Résume | We study super-commutative nonpositive DG rings. An example is the Koszul complex associated to a sequence of elements in a commutative ring. More generally such DG rings arise as semi-free resolutions of rings. They are also the affine DG schemes in derived algebraic geometry. The theme of this talk is that in many ways a DG ring $A$ resembles an infinitesimal extension, in the category of rings, of the ring $H^0(A)$.
I first discuss localization of DG rings on $Spec(H^0(A))$ and the cohomological noetherian property. Then I introduce perfect, tilting and dualizing DG $A$-modules. Existence of dualizing DG modules is proved under quite general assumptions. The derived Picard group $DPic(A)$ of $A$, whose objects are the tilting DG modules, classifies dualizing DG modules. It turns out that $DPic(A)$ is canonically isomorphic to $DPic(H^0(A))$, and that latter group is known by earlier work. A consequence is that $A$ and $H^0(A)$ have the same (isomorphism classes of) dualizing DG modules. |