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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The homological dimensions of symmetric algebras


Author: James E. Carrig
Journal: Trans. Amer. Math. Soc. 236 (1978), 275-285
MSC: Primary 13D05
MathSciNet review: 0457425
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Abstract: Let D be a Dedekind domain and M a rank-one torsion-free D-module. An analysis of $ A = {S_D}(M)$, the symmetric algebra of M, yields the following information:

Theorem. (1) Tor-dim $ A \leqslant 2\;and\; = 1\;iff\;M = K$, the quotient field of D;

(2) A is coherent;

(3) Global $ \dim A = 2$.

For higher rank modules coherence is not assured and only rough estimates of the dimensions are found.

On the other hand, if $ {S_D}(M)$ is a domain of global dimension two, then M has rank one but the dimension of D may be two. If D is local of dimension two then $ M = K$.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0457425-0
Keywords: Global dimension, Tor-dimension, coherent, symmetric algebra, Dede-kind domain, rank one flat module
Article copyright: © Copyright 1978 American Mathematical Society