Abstract
Let M be a finitely generated module over a local ring R of characteristic p > 0. If depth(R) = s, then the property that M has finite projective dimension can be characterized by the vanishing of the functor \({{\rm Ext}^i_R(M, ^{f^n}R)}\) 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 \({{\rm Ext}^i_R(M, ^{f^n}R)}\) for some i ≥ d and some n > 0.
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Nasseh, S., Tousi, M. & Yassemi, S. Characterization of modules of finite projective dimension via Frobenius functors. manuscripta math. 130, 425–431 (2009). https://doi.org/10.1007/s00229-009-0296-x
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DOI: https://doi.org/10.1007/s00229-009-0296-x