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An Extension of a Theorem on Endomorphism Rings and Equivalences
JOURNAL OF ALGEBRA ARTICLE NO.
181, 962]966 Ž1996.
0156
An Extension of a Theorem on Endomorphism Rings and Equivalences J. L. Garcıa ´ U and L. Marın ´† Department of Mathematics, Uni¨ ersity of Murcia, 30071 Murcia, Spain Communicated by Kent R. Fuller Received March 30, 1995
1. INTRODUCTION AND TERMINOLOGY Let C be a Grothendieck category, M be an object of C , and S s End C Ž M ., the endomorphism ring of M. When M is a generator of C , then we have, according to the Gabriel]Popescu theorem, that C is equivalent to a certain quotient category of Mod-S Žsee, e.g., w3, Theorem X.4.1x.. If M is arbitrary, then one may consider the hereditary torsion theory ŽT, F. of C , with the torsionfree class F consisting of all M-distinguished objects in the sense of w2x, and it is shown in w1, Theorem 1.7x that there is again an equivalence between the quotient category CM of C with respect to ŽT, F., and a quotient category of Mod-S, provided only that C be locally finitely generated. In his review of w1x, T. Kato ŽMR 91b: 16009, 1991. conjectured that this last result holds even without the condition that C be locally finitely generated. The purpose of this note is to prove that the conjecture of Kato is true, thus extending both Theorem 1.7 of w1x and the Gabriel]Popescu theorem. To this end, let us first fix some additional notation. Contrary to the use in w1x, we will now write multiplication in S as a composition of morphisms. Let us denote by F the set of right ideals I of S such that MrIM is a T-torsion object. Recall from w1, Theorem 1.6x that if F is a Gabriel filter of right ideals of S Žsee, w3, VI.5x., then CM is equivalent to the quotient category Mod-Ž S, F .. * Supported by the DGICYT ŽPB93-0515-C02-02.. Supported by the programme ‘‘Formacion ´ del Profesorado Universitario.’’
†
962 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
ENDOMORPHISM RINGS AND EQUIVALENCES
963
If L is any object of C , let G : Hom C Ž M, L. and denote by q a : M ª M ŽG. the canonical inclusion for each a g G. We shall write wG : M ŽG. ª L for the unique morphism such that wG ( q a s a, for all a g G. Note that, using this notation, a right ideal I of S belongs to F if and only if the morphism w I has a torsion cokernel. 2. THE MAIN RESULT We have to prove some preliminary lemmas. LEMMA 1. Let N be an S-submodule of Hom C Ž M, L. and let G be a family of subsets G of N such that D G g G G s N. Then ÝG g G Im wG s Im w N . Proof. For any a in N or in some G g G , let us denote the canonical inclusions by j a : M ª M Ž N . or by q a : M ª M ŽG., respectively. Also, for any G g G , there is a unique morphism jG : M ŽG. ª M Ž N . such that for all a g G, one has jG ( q a s j a . Thus, for any such G and any a g G, we have wG ( q a s a s w N ( j a s w N ( jG ( q a , from which it follows that wG s w N ( jG . Call C to the coproduct C s [G g G M ŽG. and complete our conventions by defining w : C ª M and j: C ª M Ž N . so that one has wG s w ( u G and jG s j( u G , where the u G : M ŽG. ª C are the canonical inclusions. By standard arguments, one can then see that j is an epimorphism and that w s w N ( j. It follows immediately that ÝG g G Im wG s Im w s Im w N . LEMMA 2. Let N be an S-submodule of Hom C Ž M, L. and f g Hom C Ž M, L.. For any element s g S, one has that s g Ž N : f . if and only if there exists a finite subset F of N such that if we consider the pullback diagram a
6
ZF
M
b
f
6
wF
6
6
M
ŽF.
L
then s can be factored through a . Proof. ‘‘Only if’’ part. Let s g Ž N : f ., so that f ( s s a g N. Take then F s a4 and consider the above pullback diagram with M Ž F . s M and w F s a. Since f ( s s a(1 M , we may find some d : M ª Z a such that a ( d s s. ‘‘If’’ part. Let us assume that there is a diagram as in the statement of the lemma and a morphism d : M ª ZF such that a ( d s s. Then w F ( b ( d s f ( a ( d s f ( s. If we denote by pa : M Ž F . ª M and by q a :
GARCIA ´ AND MARIN ´
964
M ª M Ž F . the canonical projections and injections, respectively, for a g F, then one has f ( s s w F (ŽÝ ag F q a ( pa .( b ( d s Ý ag F w F ( q a ( pa ( b ( d s Ý ag F a(Ž pa ( b ( d .. But each pa ( b ( d is an element of S, so that we deduce that f ( s g N and hence s g Ž N : f .. LEMMA 3. Let N be an S-submodule of Hom C Ž M, L. and f g Hom C Ž M, L.. Let us put I [ Ž N : f .. Then the cokernel of the inclusion Im w I ª f y1 ŽIm w N . is a T-torsion object. Proof. Let S be the family of all the finite subsets of N. Since N s D F g S F, we may use Lemma 1 to see that Im w N s Ý F g S Im w F . On the other hand, we can consider the family Im w F 4F g S , which is a directed family of subobjects of L. By w3, Proposition V.1.1x, we have f y1 Ž Im w N . s f y1 Ž Ý F g S Im w F . s Ý F g S f y1 Ž Im w F . Thus, we have to prove that the cokernel of the inclusion Im w I ª Ý F g S f y1 ŽIm w F . is a T-torsion object. Let us consider, for each F g S , the same pullback square of the statement of Lemma 2, and write GŽ F . [ s g S < s can be factored through a 4 . From Lemma 2, we know that I s D F g S GŽ F ., and from Lemma 1, we have that Im w I s Ý F g S Im wGŽ F . . We now claim that the inclusion Im wGŽ F . ª f y1 ŽIm w F . has T-torsion cokernel for any set F. It follows from the claim and the previous arguments that the original inclusion Im w I ª Ý F g S f y1 ŽIm w F . has also T-torsion cokernel, since the torsion class T is closed under direct sums and quotient objects. So, we next prove the claim. For each s g GŽ F ., we may obviously write s s a (h Ž s ., with h Ž s .: M ª ZF . These morphisms induce h : M ŽGŽ F .. ª ZF , such that h ( q s s h Ž s .. Let us also put X F [ Im w F : L and decompose a as an epimorphism followed by a monomorphism, so that a s g ( z . This results in a commutative diagram 6
h
M ŽGŽ F ..
YF
h g
M
6
6
f
6
XF
6
wF
6
6
ŽF.
6
M
f y1 Ž X F .
6
b
z
6
ZF
L
where YF s Im wGŽ F . and both squares at the bottom are pullbacks. So, our claim is that the cokernel of the inclusion YF ª f y1 Ž X F . is T-torsion. But suppose we had a morphism j : f y1 Ž X F . ª N, where N is Ttorsionfree, i.e., M-distinguished, and such that j annihilates on YF , so that j ( m s 0. If j / 0, then j ( z / 0, as z is an epimorphism. The fact
ENDOMORPHISM RINGS AND EQUIVALENCES
965
that N is M-distinguished implies then the existence of a morphism g: M ª ZF such that j ( z ( g / 0. But then g ( z ( g s a ( g belongs to GŽ F ., so that a ( g s a (h Ž s . for some s, from which it follows that z ( g s z ( h Ž s . s z ( h ( q s . Therefore j ( z (h ( q s / 0 and hence j ( z (h s j ( m ( h / 0, which is a contradiction because j ( m s 0. This proves the claim and the lemma. PROPOSITION 1.
F is a right Gabriel topology of the ring S.
Proof. By w3, Lemma VI.6.2x it will suffice to prove conditions ŽT3. and ŽT4.. ŽT3. Suppose I g F , so that MrIm w I is T-torsion. Take s g S, and put J [ Ž I : s .. By Lemma 3, sy1 ŽIm w I .rIm w J is T-torsion. Thus, we have the commutative diagram M
6
sy1 ŽIm w I .
s
6
6
6
Im w I
M
The arrow of the bottom row has torsion cokernel, as stated before. Consequently, the arrow of the top row has torsion cokernel, because the torsion class is closed for subobjects. But we may deduce from this and from the fact that the torsion class is closed under extensions that MrIm w J is also T-torsion. By definition, we have J g F , as we had to show. ŽT4. Now, we suppose that I, J are right ideals of S, that J g F , and that for any s g J, one has that Ž I : s . belongs to F again. We have to infer that I g F. So, we need to show that, if U s Im w I , then MrU is T-torsion. Let us set V [ Im w J , so that we know that MrV is T-torsion. Finally for each s g J, we put Ws [ Im wŽ I : s. and we know that MrWs is also T-torsion. It follows from this notation that sŽWs . : U. Since obviously Ws : sy1 ŽU ., we deduce that for any such s, Mrsy1 ŽU . is T-torsion. Moreover, by Lemma 3, sy1 ŽU .rWs is a torsion object. Now, each s gives an epimorphism from M onto sŽ M . and it is easy to see that, inasmuch as Mrsy1 ŽU . is T-torsion, sŽ M .rsŽ sy1 ŽU .. is also T-torsion for each s g J. Since the torsion class is closed under direct sums and quotients, it follows that another inclusion with torsion cokernel is Ý s g J sŽ sy1 ŽU .. ª Ý s g J sŽ M .. The second sum is nothing else than V. On the other hand, for similar reasons we have that the inclusion Ý s g J sŽWs . ª Ý s g J sŽ sy1 ŽU .. has torsion cokernel. But the first sum is contained in U, so that we may deduce that VrU is indeed a T-torsion object. Since MrV is torsion, it follows that MrU is torsion, and hence I g F.
966
GARCIA ´ AND MARIN ´
We have thus obtained the following result, conjectured by Kato. THEOREM 1. If C is any Grothendieck category, M is an object of C , and S is the endomorphism ring of M, then the functor Hom C Ž M, ] . from C to Mod-S establishes an equi¨ alence between CM and the quotient category Mod-Ž S, F .. Proof. It is a consequence of the above Proposition and w1, Theorem 1.6x.
REFERENCES 1. J. L. Garcıa ´ and M. Saorın, ´ Endomorphism rings and category equivalences, J. Algebra 127 Ž1989., 182]205. 2. T. Kato, U-Distinguished modules, J. Algebra 25 Ž1973., 15]24. 3. B. Stenstrom, ¨ ‘‘Rings of Quotients,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1975.
181, 962]966 Ž1996.
0156
An Extension of a Theorem on Endomorphism Rings and Equivalences J. L. Garcıa ´ U and L. Marın ´† Department of Mathematics, Uni¨ ersity of Murcia, 30071 Murcia, Spain Communicated by Kent R. Fuller Received March 30, 1995
1. INTRODUCTION AND TERMINOLOGY Let C be a Grothendieck category, M be an object of C , and S s End C Ž M ., the endomorphism ring of M. When M is a generator of C , then we have, according to the Gabriel]Popescu theorem, that C is equivalent to a certain quotient category of Mod-S Žsee, e.g., w3, Theorem X.4.1x.. If M is arbitrary, then one may consider the hereditary torsion theory ŽT, F. of C , with the torsionfree class F consisting of all M-distinguished objects in the sense of w2x, and it is shown in w1, Theorem 1.7x that there is again an equivalence between the quotient category CM of C with respect to ŽT, F., and a quotient category of Mod-S, provided only that C be locally finitely generated. In his review of w1x, T. Kato ŽMR 91b: 16009, 1991. conjectured that this last result holds even without the condition that C be locally finitely generated. The purpose of this note is to prove that the conjecture of Kato is true, thus extending both Theorem 1.7 of w1x and the Gabriel]Popescu theorem. To this end, let us first fix some additional notation. Contrary to the use in w1x, we will now write multiplication in S as a composition of morphisms. Let us denote by F the set of right ideals I of S such that MrIM is a T-torsion object. Recall from w1, Theorem 1.6x that if F is a Gabriel filter of right ideals of S Žsee, w3, VI.5x., then CM is equivalent to the quotient category Mod-Ž S, F .. * Supported by the DGICYT ŽPB93-0515-C02-02.. Supported by the programme ‘‘Formacion ´ del Profesorado Universitario.’’
†
962 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
ENDOMORPHISM RINGS AND EQUIVALENCES
963
If L is any object of C , let G : Hom C Ž M, L. and denote by q a : M ª M ŽG. the canonical inclusion for each a g G. We shall write wG : M ŽG. ª L for the unique morphism such that wG ( q a s a, for all a g G. Note that, using this notation, a right ideal I of S belongs to F if and only if the morphism w I has a torsion cokernel. 2. THE MAIN RESULT We have to prove some preliminary lemmas. LEMMA 1. Let N be an S-submodule of Hom C Ž M, L. and let G be a family of subsets G of N such that D G g G G s N. Then ÝG g G Im wG s Im w N . Proof. For any a in N or in some G g G , let us denote the canonical inclusions by j a : M ª M Ž N . or by q a : M ª M ŽG., respectively. Also, for any G g G , there is a unique morphism jG : M ŽG. ª M Ž N . such that for all a g G, one has jG ( q a s j a . Thus, for any such G and any a g G, we have wG ( q a s a s w N ( j a s w N ( jG ( q a , from which it follows that wG s w N ( jG . Call C to the coproduct C s [G g G M ŽG. and complete our conventions by defining w : C ª M and j: C ª M Ž N . so that one has wG s w ( u G and jG s j( u G , where the u G : M ŽG. ª C are the canonical inclusions. By standard arguments, one can then see that j is an epimorphism and that w s w N ( j. It follows immediately that ÝG g G Im wG s Im w s Im w N . LEMMA 2. Let N be an S-submodule of Hom C Ž M, L. and f g Hom C Ž M, L.. For any element s g S, one has that s g Ž N : f . if and only if there exists a finite subset F of N such that if we consider the pullback diagram a
6
ZF
M
b
f
6
wF
6
6
M
ŽF.
L
then s can be factored through a . Proof. ‘‘Only if’’ part. Let s g Ž N : f ., so that f ( s s a g N. Take then F s a4 and consider the above pullback diagram with M Ž F . s M and w F s a. Since f ( s s a(1 M , we may find some d : M ª Z a such that a ( d s s. ‘‘If’’ part. Let us assume that there is a diagram as in the statement of the lemma and a morphism d : M ª ZF such that a ( d s s. Then w F ( b ( d s f ( a ( d s f ( s. If we denote by pa : M Ž F . ª M and by q a :
GARCIA ´ AND MARIN ´
964
M ª M Ž F . the canonical projections and injections, respectively, for a g F, then one has f ( s s w F (ŽÝ ag F q a ( pa .( b ( d s Ý ag F w F ( q a ( pa ( b ( d s Ý ag F a(Ž pa ( b ( d .. But each pa ( b ( d is an element of S, so that we deduce that f ( s g N and hence s g Ž N : f .. LEMMA 3. Let N be an S-submodule of Hom C Ž M, L. and f g Hom C Ž M, L.. Let us put I [ Ž N : f .. Then the cokernel of the inclusion Im w I ª f y1 ŽIm w N . is a T-torsion object. Proof. Let S be the family of all the finite subsets of N. Since N s D F g S F, we may use Lemma 1 to see that Im w N s Ý F g S Im w F . On the other hand, we can consider the family Im w F 4F g S , which is a directed family of subobjects of L. By w3, Proposition V.1.1x, we have f y1 Ž Im w N . s f y1 Ž Ý F g S Im w F . s Ý F g S f y1 Ž Im w F . Thus, we have to prove that the cokernel of the inclusion Im w I ª Ý F g S f y1 ŽIm w F . is a T-torsion object. Let us consider, for each F g S , the same pullback square of the statement of Lemma 2, and write GŽ F . [ s g S < s can be factored through a 4 . From Lemma 2, we know that I s D F g S GŽ F ., and from Lemma 1, we have that Im w I s Ý F g S Im wGŽ F . . We now claim that the inclusion Im wGŽ F . ª f y1 ŽIm w F . has T-torsion cokernel for any set F. It follows from the claim and the previous arguments that the original inclusion Im w I ª Ý F g S f y1 ŽIm w F . has also T-torsion cokernel, since the torsion class T is closed under direct sums and quotient objects. So, we next prove the claim. For each s g GŽ F ., we may obviously write s s a (h Ž s ., with h Ž s .: M ª ZF . These morphisms induce h : M ŽGŽ F .. ª ZF , such that h ( q s s h Ž s .. Let us also put X F [ Im w F : L and decompose a as an epimorphism followed by a monomorphism, so that a s g ( z . This results in a commutative diagram 6
h
M ŽGŽ F ..
YF
h g
M
6
6
f
6
XF
6
wF
6
6
ŽF.
6
M
f y1 Ž X F .
6
b
z
6
ZF
L
where YF s Im wGŽ F . and both squares at the bottom are pullbacks. So, our claim is that the cokernel of the inclusion YF ª f y1 Ž X F . is T-torsion. But suppose we had a morphism j : f y1 Ž X F . ª N, where N is Ttorsionfree, i.e., M-distinguished, and such that j annihilates on YF , so that j ( m s 0. If j / 0, then j ( z / 0, as z is an epimorphism. The fact
ENDOMORPHISM RINGS AND EQUIVALENCES
965
that N is M-distinguished implies then the existence of a morphism g: M ª ZF such that j ( z ( g / 0. But then g ( z ( g s a ( g belongs to GŽ F ., so that a ( g s a (h Ž s . for some s, from which it follows that z ( g s z ( h Ž s . s z ( h ( q s . Therefore j ( z (h ( q s / 0 and hence j ( z (h s j ( m ( h / 0, which is a contradiction because j ( m s 0. This proves the claim and the lemma. PROPOSITION 1.
F is a right Gabriel topology of the ring S.
Proof. By w3, Lemma VI.6.2x it will suffice to prove conditions ŽT3. and ŽT4.. ŽT3. Suppose I g F , so that MrIm w I is T-torsion. Take s g S, and put J [ Ž I : s .. By Lemma 3, sy1 ŽIm w I .rIm w J is T-torsion. Thus, we have the commutative diagram M
6
sy1 ŽIm w I .
s
6
6
6
Im w I
M
The arrow of the bottom row has torsion cokernel, as stated before. Consequently, the arrow of the top row has torsion cokernel, because the torsion class is closed for subobjects. But we may deduce from this and from the fact that the torsion class is closed under extensions that MrIm w J is also T-torsion. By definition, we have J g F , as we had to show. ŽT4. Now, we suppose that I, J are right ideals of S, that J g F , and that for any s g J, one has that Ž I : s . belongs to F again. We have to infer that I g F. So, we need to show that, if U s Im w I , then MrU is T-torsion. Let us set V [ Im w J , so that we know that MrV is T-torsion. Finally for each s g J, we put Ws [ Im wŽ I : s. and we know that MrWs is also T-torsion. It follows from this notation that sŽWs . : U. Since obviously Ws : sy1 ŽU ., we deduce that for any such s, Mrsy1 ŽU . is T-torsion. Moreover, by Lemma 3, sy1 ŽU .rWs is a torsion object. Now, each s gives an epimorphism from M onto sŽ M . and it is easy to see that, inasmuch as Mrsy1 ŽU . is T-torsion, sŽ M .rsŽ sy1 ŽU .. is also T-torsion for each s g J. Since the torsion class is closed under direct sums and quotients, it follows that another inclusion with torsion cokernel is Ý s g J sŽ sy1 ŽU .. ª Ý s g J sŽ M .. The second sum is nothing else than V. On the other hand, for similar reasons we have that the inclusion Ý s g J sŽWs . ª Ý s g J sŽ sy1 ŽU .. has torsion cokernel. But the first sum is contained in U, so that we may deduce that VrU is indeed a T-torsion object. Since MrV is torsion, it follows that MrU is torsion, and hence I g F.
966
GARCIA ´ AND MARIN ´
We have thus obtained the following result, conjectured by Kato. THEOREM 1. If C is any Grothendieck category, M is an object of C , and S is the endomorphism ring of M, then the functor Hom C Ž M, ] . from C to Mod-S establishes an equi¨ alence between CM and the quotient category Mod-Ž S, F .. Proof. It is a consequence of the above Proposition and w1, Theorem 1.6x.
REFERENCES 1. J. L. Garcıa ´ and M. Saorın, ´ Endomorphism rings and category equivalences, J. Algebra 127 Ž1989., 182]205. 2. T. Kato, U-Distinguished modules, J. Algebra 25 Ž1973., 15]24. 3. B. Stenstrom, ¨ ‘‘Rings of Quotients,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1975.