We construct the Cartier duality equivalence for affine commutative group schemes whose coordinate ring is a flat Mittag-Leffler module over an arbitrary base ring . The dual of turns out to be an ind-finite ind-scheme over . When is Noetherian and admits a dualizing complex, we construct a Fourier-Mukai transform between quasicoherent derived categories of and of and also between those of and .
