It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if \({\text {Tor}}_i^R({}^{e}\!R, M)=0\) for \({\text {dim}}\,R\) consecutive positive values of i and infinitely many e. Here \({}^{e}\!R\) denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen–Macualay, it suffices that the Tor vanishing above holds for a single \(e\geqslant \log _p e(R)\), where e(R) is the multiplicity of the ring. This improves a result of Dailey et al. (J Commut Algebra), as well as generalizing a theorem due to Miller (Contemp Math 331:207–234, 2003) from finitely generated modules to arbitrary modules. We also show that if R is a complete intersection ring then the vanishing of \({\text {Tor}}_i^R({}^{e}\!R, M)\) for single positive values of i and e is sufficient to imply M has finite flat dimension. This extends a result of Avramov and Miller (Math Res Lett 8(1–2):225–232, 2001).

中文翻译：

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Frobenius与复合物的同构维

证明了一个模块中号在诺特本地环- [R素特性和正尺寸的具有有限的平面尺寸如果\（{\文本{Tor的}} _ I ^ R（{} ^ {E} \！R，M）= 0 \）为\（{\文本{暗淡}} \，R \）的连续正值我和无限多ë。在这里，\（{} ^ {e} \！R \）表示通过Frobenius同构的第e次迭代将环R视作R模块。在R为Cohen–Macualay的情况下，对于单个\（e \ geqslant \ log _p e（R）\），上面的Tor消失就足够了，其中e（R）是环的多重性。这改善了Dailey等人的结果。（J Commut Algebra），以及归纳了Miller（Contemp Math 331：207–234，2003）从有限生成的模块到任意模块的定理。我们还表明，如果R是一个完整的相交环，则对于i和e的单个正值，\（{\ text {Tor}} _ i ^ R（{} ^ {e} \！R，M）\）消失足以暗示M具有有限的平面尺寸。这扩展了Avramov和Miller的结果（Math Res Lett 8（1-2）：225-232，2001）。