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NAK for Ext and ascent of module structures


Authors: Benjamin J. Anderson and Sean Sather-Wagstaff
Journal: Proc. Amer. Math. Soc. 142 (2014), 1165-1174
MSC (2010): Primary 13B40, 13D07; Secondary 13D02
DOI: https://doi.org/10.1090/S0002-9939-2014-11862-4
Published electronically: January 28, 2014
MathSciNet review: 3162239
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Abstract: We investigate the interplay between properties of Ext modules and the ascent of module structures along local ring homomorphisms. Specifically, let $ \varphi \colon (R,\mathfrak{m},k)\to (S,\mathfrak{m} S,k)$ be a flat local ring homomorphism. We show that if $ M$ is a finitely generated $ R$-module such that $ \operatorname {Ext}_{R}^{i}(S,M)$ satisfies NAK (e.g. if $ \operatorname {Ext}_{R}^{i}(S,M)$ is finitely generated over $ S$) for $ i=1,\ldots ,\dim _{R}(M)$, then $ \operatorname {Ext}_{R}^{i}(S,M)=0$ for all $ i\neq 0$ and $ M$ has an $ S$-module structure that is compatible with its $ R$-module structure via $ \varphi $. We provide explicit computations of $ \operatorname {Ext}_{R}^{i}(S,M)$ to indicate how large it can be when $ M$ does not have a compatible $ S$-module structure.


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Additional Information

Benjamin J. Anderson
Affiliation: Department of Mathematics, North Dakota State University, Department #2750, P.O. Box 6050, Fargo, North Dakota 58108-6050
Address at time of publication: University of Wisconsin-La Crosse, 1725 State Street, La Crosse, Wisconsin 54601
Email: benjamin.j.anderson@ndsu.edu, banderson@uwlax.edu

Sean Sather-Wagstaff
Affiliation: Department of Mathematics, North Dakota State University, Department #2750, P.O. Box 6050, Fargo, North Dakota 58108-6050
Email: sean.sather-wagstaff@ndsu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-11862-4
Keywords: Ascent, Ext, flat homomorphism, NAK
Received by editor(s): November 30, 2011
Received by editor(s) in revised form: May 14, 2012
Published electronically: January 28, 2014
Additional Notes: This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442.
Dedicated: To Roger A. Wiegand on the occasion of his retirement
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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