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1、On Tor-tilting Modules1 Xiaoxiang Zhang* Lingling Yao Department of Mathematics, Southeast University Nanjing 210096, P. R. China e-mail: Abstract Let R be a ring. A right R-module U is called Tor-tilting if Cogen(U+) = U, where U+ = HomZ(U, Q/Z) and U = KerTorR 1(U, -). Some characterizations of T
2、or-tilting modules are given. Among others, it is shown that UR is Tor-tilting if and only if U+ is cotitling. Moreover, both tilting modules and completely faithful flat modules are proved to be Tor-tilting. Some properties of torsion theories induced by a Tor-tilting module are also investigated.
3、Keywords: cotilting module, tilting module, Tor-tilting module, torsion theory. 1 Introduction The relationship among projective modules, injective modules and flat modules is well known in homological algebra theory. Projective generators, injective cogenerators and completely faithful flat modules
4、 can be regarded jointly from a similar aspect. Tilting modules and cotilting modules, which generalize the projective generators and injective cogenerators respectively, have drawn more and more academic interest in representation theory and homological algebra theory. The reader is referred to 1 a
5、nd 2 for some fundamental theory on the above mentioned objects. Now we have the following three “similar triangles” Figure 1: Three “similar triangles” The motivation of the present paper is the “?” in the last triangle in Figure 1. We shall introduce the notion of Tor-tilting modules which play th
6、e role of “?”. 1 Support by the Foundation of Graduate Creative Program of JiangSu (xm04-10). Projective modules Projective generatorsTilting modules Injective modules Flat modules Injective cogeneratorCompletely faithful flat modules Cotilting modules ? http:/ 1Recall from 2 that a right R-module T
7、 is said to be tilting in case it satisfies the following three equivalent conditions: (1) GenTR = T, where GenTR denotes the class of modules generated by TR and T = KerExt1 R(T, -) = M Mod-R | Ext1 R (T, M) = 0. (2) (i) proj.dimTR 1; (ii) Ext1 R(T, T() = 0 (); (iii) KerHom R(T, -) T = 0. (3) (i) p
8、roj.dim TR 1; (ii) Ext1 R(T, T() = 0 (); (iii) there exists an exact sequence 0 RR T0 T1 0 where T0, T1 AddTR = M Mod-R | M is isomorphic to a direct summand of a direct sum of copies of TR. If the tilting module TR is finitely presented, then “Ext1 R(T, T() = 0” in the above conditions can be repla
9、ced by “Ext1 R(T, T) = 0”, simultaneously, “AddTR” can be replaced by “addTR”, which indicates the class of modules isomorphic to a direct summand of a finite direct sum of copies of TR. In this case, TR is called a classical tilting module. Dually, a left R-module RW is said to be cotilting provide
10、d the following three equivalent conditions are satisfied: (1) Cogen RW = W, where Cogen RW is the class of modules cogenerated by RW and W = KerExt1 R(-, W) = M R-Mod | Ext1 R (M, W) = 0. (2) (i) inj.dim RW 1; (ii) Ext1 R(W, W) = 0 (); (iii) KerHom R(-, W) W = 0. (3) (i) inj.dim RW 1; (ii) Ext1 R(W
11、, W) = 0 (); (iii) there exists an exact sequence 0 W1 W0 C 0 where W0, W1 ProdRW = M R-Mod | M is isomorphic to a direct summand of a direct product of copies of RW and RC is an injective cogenerator of R-Mod. It is natural to consider right R-modules U with flat.dimRU 1 such that TorR 1(U, (U ()+)
12、 = 0 (for all cardinals ) and Ker(UR -)U = 0, where U = KerTorR 1(U, -) = M R-Mod | TorR 1(U, M) = 0. Throughout R is an associative ring with identity and all modules are unitary. R-Mod and Mod-R indicate the category of left and right R-modules, respectively. The projective, injective and flat dim
13、ension of a module M are denoted, respectively, by prod.dimM, inj.dimM and flat.dimM. The reader is also referred to 1 and 2 for undefined terms and notations. 2 Main results Let us start with the following definition. Definition. A right R-module U is said to be Tor-tilting provided Cogen(U+) = Ker
14、TorR 1(U, -). Next, we give some characterizations for a Tor-tilting module UR. Theorem 1. The following are equivalent for a right R-module U. (1) UR is Tor-tilting. http:/ 2(2) U+ is cotilting. (3) UR satisfies the following three conditions: (i) flat.dim RU 1; (ii) TorR 1(U, (U ()+) = 0 for all c
15、ardinals ; and (iii) Ker(UR -) KerTorR 1(U, -) = 0. (4) UR satisfies the following three conditions: (i) flat.dim RU 1; (ii) TorR 1(U, (U ()+) = 0 for all cardinals ; and (iii) There exists an exact sequence 0 V1 V0 C 0, where V0, V1 ProdR(U+) and RC is an injective cogenerator of R-Mod. (5) TorR 1(U, U+) = 0 and (Ker(UR -), KerTorR 1(U, -) is a torsion theory. Proof. (1)(2). Note that there is a natural isomorphism (TorR 1(U, M)+ Ext1 R(M, U+) for every left R-module M (see 9, p. 360). It follows that KerTorR 1(U, -) = KerExt1 R(-, U+) which guarantees (
