Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Injective dimension of quaternion orders


Author: Mark Ramras
Journal: Proc. Amer. Math. Soc. 38 (1973), 493-498
MSC: Primary 16A18; Secondary 13H05, 16A60
DOI: https://doi.org/10.1090/S0002-9939-1973-0313293-X
MathSciNet review: 0313293
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Tarsy has shown that if $ R$ is a discrete valuation ring with quotient field $ K$, and $ \Sigma $ is a quaternion $ K$-algebra, then the finitistic global dimension of any $ R$-order in $ \Sigma $ is 1. In this paper we allow $ R$ to be any regular local ring of dimension $ n$ and study the $ R$-free orders $ \Lambda $ in $ \Sigma $. First we show that the finitistic global dimension of $ \Lambda $ is $ n$. Our main result concerns the injective dimension of $ \Lambda $ (considered as either a left or a right $ \Lambda $-module). Let $ \mathfrak{M}$ denote the maximal ideal of $ R$. Then the injective dimension of $ \Lambda $ is $ n$, unless $ \Lambda /\mathfrak{M}\Lambda $ is a commutative local ring whose socle is not principal. In this case, the injective dimension of $ \Lambda $ is $ \infty $.


References [Enhancements On Off] (What's this?)

  • [1] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367-409. MR 22 #12130. MR 0121392 (22:12130)
  • [2] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. 11, Interscience, New York, 1962, MR 26 #2519. MR 0144979 (26:2519)
  • [3] I. Kaplansky, Commutative rings, Queen Mary College Mathematics Notes, 1966.
  • [4] -, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [5] R. Tarsy, Global dimension of orders, Trans. Amer. Math. Soc. 151 (1970), 335-340. MR 42 #3125. MR 0268226 (42:3125)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16A18, 13H05, 16A60

Retrieve articles in all journals with MSC: 16A18, 13H05, 16A60


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0313293-X
Keywords: Order, quaternion algebra, injective dimension, Frobenius algebra, radical, socle
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society