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arXiv · 2010.08026

A freeness criterion without patching for modules over local rings

Sylvain Brochard·Srikanth B. Iyengar·Chandrashekhar Khare
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Abstract

It is proved that if $\varphi\colon A\to B$ is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated $B$-module $N$ whose flat dimension over $A$ is at most $\mathrm{edim}\, A - \mathrm{edim}\, B$, is free over $B$, and $\varphi$ is a special type of complete intersection. This result is motivated by a "patching method" developed by Taylor and Wiles, and a conjecture of de Smit, proved by the first author, dealing with the special case when $N$ is flat over $A$.

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Sylvain Brochard, Srikanth B. Iyengar, Chandrashekhar Khare. 2020-10-15. A freeness criterion without patching for modules over local rings. https://doi.org/10.1017/s147474802100061x

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