PROJECTIVE MODULES 9
Proof. Let 0 be the U(Go) homomorphism which sends
M ^ M ® l c M ®u{B±) U(Q) \U{B*)
It follows that cj) will extend to a U(B^) homomorphism
ri' M ®u{Go) U(B*) i—• M ®t/(s±) ^(fl) \u(B*)
This map is surjective since M ® 1 generates the module M ®C/"(BT) (7(g) IC/(B±) Hence the map
77 is an isomorphism of U(B^) modules because the dimensions of the two modules agree.•
The next proposition generalizes the standard theorem on Verma (standard cyclic) modules
[Hu-2]: A Verma module is an indecomposable (7(g) module with unique maximal submodule.
P r o p o s i t i o n 1.2.3. If M £ ob VB^ such that M has unique maximal submodule as U(GQ)
module then M
®u(B±)
U($)
nas a
unique maximal submodule and is indecomposable.
Proof. Using Frobenius reciprocity and Lemma 1.2.2 we have
Homu(BT)(M ®U{B±) (7(g) | ^
( B T )
, X ( / X ) ) * Homu(BT)(M ®u(G0) U(B*),L(fi)) (*)
= HomuiGo)(M,L(fi)).
The last expression is isomorphic to F if the head of M is isomorphic to L(fi) and 0 otherwise.
It follows that the module M ®u(B±) U($) \u(B*)
n a s a
unique maximal U(B^) submodule N.
Observe that if N' is any maximal proper (7(g) submodule of M
^u(B±)
^ ( s ) then N' |C/(BT)C N.
If N" is a proper submodule of M
®u(B±)
U($) then
(N" + N') \U{B*)C NCM ®
U ( B ± )
U(S) |
U ( B T )
.
Therefore, N" C N' and it follows that M ®u(B±) U(&)
n a s a
unique maximal proper U(#) sub-
module and thus is indecomposable.
D
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