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