Examples of ungradable algebras

Belzner, Th.; Burgess, W. D.; Fuller, K. R.; Schulz, R.

doi:10.1090/S0002-9939-1992-1062382-5

Examples of ungradable algebras
HTML articles powered by AMS MathViewer

by Th. Belzner, W. D. Burgess, K. R. Fuller and R. Schulz PDF

Proc. Amer. Math. Soc. 114 (1992), 1-4 Request permission

Abstract:

Examples are presented of finite-dimensional algebras that admit no positive grading (that is, a nontrivial grading indexed by the natural numbers). Some of these examples have finite global dimension (they are even quasihereditary), and yield a negative answer to a question of Anick and Green.

References

David J. Anick and Edward L. Green, On the homology of quotients of path algebras, Comm. Algebra 15 (1987), no. 1-2, 309–341. MR 876982, DOI 10.1080/00927878708823422
Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, DOI 10.1080/00927877408548230
V. P. Camillo and K. R. Fuller, On graded rings with finiteness conditions, Proc. Amer. Math. Soc. 86 (1982), no. 1, 1–5. MR 663852, DOI 10.1090/S0002-9939-1982-0663852-5
E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
Vlastimil Dlab and Claus Michael Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291. MR 987824
Vlastimil Dlab and Claus Michael Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1989), no. 1, 1–5. MR 943793, DOI 10.1090/S0002-9939-1989-0943793-2
Robert Gordon and Edward L. Green, Graded Artin algebras, J. Algebra 76 (1982), no. 1, 111–137. MR 659212, DOI 10.1016/0021-8693(82)90240-X
Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 674652