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This is my first MO question, so please go easy on me if you think this is too vague.

Is there anything to say about the collection of chain complexes with flat homology? Is there a name for them, or a different characterization of them? Maybe there's a way to build them out of some other simpler objects?

Specifically, I'm working in the derived category of a (non-Noetherian) commutative ring. The collection of objects with flat homology seems closed under retracts and coproducts, but not under the formation of triangles. So they don't make any triangulated subcategory, but maybe there's something else to be said about them?

My motivation: there's a spectral sequence (according to Weibel) with $E^2$ term $E^2_{p,q} = \bigoplus_{q'+q''=q} \mbox{Tor}_p^R\left( H_{q'}(A),H_{q''}(B)\right)$ that always converges to $H_{p+q}(A\otimes_L^R B)$.

I've constructed an object, say $B$, whose homology in each degree is the same - it's the vector-space dual $R^*$ of the ring $R$ (which is a truncated polynomial algebra on infinitely many generators). I'm interested in the Bousfield class of this object, i.e. the objects $A$ such that $A\otimes_L^R B=0$.

According to the spectral sequence, when A has flat homology, things collapse, and all I need to do is look at the Bousfield class of $R^*$, which is known. So that's why I care about such objects.

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up vote 4 down vote accepted

This is my first Math Overflow answer, so hopefully people will go easy on me as well. Everything I'm about to say is based on the following paper of Mark Hovey and Keir Lockridge:

You are thinking about $\mathcal{D}(R)$, the derived category of $R$. This category is equivalent to $\mathcal{D}(HR)$ where $HR$ is the Eilenberg-Maclane spectrum associated to $R$. This should explain how Ring Spectra get involved. Chain complexes with flat homology are objects $X\in \mathcal{D}(HR)$ with flat dimension zero. According to Proposition 1.4 in the paper, the following are equivalent:

1) Flat dim $X=0$

2) In the Universal Coefficient Spectral Sequence

$E_{s,t}^2 = Tor_{s,t}^R(H_*(X),H_*(Y)) \Rightarrow \mathcal{D}(HR)(X,Y)_{t-s}$

we have $E_\infty^{s,*} = 0$ for all $s>n$ and for all objects $Y\in \mathcal{D}(HR)$.

3) There is an exact triangle

$A\to X\stackrel{g}{\to}W\to \Sigma A$

with $H_*(A)$ projective and $g$ phantom (see below for a definition).

4) Every map $F\to X$ from a compact object $F\in \mathcal{D}(HR)$ factors through a compact $B$ with $H_*(B)$ projective.

Definition: A map is $g:X\to W$ phantom if for all compact $Z\in \mathcal{D}(HR)$, for all $f:Z\to X$ we have $f\circ g = 0$

That's the only other characterization I know. If you can prove more about your ring $R$, e.g. something about its weak dimension, then that could tell you a lot more about the collection of $X$ with $H_*(X)$ flat.

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