# Why do people say DG-algebras behave badly in positive characteristic?

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?

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There is the fact that over a field of positive characteristic commutative dg-algebras do not model $E_\infty$-algebras in any sense (model category, fibration category etc.) This is a pretty big deal because many objects one is naturally interested in studying are $E_\infty$-algebras and have no obvious model as a commutative dg-algebra (e.g. derived global sections of a sheaf of commutative rings on a scheme) – Denis Nardin Jan 22 at 10:54
Outside characteristic zero, you have to use $E_\infty$-dg-algebras instead of strictly commutative dg-algebras. See Tyler Lawson's answer here: mathoverflow.net/a/23885/2503. (Oops, Denis beat me to it.) – Adeel Khan Jan 22 at 10:56

In characteristic zero, the model structure on commutative dg-algebras is obtained by transfer from the projective model structure on chain complexes, along the ajunction between the free algebra functor and the forgetful functor. In particular, weak equivalences and fibrations are determined in chain complexes (quasi-isomorphims and degreewise surjections).

In positive characteristic, a model structure still exists, which is actually available for commutative dg-algebras over any commutative ring: this was proved by Don Stanley in his preprint Determining closed model category structures. However, this model structure is not nice, in the sense that fibrations are not necessarily surjective in positive degrees: weak equivalences and fibrations are not determined by the forgetful functor from commutative dg-algebras to chain complexes.

Actually this is an incarnation of a more general fact about the possibility to transfer a model structure from a model category to its category of commutative monoids. A nice criterion for this is called the commutative monoid axiom in the paper of David White Model structures on commutative monoids in general model categories, and it turns out that such a criterion fails for commutative dg-algebras in positive characteristic.

Now, going back to derived geometry, a good model that works in positive characteristic is the one of simplicial rings, which inherits a nice model structure in any characteristic. Moreover, in characteristic zero, simplicial rings are Quillen equivalent to commutative dg-algebras (equipped with the model structure induced by the one of chain complexes) via the Dold-Kan correspondence.

I would like also to point out that, in characteristic zero, using commutative dg-algebras as affine derived stacks can be very useful to, for instance, study geometric structures on derived stacks such as shifted symplectic structures. The paper Shifted symplectic structures by Pantev-Toen-Vaquié-Vezzosi is written in this context.

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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.

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Or the work of Mike Mandell. – Sean Tilson Jan 26 at 17:11