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This work builds on earlier work of the first three authors where a notion ofcongruence modules in higher codimension is introduced. The main new resultsare a criterion for detecting regularity of local rings in terms of congruencemodules, and a more refined version of a result tracking the change ofcongruence modules under deformation is proved. Number theoretic applicationsinclude the construction of canonical lines in certain Galois cohomology groupsarising from adjoint motives of Hilbert modular forms.

Congruence modules in higher codimension and zeta lines in Galois cohomology