# A freeness criterion without patching for modules over local rings

Seminar
Speaker
Dr. Sylvain Brochard (Montpellier)
Date
02/12/2020 - 11:30 - 10:30Add to Calendar 2020-12-02 10:30:00 2020-12-02 11:30:00 A freeness criterion without patching for modules over local rings  Let A->B be a local homomorphism of (commutative) Noetherian local rings. Bart de Smit conjectured in the late 1990's that if A and B have the same embedding dimension, then any finitely generated B-module that is flat over A, is flat over B. This conjecture was proved in 2017 and allows one in some situations to dispense with patching in the techniques à la Wiles to prove modularity lifting theorems (e.g. in the proof of FLT). We generalize this result as follows: if M is a finitely generated B-module whose flat dimension over A satisfies flat dim_A(M) \leq edim(A)-edim(B), then M is free as a B-module. If moreover M is nonzero this forces the morphism A->B to be a special type of complete intersection. This provides by the way a new and simpler proof of de Smit's conjecture. This is joint work with Srikanth Iyengar and Chandrashekhar Khare.   Zoom -- see e-mail or contact organizer for details אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Zoom -- see e-mail or contact organizer for details
Abstract

Let A->B be a local homomorphism of (commutative) Noetherian local rings. Bart de Smit conjectured in the late 1990's that if A and B have the same embedding dimension, then any finitely generated B-module that is flat over A, is flat over B. This conjecture was proved in 2017 and allows one in some situations to dispense with patching in the techniques à la Wiles to prove modularity lifting theorems (e.g. in the proof of FLT). We generalize this result as follows: if M is a finitely generated B-module whose flat dimension over A satisfies flat dim_A(M) \leq edim(A)-edim(B), then M is free as a B-module. If moreover M is nonzero this forces the morphism A->B to be a special type of complete intersection. This provides by the way a new and simpler proof of de Smit's conjecture. This is joint work with Srikanth Iyengar and Chandrashekhar Khare.

תאריך עדכון אחרון : 23/11/2020