| Representation theory of algebra as a new branch of algebra started in 1970's. With the development of the last thirty years, this theory has maturated gradually.In this paper I emphasize on Cartan Determinant Conjecture and the property of the cotilting modules.In 1954 S.Eilenberg showed that the determinant of the Cartan matrix of an artinian ring R of finite global dimension must be ±1,and conjecture the determinant of the Cartan matrix is 1.In the last 50 years many mathematicians have got much achievement on this problem.In 1974 Donovan and Freislich have shown that if A is a group algebra of finite representation type,then the determinant of the Cartan matrix must be 1.In 1983 Zacharia proved that the determinant is 1 when gldimΛ ≤ 2.In 1985 Burgess,Fuller and Zimmermann have proved that if R is a serial ring,then In 1989 Burgess and Fuller proved that if R is a quasihereditary ring,then the Cartan determinant conjecture is true. In 1998 Burgess and Fuller discuss the problem again and showed the more inclusive result.On the basis of the proved results,I have showed the following results: R is a left artinian ring, (1) if R is regular ring,then det C(R) = 1; (2) if Jthe radical of R is a projective R-module,then det C(R) = 1;(3)if the global dimension of R is 4,then R has a simple module S_i with the projective dimension 0 or 2 or 3.In 1982, D.Happel and C.Ringel introduced tilted algebras as a generalization of hereditary algebras. The tilted algebras is now considered to be one of the most useful algebras in representation theory of algebras. The dual of the tilting module is the cotilting module which have several interesting characterizations.In the paper,I discuss the following questions: a correspondence between cotilting Λ—modules and torsion pairs for modΛ;The perpendicular category of a cotilting module is closed under product;The perpendicular category of a pure-injective module must have no nonzero preinjective module; The classification of the cotilting module. |
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