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We continue our study of the Wiles defect of deformation rings $R$ and Heckerings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. TheWiles defect at an augmentation $\lambda_f:T \to O$ measures the failure of$R,T$ to be complete intersections locally at $\lambda_f$. In situations westudy here the Taylor-Wiles-Kisin patching method gives an isomorphism $R=T$without the rings being complete intersections. Using novel arguments incommutative algebra and patching, we generalize significantly and givedifferent proofs of our earlier results that compute the Wiles defect at$\lambda_f: R=T \to O$, and explain in an a priori manner why the answer is asum of local defects. As a curious application of our work we give a new andmore robust approach to the result of Ribet--Takahashi that computes change ofdegrees of optimal parametrizations of elliptic curves by Shimura curves as wevary the Shimura curve.

Wiles defect of Hecke algebras via local-global arguments