Hom and Ext, Revisited
Abstract
Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated $R$-modules. We prove a number of results of the form: if $\mbox{Hom}_R(M,N)$ has some nice properties and $\mbox{Ext}^{1 \leq i \leq n}_R(M,N)=0$ for some $n$, then $M$ (and sometimes $N$) must be be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.05123
- arXiv:
- arXiv:1710.05123
- Bibcode:
- 2017arXiv171005123D
- Keywords:
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- Mathematics - Commutative Algebra
- E-Print:
- Some typos fixed. Remark 2.1 and Lemma 3.1 were expanded to cover semi-dualizing modules. Remark 3.17 was added