We study supercommutative 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 semifree 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 Amodules. 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.
If there is time I will also talk about CM DG modules and an application.
