h : S(@A.) - » B such that hk. = f . for all j, then clearly
(h-g)(@A.) = 0. Inasmuch as E(@A.)/(@A.) is singular while B
is nonsingular, it follows that h - g = 0. Therefore E(@A.) is a
coproduct of the A. in 7((R).
DEFINITION. Following Gabriel-Oberst [3], we define a spectral
category to be an abelian category with exact direct limits and a
generator in which every morphism splits, i.e., the kernel and image
of any morphism f are direct summands of the domain and range of f.
PROPOSITION 1.13. 7?(R) is a spectral category.
Proof. According to 1.11, 7?(R) is an abelian category. It is
clear from 1.3 and 1.5 that every morphism in 7?(R) splits.
Let G be the injective hull of the direct sum of all cyclic
modules R/I, where I c L*(RR), and note that G e ??(R).
Since any cyclic submodule of any A e T(CR) is isomorphic to R/I
for suitable I e L*(RR), it follows that G is a generator in ??(R).
Taking account of the form of coproducts in 71(B) as given by
1.12, we see from [3, p. 389] that 7?(R) has exact direct limits if
and only if the following condition holds: for any set {A. | ie 1}
of objects in 7?(R) and any subobject B of E(©A.), B is the
supremum of all subobjects B f l E( 0 A.), where J ranges over all
i e J 1
f i n i t e s u b s e t s of I . The sum of a l l 3 0 ( e A.) i s BO ( ® A . )
1 x
which is essential in B, hence we see that the sum of all
B0E( © A.) is essential in B. According to 1.1, B is the
i e J
S-closure in E(@A.) of the sum of all B ( 1 E( © A.), whence 1.6
1 ieJ 1
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