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Co-point modules over Koszul algebras. (English) Zbl 1115.16013
Let \(k\) be a field and \(A\) be a positively graded \(k\)-algebra, which is generated in degree \(1\) over \(A_0=k\). A graded module \(M\) over \(A\) is called a ‘point’ module if it is cyclic and \[ \dim_kM_i=\begin{cases} 0,&i<0;\\ 1,&i\geq 0.\end{cases} \] In the paper under review the author defines the notion of a ‘co-point’ module as follows: a graded module \(M\) over \(A\) is called ‘co-point’ provided that it is cyclic and the \(i\)-th term of its minimal projective resolution is isomorphic to \(A\) shifted by \(i\) in degree. In some sense co-point modules are point modules for the quadratic dual of \(A\), the module category of which can be realized via the category of linear complexes of projective \(A\)-modules.
Using the notion of co-point modules the author constructs counter-examples to the following condition on the existence of a uniformal bound for non-vanishing extensions, formulated by Auslander: For every finitely generated right module \(M\) over ring \(R\) there is a natural number \(n_M\) such that for any finitely generated right module \(N\) over \(R\) the condition \(\text{Ext}^i_R(M,N)=0\) for all \(i\) big enough implies \(\text{Ext}^i_R(M,N)=0\) for all \(i>n_M\).

MSC:
16S37 Quadratic and Koszul algebras
16S38 Rings arising from noncommutative algebraic geometry
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16E05 Syzygies, resolutions, complexes in associative algebras
16W50 Graded rings and modules (associative rings and algebras)
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