One possible answer is in Toën-Vezzozi paper [From HAG to DAG][1], who were themselves inspired by Ciocan-Fontanine and Kapranov (Derived Quot schemes and [Derived Hilbert schemes][2]). 

This approach works well in characteristic zero (otherwise one has to deal with simplicial commutative rings or $E_\infty$-ring spectra, like in Lurie's work). 


  [1]: https://arxiv.org/abs/math/0210407
  [2]: http://arxiv.org/abs/math/0005155