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 $\operatorname {edim} A - \operatorname {edim} B$ 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 .