TITLE: Why do people say DG-algebras behave badly in positive characteristic?
QUESTION [23 upvotes]: It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive characteristic. 
What are some examples for this bad behaivor? I know that it is unknown if the category of commutative DG-algebras has a nice model structure, but is there more to this saying?

REPLY [8 votes]: In characteristic zero, for example, it's possible to find cdgas describing the rational cohomology $H^{\bullet}(X, \mathbb{Q})$ of a (say simply connected) topological space; this is the starting point for Sullivan's approach to rational homotopy theory. In positive characteristic there are obstructions to doing this coming from cohomology operations (e.g. the Steenrod operations). It's known that one can fix this problem by using $E_{\infty}$ algebras instead; see, for example, DAG XIII.