Coherent MU_*-Modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:52:28Z http://mathoverflow.net/feeds/question/107874 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107874/coherent-mu-modules Coherent MU_*-Modules Jon Beardsley 2012-09-23T01:34:11Z 2012-09-24T16:32:28Z <p>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?</p> <p>Thanks!</p> http://mathoverflow.net/questions/107874/coherent-mu-modules/107880#107880 Answer by Beren Sanders for Coherent MU_*-Modules Beren Sanders 2012-09-23T04:36:53Z 2012-09-24T16:32:28Z <p>I believe that the result holds quite generally. The specific case of complex bordism is discussed in the following two papers:</p> <ul> <li>Larry Smith - On the finite generation of $\Omega_\ast^U(X)$ (<em>J. Math. Mech.</em>, 1969)</li> <li>Pierre Conner &amp; Larry Smith - On the complex bordism of finite complexes (<em>Publications Mathématiques de l'IHÉS</em>, 1969)</li> </ul> <p>The first paper can be found at the webpage for the Indiana University Mathematics Journal (<a href="http://www.iumj.indiana.edu/" rel="nofollow">http://www.iumj.indiana.edu/</a>) while the second paper can be found on NUMDAM (<a href="http://www.numdam.org/" rel="nofollow">http://www.numdam.org/</a>). Note that there is no projective dimension requirement: $\Omega_\ast^U(X)$ is a coherent $\Omega_\ast^U$-module for any finite complex $X$.</p> <p>But more generally: any ring spectrum $\mathbb{E}$ induces a homological functor</p> <p>$$\mathbb{E}_*(-) : \text{SH}^\text{fin} \rightarrow \mathbb{E}_\ast\text{-grMod}$$</p> <p>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 $\mathbb{E}$).</p> <p>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 $\text{SH}^\text{fin}$.</p> <p>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 $X$.</p> <p>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 $X$.</p> <p>(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.)</p> http://mathoverflow.net/questions/107874/coherent-mu-modules/107937#107937 Answer by Peter May for Coherent MU_*-Modules Peter May 2012-09-23T21:38:00Z 2012-09-23T21:38:00Z <p>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''.</p>