NONSINGULAR INJECTIVS MODULES 7

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 . )

f

ieJ

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