Toward a definition of moduli of complexes of coherent sheaves on a projective scheme.
type: Item
instance of: scholarly article
title: Toward a definition of moduli of complexes of coherent sheaves on a projective scheme.
zbMATH Open document ID: 1063.14013
author: Michi-Aki Inaba
published in: Journal of Mathematics of Kyoto University
publication date: 2003-12-15T00:00:00Z
review text: 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}].
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1998394
zbMATH Keywords: projective scheme
zbMATH Keywords: moduli space
zbMATH Keywords: derived category
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283873
OpenAlex ID: W1591547642
DOI: 10.1215/KJM/1250283873
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description: scientific article; zbMATH DE number 1998394