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LOCALLY Λ-PROJECΠVE ABELIAN GROUPS AND GENERALIZATIONS 221
of / such that p.d. ((& n<ω
In) = gl. dimi? - 1 where oo - 1 is defined
to be oo. The inclusion map /: ® n<ω
I n
—• ® ω
E(A) yields the
commutative diagram
0 . H λ
T λ
(©/„) J&®. H Λ
T
Hence φ® n<ω
i n
is a monomorphism.
Choose a free right £ l (^4)-module F which admits an epimorphism
π:F -> ® n<ω
I n
. Since T A
(φ n<ω
I n
) c T A
(φ ω
E(A)) yields that
the sequence T A
(F) -> T A
(φ n<ω
I n
) -• 0 is ^-balanced, because ^ is
faithfully flat, the top-row of the commutative diagram
H A
T A
{F) ^ Ά H A
T A
(Q n<
J n
) • 0
UΦF
Φ@n<ωln
is exact. Hence 0 w < ω
I n
= H A
T A
(@ n<ω
I n
) and has projective dimension
equal to (gl.dimi?) - 1. By Proposition 3.4,
Λ-p.d.
4. Locally ^4-projective groups. The combination of the results of
Sections 2 and 3 allows to give a more precise estimate of the A-
projective dimension of locally ^4-projective groups than we can obtain
from Corollary 3.5:
THEOREM 4.1. Let A be an abelian group which is flat as an E(A)-
module and has an endomorphism ring which is discrete in the finite
topology. A locally A-projective group G has A-projective dimension at
most n if it is an epimorphic image ofφ ωn
A.
Proof. Since G is an epimorphic image of 0 ω^ A, there exists a
family {φv}v<ω n
Q H A
{G) such that G = (φ v
(A)\v "< ω n
). Moreover,
we can find a finite subset X of A such that {a e E(A)\a(X) = 0} =
0. We show that G is an ^-balanced epimorphic image of a group
isomorphic to a direct summand of 0 ω^ A.
,n<ω