Cotilting dualities

The paper surveys several generalizations of the notion of a cotilting module over an Artin algebra to more general rings. Classically, a module U over an Artin algebra Λ is a cotilting module if it satisfies inj.dim. U≤1, Ext1Λ(U,U)=0 and there exists a short exact sequence 0→U1→U0→DΛ→0 with U1 and U0 in add(U). For a cotilting module U with Γ=End(UΛ), S. Brenner and M. C. R. Butler [in Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), 103--169, Lecture Notes in Math., 832, Springer, Berlin, 1980; MR0607151 (83e:16031)] established dualities between certain pairs of subcategories of mod-Λ and mod-Γ given by the functors Hom(−,U) and Ext1(−,U). With this result as a guide the authors define the notion of a cotilting theorem between two abelian subcategories of module categories. The paper then reviews several instances of such theorems, including generalized and weak Morita dualities studied by R. R. Colby and K. R. Fuller in [R. R. Colby, Comm. Algebra 17 (1989), no. 7, 1709--1722; MR1006521 (90i:16021)] and [R. R. Colby and K. R. Fuller, Comm. Algebra 31 (2003), no. 4, 1859--1879; MR1972896 (2004b:16007)], respectively, and describes the properties that characterize the corresponding cotilting module in each case.