dimension of Aη := A(1 − η, η, 1) is 2. (Praton 2004): Z(Aη) = C[w] ⇒ Aη/mAη has Krull dimension 1 for m = w . Christian Lomp Indecomposable injective modules over Down-Up algebras
40. 40. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Conclusion Theorem (Carvalho, Pusat-Yilmaz,L.) Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if η is a root of unity. (Bavula+Lenagan, 2001): Krull dimension of Aη := A(1 − η, η, 1) is 2. (Praton 2004): Z(Aη) = C[w] ⇒ 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
41. 41. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Progress ... Theorem (Carvalho,Musson 2010) 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. Christian Lomp Indecomposable injective modules over Down-Up algebras
42. 42. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Progress ... Theorem (Carvalho,Musson 2010) 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. 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 Christian Lomp Indecomposable injective modules over Down-Up algebras
43. 43. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras Progress ... Theorem (Carvalho,Musson 2010) 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. 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 ! Christian Lomp Indecomposable injective modules over Down-Up algebras