generalized universal coefficient sequence

Question

Take the familiar Universal Coefficient Theorem for ordinary homology with $\mathbb{Z}$-coefficients and ordinary cohomology with coefficients in some abelian group $A$:$$0\rightarrow \text{Ext}_\mathbb{Z}^1(H_{i-1}(X;\mathbb{Z}),A)\rightarrow H^i(X;A)\rightarrow \text{Hom}_\mathbb{Z}(H_i(X;\mathbb{Z}),A)\rightarrow 0$$This exact sequence can be rewritten in the language of topological spectra:$$0\rightarrow\text{ext}_{H\mathbb{Z}}^1([\Sigma^{i-1}S,H\mathbb{Z}\wedge X],HA)\rightarrow [X,\Sigma^iHA]\rightarrow [[\Sigma^iS,H\mathbb{Z}\wedge X],HA]\rightarrow 0$$where $[-,-]$ denotes the internal hom $\text{hom}_{H\mathbb{Z}}(-,-)$ of $H\mathbb{Z}$-module spectra. (I am confused about some basic points [see my question], but I presume that $\text{ext}$ makes sense in this context.)

How can one generalize this sequence to generalized (co)homology theories? One guess is to replace $H\mathbb{Z}$ with some commutative ring spectrum $E$ and $HA$ with an $E$-module spectrum $F$. I imagine this guess is far too ambitious and a more conservative theorem is the correct generalization.

Answer 1

There is more than one possible generalization. The most common is the universal coefficient spectral sequence. Given a (homotopy) commutative ring spectrum $E$ and a spectrum $X$, there is under certain conditions a spectral sequence $$ Ext^{p,q}_{E_*}(E_*(X), E_*) \Rightarrow E^{q-p}(X) $$ This is true for example if

• $E$ is an $A_\infty$-ring spectrum (e.g. if $E =ko, ku, KO, KU, TMF, MO, MSO, MSpin \dots$) [EKMM, IV.4]
• $E$ is even and Landweber exact (e.g. if $E = MU, E(n), E_n, \dots$) [Adams' lectures on generalized cohomology, which Peter alluded to, and Devinatz: Morava Modules and Brown-Comenetz Duality, Prop 1.3, and the discussion thereafter]

Sometimes, this spectral sequence is not extremely useful though. For example, take $E = KO$. In general, $KO_*X$ might have infinite cohomological dimension over $KO_*$ so that the spectral sequence might be difficult to control. In this case another perspective is more useful: Anderson duality.

Consider the functor $E \mapsto Hom(\pi_*E, \mathbb{Q}/\mathbb{Z})$ from the homotopy category of spectra to graded abelian groups. As $\mathbb{Q}/\mathbb{Z}$ is injective, this is a cohomological functor, by Brown representability represented by a spectrum $I_{\mathbb{Q}/\mathbb{Z}}$. There is an evident map $H\mathbb{Q} \to I_{\mathbb{Q}/\mathbb{Z}}$ whose fiber we denote by $I$. For a spectrum $E$, we define its Anderson dual $IE$. to be the function spectrum $F(E, I)$. It is easy to show that we get for every spectrum $X$ a short exact sequence $$ 0 \to Ext^1_{\mathbb{Z}}(E_{k-1}X, \mathbb{Z}) \to (IE)^kX \to Hom_{\mathbb{Z}}(E_kX, \mathbb{Z}) \to 0.$$ That means that there exists always a short exact sequence computing from $E$-homology the $IE$-cohomology. This is, of course, only useful if we can identify $IE$. Luckily, this has been done for a few spectra:

• $IH\mathbb{Z} \simeq H\mathbb{Z}$
• $IKU \simeq KU$
• $IKO \simeq \Sigma^4 KO$ [see e.g. Heard, Stojanoska]
• $ITmf \simeq \Sigma^{21}Tmf$ (at least at primes $>2$) [see Stojanoska]

For example, for $KO$ this means that we get a universal coefficient sequence $$ 0 \to Ext^1_{\mathbb{Z}}(KO_{k-1}X, \mathbb{Z}) \to KO^{k+4}X \to Hom_{\mathbb{Z}}(KO_kX, \mathbb{Z}) \to 0.$$