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On the lattice of cotilting modules. (English) Zbl 1012.16014

The authors study here the collection \(\text{Cotilt }\Lambda\) of the cotilting modules over an associative ring \(\Lambda\) and show that it carries the structure of a lattice. By a cotilting \(\Lambda\)-module it is meant a pure injective module \(T\) such that: (i) the injective dimension of \(T\) is finite; (ii) \(\text{Ext}^i_\Lambda(T^\alpha,T)=0\) for all \(i>0\) and every cardinal \(\alpha\); (iii) there exists an injective cogenerator \(Q\) and a long exact sequence \(0\to T_n\to\cdots\to T_0\to Q\to 0\) with \(T_i\) in \(\text{Prod }T\) for each \(i\). The authors also look at the so-called minimal cotilting module which is the unique minimal element of \(\text{Cotilt }\Lambda\) and characterize it.

MSC:

16G10 Representations of associative Artinian rings
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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