Wiles defect for Hecke algebras that are not complete intersections

In his work on modularity theorems, Wiles proved a numerical criterion for map of rings $R\to T$ to be an isomorphism complete intersections. He used this show that certain deformation and Hecke algebras associated mod $p$ Galois representation at non-minimal level are isomorphic intersections, provided the same is true minimal level. paper we study acting cohomology Shimura curves arising from maximal orders in indefinite quaternion over rationals localized semistable irreducible $\bar {\rho }$ . If scalar some primes dividing discriminant algebra, then algebra still ring, but not intersection, or even Gorenstein, so cannot apply. We consider weight-2 newform $f$ which contributes curve gives rise augmentation $\lambda _f$ algebra. quantify failure by computing defect purely terms local behavior global $\rho Eichler–Shimura construction. One main tools proof Taylor–Wiles–Kisin patching.