Rank-two commutative Frobenius algebras over Dedekind domains can be projective but not free, realized as A = O ⊕ µX with µ² = (z), for example over Z[√-5].
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Two-dimensional topological quantum field theories of rank two over Dedekind domains
Rank-two commutative Frobenius algebras over Dedekind domains can be projective but not free, realized as A = O ⊕ µX with µ² = (z), for example over Z[√-5].