Let A be a commutative simple algebra in a braided finite tensor category B. We identify the largest transparent subalgebra of A as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local A-modules. These formulas give criteria for nondegeneracy, symmetry, and modularity, together with sharp bounds on FPdimB(A)
Nearby in the stack
. We also realize the Müger center of
B
as a category of local modules over an adjoint algebra. Finally, we prove a relative-center factorization and deduce that taking the category of local modules preserves the relative Witt class.