Abelian Groups - László Fuchs [Springer]

6 Groups as Modules Over Their Endomorphism Rings 647
of order p n ,wehaveA 0 Š Hom.C; A/ as E-modules, so p.d.A 0 D 0.Ifweprove
that p.d..A k =A k 1 / D 1, then another form of Auslander’s lemma on projective
dimensions will assure that p.d.A cannot exceed 1. It is immediately seen that
the factor group A k =A k 1 is generated over E by any element d 2 D of order
p nCk , so it is isomorphic to E=L where L Df 2 E j d 2 A k 1 g is a left
ideal of E. Write A D D 0 ˚ A 0 where d 2 D 0 Š Z.p 1 /, and define 2 E as
multiplication by p on D 0 and the identity on A 0 . It follows that L D E Š E,
so L is E-projective. Hence p.d..A k =A k 1 / D p.d..E=L/ D 1, and p.d.A 1.
(ii) If D is a divisible group that is not torsion, then D Š E where denotes the
projection onto a summand Š Q,sinceD D Ed for any d 2 D of infinite order.
In this case, D is evidently endo-projective.
If D is a divisible p-group, then form a projective resolution of Z.p 1 / as a
J p -module: 0 ! H ! F ! Z.p 1 / ! 0 where F; H are free J p -modules
(submodules of free are free as J p is a PID). Now E D End D is a (torsion-free,
hence) flat J p -module, so the tensored sequence
0 ! E ˝Jp H ! E ˝Jp F ! E ˝Jp Z.p 1 / D D ! 0
is exact. Moreover, it is exact even as an E-sequence. The first two tensor
products are free E-modules, whence p.d.D 1 is the consequence of a
Kaplansky inequality for projective dimensions in an exact sequence. ut
In contrast, the endo-projective dimension of a torsion-free group can be any
integer or 1; see the Notes.
Endo-Quasi-Projective Groups Turning our attention to the quasi-projective
case, we can prove the following characterization for p-groups.
Theorem 6.11 (Fuchs [19]). A p-group is endo-quasi-projective if and only if it is
bounded or has an unbounded basic subgroup.
Proof. Evidently, a group A is quasi-projective over E if and only if, for each fully
invariant subgroup H of A,everyE-homomorphism ˛ W A ! A=H factors as ˛ D
where W A ! A=H is the canonical map, and W A ! A is a suitable E-map, i.e.
multiplication by some 2 J p .
To prove necessity, we have to rule out the case when A D B˚D where p m B D 0
for some m 2 N and D ¤ 0 is divisible. Let W A ! A=H be the canonical map
with H D AŒp m . Choose ˛ W A ! A=H so as to satisfy ˛B D 0 and ˛ D W D !
D=.DŒp m / an isomorphism; clearly, ˛ is an E-map. There is no 2 J p such that
˛ D , since˛DŒp m ¤ 0, butDŒp m D 0 for all 2 J p . Consequently, A must
be as stated.
For sufficiency, note that a bounded group is by Theorem 6.8 endo-projective,
while for groups with unbounded basic subgroups, an appeal to Lemma 6.1(ii)
completes the proof.
ut
Endo-Flat Groups In order to describe the endo-flat torsion groups, we may
again restrict our consideration to p-groups.