Finiteness of representation dimension

Iyama, Osamu

doi:10.1090/S0002-9939-02-06616-9

Finiteness of representation dimension
HTML articles powered by AMS MathViewer

by Osamu Iyama PDF

Proc. Amer. Math. Soc. 131 (2003), 1011-1014 Request permission

Abstract:

We will show that any module over an artin algebra is a direct summand of some module whose endomorphism ring is quasi-hereditary. As a conclusion, any artin algebra has a finite representation dimension.

References

M. Auslander: Representation dimension of Artin algebras, Lecture notes, Queen Mary College, London, 1971.
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
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 M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR 0389981
P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
O. Iyama: $\tau$-categories II: Nakayama pairs and rejective subcategories, to appear in Algebras and Representation theory.
O. Iyama: $\tau$-categories III: Auslander orders and Auslander-Reiten quivers, to appear in Algebras and Representation theory.
O. Iyama: A proof of Solomon’s second conjecture on local zeta functions of orders, to appear in J. Algebra.
K. Igusa, G. Todorov: On the finitistic global dimension conjecture, preprint.
Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740
Changchang Xi, On the representation dimension of finite dimensional algebras, J. Algebra 226 (2000), no. 1, 332–346. MR 1749892, DOI 10.1006/jabr.1999.8177
C. C. Xi: Representation dimension and quasi-hereditary algebras, to appear in Adv. Math.