Abstract
AbstractIt 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.
Highlights
The work of Wiles and Taylor-Wiles [18, 20] on modularity lifting theorems relies on a patching method that has been generalised to prove a series of remarkable results that verify in many cases the Fontaine–Mazur conjecture
The spectacular proof by Newton and Thorne [17] of the automorphy of all symmetric powers of Galois representations associated to classical newforms is a recent example
The assumptions there are used to reduce the proof to a statement about modules over regular local rings, which follows from the Auslander–Buchsbaum formula
Summary
The work of Wiles and Taylor-Wiles [18, 20] on modularity lifting theorems relies on a patching method that has been generalised to prove a series of remarkable results that verify in many cases the Fontaine–Mazur conjecture. Bart de Smit made the remarkable conjecture that if A → B is a local homomorphism of (commutative) Artinian local rings of the same embedding dimension, any B -module that is flat as an A-module is flat as a B -module This strengthens Proposition 1.1 in the case r = s and allows one in principle to dispense with patching in the techniquesa la Wiles to prove modularity lifting theorems. Suppose φ : A → B is a local homomorphism of noetherian local rings and N is a nonzero finitely generated B-module whose flat dimension over A satisfies flat dimA N ≤ edim A − edim B; N is free as a B module and φ is an exceptional complete intersection map.
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