Multiplicativity in the descent spectral sequence

Question

For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U_i$, we can build a "descent spectral sequence" with signature $$E^1_{p, q} = \pi_{p+q} \mathcal{F}\left(\coprod_{|I| = q} U_I \right) \Rightarrow \pi_{p+q} \mathcal{F}(X).$$ This comes about by building the simplicial object associated to the cover, applying the sheaf to get a cosimplicial object, and then instantiating a filtration-type spectral sequence given by looking at varying truncations of totalizations --- the standard spectral sequence for a cosimplicial object. The $E^2$-page of the spectral sequence can be identified with Cech cohomology, and so the spectral sequence is meant to provide an intermediary between that homological object and the homotopy-sensitive information in the sheaf of spectra. This construction is natural enough that...

1. ...a refinement of the covering induces a map of spectral sequences. Just like limiting Cech cohomologies over cover refinements gives sheaf cohomology, limiting Cech descent spectral sequences gives a descent spectral sequence with $E^2$-page described by sheaf cohomology.
2. ...a map of sheaves induces a map of spectral sequences.

This construction doesn't really use the fact that $\mathcal{F}$ takes values in rings. I feel that this must appear in the spectral sequence, that we should expect some kind of multiplicativity --- maybe one that mixes the products of the ring spectra and the Cech product.

So, my question is: Is there a multiplicative structure in any of these spectral sequences? What is its signature? How might I compute with it? Better yet, are there examples of other people computing with it that I can read about?

For what it's worth, I'm interested most in starting with a sheaf of ring spectra $\mathcal{F}$ and computing $\mathcal{F}(X)^*(Y)$ for my favorite space $Y$ by augmenting $\mathcal{F}$ to $F(\Sigma^\infty_+ Y, \mathcal{F})(U) := F(\Sigma^\infty_+ Y, \mathcal{F}(U))$ and working with that. This comes with a map $F(\Sigma^\infty_+ (Y \times Y), \mathcal{F}) \to F(\Sigma^\infty_+ Y, \mathcal{F})$, which is in the vein of the usual construction of the cup product.

(There are a lot of details I'm eliding past, like conditional convergence, $\lim^1$ problems, when my augmented guy is actually a homotopy sheaf, when the analogy with sheaf cohomology can be made, so on and so forth. I don't think I've said anything not true in the nicest of settings, which is where I'd like to start learning, at least. Sorry if this is off-putting.)

Answer 1

Using your $X$ and $Y$, you get an augmented simplicial object as follows: $$ \cdots Y \times_X Y \times_X Y \Rrightarrow Y \times_X Y \Rightarrow Y $$ Applying $\cal F$ to this diagram, you get a coaugmented cosimplicial ring spectrum. The spectral sequence for the homotopy groups of Tot of this which realizes your Cech cohomology.

So this reduces you to a question: Given a cosimplicial object in ring spectra, do you get a multiplication on the associated spectral sequence converging to the multiplication on Tot?

This is true; most types of multiplicative structure carry over like this (although it may turn into "coherent" multiplicative structure). However, I've had a little trouble chasing this through the literature this morning.

One method you could use is a method of universal example, where $E_r$-cycles are carried by certain maps of cosimplicial objects. In Bousfield and Kan's "A second quadrant homotopy spectral sequence," they do this for the smash product in the homotopy spectral sequence of cosimplicial spaces (which is harder because you have to worry about basepoints!), and other authors (James Turner, and recently Philip Hackney) have studied operations arising on cosimplicial chain complexes by methods that should translate to the context of ring spectra.

Wish I had a more definitive reference for you.