Derived quot schemes

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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.

Résumé

On construit une version “dérivée” du schéma Quot de Grothendieck. C'est un dg-schéma, i.e., un objet RQuot d'une certaine catégorie dérivée non-abélienne à droite des schémas usuels. Il est toujours lisse en un sens convenable (tandis que le schéma Quot classique peut être singulier). On peut obtenir le schéma Quot de RQuot par troncature en degré 0. La construction de RQuot peut être vue comme réalisation d'une part du programme des déformations dérivées qui cherche à remplacer tous les espaces de modules en géométrie par leurs versions dérivées qui font usage de la cohomologie supérieure, même au niveau des espaces tangents qui en théorie traditionnelle s'interprètent comme certains H1.

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