On when a graded ring is graded equivalent to a crossed product
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- by Jeremy Haefner
- Proc. Amer. Math. Soc. 124 (1996), 1013-1021
- DOI: https://doi.org/10.1090/S0002-9939-96-03138-3
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Abstract:
Let $R$ be a ring graded by a group $G$. We are concerned with describing those $G$-graded rings that are graded equivalent to $G$-crossed products. We give necessary and sufficient conditions for when a strongly graded ring is graded equivalent to a crossed product, provided that the 1-component is either Azumaya or semiperfect. Our result uses the torsion product theorem of Bass and Guralnick. We also construct various examples of such rings.References
- David F. Anderson, The kernel of $\textrm {Pic}(R_0)\to \textrm {Pic}(R)$ for $R$ a graded domain, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), no.ย 6, 248โ252. MR 1145116
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223
- Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
- Hyman Bass and Robert Guralnick, Projective modules with free multiples and powers, Proc. Amer. Math. Soc. 96 (1986), no.ย 2, 207โ208. MR 818444, DOI 10.1090/S0002-9939-1986-0818444-5
- H. Bass and R. Guralnick, Torsion in the Picard group and extension of scalars, J. Pure Appl. Algebra 52 (1988), no.ย 3, 213โ217. MR 952079, DOI 10.1016/0022-4049(88)90092-8
- Margaret Beattie, A generalization of the smash product of a graded ring, J. Pure Appl. Algebra 52 (1988), no.ย 3, 219โ226. MR 952080, DOI 10.1016/0022-4049(88)90093-X
- M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no.ย 1, 237โ258. MR 728711, DOI 10.1090/S0002-9947-1984-0728711-4
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
- Everett C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), no.ย 3, 241โ262. MR 593823, DOI 10.1007/BF01161413
- Robert M. Guralnick, Bimodules over PI rings, Methods in module theory (Colorado Springs, CO, 1991) Lecture Notes in Pure and Appl. Math., vol. 140, Dekker, New York, 1993, pp.ย 117โ134. MR 1203803
- Robert M. Guralnick and Susan Montgomery, On invertible bimodules and automorphisms of noncommutative rings, Trans. Amer. Math. Soc. 341 (1994), no.ย 2, 917โ937. MR 1150014, DOI 10.1090/S0002-9947-1994-1150014-3
- Robert Gordon and Edward L. Green, Graded Artin algebras, J. Algebra 76 (1982), no.ย 1, 111โ137. MR 659212, DOI 10.1016/0021-8693(82)90240-X
- Jeremy Haefner, Graded Morita theory for infinite groups, J. Algebra 169 (1994), no.ย 2, 552โ586. MR 1297162, DOI 10.1006/jabr.1994.1297
- โ, Graded equivalence theory with applications, J. Algebra 172 (1995), 385โ423.
- Jeremy Haefner, A strongly graded ring that is not graded equivalent to a skew group ring, Comm. Algebra 22 (1994), no.ย 12, 4795โ4799. MR 1285709, DOI 10.1080/00927879408825104
- C. Menini and C. Nฤstฤsescu, When is $R$-$\textbf {gr}$ equivalent to the category of modules?, J. Pure Appl. Algebra 51 (1988), no.ย 3, 277โ291. MR 946579, DOI 10.1016/0022-4049(88)90067-9
- Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR 979094
- Declan Quinn, Group-graded rings and duality, Trans. Amer. Math. Soc. 292 (1985), no.ย 1, 155โ167. MR 805958, DOI 10.1090/S0002-9947-1985-0805958-0
- Angel del Rรญo, Graded rings and equivalences of categories, Comm. Algebra 19 (1991), no.ย 3, 997โ1012. MR 1103000, DOI 10.1080/00927879108824184
Bibliographic Information
- Jeremy Haefner
- Email: haefner@math.uccs.edu
- Received by editor(s): April 26, 1994
- Received by editor(s) in revised form: September 6, 1994
- Additional Notes: The authorโs research was partially supported by the National Security Agency
- Communicated by: Ken Goodearl
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1013-1021
- MSC (1991): Primary 16D90, 16S35, 16S40, 16S50, 16W50
- DOI: https://doi.org/10.1090/S0002-9939-96-03138-3
- MathSciNet review: 1301027