Homological properties of balanced Cohen-Macaulay algebras
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- by Izuru Mori
- Trans. Amer. Math. Soc. 355 (2003), 1025-1042
- DOI: https://doi.org/10.1090/S0002-9947-02-03166-5
- Published electronically: October 24, 2002
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Abstract:
A balanced Cohen-Macaulay algebra is a connected algebra $A$ having a balanced dualizing complex $\omega _A[d]$ in the sense of Yekutieli (1992) for some integer $d$ and some graded $A$-$A$ bimodule $\omega _A$. We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem:
Theorem. Let $A$ be a Noetherian balanced Cohen-Macaulay algebra, and $M$ a nonzero finitely generated graded left $A$-module. Then:
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$M$ has a finite resolution of the form \[ 0\to \bigoplus ^{r_m}_{j=1}\omega _A(-l_{mj})\to \cdots \to \bigoplus ^{r_1} _{j=1}\omega _A(-l_{1j})\to H\to M\to 0,\] where $H$ is a finitely generated maximal Cohen-Macaulay graded left $A$-module.
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$M$ has finite injective dimension if and only if $M$ has a finite resolution of the form \begin{align*} 0&\to \bigoplus ^{r_m}_{j=1}\omega _A(-l_{mj})\to \cdots \to \bigoplus ^{r_1} _{j=1}\omega _A(-l_{1j}) &\to \bigoplus ^{r_0}_{j=1} \omega _A(-l_{0j})\to M\to 0. \end{align*}
As a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras:
Corollary. Let $A$ be a Noetherian balanced Cohen-Macaulay algebra.
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$A$ is AS Gorenstein if and only if $\omega _A$ has finite projective dimension as a graded left $A$-module.
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$A$ is AS regular if and only if every finitely generated maximal Cohen-Macaulay graded left $A$-module is free.
References
- M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
- Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37 (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044344
- 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 and Hans-Bjørn Foxby, Ring homomorphisms and finite Gorenstein dimension, Proc. London Math. Soc. (3) 75 (1997), no. 2, 241–270. MR 1455856, DOI 10.1112/S0024611597000348
- Peter Jørgensen, Local cohomology for non-commutative graded algebras, Comm. Algebra 25 (1997), no. 2, 575–591. MR 1428799, DOI 10.1080/00927879708825875
- Peter Jørgensen, Non-commutative graded homological identities, J. London Math. Soc. (2) 57 (1998), no. 2, 336–350. MR 1644217, DOI 10.1112/S0024610798006164
- Peter Jørgensen, Properties of AS-Cohen-Macaulay algebras, J. Pure Appl. Algebra 138 (1999), no. 3, 239–249. MR 1691464, DOI 10.1016/S0022-4049(98)00086-3
- Peter Jørgensen, Gorenstein homomorphisms of noncommutative rings, J. Algebra 211 (1999), no. 1, 240–267. MR 1656580, DOI 10.1006/jabr.1998.7608
- Peter Jørgensen and James J. Zhang, Gourmet’s guide to Gorensteinness, Adv. Math. 151 (2000), no. 2, 313–345. MR 1758250, DOI 10.1006/aima.1999.1897
- I. Mori, Intersection Multiplicity over Noncommutative Algebras, J. Algebra 252 (2002), 241–257.
- S. P. Smith, Non-commutative Algebraic Geometry, lecture notes, University of Washington, (1994).
- Michel van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), no. 2, 662–679. MR 1469646, DOI 10.1006/jabr.1997.7052
- Amnon Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), no. 1, 41–84. MR 1195406, DOI 10.1016/0021-8693(92)90148-F
Bibliographic Information
- Izuru Mori
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Address at time of publication: Department of Mathematics, Syracuse University, Syracuse, New York, 13244-1150
- Email: mori@math.purdue.edu, imori@syr.edu
- Received by editor(s): October 10, 2001
- Received by editor(s) in revised form: February 5, 2002
- Published electronically: October 24, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 1025-1042
- MSC (2000): Primary 16W50, 16E05, 16E65, 16E10
- DOI: https://doi.org/10.1090/S0002-9947-02-03166-5
- MathSciNet review: 1938744