A freeness criterion without patching for modules over local rings

Abstract

It is proved that if $\phi :A\to B$ 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 $\phi$ 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 $A$.

11 page; minor changes in version 2. To appear in J. Inst. Math. Jussieu

Cited by in corpus (1)

• Exceptional complete intersection maps of local rings