Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?

Question

In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this definition used in talks and the like as well, but I'm confused about where it comes from. I know two meanings of the word "flat" in this context. Firstly, Lurie's version (2): a spectrum $E$ is flat (over the sphere spectrum) if $\pi_0E$ is flat as an abelian group and $\pi_0E\times\pi_n\mathbb{S}\to\pi_nE$ is an isomorphism for all $n$. Second, the version defined in e.g. the nLab page on the Adams spectral sequence (3): a ring spectrum $E$ is flat if $E_*E$ is flat over $\pi_*E$.

As far as I can tell, these three definitions are all different, though I guess (1) implies (3). What I would like to know is, why did Ravenel call this condition "flat", and what is its precise relationship to the other two definitions?

Answer 1

Here is a little context to maybe complement Tom's and Nick's answers.

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The definition (2) in terms of being flat over $\pi_* \Bbb S$ is new - it's a specialization of a definition of flatness over $R$ that comes from derived / spectral algebraic geometry. Over $\Bbb S$ this is so rare that most examples are either in characteristic zero, or they are tailor-made to satisfy it (eg: localizations). The main utility would be to get a Kunneth theorem, identifying $E_* X$ with $E_* \otimes_{\pi_* \Bbb S} \pi_* X$. It has very specific goals, such as making base-change and descent effectively calculable.

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As Nick says, Ravenel's interest is in the general Adams-Novikov spectral sequence, and specifically getting better than the $E_1$-term. (In any case where the version (2) definition applies, the Adams-Novikov spectral sequence is pretty degenerate.) Slightly more explicitly, the $E_1$-term starts with the homotopy groups of $E \wedge X$, $E \wedge E \wedge X$, and so on, and produces a spectral sequence for the homotopy groups of $X$. We would ideally like to understand $\pi_*(E \wedge E \wedge \dots \wedge E \wedge X)$ in terms of $E_* X$.

This leads to definitions (1) and (3). Both of these are geared at getting a Kunneth isomorphism $$\pi_*(E \wedge E \wedge Y) \cong \pi_*(E \wedge E) \otimes_{\pi_* E} E_* Y$$ which we can then apply inductively to $X$. Definition (1) gets at this Kunneth isomorphism explicitly, because the splitting of $E \wedge E$ tells you that $$ E \wedge E \wedge Y \simeq \bigvee_i \Sigma^{n_i} (E \wedge Y). $$ Definition (3), by contrast, gets at this isomorphism with the Kunneth spectral sequence. You re-express $$ E \wedge E \wedge Y \simeq (E \wedge E) \wedge_E (E \wedge Y) $$ and this has a spectral sequence $$ Tor_{**}^{E_*} (E_* E, E_* Y) \Rightarrow \pi_*((E \wedge E) \wedge_E (E \wedge Y)). $$ Definition (3) implies that this degenerates to a Kunneth isomorphism.

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So why would we choose version (1), when version (3) is usually strictly stronger? There are two reasons.

• Maybe $E$ isn't good enough. Version (3) has an assumption - it only applies if $E$ is a highly structured (associative) ring spectrum, so that modules over $E$ and smash products over $E$ make sense. This is a problem if you're working with a ring spectrum that doesn't admit that nice of a multiplication (or you simply don't know that it does). By contrast, version (1) applies to not-associative-but-homotopy-associative things -- for example, the mod-$p$ Moore spectrum when $p$ is a prime greater than 3.

• More seriously, maybe it is 1992, when the orange book is being published. In 1992, categories of module spectra over highly structured ring spectra don't really exist, and there is no high-powered Kunneth spectral sequence available to you. Version (1) is what you've got.

Answer 2

I will propose a definition (4) and then discuss what it has to do with (1) and (2).

In algebra an $R$-module is flat if and only if it is a filtered colimit of free $R$-modules. (I don't know how widely known this is. I learned it from Mumford's "Little Red Book".)

So for a ring spectrum $E$ we could make a definition (4): Let's call an $E$-module flat if it is a filtered homotopy colimit of free $E$-modules, where we call an $E$-module free if it is a coproduct of modules each of which is $S^n\wedge E$ for some $n\in \mathbb Z$.

It seems to me that Lurie's definition (2) means that the $\mathbb S$-module $E$ is flat if it is a filtered homotopy colimit of $\mathbb S$-modules that are free in a narrower sense: each of them is a coproduct of copies of $\mathbb S$.

Definition (1) says that $E\wedge E$ is a free $E$-module (in the not narrow sense) and therefore is a flat $E$-module in sense (4). This is the case whenever $E$ is a free $\mathbb S$-module. Also, $E\wedge E$ will be a flat $E$-module in sense (4) if $E$ is a flat $\mathbb S$-module in sense (4). (But the converse is false, so this seems like a funny definition. Do we really want to call $H\mathbb F_2$ flat (over $\mathbb S$) just because $H\mathbb F_2\wedge H\mathbb F_2$ is a free $H\mathbb F_2$-module?)

Answer 3

In Ravenel's green book, he says (in his discussion of the $E$-based ASS) that $E$ is flat if $\pi_*(E \wedge E)$ is a flat $E_*$ module. I suspect that for lots of the $E$'s he cares about flat will equal free: $E_*$ tends to be a friendly graded ring finitely generated in each degree. So the more restrictive definition in the orange book doesn't rule out anything he cares about.

And what does he care about? $E$ is likely a complex oriented theory, and he wants to be able to identify the $E_2$ term of the $E$-based Adams spectral sequence as an algebraic Ext of some sort. So applying $\pi_*$ to the canonical resolution $S \rightarrow E \rightarrow \dots$ should yield an appropriately flat algebraic resolution of some sort. Things are really obvious with the orange book definition, as it can be easily iterated, to learn about $E^{\wedge s}$ for any $s$.