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A note on the global dimension of polynomial rings

Published online by Cambridge University Press:  24 October 2008

D. G. Northcott
D. G. Northcott
Affiliation:
The University Sheffield
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Extract

It is well known that if K is a commutative field then the polynomial ring K[x1, x2, …, xn] has global dimension (in the sense of homology theory) equal to the number of variables. This, of course, is not an isolated result but, in view of the special interest which attaches to such a familiar object as a polynomial ring, a short and comparatively elementary proof of this particular result may be of interest. The present paper is devoted to such a proof and therefore it is proper that this first section should indicate, in some detail, what knowledge of homological algebra is presupposed.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 4 , October 1957 , pp. 796 - 799
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1) Auslander, M. On the dimension of modules and algebras. III. Global dimension. Nagoya Math. J. 9 (1956), 67–77.Google Scholar
(2) Cartan, H. and Eilenberg, S. Homological algebra (Princeton, 1956).Google Scholar
(3) Rees, D. A theorem of homological algebra. Proc. Camb. Phil. Soc. 52 (1956), 605–10.Google Scholar