Can edge detection be done in the frequency domain?
Since convolution in spatial domain is multiplication in the Fourier (frequency) domain, you can perform edge detection in Fourier domain by multiplying the spectra of image and the edge detection kernel and then perform IFFT on the result.

I think the high-pass filter alone is not appropriate for edge detection since it keeps all high-frequency features (e.g. sharp peaks and corners) which are usually not classified as edges.

More advanced edge detection methods would be tricky in frequency domain since edges are best described in spatial domain (in my opinion).

The question is why to do edge detection using FFT in the first place? Is it because performance considerations? If so, maybe the high-pass filtered image (produced quickly by FFT) can be quickly filtered again to remove non-edge parts.