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Proceedings of the American Mathematical Society
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
MathSciNet review: 0313293
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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 $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0313293-X
PII: S 0002-9939(1973)0313293-X
Keywords: Order, quaternion algebra, injective dimension, Frobenius algebra, radical, socle
Article copyright: © Copyright 1973 American Mathematical Society