Motivic symmetric spectra, by J.F. Jardine

This paper replaces A^1-local symmetric spectra (No. 302). The present version is quite different from the original in terms of technical details, language, and length. Some errors have also been corrected.

The paper demonstrates the existence of a theory of symmetric spectra for the motivic stable category. The main results imply the existence of a categorical model for the motivic stable category which has an internal symmetric monoidal smash product.

More explicitly, it is shown that there is a proper closed simplicial model category structure for the category of symmetric T-spectra, suitably defined, on the smooth Nisnevich site of a noetherian scheme of finite type. The weak equivalences for this structure are stable equivalences, defined by analogy with the definitions given by Hovey, Shipley and Smith for ordinary symmetric spectra and by Jardine for presheaves of symmetric spectra, except that one suspends by the Morel-Voevodsky object T, and the underlying unstable category is the motivic closed model structure for simplicial presheaves on the Nisnevich site. The homotopy category obtained from the category of symmetric T-spectra is equivalent to the motivic stable category.

The details of the basic construction of the original proper closed simplicial model structure underlying the motivic stable category are required to handle the symmetric case, and are displayed in the first three sections of this paper.

The paper can be found, for now, in a variety of file formats (dvi, ps, pdf) at Jardine's web site. It is to appear in Documenta Mathematica. In that event, the files on Jardine's web page will be replaced by a link to the journal.

J.F. Jardine <>