It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega\_\ast^U(X)$, is a coherent module over $\Omega\_\ast^U$ if $\Omega\_\ast^U(X)$ has projective dimension 0 or 1 over $\Omega\_\ast^U$. It is stated that in a series of lecture notes by Larry Smith that this result can probably be extended to other complexes (and spectra...). However, these lecture notes are from 1970. Does anyone know if this result has been fully generalized? I guess I mean, is it known precisely how far this result can be extended?


Thanks!



Coherent MU_*-Modules
I believe that the result holds quite generally. The specific case of complex bordism is discussed in the following two papers:


* Larry Smith - On the finite generation of $\Omega\_\ast^U(X)$ (*J. Math. Mech.*, 1969)
* Pierre Conner & Larry Smith - On the complex bordism of finite complexes (*Publications Mathématiques de l'IHÉS*, 1969)


The first paper can be found at the webpage for the Indiana University Mathematics Journal (<http://www.iumj.indiana.edu/>)
while the second paper can be found on NUMDAM (<http://www.numdam.org/>). Note that there is no projective dimension requirement: $\Omega\_\ast^U(X)$ is a coherent $\Omega\_\ast^U$-module for any finite complex $X$.


But more generally: any ring spectrum $\mathbb{E}$ induces a homological functor


$$ \mathbb{E}\_\*(-) : \text{SH}^\text{fin} \rightarrow \mathbb{E}\_\ast\text{-grMod}$$


from the stable homotopy category of finite spectra to the category of graded $\mathbb{E}\_\*$-modules (where $\mathbb E\_\ast = \pi\_\ast(\mathbb E) = \mathbb E\_\ast(\mathbb S)$ is the coefficient ring of $\mathbb{E}$).


It follows from basic properties of coherent modules and the fact that $\mathbb E\_\ast(-)$ is a homological functor that the collection of those finite spectra $X$ such that $\mathbb{E}\_\*(X)$ is coherent as a graded $E\_\ast$-module is a thick triangulated subcategory of $\text{SH}^\text{fin}$.


If $\mathbb E\_{\ast}$ is a coherent ring then this thick triangulated subcategory contains the sphere spectrum. But since the thick triangulated subcategory generated by the sphere spectrum is the whole of $\text{SH}^\text{fin}$ it follows that $\mathbb E\_\ast(X)$ is a coherent $\mathbb E\_\*$-module for every finite spectrum $X$.


In other words, if $\mathbb{E}$ is a ring spectrum whose coefficient ring is coherent then $\mathbb E\_\ast(X)$ is a coherent graded $\mathbb{E}\_\\*$-module for any finite spectrum $X$.


(In the case of complex bordism, the coefficient ring $\Omega\_\ast^U$ is a polynomial algebra over the integers in an countably infinite number of generators. [Smith69] includes a proof that a polynomial algebra in a countable number of variables over a noetherian ring is coherent. On the other hand, it might be worth mentioning that if the generating hypothesis is true then the stable homotopy groups of the spheres are "totally non-coherent" in a certain precise sense.)


