“Geometric” vs Homotopical completion

Question

There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.

The first one is the “homotopical” (or maybe it should be called algebraic) completion of a morphism in a sense of, e.g. the paper of Kathryn Hess (section 4), i.e. totalization of a monadic bar-construction.

The second one is the “geometric completion” of a morphism of schemes in the sense of Gaitsgory—Rozenblyum, see for instance this paper section 6. Here the completion is calculated using the fiber product $X^{\wedge}_Y= X\times_{X_{\textrm{dR}}} Y_{\textrm{dR}}.$

Is there a way to relate the two in the case when the morphism of derived schemes is representable? More precisely, I am looking for a statement of the type: functions on the geometric completion is the homotopical completion of the morphism of underlying rings. I would also really appreciate a reference if there exists one.

Answer 1

Yes, there's a way to relate the two. First, it's helpful to think of both in terms of universal properties. Since geometric completion is a fiber product, it's a pullback. Meanwhile, the homotopical completion can be defined as a (homotopy) pushout. As Hess remarks, Bousfield's $p$-completion is a special case, and that's a special case of Bousfield localization, and hence can be computed by a particular homotopy pushout. Completion in algebra is also a special case of this machinery. Applying $Hom$ converts homotopy pullbacks to homotopy pushouts and vice versa.

Both of your settings are treated together, in a series of papers by Barthel, Heard, and Valenzuela. One has "derived completion" in the title. On page 4 they have a table relating the algebraic geometry side (geometric completion) with the commutative algebra side (algebraic completion). They introduce what they call a local duality framework that lets them treat completion (and localization, torsion, etc.) in both settings simultaneously. They then use recollements to relate torsion and completion, and they relate derived completion of rings to certain algebraic stacks. Moreover, they relate comodules over Hopf algebroids (on the algebra side) to quasi-coherent sheaves over algebraic stacks (on the algebraic geometry side).

This work builds on earlier work of Dwyer-Greenlees and Greenlees-May. Some of that is summarized at the Stacks project (but, I've not read through that treatment fully), and that might save you from the machinery of local duality frameworks.

Answer 2

In the affine case, this is more-or-less proved in [Bhargav Bhatt, Completions and derived de Rham cohomology]. More precisely, Kathryn Hess' completion is more akin to Carlsson's Adams completion, and the relative de Rham stack is closely related to the derived de Rham cohomology (in some sense, the former is represented by the later). Then the precise statement is Proposition 4.16 loc. cit.

PS: In fact, some variant of this also holds for general morphisms, not necessarily closed immersions. It is Theorem 4.10 loc. cit., whose proof depends on the statement for closed immersions.