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Characterization of modules of finite projective dimension via Frobenius functors. (English) Zbl 1222.13013

Summary: Let \(M\) be a finitely generated module over a local ring \(R\) of characteristic \(p>0\). If \(\text{depth}(R)=s\), then the property that \(M\) has finite projective dimension can be characterized by the vanishing of the functor \(\text{Ext}^i_R(M,{^{f^n}R})\) for \(s+1\) consecutive values \(i>0\) and for infinitely many \(n\). In addition, if \(R\) is a \(d\)-dimensional complete intersection, then \(M\) has finite projective dimension can be characterized by the vanishing of the functor \(\text{Ext}^i_R(M,{^{f^n}R})\) for some \(i\geq d\) and some \(n>0\).

MSC:

13D05 Homological dimension and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
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