projective modules over universal enveloping...
TRANSCRIPT
Projective modules over universal enveloping algebras
September 27, 2011
() Projective modules over universal enveloping algebras September 27, 2011 1 / 14
Intro
R is an associative ring with unit
R-module means a right module over R
A module P is said to be projective if it is isomorphic to a direct
summand of R(�)R
Problem: For a (speci�ed) ring give a classi�cation of projective modules.By a theorem of Kaplansky every projective module is a direct sum ofcountably generated ones.
() Projective modules over universal enveloping algebras September 27, 2011 2 / 14
Intro
R is an associative ring with unit
R-module means a right module over R
A module P is said to be projective if it is isomorphic to a direct
summand of R(�)R
Problem: For a (speci�ed) ring give a classi�cation of projective modules.By a theorem of Kaplansky every projective module is a direct sum ofcountably generated ones.
() Projective modules over universal enveloping algebras September 27, 2011 2 / 14
Why?
why not?
Projective modules are relevant in homological algebra, geometry anddirect sum decompositions problems in module theory.
Theorem
(Swan) Let X be a compact topological space. The category of real vector
bundles on X is equivalent to the category of �nitely generated projective
modules over C (X ).
Add(M) consists of direct summands of M(�) for some �.
Theorem
(Dress) Let M be a �nitely generated module over R. The category
Add(M) is equivalent to the category of projective modules over
EndR(M).
() Projective modules over universal enveloping algebras September 27, 2011 3 / 14
Why?
why not?
Projective modules are relevant in homological algebra, geometry anddirect sum decompositions problems in module theory.
Theorem
(Swan) Let X be a compact topological space. The category of real vector
bundles on X is equivalent to the category of �nitely generated projective
modules over C (X ).
Add(M) consists of direct summands of M(�) for some �.
Theorem
(Dress) Let M be a �nitely generated module over R. The category
Add(M) is equivalent to the category of projective modules over
EndR(M).
() Projective modules over universal enveloping algebras September 27, 2011 3 / 14
Why?
why not?
Projective modules are relevant in homological algebra, geometry anddirect sum decompositions problems in module theory.
Theorem
(Swan) Let X be a compact topological space. The category of real vector
bundles on X is equivalent to the category of �nitely generated projective
modules over C (X ).
Add(M) consists of direct summands of M(�) for some �.
Theorem
(Dress) Let M be a �nitely generated module over R. The category
Add(M) is equivalent to the category of projective modules over
EndR(M).
() Projective modules over universal enveloping algebras September 27, 2011 3 / 14
Noetherian setting
Typical situation over noetherian rings is that �nitely generated projectivemodules are harder to classify than non�nitely generated ones.
Evolution of so called Serre's problem (1955):
In 1957 Serre proved that �nitely generated projectives overk[x1; : : : ; xn] are stably free.
In 1963 Bass proved that non�nitely generated projectives overk[x1; : : : ; xn] are free.
In 1976 Suslin and Quillen independently proved that all projectivemodules over k[x1; : : : ; xn] are free.
() Projective modules over universal enveloping algebras September 27, 2011 4 / 14
Noetherian setting
Typical situation over noetherian rings is that �nitely generated projectivemodules are harder to classify than non�nitely generated ones.Evolution of so called Serre's problem (1955):
In 1957 Serre proved that �nitely generated projectives overk[x1; : : : ; xn] are stably free.
In 1963 Bass proved that non�nitely generated projectives overk[x1; : : : ; xn] are free.
In 1976 Suslin and Quillen independently proved that all projectivemodules over k[x1; : : : ; xn] are free.
() Projective modules over universal enveloping algebras September 27, 2011 4 / 14
Noetherian setting cont.
On the other hand sometimes there is a nice connection between propertiesof the ring R and the structure theory for non�nitely generated projectivemodules. Let us discuss the case of an integral group ring of a �nite group.
Theorem
(Swan, Akasaki, Linnell) If G is a �nite group then G is solvable if and
only if every non�nitely generated projective module over ZG is free.
If P is a projective module Tr(P) =P
f : P!R f (P). This is an idempotentideal in R (trace ideal of P) and PTr(P) = P
Theorem
(Bass) If R is a connected commutative noetherian ring then every
non�nitely generated projective module over R is free.
() Projective modules over universal enveloping algebras September 27, 2011 5 / 14
Noetherian setting cont.
On the other hand sometimes there is a nice connection between propertiesof the ring R and the structure theory for non�nitely generated projectivemodules. Let us discuss the case of an integral group ring of a �nite group.
Theorem
(Swan, Akasaki, Linnell) If G is a �nite group then G is solvable if and
only if every non�nitely generated projective module over ZG is free.
If P is a projective module Tr(P) =P
f : P!R f (P). This is an idempotentideal in R (trace ideal of P) and PTr(P) = P
Theorem
(Bass) If R is a connected commutative noetherian ring then every
non�nitely generated projective module over R is free.
() Projective modules over universal enveloping algebras September 27, 2011 5 / 14
Noetherian setting cont.
On the other hand sometimes there is a nice connection between propertiesof the ring R and the structure theory for non�nitely generated projectivemodules. Let us discuss the case of an integral group ring of a �nite group.
Theorem
(Swan, Akasaki, Linnell) If G is a �nite group then G is solvable if and
only if every non�nitely generated projective module over ZG is free.
If P is a projective module Tr(P) =P
f : P!R f (P). This is an idempotentideal in R (trace ideal of P) and PTr(P) = P
Theorem
(Bass) If R is a connected commutative noetherian ring then every
non�nitely generated projective module over R is free.
() Projective modules over universal enveloping algebras September 27, 2011 5 / 14
Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .
countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.
(0;P) corresponds to a �nitely generated projective R-module P
(R; 0) corresponds to R(!)R
(I ;P) are decomposable whenever 0 6= I .
Theorem
(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free
module over Z[pn] is a direct sum of �nitely generated modules if and
only if n 6� 1 mod 8.
() Projective modules over universal enveloping algebras September 27, 2011 6 / 14
Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .
countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.
(0;P) corresponds to a �nitely generated projective R-module P
(R; 0) corresponds to R(!)R
(I ;P) are decomposable whenever 0 6= I .
Theorem
(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free
module over Z[pn] is a direct sum of �nitely generated modules if and
only if n 6� 1 mod 8.
() Projective modules over universal enveloping algebras September 27, 2011 6 / 14
Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .
countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.
(0;P) corresponds to a �nitely generated projective R-module P
(R; 0) corresponds to R(!)R
(I ;P) are decomposable whenever 0 6= I .
Theorem
(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free
module over Z[pn] is a direct sum of �nitely generated modules if and
only if n 6� 1 mod 8.
() Projective modules over universal enveloping algebras September 27, 2011 6 / 14
Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .
countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.
(0;P) corresponds to a �nitely generated projective R-module P
(R; 0) corresponds to R(!)R
(I ;P) are decomposable whenever 0 6= I .
Theorem
(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free
module over Z[pn] is a direct sum of �nitely generated modules if and
only if n 6� 1 mod 8.
() Projective modules over universal enveloping algebras September 27, 2011 6 / 14
Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .
countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.
(0;P) corresponds to a �nitely generated projective R-module P
(R; 0) corresponds to R(!)R
(I ;P) are decomposable whenever 0 6= I .
Theorem
(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free
module over Z[pn] is a direct sum of �nitely generated modules if and
only if n 6� 1 mod 8.
() Projective modules over universal enveloping algebras September 27, 2011 6 / 14
Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .
countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.
(0;P) corresponds to a �nitely generated projective R-module P
(R; 0) corresponds to R(!)R
(I ;P) are decomposable whenever 0 6= I .
Theorem
(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free
module over Z[pn] is a direct sum of �nitely generated modules if and
only if n 6� 1 mod 8.
() Projective modules over universal enveloping algebras September 27, 2011 6 / 14
Lie Algebras
Let k be a �eld, an algebra (L; [�;�]) is called a Lie algebra if
[�;�] is bilinear[x ; x ] = 0 for every x 2 L
[[a; b]; c] + [[b; c]; a] + [[c ; a]; b] = 0 for every a; b; c
A representation of L is a Lie algebra homomorphism of L to gln(k)Any associative algebra give a Lie structure via [a; b] := ab � ba.De�ne L(1) = L, L(2) = [L; L],. . . ,L(i+1) = [L(i); L(i)],. . .A Lie algebra L is said to be solvable if there exists i 2 N such thatL(i) = 0.A radical of a Lie algebra L is the greatest solvable ideal in L.A Lie algebra L is called semisimple if has zero radical.A Lie algebra L is called simple if [L; L] 6= 0 and there are no nontrivialideals of L.
() Projective modules over universal enveloping algebras September 27, 2011 7 / 14
Basic structure theorems for Lie algebras
Theorem
(Cartan) If L is a �nite dimensional semisimple Lie algebra over a �eld of
characteristic zero, then L = L1 � � � � � Ln, where every Li is an ideal of L
which is a simple Lie algebra.
Theorem
(Levi) Let B be a �nite-dimensional Lie algebra over a �eld of
characteristic zero and let R be the radical of B. Then B contains a
�nite-dimensional semisimple subalgebra L such that B = L� R.
() Projective modules over universal enveloping algebras September 27, 2011 8 / 14
Universal enveloping algebras
Let L be a Lie algebra with basis fx1; : : : ; xng over k. Let
[xi ; xj ] =nX
k=1
ci ;j ;kxk :
The algebra U(L) = khx1; : : : ; xni=hxixj � xjxi �Pn
k=1 ci ;j ;kxki is calledthe universal enveloping algebra of L
Monomials xk11 xk2
2 � � � xknn form a basis of U(L), so L is contained in
U(L).
If A is an associative k-algebra and ' : L! A a linear map such that'([a; b]) = '(a)'(b)� '(b)'(a) then there is a unique extension of' to a homomorphism �: U(L)! A.
The category of representations of L is equivalent to the category of�nite-dimensional right U(L)-modules.
U(L) is a noetherian domain.
Krull dimension of U(L) is at most dimk(L).
() Projective modules over universal enveloping algebras September 27, 2011 9 / 14
Universal enveloping algebras
Let L be a Lie algebra with basis fx1; : : : ; xng over k. Let
[xi ; xj ] =nX
k=1
ci ;j ;kxk :
The algebra U(L) = khx1; : : : ; xni=hxixj � xjxi �Pn
k=1 ci ;j ;kxki is calledthe universal enveloping algebra of L
Monomials xk11 xk2
2 � � � xknn form a basis of U(L), so L is contained in
U(L).
If A is an associative k-algebra and ' : L! A a linear map such that'([a; b]) = '(a)'(b)� '(b)'(a) then there is a unique extension of' to a homomorphism �: U(L)! A.
The category of representations of L is equivalent to the category of�nite-dimensional right U(L)-modules.
U(L) is a noetherian domain.
Krull dimension of U(L) is at most dimk(L).
() Projective modules over universal enveloping algebras September 27, 2011 9 / 14
Universal enveloping algebras
Let L be a Lie algebra with basis fx1; : : : ; xng over k. Let
[xi ; xj ] =nX
k=1
ci ;j ;kxk :
The algebra U(L) = khx1; : : : ; xni=hxixj � xjxi �Pn
k=1 ci ;j ;kxki is calledthe universal enveloping algebra of L
Monomials xk11 xk2
2 � � � xknn form a basis of U(L), so L is contained in
U(L).
If A is an associative k-algebra and ' : L! A a linear map such that'([a; b]) = '(a)'(b)� '(b)'(a) then there is a unique extension of' to a homomorphism �: U(L)! A.
The category of representations of L is equivalent to the category of�nite-dimensional right U(L)-modules.
U(L) is a noetherian domain.
Krull dimension of U(L) is at most dimk(L).
() Projective modules over universal enveloping algebras September 27, 2011 9 / 14
Universal enveloping algebras
Let L be a Lie algebra with basis fx1; : : : ; xng over k. Let
[xi ; xj ] =nX
k=1
ci ;j ;kxk :
The algebra U(L) = khx1; : : : ; xni=hxixj � xjxi �Pn
k=1 ci ;j ;kxki is calledthe universal enveloping algebra of L
Monomials xk11 xk2
2 � � � xknn form a basis of U(L), so L is contained in
U(L).
If A is an associative k-algebra and ' : L! A a linear map such that'([a; b]) = '(a)'(b)� '(b)'(a) then there is a unique extension of' to a homomorphism �: U(L)! A.
The category of representations of L is equivalent to the category of�nite-dimensional right U(L)-modules.
U(L) is a noetherian domain.
Krull dimension of U(L) is at most dimk(L).
() Projective modules over universal enveloping algebras September 27, 2011 9 / 14
Universal enveloping algebras
Let L be a Lie algebra with basis fx1; : : : ; xng over k. Let
[xi ; xj ] =nX
k=1
ci ;j ;kxk :
The algebra U(L) = khx1; : : : ; xni=hxixj � xjxi �Pn
k=1 ci ;j ;kxki is calledthe universal enveloping algebra of L
Monomials xk11 xk2
2 � � � xknn form a basis of U(L), so L is contained in
U(L).
If A is an associative k-algebra and ' : L! A a linear map such that'([a; b]) = '(a)'(b)� '(b)'(a) then there is a unique extension of' to a homomorphism �: U(L)! A.
The category of representations of L is equivalent to the category of�nite-dimensional right U(L)-modules.
U(L) is a noetherian domain.
Krull dimension of U(L) is at most dimk(L).
() Projective modules over universal enveloping algebras September 27, 2011 9 / 14
Some results on �nitely generated projectives
Theorem
Let L be a Lie algebra of �nite dimension. Then
(Quillen) Every �nitely generated projective module over U(L) isstably free.
(Sta�ord) If P is a �nitely generated projective module over U(L) ofrank bigger than dimk(L), then P is free. In particular, Tr(P) = U(L)for every nonzero �nitely generated projective U(L)-module.
(Sta�ord) If [L; L] 6= 0 then U(L) contains a right ideal which is stably
free but not free.
() Projective modules over universal enveloping algebras September 27, 2011 10 / 14
Some results on �nitely generated projectives
Theorem
Let L be a Lie algebra of �nite dimension. Then
(Quillen) Every �nitely generated projective module over U(L) isstably free.
(Sta�ord) If P is a �nitely generated projective module over U(L) ofrank bigger than dimk(L), then P is free. In particular, Tr(P) = U(L)for every nonzero �nitely generated projective U(L)-module.
(Sta�ord) If [L; L] 6= 0 then U(L) contains a right ideal which is stably
free but not free.
() Projective modules over universal enveloping algebras September 27, 2011 10 / 14
Some results on �nitely generated projectives
Theorem
Let L be a Lie algebra of �nite dimension. Then
(Quillen) Every �nitely generated projective module over U(L) isstably free.
(Sta�ord) If P is a �nitely generated projective module over U(L) ofrank bigger than dimk(L), then P is free. In particular, Tr(P) = U(L)for every nonzero �nitely generated projective U(L)-module.
(Sta�ord) If [L; L] 6= 0 then U(L) contains a right ideal which is stably
free but not free.
() Projective modules over universal enveloping algebras September 27, 2011 10 / 14
Solvable case
Theorem
Let k be a �eld of characteristic 0 and let L be a solvable Lie algebra of
�nite dimension over k. If I1; I2; : : : is a sequence of ideals in U(L) suchthat Ik+1Ik = Ik+1; k 2 N. Then either Ik = U(L); k 2 N or there exists l
such that Il = 0.
Corollary
If L is a solvable Lie algebra of �nite dimension over a �eld of characteristic
zero then every non-�nitely generated projective U(L)-module is free.
If [L; L] = 0 we get the Bass' result for k[x1; : : : ; xn].
() Projective modules over universal enveloping algebras September 27, 2011 11 / 14
Solvable case
Theorem
Let k be a �eld of characteristic 0 and let L be a solvable Lie algebra of
�nite dimension over k. If I1; I2; : : : is a sequence of ideals in U(L) suchthat Ik+1Ik = Ik+1; k 2 N. Then either Ik = U(L); k 2 N or there exists l
such that Il = 0.
Corollary
If L is a solvable Lie algebra of �nite dimension over a �eld of characteristic
zero then every non-�nitely generated projective U(L)-module is free.
If [L; L] = 0 we get the Bass' result for k[x1; : : : ; xn].
() Projective modules over universal enveloping algebras September 27, 2011 11 / 14
Simple case
Theorem
(Weyl) Let L be a semisimple Lie algebra of �nite dimension over a �eld of
characteristic zero. Then every U(L)-module of �nite dimension is
completely reducible. In particular, if I is an ideal of �nite codimension
over U(L), then U(L)=I is semisimple artinian.
Corollary
(Kraft, Small, Wallach) If L is as above then every ideal in U(L) of �nitecodimension is idempotent.
Put R = U(sl2(C)). For every k there is a simple R-module Mk ofdimension k . Every Ik = Ann(Mk) is an ideal of codimension k2 thereforeis idempotent. By a result of Whitehead every Ik is a trace ideal of aprojective module which cannot contain a �nitely generated directsummand.
() Projective modules over universal enveloping algebras September 27, 2011 12 / 14
Simple case
Theorem
(Weyl) Let L be a semisimple Lie algebra of �nite dimension over a �eld of
characteristic zero. Then every U(L)-module of �nite dimension is
completely reducible. In particular, if I is an ideal of �nite codimension
over U(L), then U(L)=I is semisimple artinian.
Corollary
(Kraft, Small, Wallach) If L is as above then every ideal in U(L) of �nitecodimension is idempotent.
Put R = U(sl2(C)). For every k there is a simple R-module Mk ofdimension k . Every Ik = Ann(Mk) is an ideal of codimension k2 thereforeis idempotent. By a result of Whitehead every Ik is a trace ideal of aprojective module which cannot contain a �nitely generated directsummand.
() Projective modules over universal enveloping algebras September 27, 2011 12 / 14
Problems in U(sl2(C))
One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.
There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.
Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?
Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)
Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?
Is there a way how to classify projective modules over U(sl2(C))?
() Projective modules over universal enveloping algebras September 27, 2011 13 / 14
Problems in U(sl2(C))
One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.
There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.
Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?
Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)
Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?
Is there a way how to classify projective modules over U(sl2(C))?
() Projective modules over universal enveloping algebras September 27, 2011 13 / 14
Problems in U(sl2(C))
One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.
There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.
Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?
Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)
Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?
Is there a way how to classify projective modules over U(sl2(C))?
() Projective modules over universal enveloping algebras September 27, 2011 13 / 14
Problems in U(sl2(C))
One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.
There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.
Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?
Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)
Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?
Is there a way how to classify projective modules over U(sl2(C))?
() Projective modules over universal enveloping algebras September 27, 2011 13 / 14
Problems in U(sl2(C))
One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.
There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.
Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?
Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)
Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?
Is there a way how to classify projective modules over U(sl2(C))?
() Projective modules over universal enveloping algebras September 27, 2011 13 / 14
Problems in U(sl2(C))
One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.
There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.
Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?
Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)
Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?
Is there a way how to classify projective modules over U(sl2(C))?
() Projective modules over universal enveloping algebras September 27, 2011 13 / 14
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Thanks for your attention.
() Projective modules over universal enveloping algebras September 27, 2011 14 / 14