### 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>