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Finite Cohen–Macaulay Type and Smooth Non-Commutative Schemes

Published online by Cambridge University Press:  20 November 2018

Peter Jørgensen
Peter Jørgensen*
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, U.K. e-mail: peter.jorgensen@ncl.ac.uk
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Abstract

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A commutative local Cohen–Macaulay ring $R$ of finite Cohen–Macaulay type is known to be an isolated singularity; that is, $\text{Spec}(R)\backslash \{m\}$ is smooth. This paper proves a non-commutative analogue. Namely, if $A$ is a (non-commutative) graded Artin–Schelter Cohen–Macaulay algebra which is fully bounded Noetherian and has finite Cohen–Macaulay type, then the non-commutative projective scheme determined by $A$ is smooth.

Keywords

14A2216E6516W50Artin–Schelter Cohen–Macaulay algebraArtin–Schelter Gorenstein algebraAuslander’s theorem on finite Cohen–Macaulay typeCohen–Macaulay ringfully bounded Noetherian algebraisolated singularitymaximal Cohen–Macaulay modulenon-commutative projective schemepunctured spectrum
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 379 - 390
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Artin, M. and Zhang, J. J., Noncommutative projective schemes. Adv. Math. 109(1994), no. 2, 228–287.Google Scholar
[2] Auslander, M., Isolated singularities and existence of almost split sequences. In: Representation Theory. II. Lecture Notes in Math. 1178, Springer, Berlin, 1986, pp. 194–242.Google Scholar
[3] Auslander, M., Rational singularities and almost split sequences. Trans. Amer. Math. Soc. 293(1986), no. 2, 511–531.Google Scholar
[4] Brown, K. A., Fully bounded Noetherian rings of finite injective dimension. Quart. J. Math. Oxford 41(1990), no. 161, 1–13.Google Scholar
[5] Goodearl, K. and Warfield, R., An introduction to noncommutative Noetherian rings. London Mathematical Society Student Texts 16, Cambridge University Press, Cambridge, 1989.Google Scholar
[6] Herzog, J., Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln Math. Ann. 233(1978), no. 1, 21–34.Google Scholar
[7] Huneke, C. and Leuschke, G., Two theorems about maximal Cohen-Macaulay modules. Math. Ann. 324(2002), no. 2, 391–404.Google Scholar
[8] Jørgensen, P., Local cohomology for non-commutative graded algebras. Comm. Algebra 25(1997), no. 2, 575–591.Google Scholar
[9] Jørgensen, P., Non-commutative graded homological identities J. LondonMath. Soc. 57(1998), no. 2, 336–350.Google Scholar
[10] Jørgensen, P. and Zhang, J. J., Gourmet's Guide to Gorensteinness. Adv. Math. 151(2000), no. 2, 313–345.Google Scholar
[11] Mori, I., Homological properties of balanced Cohen-Macaulay algebras. Trans. Amer.Math. Soc. 355(2003), no. 3, 1025–1042.Google Scholar
[12] Năstăsescu, C. and van Oystaeyen, F., Graded Ring Theory. North-Holland Mathematical Library 28, North-Holland Publishing, Amsterdam, 1982.Google Scholar
[13] Stafford, J. T. and Zhang, J. J., Homological properties of (graded) Noetherian PI rings. J. Algebra 168(1994), no. 3, 988–1026.Google Scholar
[14] Van den Bergh, M., Existence theorems for dualizing complexes over non-commutative graded and filtered rings. J. Algebra 195(1997), no. 2, 662–679.Google Scholar
[15] Yekutieli, A., Dualizing complexes over noncommutative graded algebras. J. Algebra 153(1992), no. 1, 41–84.Google Scholar
[16] Zhang, J. J., Twisted graded algebras and equivalences of graded categories. Proc. LondonMath. Soc. 72(1996), no. 2, 281–311.Google Scholar