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Adic approximation of complexes, and multiplicities

Published online by Cambridge University Press:  22 January 2016

David Eisenbud
David Eisenbud*
Affiliation:
Brandeis University
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In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin’s Approximation Theorem [1], allows them to “descend” modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite étale extensions of those rings. In this note we will prove a more general result, which deals with the change in homology under an adic approximation of any complex of finitely generated modules over a noetherian ring, and which allows one to descend not only modules of finite projective dimension, but also the Euler characteristic or intersection multiplicity of two such modules.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 54 , July 1974 , pp. 61 - 67
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1974

References

[1] Artin, M.: Algebraic approximation of structures over complete local rings; Publ. Math. I.H.E.S. 36 (1969).CrossRefGoogle Scholar
[2] Peskine, C. and Szpiro, L.: Dimension projective finie et cohomologie locale; Publ. Math. I.H.E.S. 42 (1973).CrossRefGoogle Scholar
[3] Serre, J-P.: “Algèbre locale-multiplicités”; Springer Lecture Notes in Math. 11 (1965).Google Scholar