Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions

We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra regular at $\mathfrak{p}$, the kernel of $\lambda$, and $M$ a finitely generated $A$-module. We establish a numerical criterion for $M$ to have a free direct summand over $A$ of positive rank. It is in terms of the lengths of $\Psi_A(M)$ and the torsion part of $\mathfrak{p}/\mathfrak{p}^2$. It generalizes results of Wiles, Lenstra, and Diamond, that deal with the case when the codimension of $\mathfrak{p}$ is zero. Number theoretic applications include integral (non-minimal) $R=\mathbb T$ theorems in situations of positive defect conditional on certain standard conjectures. Here $R$ is a deformation ring parametrizing certain Galois representations and $\mathbb T$ is a Hecke algebra. An example is a modularity lifting for 2-dimensional $\ell$-adic Galois representations over an imaginary quadratic field. The proofs combine our commutative algebra results with a generalization due to Calegari and Geraghty of the patching method of Wiles and Taylor--Wiles and level raising arguments that go back to Ribet. The results provide new evidence in favor of the intriguing, and as yet fledgling, torsion analog of the classical Langlands correspondence. We also prove unconditional integral $R=\mathbb T$ results for Hecke algebras $\mathbb T$ acting on weight one cohomology of Shimura curves over $\mathbb Q$. This leads to a torsion Jacquet--Langlands correspondence comparing integral Hecke algebras acting on weight one cohomology of Shimura curves and modular curves. In this case the cohomology has abundant torsion and so our correspondence cannot be deduced by means of the classical Jacquet--Langlands correspondence.