## Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism.(English)Zbl 1442.13041

Summary: It is proved that a module $$M$$ over a Noetherian ring $$R$$ of positive characteristic $$p$$ has finite flat dimension if there exists an integer $$t \geqslant 0$$ such that $$\operatorname{Tor}_i^R (M,^{f^{e}}R)=0$$ for $$t \leqslant i \leqslant t + \operatorname{dim} R$$ and infinitely many $$e$$. This extends results of Herzog, who proved it when $$M$$ is finitely generated. It is also proved that when $$R$$ is a Cohen-Macaulay local ring, it suffices that the $$\operatorname{Tor}$$ vanishing holds for one $$e \geqslant \operatorname{log}_p e(R)$$, where $$e(R)$$ is the multiplicity of $$R$$.

### MSC:

 13D05 Homological dimension and commutative rings 13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.) 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure

### Keywords:

Frobenius map; flat dimension; homotopical Loewy length
Full Text:

### References:

 [1] L. L. Avramov and H.-B. Foxby, “Homological dimensions of unbounded complexes”, J. Pure Appl. Algebra 71:2-3 (1991), 129-155. · Zbl 0737.16002 [2] L. L. Avramov and S. B. Iyengar, “Ghost homomorphisms in local algebra”, in preparation. [3] L. L. Avramov, S. Iyengar, and C. Miller, “Homology over local homomorphisms”, Amer. J. Math. 128:1 (2006), 23-90. · Zbl 1102.13011 [4] L. L. Avramov, M. Hochster, S. B. Iyengar, and Y. Yao, “Homological invariants of modules over contracting endomorphisms”, Math. Ann. 353:2 (2012), 275-291. · Zbl 1241.13013 [5] N. Bourbaki, Éléments de mathématique: Algèbre commutative, chapitres 8 et 9, Springer, 2006. · Zbl 1103.13003 [6] L. W. Christensen, S. B. Iyengar, and T. Marley, “Rigidity of Ext and Tor with coefficients in residue fields of a commutative Noetherian ring”, Proc. Edinb. Math. Soc. $$(2) 62$$:2 (2019), 305-321. · Zbl 1412.13020 [7] J. A. Eagon and M. M. Fraser, “A note on the Koszul complex”, Proc. Amer. Math. Soc. 19 (1968), 251-252. · Zbl 0162.05601 [8] E. Enochs and J. Xu, “On invariants dual to the Bass numbers”, Proc. Amer. Math. Soc. 125:4 (1997), 951-960. · Zbl 0868.13011 [9] H.-B. Foxby and S. Iyengar, “Depth and amplitude for unbounded complexes”, pp. 119-137 in Commutative algebra (Grenoble/Lyon, 2001), edited by L. L. Avramov et al., Contemp. Math. 331, Amer. Math. Soc., Providence, RI, 2003. · Zbl 1096.13516 [10] J. Herzog, “Ringe der Charakteristik $$p$$ und Frobeniusfunktoren”, Math. Z. 140 (1974), 67-78. · Zbl 0278.13006 [11] C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series 336, Cambridge University Press, 2006. · Zbl 1117.13001 [12] J. Koh and K. Lee, “Some restrictions on the maps in minimal resolutions”, J. Algebra 202:2 (1998), 671-689. · Zbl 0909.13008 [13] E. Kunz, “Characterizations of regular local rings of characteristic $$p$$”, Amer. J. Math. 91 (1969), 772-784. · Zbl 0188.33702 [14] C. Lech, “Inequalities related to certain couples of local rings”, Acta Math. 112 (1964), 69-89. · Zbl 0123.03602 [15] T. Marley and M. Webb, “The acyclicity of the Frobenius functor for modules of finite flat dimension”, J. Pure Appl. Algebra 220:8 (2016), 2886-2896. · Zbl 1375.13019 [16] C. · Zbl 0268.13008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.