In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra with an action of an $A\_\infty$-operad (if I understand correctly) and commutative rings can be identified with spectra with an action of the $E\_\infty$-operad. Has anyone written about $E\_n$-rings in this category? In particular, are there nice model category structures on categories of $A$-modules and $A$-algebras if $A$ is only an $E\_n$-algebra? Can this perhaps be shown by somehow trapping this model structure between the nice model category structures on commutative rings and associative rings?


Thanks!



Higher coherent multiplicative structures on S-algebras
You can pick a model for the operad $E\_n$ which receives a map from the associative operad. For instance, the Boardman-Vogt tensor product of the associative operad with $E\_{n-1}$ has this property. Then, if you have an algebra $A$ over that operad, it is in particular an associative algebra and you can put a model structure on its category of modules using EKMM's method. The only problem with this method is that it is not clear that the resulting model category has a monoidal structure. According to Lurie, it should have and $E\_{n-1}$-monoidal structure but this is quite hard to prove using model categories. 


To put a model structure on $E\_n$-algebras over $A$ is easy. It suffices to observe that an $E\_n$-algebra over $A$ is an $E\_n$-algebra in spectra together with a map of $E\_n$-algebra from $A$. You can put a model structure on that category by taking the model structure on $E\_n$-algebras and then taking the under-category model structure.


Constructing the model structure on $E\_n$-algebras is a bit technical and I don't know of a reference for EKMM spectra. However, it has been worked out in details by John Harper in symmetric spectra. In fact John Harper proves it for any operad (see <http://www.msp.warwick.ac.uk/agt/2009/09-03/p057.xhtml>).


