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Keywords:
Bass classes; contracting endomorphisms; dualizing complex; Frobenius endomorphisms; ${\rm G}_{C}$-dimension; semidualizing complex
Bass classes; contracting endomorphisms; dualizing complex; Frobenius endomorphisms; ${\rm G}_{C}$-dimension; semidualizing complex
Summary:
We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim {\bold R}{\rm Hom}_R(^nR,C)$ for some $n>0$; (iii) ${\rm G}_C\text {\rm -dim} ^nR <\infty $ and $C$ is derived ${\bold R}{\rm Hom}_R(^nR,C)$-reflexive for some $n>0$; and (iv) ${\rm G}_C\text {\rm -dim} ^nR <\infty $ for infinitely many $n>0$.
References:
[1] André, M.: Homologie des algèbres commutatives. Die Grundlehren der mathematischen Wissenschaften 206 Springer, Berlin French (1974). MR 0352220 | Zbl 0284.18009
[2] Auslander, M., Bridger, M.: Stable Module Theory. Memoirs of the American Mathematical Society 94 American Mathematical Society, Providence (1969). MR 0269685 | Zbl 0204.36402
[3] Auslander, M., Buchsbaum, D. A.: Homological dimension in local rings. Trans. Am. Math. Soc. 85 (1957), 390-405. DOI 10.1090/S0002-9947-1957-0086822-7 | MR 0086822 | Zbl 0078.02802
[4] Avramov, L. L., Foxby, H.-B.: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc. (3) 75 (1997), 241-270. DOI 10.1112/S0024611597000348 | MR 1455856 | Zbl 0901.13011
[5] Avramov, L. L., Foxby, H.-B.: Locally Gorenstein homomorphisms. Am. J. Math. 114 (1992), 1007-1047. DOI 10.2307/2374888 | MR 1183530 | Zbl 0769.13007
[6] Avramov, L. L., Foxby, H.-B.: Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71 (1991), 129-155. DOI 10.1016/0022-4049(91)90144-Q | MR 1117631 | Zbl 0737.16002
[7] Avramov, L. L., Foxby, H.-B., Herzog, B.: Structure of local homomorphisms. J. Algebra 164 (1994), 124-145. DOI 10.1006/jabr.1994.1057 | MR 1268330 | Zbl 0798.13002
[8] Avramov, L. L., Hochster, M., Iyengar, S. B., Yao, Y.: Homological invariants of modules over contracting endomorphisms. Math. Ann. 353 (2012), 275-291. DOI 10.1007/s00208-011-0682-z | MR 2915536 | Zbl 1241.13013
[9] Avramov, L. L., Iyengar, S. B., Lipman, J.: Reflexivity and rigidity for complexes. I. Commutative rings. Algebra Number Theory 4 (2010), 47-86. DOI 10.2140/ant.2010.4.47 | MR 2592013 | Zbl 1194.13017
[10] Avramov, L. L., Iyengar, S., Miller, C.: Homology over local homomorphisms. Am. J. Math. 128 (2006), 23-90. DOI 10.1353/ajm.2006.0001 | MR 2197067 | Zbl 1102.13011
[11] Christensen, L. W.: Semi-dualizing complexes and their Auslander categories. Appendix: Chain defects Trans. Am. Math. Soc. 353 (2001), 1839-1883. MR 1813596 | Zbl 0969.13006
[12] Christensen, L. W., Frankild, A., Holm, H.: On Gorenstein projective, injective and flat dimensions---a functorial description with applications. J. Algebra 302 (2006), 231-279. DOI 10.1016/j.jalgebra.2005.12.007 | MR 2236602 | Zbl 1104.13008
[13] Christensen, L. W., Holm, H.: Ascent properties of Auslander categories. Can. J. Math. 61 (2009), 76-108. DOI 10.4153/CJM-2009-004-x | MR 2488450 | Zbl 1173.13016
[14] Foxby, H.-B.: Isomorphisms between complexes with applications to the homological theory of modules. Math. Scand. 40 (1977), 5-19. DOI 10.7146/math.scand.a-11671 | MR 0447269 | Zbl 0356.13004
[15] Foxby, H.-B., Frankild, A. J.: Cyclic modules of finite Gorenstein injective dimension and Gorenstein rings. Ill. J. Math. 51 (2007), 67-82. DOI 10.1215/ijm/1258735325 | MR 2346187 | Zbl 1121.13015
[16] Foxby, H.-B., Thorup, A.: Minimal injective resolutions under flat base change. Proc. Am. Math. Soc. 67 (1977), 27-31. DOI 10.1090/S0002-9939-1977-0453724-1 | MR 0453724 | Zbl 0381.13006
[17] Frankild, A., Sather-Wagstaff, S.: Reflexivity and ring homomorphisms of finite flat dimension. Commun. Algebra 35 (2007), 461-500. DOI 10.1080/00927870601052489 | MR 2294611 | Zbl 1118.13015
[18] Gelfand, S. I., Manin, Y. I.: Methods of Homological Algebra. Springer Monographs in Mathematics Springer, Berlin (1996), translated from the Russian Nauka Moskva (1988). Zbl 0668.18001
[19] Goto, S.: A problem on Noetherian local rings of characteristic $p$. Proc. Am. Math. Soc. 64 (1977), 199-205. MR 0447212 | Zbl 0408.13008
[20] Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents, Première partie (I). Publ. Math., Inst. Hautes Étud. Sci. 11 French (1961), 349-511.
[21] Hartshorne, R.: Residues and Duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/1964 Lecture Notes in Mathematics 20 Springer, Berlin (1966). DOI 10.1007/BFb0080482 | MR 0222093 | Zbl 0212.26101
[22] Hungerford, T. W.: Algebra. Graduate Texts in Mathematics 73 Springer, New York (1980). MR 0600654 | Zbl 0442.00002
[23] Iyengar, S., Sather-Wagstaff, S.: G-dimension over local homomorphisms. Applications to the Frobenius endomorphism. Ill. J. Math. 48 (2004), 241-272. DOI 10.1215/ijm/1258136183 | MR 2048224 | Zbl 1103.13009
[24] Kunz, E.: On Noetherian rings of characteristic $p$. Am. J. Math. 98 (1976), 999-1013. DOI 10.2307/2374038 | MR 0432625 | Zbl 0341.13009
[25] Kunz, E.: Characterizations of regular local rings for characteristic $p$. Am. J. Math. 91 (1969), 772-784. DOI 10.2307/2373351 | MR 0252389
[26] Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8 Cambridge University Press, Cambridge (1989). MR 1011461 | Zbl 0666.13002
[27] Nasseh, S., Sather-Wagstaff, S.: Cohen factorizations: Weak functoriality and applications. J. Pure Appl. Algebra 219 (2015), 622-645. DOI 10.1016/j.jpaa.2014.05.017 | MR 3279378 | Zbl 1304.13014
[28] Nasseh, S., Tousi, M., Yassemi, S.: Characterization of modules of finite projective dimension via Frobenius functors. Manuscr. Math. 130 (2009), 425-431. DOI 10.1007/s00229-009-0296-x | MR 2563144 | Zbl 1222.13013
[29] Rodicio, A. G.: On a result of Avramov. Manuscr. Math. 62 (1988), 181-185. DOI 10.1007/BF01278977 | MR 0963004 | Zbl 0657.13012
[30] Sather-Wagstaff, S.: Bass numbers and semidualizing complexes. Commutative Algebra and Its Applications M. Fontana et al. Conf. Proc. Fez, Morocco, 2009. Walter de Gruyter Berlin (2009), 349-381. MR 2640315 | Zbl 1184.13045
[31] Sather-Wagstaff, S.: Complete intersection dimensions and Foxby classes. J. Pure Appl. Algebra 212 (2008), 2594-2611. DOI 10.1016/j.jpaa.2008.04.005 | MR 2452313 | Zbl 1156.13003
[32] Serre, J.-P.: Sur la dimension homologique des anneaux et des modules noethériens. Proc. of the International Symposium on Algebraic Number Theory, Tokyo & Nikko, 1955 Science Council of Japan Tokyo French (1956), 175-189. MR 0086071 | Zbl 0073.26004
[33] Takahashi, R., Yoshino, Y.: Characterizing Cohen-Macaulay local rings by Frobenius maps. Proc. Am. Math. Soc. 132 (2004), 3177-3187. DOI 10.1090/S0002-9939-04-07525-2 | MR 2073291 | Zbl 1094.13007
[34] Verdier, J.-L.: On derived categories of abelian categories. G. Maltsiniotis Astérisque 239 Société Mathématique de France, Paris French (1996). MR 1453167
[35] Verdier, J.-L.: Catégories dérivées. Quelques résultats (Etat O). Cohomologie étale. Séminaire de géométrie algébrique du Bois-Marie SGA 4 1/2; Lecture Notes in Mathematics 569 Springer, Berlin French 262-311 (1977). MR 0463174 | Zbl 0407.18008
[36] Yassemi, S.: G-dimension. Math. Scand. 77 (1995), 161-174. MR 1379262 | Zbl 0864.13010
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