Local projective model structures on simplicial presheaves

We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Gro-then-dieck site T and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)(Nis),A(1)) and SH((Sch/k)(cdh),A(1)) are equivalent. This result was first proven by Voevodsky and our proof uses many of his techniques, but it does not use his theory of Delta -closed classes.