Local Projective Model Structures on Simplicial Presheaves

We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in (MV), but with fewer cobrations and consequently more brations. This allows a simpler proof of the comparison theorem of (V2), one which makes no use of -closed classes. The purpose of this note is to introduce dieren t model structures on the categories of simplicial presheaves and simplicial sheaves on some es- sentially small Grothendieck site T and to give some applications of these simplied model categories. In particular, we prove that the stable homo- topy categories SH((Sm=k)Nis; A1) and SH((Sch=k)cdh; A1) are equivalent. This result was rst proven by Voevodsky in (V2) and our proof uses many of his techniques, but it does not use his theory of -closed classes developed in (V3).