The notion of dg-schemes was introduced by Maxim Kontsevich as the first approach to derived algebraic geometry, and was further developed by Mikhail Kapranov and Ionut Ciocan-Fontanine.


A differential graded scheme (dg-scheme) is a scheme (X,O X)(X, O_X) together with a sheaf O X O_X^\bullet of nonnegatively graded commutative differential graded O XO_X-algebras, such that O XH 0(O X )O_X \to H^0(O_X^\bullet) is surjective.


Ciocan-Fontanine and Kapranov construct RQuot(X,F)RQuot(X, F), a derived enhancement of the classical Quot scheme parametrizing subsheaves of a given coherent sheaf FF on a smooth projective variety XX (1999). Similarly they also construct a dg-scheme RHilb h(X)RHilb_h(X), a derived enhancement of the Hilbert scheme parametrizing subschemes? of a given projective scheme? XX with Hilbert polynomial? hh (2000). As an application they construct the derived moduli stack of stable maps of curves to a given projective variety.

Relation with derived stacks

There is a functor from the category of dg-schemes to the category of derived stacks of Bertrand Toen and Gabriele Vezzosi. It takes values in the full subcategory of 1-geometric derived stacks, but is not known (or expected) to be fully faithful.

In particular, the dg-schemes RQuot(X,F)RQuot(X, F) and RHilb h(X)RHilb_h(X), discussed above, also induce derived stacks in the modern sense.


A prediction of derived moduli spaces is spelled out (in a bit different language) in

  • M. Kontsevich, Enumeration of rational curves via torus actions. The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math. 129, Birkhäuser 1995. MR1363062 (97d:14077), hep-th/9405035.

The first examples of derived moduli spaces, using dg-schemes, are constructed in

  • M. Kapranov, Injective resolutions of BG and derived moduli spaces of local systems, J. Pure Appl. Algebra 155 (2001), no. 2-3, 167–179; math/alg-geom/9710027, MR1801413 (2002b:18017)

Last revised on November 29, 2013 at 06:21:26. See the history of this page for a list of all contributions to it.