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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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