Résumé
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$A$ .