Characterizing Cohen-Macaulay local rings by Frobenius maps

Takahashi, Ryo; Yoshino, Yuji

doi:10.1090/S0002-9939-04-07525-2

Characterizing Cohen-Macaulay local rings by Frobenius maps
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by Ryo Takahashi and Yuji Yoshino PDF

Proc. Amer. Math. Soc. 132 (2004), 3177-3187 Request permission

Abstract:

Let $R$ be a commutative noetherian local ring of prime characteristic. Denote by ${{}^e\hspace {-1.6pt}{}} R$ the ring $R$ regarded as an $R$-algebra through $e$-times composition of the Frobenius map. Suppose that $R$ is F-finite, i.e., ${{}^1\hspace {-2pt}{}} R$ is a finitely generated $R$-module. We prove that $R$ is Cohen-Macaulay if and only if the $R$-modules ${{}^e\hspace {-1.6pt}{}} R$ have finite Cohen-Macaulay dimensions for infinitely many integers $e$.

References

Anneaux de Gorenstein, et torsion en algèbre commutative, École Normale Supérieure de Jeunes Filles, Secrétariat mathématique, Paris, 1967 (French). Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, 1966/67; Texte rédigé, d’après des exposés de Maurice Auslander, Marguerite Mangeney, Christian Peskine et Lucien Szpiro. MR 0225844
Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
L. L. Avramov, Homological dimensions and related invariants of modules over local rings, Proceedings of the 9th International Conference on Representations of Algebras (Beijing, 2000), to appear.
Luchezar L. Avramov and Hans-Bjørn Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129–155. MR 1117631, DOI 10.1016/0022-4049(91)90144-Q
Luchezar L. Avramov, Hans-Bjørn Foxby, and Bernd Herzog, Structure of local homomorphisms, J. Algebra 164 (1994), no. 1, 124–145. MR 1268330, DOI 10.1006/jabr.1994.1057
Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva, Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114 (1998). MR 1608565
T. Araya, R. Takahashi, and Y. Yoshino, Homological invariants associated to semi-dualizing bimodules, Preprint 2002.
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
Amalia Blanco and Javier Majadas, Sur les morphismes d’intersection complète en caractéristique $p$, J. Algebra 208 (1998), no. 1, 35–42 (French). MR 1643971, DOI 10.1006/jabr.1998.7519
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Lars Winther Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839–1883. MR 1813596, DOI 10.1090/S0002-9947-01-02627-7
Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866, DOI 10.1007/BFb0103980
Hans-Bjørn Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284 (1973). MR 327752, DOI 10.7146/math.scand.a-11434
A. A. Gerko, On homological dimensions, Mat. Sb. 192 (2001), no. 8, 79–94 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 7-8, 1165–1179. MR 1862245, DOI 10.1070/SM2001v192n08ABEH000587
E. S. Golod, $G$-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov. 165 (1984), 62–66 (Russian). Algebraic geometry and its applications. MR 752933
Jürgen Herzog, Ringe der Charakteristik $p$ und Frobeniusfunktoren, Math. Z. 140 (1974), 67–78 (German). MR 352081, DOI 10.1007/BF01218647
Jee Koh and Kisuk Lee, New invariants of Noetherian local rings, J. Algebra 235 (2001), no. 2, 431–452. MR 1805466, DOI 10.1006/jabr.2000.8506
Ernst Kunz, Characterizations of regular local rings of characteristic $p$, Amer. J. Math. 91 (1969), 772–784. MR 252389, DOI 10.2307/2373351
Vladimir Maşek, Gorenstein dimension and torsion of modules over commutative Noetherian rings, Comm. Algebra 28 (2000), no. 12, 5783–5811. Special issue in honor of Robin Hartshorne. MR 1808604, DOI 10.1080/00927870008827189
C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119 (French). MR 374130
Antonio G. Rodicio, On a result of Avramov, Manuscripta Math. 62 (1988), no. 2, 181–185. MR 963004, DOI 10.1007/BF01278977
Oana Veliche, Construction of modules with finite homological dimensions, J. Algebra 250 (2002), no. 2, 427–449. MR 1899298, DOI 10.1006/jabr.2001.9100
Siamak Yassemi, G-dimension, Math. Scand. 77 (1995), no. 2, 161–174. MR 1379262, DOI 10.7146/math.scand.a-12557