Apparently there is a thing called Fourier-Mukai equivalence in which the derived categories of coherent sheaves of two distinct schemes (e.g. an abelian variety and its dual) can be equivalent. On the other hand there are lots of reconstruction theorems whose gist tend to be that a scheme is determined by its (derived) category of sheaves, e.g. Tannaka dualities and the Gabriel-Rosenberg reconstruction theorem. How do these not contradict each other?
2026-03-25 07:48:38.1774424918
How do Fourier-Mukai equivalences not contradict reconstruction theorems?
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Fourier-Mukai equivalences are derived, they don't preserve the abelian categories of (quasi)coherent sheaves, unless they are given by isomorphisms composed with line bundle twists. So, there is no contradiction with the Gabriel-Rosenberg theorem.
Furthermore, non-trivial Fourier-Mukai equivalences exist only in the case when the canonical class is neither ample, nor anti-ample. So, there is no contradiction with Bondal-Orlov reconstruction theorem either.