A^1-local symmetric spectra, by J.F. Jardine

Revised: June 6, 1999. This version has been rewritten more than once, but the only change of mathematical substance from the former version on this server is that the original Lemma 4.9 was incorrect and has been removed.

This paper demonstrates the existence of a theory of symmetric spectra for the Morel-Voevodsky stable category. The main results imply the existence of a categorical model for the Morel-Voevodsky 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 presheaves of symmetric T-spectra, suitably defined, on the smooth Nisnevich site of a field. 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 A^1-local category of simplicial presheaves. The homotopy category obtained from the category of presheaves of symmetric T-spectra is equivalent to the Morel-Voevodsky stable category.

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

This paper was written in lamstex, and the dvi file requires the lamstex fonts to view or print. Postscript and hyperlinked pdf versions are available at Jardine's home page.


J.F. Jardine <jardine@uwo.ca>