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CANONICAL AND $n$ -CANONICAL MODULES OF A NOETHERIAN ALGEBRA

Published online by Cambridge University Press:  20 October 2016

MITSUYASU HASHIMOTO*
Affiliation:
Department of Mathematics, Okayama University, Okayama 700-8530, Japan email mh@okayama-u.ac.jp

Abstract

We define canonical and $n$ -canonical modules of a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$ -canonical modules, we generalize a theorem on $(n,C)$ -syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a noncommutative version of Aoyama’s theorem which states that a canonical module descends with respect to a flat local homomorphism.

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Article
Copyright
© 2016 by The Editorial Board of the Nagoya Mathematical Journal  

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References

Aoyama, Y., Some basic results on canonical modules , J. Math. Kyoto Univ. 23 (1983), 8594.CrossRefGoogle Scholar
Aoyama, Y. and Goto, S., On the endomorphism ring of the canonical module , J. Math. Kyoto Univ. 25 (1985), 2130.CrossRefGoogle Scholar
Araya, T. and Iima, K.-i., Locally Gorensteinness over Cohen–Macaulay rings, arXiv:1408.3796v1.Google Scholar
Assem, I., Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
Avramov, L. L. and Foxby, H.-B., Locally Gorenstein homomorphisms , Amer. J. Math. 114 (1992), 10071047.CrossRefGoogle Scholar
Brodmann, M. P. and Sharp, R. Y., Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
Evans, E. G. and Griffith, P., Syzygies, London Mathematical Society Lecture Note Series 106 , Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
Grothendieck, A., Eléments de Géométrie Algébrique IV, 3e Partie, Publ. Math. Inst. Hautes Études Sci. 28 , 1966.Google Scholar
Hartshorne, R., Residues and Duality, Lecture Notes in Mathematics 20 , Springer, Berlin–New York, 1966.CrossRefGoogle Scholar
Hartshorne, R., Generalized divisors on Gorenstein schemes , K-Theory 8 (1994), 287339.CrossRefGoogle Scholar
Hashimoto, M., Equivariant class group. III. Almost principal fiber bundles, arXiv:1503.02133v1.Google Scholar
Iyama, O. and Wemyss, M., On the noncommutative Bondal–Orlov conjecture , J. Reine Angew. Math. 683 (2013), 119128.Google Scholar
Leuschke, G. J. and Wiegand, R., Cohen–Macaulay Representations, American Mathematical Society, Providence, 2012.CrossRefGoogle Scholar
Matsumura, H., Commutative Ring Theory, First paperback edition, Cambridge University Press, Cambridge, 1989.Google Scholar
Ogoma, T., Existence of dualizing complexes , J. Math. Kyoto Univ. 24 (1984), 2748.CrossRefGoogle Scholar
Peskine, C. and Szpiro, L., Dimension projective finie et cohomologie locale , Publ. Math. Inst. Hautes Études Sci. 42 (1973), 47119.CrossRefGoogle Scholar
Takahashi, R., A New approximation theory which unifies spherical and Cohen–Macaulay approximations , J. Pure Appl. Algebra 208 (2007), 617634.CrossRefGoogle Scholar
Thomason, R. W. and Trobaugh, T., “ Higher algebraic K-theory of schemes and of derived categories ”, in The Grothendieck Festschrift. III, Birkhäuser, Boston, 1990, 247435.CrossRefGoogle Scholar
Yekutieli, A., Dualizing complexes over noncommutative graded algebras , J. Algebra 153 (1992), 4184.CrossRefGoogle Scholar

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