On colimits of injectives in Grothendieck categories
DANIEL SIMSO~r
Nieolaus Copernicus University, TormS, Poland
1. Colimits of injectives
A Grothendieck category is an Abelian category d which has a set of generators and admits colimits that are exact functors when taken over directed sets. A n y Grotendieck category has arbitrary products and sufficiently m a n y injective objects (see [l]).
An object D of ~9~ is said to be a strict cogenerator if every object of ~4 admits an embedding in a suitable coproduct of copies of D.
In [6] (see also [2] and [5]) J. E. Roos has proved t h a t the following conditions are equivalent for a Grothendieck category :A
(i) There exists an injective object I of :A such that every injective in ~ is a coproduct of direct summands of I.
(ii) Every coproduct of injectives in ~ is still injective.
(iii) ~ admits a strict cogenerator.
The aim of this paper is to prove t h a t in a Grothendieck category ~ each of the above equivalent conditions (i)--(iii) implies t h a t a directed colimit of injectives is injective. This answers a question of J. E. l~oos [6, p. 202].
Throughout this paper colim is a colimit functor, [ [ and " ~ denote coproduct and product respectively.
We start with the following
L~M~A. Let {M~, hz~}~_<z<~ be a well-ordered directed system in a Grothendieelc category c54 such that M o = O, h~ are monomorphisms and M,~ : colim~<,j M~
whenever ~ is a limit ordinal number. I f h~+l, ~: M s ---> M~+ 1 splits for any ~ < y then
colim~<y XI _~ 1_1 M~+I/M~"
162 DANIEL 813{8 ON-
_p,-oof. Let G = IL< and let G , -< fi < be natural monomorphism. For any ~ < y we fix such splitting maps d~ and t of h~+:,~ and v : M~+:-+M~+:/M~ respectively, that the identity map on M + : is equal to h +1, d + t v .
Let fr B ~ - ~ 2k1r be a map defined b y commutative diagrams
.f,
h~, ~+:
I t is easy to verify that {f~}: {B~, s ~ } - ~ {M~, h~} is a map of directed systems and t h a t the following diagram commutes
O - - * B B +: ____~-
fo~+: id
8c~+ 1, o~
fc~ dc~ * -~ tc~ *
0 - - ~ - M _ ~ - - ~ M~+: _ _ + M~+:/M~ - - - + 0 ho~q-1
where j~ is a natural projection and p~ is a natural injection. I t follows that f~+: is an isomorphism whenever so is f . Clearly f0 is an isomorphism. I f ~1 is
a limit ordinal and f~ are isomorphisms for ~ < ~ then so is f , = eolim~<,~ f~: B, ~-~ colim~<,~ B; -+ M~.
Consequently, colim <rf~ is an isomorphism and the lemma is proved.
T~ino~aE~ 1. I f a Grothendiec]c category d admits a strict cogenerator then any directed colimit of injectives in o i is injective.
Pro@ Consider an exact sequence
0 ~ K - - ~ - - ~ ~ 2k/i - - + h coliml e IM~ -- ~ 0 (*) iEI
where 3f~, i C I, is a direct system of injeetive objects in d and h is the natural colimit morphism. We shall prove b y transfinite induction on the eardinMity [I[
of ] that any sequence of the form (*) splits. Clearly it is true if xr is finite. Now suppose t h a t I is arbitrary and that our statement holds for any directed set of cardinMity less than [II. I t follows from [4, Lemma 1.4] that I is a union of an ascending transfinite sequence of directed subsets I~ such that [I~[ < I I [ for any and /~j = I.Js<, I~ whenever ~ is a limit ordinal number. Then for any ~ we have a commutative diagram
O N C O L I M I T S O:F I N J E C T I V E S I N G I ~ O T H E N D I E C K C A T E G O I ~ I E S 163 0- ~ K~ - - - - > J~.~e~21~ ~ eolim~et~M/ + 0
0 ~ K~+ 1 - - + IA~eI~+121 i ~ colim~ei~.iMi + 0
w i t h splitting rows b y t h e inductive assumption. I t follows t h a t Ke c_+ K~+~ splits for a n y ~ a n d therefore K~+~/K~ is injective. Since t h e exact sequence
0 - ~ colim K~ - - - + eolim 1_[~ 9 x~M~ - -~ colim colim/e ~M~ --+ 0 is isomorphic to (*) t h e n b y L e m m a K _~ J~_ K~+~/K~. Hence K is injeetive a n d our conclusion follows.
C0~0LLARu Let r
for any directed system
be a Grothendiect: category with a strict cogenerator. Then inj dim colim M~ _< sup inj dim 21i
21, i E I , in J .
Pro@ I t follows from our a s s u m p t i o n t h a t ~ has a n injective cogenerator E.
F o r each object 21 of o~4 we set Q ( 2 1 ) = ]-VEx where E I = E a n d f runs t h r o u g h all morphisms from M to E. I t is clear t h a t t h e n a t u r a l m o r p h i s m 21 - ~ Q(21) is a m o n o m o r p h i s m a n d a n y morphism h: M --~ N induces a n a t u r a l morphism Q(h): Q(M) --> Q(N) such t h a t Q becomes a covariant e n d o f u n c t o r of r
We shall prove t h e corollary b y i n d u c t i o n on d = sup inj dim M~, since there is n o t h i n g to prove provided d = ~ . I n v i r t u e of Theorem ] it is sufficient to pass from d -- 1 to d. F o r this purpose observe t h a t from a directed s y s t e m of exact sequences
0 - + 21i ~ Q(Md -+ Ki -+ 0 we derive the exact sequence
0 ~ colim M ~ - ~ colim Q ( M d - ~- colim Ki ~ 0
where t h e middle object is injective b y Theorem 1 a n d the last one has injective dimension less t h e n d b y the inductive assumption. Hence t h e corollary follows.
2. Pure-i~jectives and pure-projectives
L e t ,v4 be a G r o t h e n d i e e k category. Recall t h a t an object M of ~9/ is finitely generated if for each directed f a m i l y 2 1 , i E I , of subobjects of M w i t h M = [J/e i M~
there is an j E I w i t h 2Ij -- M. M is finitely presented if it is f i n i t e l y generated a n d every epimorphism N--~ M , where N is f i n i t e l y generated, has a f i n i t e l y generated kernel. ~ is said to be a locally finitely presented category if it has a f a m i l y
164 D A N I E L SI]VIS O N
of finitely presented generators. A short exact sequence in ~-4 is pure if every finitely presented object is relatively projective for it.
By [8], pure sequences in a locally finitely presented category form a proper class which is closed under directed colimits. 3/[oreover, in this case there exist enough pure-projective objects.
The following result is an extension of the Theorem 4.2 in [3] from module categories to locally finitely presented ones.
T ~ o g ~ M 2. Let ~ be a locally finitely presented category. The following statements are equivalent:
(a) All objects of r are pure-projective.
(b) Every pure-projective object of ~4 is pure-injeetive.
Moreover, i f ~i has enough pure-injectives then (a) is equivalent to the following statement
(e) Every pure-injeetive object in r is pure-projective.
Proof. The implications (a) ~ (b) and (a) ~ (c) are trivial, and (c) ~ (a) m a y be proved as the one in [3, Theorem 4.2].
To prove (b) ~ (a) observe t h a t every sequence of the form (*) is pure. I t is trivial if the set I is finite. The general case follows by transfinite induction on the eardinality ]II using the fact t h a t (*) is a directed colimit of sequences of the form (**). Assume (b). By [8, L e m m a 4] every object is a directed colimit of finitely presented (so pure-injeetive) objects. Then it is sufficient to show t h a t every sequence of the form (*), with Ms pure-injective, splits. But this follows from the above remarks and the arguments from the proof of Theorem 1. This completes the proof.
After not hard modifications of arguments from [3, Sec. 5] and [7, Sec. 1] one obtains
THEOREM 3. Let ~4 be a locally finitely presented category. I f O-- ~- K + Pn- ~ .. . + P o - - ~ M ~ 0
is a pure exact ~'equence in +/ and Po . . . P , are pure-projective objects then K is an lt,,-directed union of pure-projective subobjects (which are ~ ,,_1G-pure sub- objects of K in the sense of [3, Sec. 5]).
Note Added in P r o @ Theorem 1 and Corollary have been proved by J a n - E r i k Roos in the paper >)On the structure of abelian categories with generators and exact direct limits)> (to appear).
References
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Received April 10, 1973 Dr. Daniel S i m s o n
Institute of M a t h e m a t i c s Nicolaus Copernicus University 87--100 Toruh, Polandd ul. Gruclzis~dzka 5.
