On equivalence of some subcategories of modules in Morita contexts

On equivalence of some subcategories of modules in Morita contexts

Gespeichert in:
Bibliographische Detailangaben
Datum:2003
1. Verfasser: Kashu, A.I.
Format: Artikel
Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 2003
Schriftenreihe:Algebra and Discrete Mathematics
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/155727
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren: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.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-155727
record_format dspace
spelling irk-123456789-1557272019-06-18T01:25:56Z On equivalence of some subcategories of modules in Morita contexts Kashu, A.I. 2003 Article 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. http://dspace.nbuv.gov.ua/handle/123456789/155727 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
format Article
author Kashu, A.I.
spellingShingle Kashu, A.I.
On equivalence of some subcategories of modules in Morita contexts
Algebra and Discrete Mathematics
author_facet Kashu, A.I.
author_sort Kashu, A.I.
title On equivalence of some subcategories of modules in Morita contexts
title_short 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_sort on equivalence of some subcategories of modules in morita contexts
publisher Інститут прикладної математики і механіки НАН України
publishDate 2003
url http://dspace.nbuv.gov.ua/handle/123456789/155727
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.
series Algebra and Discrete Mathematics
work_keys_str_mv AT kashuai onequivalenceofsomesubcategoriesofmodulesinmoritacontexts
first_indexed 2025-07-14T07:57:45Z
last_indexed 2025-07-14T07:57:45Z
_version_ 1837608328607301632
fulltext Jo u rn al A lg eb ra D is cr et e M at h . 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. Jo u rn al A lg eb ra D is cr et e M at h .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) Jo u rn al A lg eb ra D is cr et e M at h .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). Jo u rn al A lg eb ra D is cr et e M at h .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)) Jo u rn al A lg eb ra D is cr et e M at h .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 Jo u rn al A lg eb ra D is cr et e M at h .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), Jo u rn al A lg eb ra D is cr et e M at h .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. Jo u rn al A lg eb ra D is cr et e M at h .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.

Ähnliche Einträge