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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?


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up vote 8 down vote accepted

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.)

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Oh thanks!! yeah, I got the projective dimension situation from another theorem I was getting this mixed up with. –  Jon Beardsley Sep 23 '12 at 23:44
Also thanks for that reference for the Smith paper! I couldn't find a copy of it anywhere. –  Jon Beardsley Sep 23 '12 at 23:46
But one more question: you seem to be talking about only the finite spectra. Can this be extended to say, suspension spectra, or harmonic spectra? –  Jon Beardsley Sep 23 '12 at 23:50
@Jon: Consider the suspension spectrum of an infinite wedge of copies of $S^0$... –  Tyler Lawson Sep 24 '12 at 4:52
Haha, okay yeah, obvious. (Wondering what Ind category of coherent modules looks like....) –  Jon Beardsley Sep 25 '12 at 14:26
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The question is also addressed in Lecture 5 of J.F. Adams ``Lectures on generalized cohomology'' in Springer Lecture Notes in Mathematics Vol 99(1969). Again ancient, but none the worse for that. The paper is reprinted in Volume I of "The selected works of J. Frank Adams''.

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