Derived quot schemes

Ionuţ Ciocan-Fontanine, Mikhail Kapranov

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Abstract

We construct a "derived" version of Grothendieck's Quot scheme which is a dg-scheme, i.e., an object RQuot of a certain nonabelian right derived category of schemes. It has the property of being manifestly smooth in an appropriate sense (whereas the usual Quot scheme is often singular). The usual scheme Quot is obtained from RQuot by degree 0 truncation. The construction of RQuot can be seen as realization of a part of the Derived Deformation Theory program, which proposes to replace all the moduli spaces arising in geometry by their derived versions by retaining the information about all the higher cohomology instead of H1 in the classical theory.

Original languageEnglish (US)
Pages (from-to)403-440
Number of pages38
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume34
Issue number3
DOIs
StatePublished - 2001

Bibliographical note

Funding Information:
0.6. The first published reference for the DDT program seems to be the paper [12] by M. Kontsevich, who gave an exposition of the ensuing “hidden smoothness philosophy” in a lecture course in Berkeley in 1994. We are also aware of earlier unpublished suggestions of P. Deligne and V. Drinfeld containing very similar basic ideas. We gladly acknowledge our intellectual debt to our predecessors. We are also grateful to participants of the deformation theory seminar at Northwestern, where this work originated and was reported. Both authors were partially supported by NSF.

Copyright:
Copyright 2005 Elsevier Science B.V., Amsterdam. All rights reserved.

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