Moduli Problems in Abelian Categories and the Reconstruction Theorem

Understand

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it.

• The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces.
• Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles.
• We apply our strategy to prove analogous statements for categories of sheaves twisted by a Gm-gerbe.