# Milnor descent for ring spectra

Suppose given a homotopy cartesian square of (commutative) ring spectra (or (c)dgas)

$\begin{matrix}A & \to & A_1 \\ \downarrow & & \downarrow \\ A_2 & \to &A'\end{matrix}.$

Here the morphisms are morphisms of ring spectra (dgas) and by homotopy cartesian I mean that the underlying diagram of spectra (chain complexes) is homotopy cartesian (equivalently, cocartesian).

Let me write $Mod(A)$ for the stable $\infty$-category (dg category) of $A$-module spectra, and similarly for the other rings. There is an induced diagram of infinity categories

$\begin{matrix}Mod(A) & \to & Mod(A_1) \\ \downarrow & & \downarrow \\ Mod(A_2) & \to &Mod(A')\end{matrix},$

where the morphisms are given by base change. Recall the homotopy fibre product $Mod(A_1) \times^h_{Mod(A')} Mod(A_2)$; we obtain a functor $F: Mod(A) \to Mod(A_1) \times^h_{Mod(A')} Mod(A_2)$.

I want to know if there are (tractable) criteria for $F$ to be a quasi-equivalence?

Here are some remarks:

• In http://arxiv.org/abs/1201.6118 the authors provide a nice explicit model for the homotopy fibre product of dg categories, see section 4. They also state that $F$ is a quasi-equivalence if the diagram is actually cartesian, the dgas have no positive homotopy groups (in topologists' notation), and the degree zero chains satisfy milnor descent (c/f below). I'd be interested in a more "homotopical" criterion.

• Suppose all the rings are commutative. Let me write $P(A) \subset Ho(Mod(A))$ for the additive karoubi-closed subcategory generated by $A$. If the diagram

$\begin{matrix}\pi_0(A) & \to & \pi_0(A_1) \\ \downarrow & & \downarrow \\ \pi_0(A_2) & \to &\pi_0(A')\end{matrix}$

is cartesian and one of the lower/right maps is surjective, then we have that $P(A) \to P(A_1) \times_{P(A')} P(A_2)$ is an equivalence. This is milnor descent.

• Using the explicit model for $Mod(A_1) \times^h_{Mod(A')} Mod(A_2)$, namely objects being triples $(M_1, M_2, \phi)$ where $M_i \in Mod(A_i)$ and $\phi: M_1 \otimes_{A_1} A' \to M_2 \otimes A_2 A'$ is a closed equivalence of degree zero, one defines a functor $R: Mod(A_1) \times^h_{Mod(A')} Mod(A_2) \to Mod(A)$ mapping $(M_1, M_2, \phi)$ to the homotopy fibre of $M_1 \oplus M_2 \to M_2 \otimes_{A_2} A'$ (viewed as a diagram of $A$-modules). I think functors $Ho(F), Ho(R)$ are adjoints and $RF$ is equivalent to the identity functor. I also think that $FR$ is equivalent to the identity functor if (and only if) $F$ has dense image, i.e. iff $Mod(A_1) \times^h_{Mod(A')} Mod(A_2)$ is generated by $F(A)$. I do not know how to make use of this criterion.