Gorenstein test modules. (English) Zbl 1086.13008
Let \((R,m)\) be a Noetherian local ring. An \(R\)-module \(T\) is a Gorenstein test module if for each finite \(R\)-module \(M\) and for each \(i>0\), if Ext\(_{\mathcal G}^j(M,T)=0\) for all \(j\geq i\) then G-dim\(_RM<i\), \({\mathcal G}\) being the full subcategory of modules of G-dimension zero. The aim of this note is to study the Gorenstein test modules and to give such examples including \(k=R/m\).
Reviewer: Dorin-Mihail Popescu (Bucureşti)
MSC:
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13D05 | Homological dimension and commutative rings |
| 13C14 | Cohen-Macaulay modules |
| 13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 13D07 | Homological functors on modules of commutative rings (Tor, Ext, etc.) |
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