Contents

Idea

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.

Definition

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

Examples

Ciocan-Fontanine and Kapranov construct $RQuot(X, F)$, a

derived enhancement of the classical Quot scheme parametrizing subsheaves of a given coherent sheaf $F$ on a smooth projective variety $X$ (1999). Similarly they also construct a dg-scheme $RHilb_h(X)$, a derived enhancement of the Hilbert scheme parametrizing subschemes? of a given projective scheme $X$ with Hilbert polynomial? $h$ (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 $RQuot(X, F)$ and $RHilb_h(X)$, discussed above, also induce derived stacks in the modern sense.

Related concepts

• dg-manifold
• dg-geometry
• derived stack

References

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

• Ionut Ciocan-Fontanine, Mikhail Kapranov, Derived Quot

  schemes_, 1999, arXiv:math/9905174.

• Ionut Ciocan-Fontanine, Mikhail Kapranov, Derived Hilbert

  schemes_, 2000, arXiv:math/0005155.

• 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)

• Kai Behrend, Differential graded schemes I: prefect resolving

  algebras_ (arXiv:0212225)

• Kai Behrend, Differential Graded Schemes II: The 2-category of

  Differential Graded Schemes_ (arXiv:0212226)