Toward a definition of moduli of complexes of coherent sheaves on a projective scheme.
Let \(f:X\to S\) be a flat projective morphism of noetherian schemes. The author constructs the moduli space of bounded complexes \(E^{\bullet}\) of flat coherent sheaves on \(X\) such that \(\text{Ext}^0(E^{\bullet}(s),E^{\bullet}(s))\simeq k(s)\) and \(\text{Ext}^{-1}(E^{\bullet}(s),E^{\bullet}(s))\simeq k(s)\) for each \(s\in S\). The constructed moduli space is a locally separated algebraic space but otherwise it has very bad properties. This moduli space contains as an open subset the moduli space of simple sheaves constructed by \textit{A. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50--112 (1980; Zbl 0427.14015)]. One of the motivations for constructing the moduli space of complexes is that the Fourier--Mukai transform usually transforms sheaves into complexes and not sheaves, so it induces a morphism of moduli spaces of sheaves only under some additional assumptions. However, it always induces a morphism of moduli spaces of complexes constructed in the paper. It would be very desirable to construct projective moduli schemes of objects in the derived category of coherent sheaves satisfying some sort of stability conditions in the sense of Douglas and Bridgeland [see \textit{T. Bridgeland}, Stability conditions on triangulated categories, preprint, \texttt{http://arxiv.org/abs/math.AG/0212237}].
scientific article; zbMATH DE number 1998394
scientific article