A FREENESS CRITERION WITHOUT PATCHING FOR MODULES OVER LOCAL RINGS

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