On equivalence of some subcategories of modules in Morita contexts
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2003 |
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Інститут прикладної математики і механіки НАН України
2003
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| Цитувати: | On equivalence of some subcategories of modules in Morita contexts / A.I. Kashu // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 46–53. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860242583797628928 |
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| author | Kashu, A.I. |
| author_facet | Kashu, A.I. |
| citation_txt | On equivalence of some subcategories of modules in Morita contexts / A.I. Kashu // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 46–53. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2003). pp. 46 – 53
c© Journal “Algebra and Discrete Mathematics”
On equivalence of some subcategories of modules
in Morita contexts
A. I. Kashu
Abstract. A Morita context (R, RVS , SWR, S) defines the
isomorphism L0(R) ∼= L0(S) of lattices of torsions r ≥ rI of R-Mod
and torsions s ≥ rJ of S-Mod, where I and J are the trace ideals
of the given context. For every pair (r, s) of corresponding torsions
the modifications of functors TW = W⊗R- and TV = V ⊗S- are
considered:
R-Mod ⊇ P(r)
T̄W = (1/s) · T
W
−−−−−−−−−−−−−−−−−−−→←−−−−−−−−−−−−−−−−−−
T̄V = (1/r) · T
V
P(s) ⊆ S-Mod,
where P(r) and P(s) are the classes of torsion free modules. It is
proved that these functors define the equivalence
P(r) ∩ JI ≈ P(s) ∩ JJ ,
where P(r) = {RM | r(M) = 0} and JI = {RM | IM = M}.
Let (R, RVS , SWR, S) be an arbitrary Morita context with the bimod-
ule morphisms
(, ) : V ⊗S W −→ R, [, ] : W ⊗R V −→ S,
satisfying the conditions of associativity:
(v, w)v1 = v[w, v1], [w, v]w1 = w(v, w1) (1)
for v, v1 ∈ V and w, w1 ∈ W . We denote by I = (V, W ) and J = [W, V ]
the trace ideals of this context, where I is ideal of R and J is ideal of S.
2000 Mathematics Subject Classification: 16S90, 16D90.
Key words and phrases: torsion (torsion theory), Morita context, torsion free
module, accessible module, equivalence.
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.A. I. Kashu 47
They define the torsions rI in R-Mod and rJ in S-Mod such that the
classes of torsion free modules are:
P(rI) = {RM | I m = 0, m ∈ M =⇒ m = 0},
P(rJ) = {SN | J n = 0, n ∈ N =⇒ n = 0},
i.e. rI and rJ are determined by the smallest Gabriel filters, containing
I and J , respectively [7].
In the lattices L(R) and L(S) of all torsions of R-Mod and S-Mod,
respectively, we distinguish the following sublattices:
L0(R) = {r ∈ L(R) | r ≥ rI}, (2)
L0(S) = {s ∈ L(S) | s ≥ rJ}.
The following result is well known ([1], [4], [5], [7]).
Theorem 1. There exists a preserving order bijection between the tor-
sions of R-Mod containing rI and torsions of S-Mod containing rJ , i.e.
L0(R) ∼= L0(S).
This bijection is obtained with the help of the functors:
R-Mod
HV = HomR(V,−)
−−−−−−−−−−−−−−−−−−−−−→←−−−−−−−−−−−−−−−−−−−−−
HW = HomS(W,−)
S-Mod, (3)
acting by HV and HW to the injective cogeneratots of torsions [4]. From
the definitions it follows
Lemma 2. ([4], Lemma 4). If (r, s) is a pair of corresponding torsions
in the sense of Theorem 1 (i.e. HV (r) = s and HW (s) = r), then
HV (P(r)) ⊆ P(s) and HW (P(s)) ⊆ P(r), where P(r) and P (s) are (P)
the classes of torsion free modules.
Now we consider the following functors accompanying the given Mori-
ta context:
R-Mod
TW = W⊗R−−−−−−−−−−−−−−−−−−→←−−−−−−−−−−−−−−−−
T V = V ⊗S−
S-Mod (4)
with the natural transformations
η : T V TW −→ 1R−Mod, ρ : TW T V −→ 1S−Mod,
defined by the rules:
ηM(v ⊗ w ⊗ m) = (v, w)m, ρN(w ⊗ v ⊗ n) = [w, v]n, (5)
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.48 On equivalence of some subcategories of modules...
for v ⊗ w ⊗ m ∈ T V TW (M), M ∈ R-Mod and w ⊗ v ⊗ n ∈ TW T V (N),
N ∈ S-Mod. By definitions it follows:
ImηM = IM, ImρN = JN.
It is easy to verify the following relations:
TW (ηM) = ρ
TW (M), (6)
T V (ρN) = η
TV (N), (7)
for every M ∈ R-Mod and N ∈ S-Mod (i.e. (TW , T V ) and (η, ρ) define
a wide Morita context in the sense of [3]).
For an arbitrary class of modules K ⊆ R-Mod we denote:
K↑ = {X ∈ R-Mod |HomR(X, Y ) = 0 ∀Y ∈ K},
K↓ = {Y ∈ R-Mod |HomR(X, Y ) = 0 ∀X ∈ K}.
If r is a torsion of R-Mod, R(r) = {M ∈ R-Mod | r(M) = M} and
P(r) = {M ∈ R-Mod | r(M) = 0}, then R(r) = P(r)↑ and P(r) = R(r)↓
([5], [7], [8]).
The following statement is known ([6], lemma 3), but for convenience
we give the proof.
Lemma 3. If (r, s) is a pair of corresponding torsions in the sense of
Theorem 1, then TW (R(r)) ⊆ R(s) and T V (R(s)) ⊆ R(r).
Proof. Let SN ∈ R(s) = P(s)↑, i.e. HomS(N, Y ) = 0 for every Y ∈
P(s). If M ∈ P(r), then by Lemma 2 SHV (M) = HomR(V, M) ∈ P(s).
Now from N ∈ R(s) it follows that HomS(N, HomR(V, M)) = 0. By
adjunction
HomR(V ⊗S N, M) ∼= HomS(N, HomR(V, M)) = 0
for every M ∈ P(r), therefore V ⊗S N ∈ P(r)↑ = R(r), i.e. T V (R(s)) ⊆
R(r). By symmetry the relation TW (R(r)) ⊆ R(s)(R) is true.
In continuation we mention some facts about the classes of modules
determined by trace ideals I C R and J C S in the categories R-Mod and
S-Mod, respectively. The ideal I C R defines in R-Mod the following
classes of modules:
A(I) = {M ∈ R-Mod | IM = 0},
JI = {M ∈ R-Mod | IM = M},
FI = {M ∈ R-Mod | I m = 0, m ∈ M =⇒ m = 0} = P(rI).
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.A. I. Kashu 49
The modules of JI are called I-accessible and
JI = {M ∈ R-Mod | ImηM = M}.
The following relations are known ([7], [8]):
JI = A(I)↑, FI = A(I)↓. (8)
Similarly we define the classes A(J), JJ and FJ in S-Mod with the re-
lations JJ = A(J)↑ and FJ = A(J)↓, where FJ = P(rJ).
Lemma 4. Let (r, s) be a pair of corresponding torsions (Theorem 1).
Then A(I) ⊆ R(r) and A(J) ⊆ R(s).
Proof. From r ≥ rI it follows P(r) ⊆ P(rI) = FI and by (8) we obtain
R(r) = P(r)↑ ⊇ P(rI)
↑ = F↑
I
= A(I)↓↑ ⊇ A(I).
Similarly, R(s) ⊇ A(J).
From now on we fix an arbitrary pair (r, s) of corresponding torsions,
i.e. r ≥ rI , s ≥ rJ , s = HV (r) and r = HW (s) (Theorem 1). We
consider the following modifications of the functors TW and T V :
-¾
¾ -
??
1/r 1/s
S-Mod
T W
T V
T̄ W
T̄ V
R-Mod
R-Mod S-Mod,
where (1/r)(M) = M/r(M), (1/s)(N) = N/s(N), T̄W = (1/s) · TW
and T̄ V = (1/r) · T V . So, by definition:
T̄W (RM) = (W ⊗R M)/s(W ⊗R M), T̄V (SN) = (V ⊗S N)/r(V ⊗S N) (9)
for M ∈ R-Mod and N ∈ S-Mod. Denote by α and β the natural
transformations:
α : TW −→ T̄W , β : T V −→ T̄ V ,
where
αM : TW (M) −→ TW (M)/s(TW (M))
and
βN : T V (N) −→ T V (N)/r(T V (N))
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.50 On equivalence of some subcategories of modules...
are the natural epimorphisms. Since the functors TW and T V are right
exact, it is clear that the functors T̄W and T̄ V preserve epimorphisms. By
definitions of T̄W and T̄ V it follows that T̄W (M) ∈ P(s) and T̄ V (N) ∈
P(r) for every M ∈ R-Mod and N ∈ S-Mod, therefore we can consider
the restrictions of these functors on the subcategories P(r) and P(s):
P(r)
T̄W
−−−−−−−−−→←−−−−−−−−
T̄ V
P(s). (10)
In the situation (10) there exist the modifications of natural transfor-
mations η and ρ:
η̄ : T̄ V T̄W −→ 1P(r), ρ̄ : T̄W T̄ V −→ 1P(s),
which are defined (see [3]) as follows. For every M ∈ P(r) applying T V
to the exacte sequence
0 → s(TW (M)) iM−→
⊆
TW (M)
αM−−−→
nat
TW (M)/s(TW (M)) → 0, (11)
we obtain the diagram:
?
M
r(T V T̄ W (M))
⋂
|i
η′
M
@
@
@
@
@@R
η
M
ηM
?
ª
T V (s(T W (M)))
T V (iM )
−−−−−−→T V T W (M)
T V (αM )
−−−−−−→T V T̄ W (M)
β
T̄
W (M)
−−−−−−→ T̄ V T̄ W (M) → 0 (12)
Since s(TW (M)) ∈ R(s), by Lemma 3 T V (s(TW (M))) ∈ R(r), so
from M ∈ P(r) it follows HomR(T V (s(TW (M))), M) = 0, therefore
ηM · T V (iM) = 0. Since ImT V (iM) = Ker T V (αM) ⊆ Ker ηM and
T V (αM) is an epimorphism, there exists an unique morphism η′
M
such
that η′
M
· T V (αM) = ηM . The following step: from M ∈ P(r) and
r(T V T̄W (M)) ∈ R(r) it follows η′
M
· i = 0 and there exists an unique
morphism η̄M such that η̄M · β
T̄W (M) = η′
M
. So, by definitions we have:
ηM = ηM · β
T̄W (M) · T
V (αM). (13)
In such a way it is obtained a natural transformations η ([3]) and
symmetrically ρ̄ is defined. From these definitions follows immediately
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.A. I. Kashu 51
Lemma 5. a) If the module M ∈ P(r) is I-accessible (i.e. ηM is epi),
then η̄M is an epimorphism.
b) If the module N ∈ P(s) is J-accessible, then ρ̄N is an epimorphism.
Now we consider in P(r) and P(s) the following subcategories of
torsion free and accessible modules:
A = P(r) ∩ JI ⊆ R-Mod, B = P(s) ∩ JJ ⊆ S-Mod.
Lemma 6. The functors T̄W and T̄ V transfer subcategories A and B
each one in another, i.e. T̄W (A) ⊆ B and T̄ V (B) ⊆ A.
Proof. Let M ∈ A. Since T̄W (M) ∈ P(s), it is sufficient to check that
T̄W (M) ∈ JJ . For that we consider the following commutative diagram:
-
TW T V (αM) αM
TW (M)
ρ
TW (M)
ρ
T̄W (M)
TW T V TW (M)
TW T V T̄W (M) T̄W (M)-
? ?
(14)
Since M ∈ JI , ηM is epi, therefore TW (ηM) is epi. From (6) ρ
T W (M) =
TW (ηM), so ρ
T W (M) is epi, therefore αM ·ρ
T W (M) also is epi. Now diagram
(14) shows that ρ
T̄ W (M) is epimorphism, i.e. T̄W (M) ∈ JJ . This proves
that T̄W (A) ⊆ B. By symmetry T̄ V (B) ⊆ A.
Another proof of Lemma 6 follows from the remark that
TW (JI) ⊆ JJ , T V (JJ) ⊆ JI . (15)
Indeed, if M ∈ JI then:
J(W ⊗R M) = [W, V ]W ⊗R M = W (V, W ) ⊗R M =
= W ⊗R (V, W )M = W ⊗R IM = W ⊗R M,
i.e. TW (M) ∈ JJ , and similarly for the second relation.
Now from (15) for every M ∈ JI we obtain:
J · T̄W (M) = J · [(W ⊗R M)/s(W ⊗R M)] =
= [J(W ⊗R M) + s(W ⊗R M)]/s(W ⊗R M)
(15)
=
= [W ⊗R M + s(W ⊗R M)]/s(W ⊗R M) =
= (W ⊗R M)/s(W ⊗R M) = T̄W (M),
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.52 On equivalence of some subcategories of modules...
therefore T̄W (M) ∈ JJ .
Lemma 6 permits to obtain by restriction the functors:
A
T̄W
−−−−−−−−−→←−−−−−−−−
T̄ V
B (16)
with the natural transformations η̄ and ρ̄.
Lemma 7. a) For every M ∈ P(r), I · Ker η̄M = 0, i.e. Ker η̄M ∈
A(I) ⊆ R(r).
b) For every N ∈ P(s), J · Ker ρ̄N = 0, i.e. Ker ρ̄N ∈ A(J) ⊆ R(s).
Proof. From definition of η̄M (see (12), (13)) it is clear that η̄M acts as
follows:
η̄M(v ⊗ (w ⊗ m + s(W ⊗R M)) = ηM(v ⊗ w ⊗ m) = (v, w)m,
where (v ⊗ (w ⊗ m + s(W ⊗R M)) = β
T̄W (M)T
V (αM)(v ⊗ w ⊗ m).
If (v ⊗ (w ⊗ m + s(W ⊗R M)) ∈ Ker η̄M ,
then ηM(v ⊗ w ⊗ m) = (v, m)m = 0 and for every (v′, w′) ∈ I we obtain:
(v′, w′)(v ⊗ (w ⊗ m + s(W ⊗R M)) =
= (v′, w′)v ⊗ (w ⊗ m + s(W ⊗R M)) =
= v′[w′, v] ⊗ (w ⊗ m + s(W ⊗R M)) =
= v′ ⊗ ([w′, v]w ⊗ m + s(W ⊗R M)) =
= v′ ⊗ (w′(v, w) ⊗ m + s(W ⊗R M)) =
= v′ ⊗ (w′ ⊗ (v, w)m + s(W ⊗R M)) = 0,
because (v, w)m = 0. From this we can conclude that I ·Ker η̄M = 0 and
by Lemma 4 Ker η̄M ∈ A(I) ⊆ R(r). The statement (b) follows from
symmetry.
Lemma 8. a) Ker η̄M = 0 for every M ∈ P(r). b) Ker ρ̄N = 0 for every
N ∈ P(s).
Proof. Since Ker η̄M ⊆ T̄ V T̄W (M) ∈ P(s), we have Ker η̄M ∈ P(r).
By Lemma 7 Ker η̄M ∈ R(r), therefore Ker η̄M ∈ R(r) ∩ P(r) = {0}.
Similarly Ker ρ̄N = 0 for N ∈ P(s).
Theorem 9. For every pair (r, s) of corresponding torsions (in the sense
of Theorem 1) the functors T̄W and T̄ V (see (10)) with natural trans-
formations η̄ and ρ̄ define an equivalence between the subcategories of
torsion free and accessible modules A = P(r) ∩ JI ⊆ R-Mod and B =
P(s) ∩ JJ ⊆ S-Mod.
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.A. I. Kashu 53
Proof. If M ∈ A, then by Lemma 5 a) η̄M is epi. Moreover, from M ∈
P(r) by Lemma 8 a) we conclude that η̄M is mono, so η̄M is an ismorphism.
Symmetrically, for every N ∈ B we obtain that ρ̄N is an isomorphism.
Therefore the functors T̄W and T̄ V with the natural transformations η̄
and ρ̄ establish the equivalence A ≈ B.
The more general situation of wide Morita contexts is studied in [3].
The equivalence of Theorem 9 can be proved by [3, Theorem 2.6], using
the preceding lemmas. We exposed the direct proof of this result.
For the particular case of the smallest pair (rI , rJ) of corresponding
torsions we have
Corollary 10. ([2], [3]). The subcategories of torsion free and accessible
modules P(rI) ∩ JI and P(rJ) ∩ JJ are equivalent.
References
[1] Müller B. J., The quotient category of a Morita context. J. Algebra, 28 (1974),
pp. 389-407.
[2] Nicholson W. K., Watters J. F., Morita context functors. Math. Proc. Camb. Phil.
Soc., 103 (1988), pp. 399-408.
[3] Iglesias F. C., Torrecillas J. G., Wide Morita contexts. Commun. Algebra, 23, No.
2, 1995, pp. 601-622.
[4] Kashu A.I., Morita contexts and torsions of modules. Matem. Zametki, 28, No 4,
1980, pp. 491-499 (in Russian).
[5] Golan J. S., Torsion theories. Longman Sci. Techn., New York, 1986.
[6] Kashu A. I., Torsion theories and subcategories of divisible modules. “Bulet. A.S.R.
Moldova. Matematica”, No. 2 (15), 1994, pp. 86-89.
[7] Kashu A.I., Radicals and torsions in modules. Kishinev, Stiinta, 1983 (in Russian).
[8] Kashu A.I., Functors and torsions in categories of modules. Acad. of Sciences of
Mold., Inst. of Mathem., Kishinev, 1997 (in Russian).
Contact information
A. I. Kashu Str. Academiei, 5, Inst. of Mathematics
and Computer Science, MD-2028 Chisinau,
Rep. of Moldova
E-Mail: kashuai@math.md
Received by the editors: 04.06.2003
and final form in 27.10.2003.
|
| id | nasplib_isofts_kiev_ua-123456789-155697 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:31:44Z |
| publishDate | 2003 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Kashu, A.I. 2019-06-17T10:54:06Z 2019-06-17T10:54:06Z 2003 On equivalence of some subcategories of modules in Morita contexts / A.I. Kashu // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 46–53. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16S90, 16D90. https://nasplib.isofts.kiev.ua/handle/123456789/155697 en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On equivalence of some subcategories of modules in Morita contexts Article published earlier |
| spellingShingle | On equivalence of some subcategories of modules in Morita contexts Kashu, A.I. |
| title | On equivalence of some subcategories of modules in Morita contexts |
| title_full | On equivalence of some subcategories of modules in Morita contexts |
| title_fullStr | On equivalence of some subcategories of modules in Morita contexts |
| title_full_unstemmed | On equivalence of some subcategories of modules in Morita contexts |
| title_short | On equivalence of some subcategories of modules in Morita contexts |
| title_sort | on equivalence of some subcategories of modules in morita contexts |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155697 |
| work_keys_str_mv | AT kashuai onequivalenceofsomesubcategoriesofmodulesinmoritacontexts |