Abstract
We review the definition and some basic properties of the dualizing complex and present three applications to noncommutative algebras.
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References
F. W. Anderson and K. R. Fuller, “Rings and Categories of Modules,” Second edition, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1992.
J. E. Björk, The Auslander condition on noetherian rings, “Séminaire Dubreil-Malliavin
“ Lecture Notes in Math. 1404, Springer, Berlin, 1989, 137–173.
K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997 ), 49–79, Contemp. Math., 229, AMS, Providence, RI, 1998.
K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension, J. Algebra 198 (1997), no. 1, 240–265.
D. Chan, Q. S. Wu and J. J. Zhang, Pre-balanced Dualizing Complexes, Israel J. Math., to appear.
B. Conrad, “Grothendieck duality and base change”, Lecture Notes in Mathematics, 1750. Springer-Verlag, Berlin, 2000.
V. G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley, Vol. 1, 798–820, Amer. Math. Soc., Providence, RI, 1987.
H. -B. Foxby, A homological theory of complexes of modules, Preprint series no 19a & 19b, Copenhagen Univ. Math. Inst., 1981.
O. Gabber, Equidimensionalité de la variété caractéristique, Exposé de O. Gabber rédigé par T. Levasseur, Univesité de Paris V I, 1982
K. R. Goodearl and T. H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra, 111 (1996), 123–142.
R. Hartshorne, “Residues and Duality,” Lecture Notes in Math. 20, Springer-Verlag, Berlin, 1966.
F. Ischebeck, Eine Dualität zwischen den Funktoren Ext und Tor (German), J. Algebra 11 (1969), 510–531.
M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69.
M. Jimbo, A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252.
P. Jorgensen, Local cohomology for non-commutative graded algebras, Comm. Algebra 25 (1997), 575–591.
P. Jorgensen, Non-commutative graded homological identities, J. London Math. Soc. 57 (1998), no. 2, 336–350.
G. R. Krause and T. H. Lenagan, “Growth of algebras and Gelfand-Kirillov dimension,” Research Notes in Mathematics 116, Pitman, Boston-London, 1985.
T. H. Lenagan, Enveloping algebras of solvable Lie superalgebras are catenary, “Abelian groups and noncommutative rings,” Contemp. Math. 130, Amer. Math. Soc., Providence, RI, 1992, 231–236.
E. Letzter and M. Lorenz, Polycyclic-by-finite group algebras are catenary, Math. Res. Lett. 6 (1999), no. 2, 183–194.
J. Miyachi, Derived categories and Morita duality theory, J. Pure Appl. Algebra, 128 (1998), 153–170.
J. C. McConnell and J. C. Robson, “Noncommutative Noetherian Rings,” Wiley, Chichester, 1987.
S. Montgomery, “Hopf algebras and their actions on rings”, CBMS Regional Conference Series in Mathematics, 82, Providence, RI, 1993.
M. Nagata, “Local Rings,” Wiley, New York, 1962.
P. Roberts, Two applications of dualizing complexes over local rings, Ann. scient. Éc. Norm. Sup. 4. série 9 (1976), 103–106.
M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), no. 2, 662–679.
J.-L. Verdier, Des catégories dérivées des catégories abéliennes. (French) [On derived categories of abelian categories] Astérisque No. 239 (1996).
Q.-S. Wu and J. J. Zhang, Some homological invariants of PI local algebras, J. Algebra, 225 (2000) 904–935.
Q.-S. Wu and J. J. Zhang, Dualizing complexes over noncommutative local rings, J. Algebra, 239 (2001) 513–548.
Q.-S. Wu and J. J. Zhang, Homological identities for noncommutative rings, J. Algebra, 242 (2001) 516–535.
Q.-S. Wu and J. J. Zhang, Noetherian PI Hopf algebras are Gorenstein, preprint, (2002).
W. Xue, “Rings with Morita Duality”, Lecture Notes in Math. 1523, Springer-Verlag, Berlin, 1992.
A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), 41–84.
A. Yekutieli, The residue complex of a noncommutative graded algebra, J. Algebra 186 (1996), 522–543.
A. Yekutieli, Dualizing complexes, Morita equivalence and the derived Picard group of a ring, J. London Math. Soc. (2) 60 (1999), no. 3, 723–746.
A. Yekutieli and J. J. Zhang, Serre duality for noncommutative projective schemes, Proc. Amer. Math. Soc. 125 (1997), 697–707.
A. Yekutieli and J. J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), no. 1, 1–51.
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Wu, QS., Zhang, J.J. (2003). Applications of Dualizing Complexes. In: Proceedings of the Third International Algebra Conference. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0337-6_12
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DOI: https://doi.org/10.1007/978-94-017-0337-6_12
Publisher Name: Springer, Dordrecht
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