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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 thelengths 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 casewhen 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 standardconjectures. Here $R$ is a deformation ring parametrizing certain Galoisrepresentations and $\mathbb T$ is a Hecke algebra. An example is a modularitylifting for 2-dimensional $\ell$-adic Galois representations over an imaginaryquadratic field. The proofs combine our commutative algebra results with ageneralization due to Calegari and Geraghty of the patching method of Wiles andTaylor--Wiles and level raising arguments that go back to Ribet. The resultsprovide new evidence in favor of the intriguing, and as yet fledgling, torsionanalog 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 integralHecke algebras acting on weight one cohomology of Shimura curves and modularcurves. In this case the cohomology has abundant torsion and so ourcorrespondence cannot be deduced by means of the classical Jacquet--Langlandscorrespondence.

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