Suppose $F$ is a field. I want to know whether the map $GL\_n(GW(F))\to GL\_n(W(F))$ is surjective, where $GW$ means Grothendieck-Witt and $W$ means Witt. In the case $F$ is algebraic closed, it reduces to the surjectivity of $GL\_n(\mathbb{Z})\to GL\_n(\mathbb{Z}/2\mathbb{Z})$. I know the case $n=1$ is true.



Is the map $GL_n(\mathbb{Z})\to GL_n(\mathbb{Z}/2\mathbb{Z})$ surjective?
For the question in your title, yes: $\operatorname{GL}\_n(\mathbb Z/2\mathbb Z) = \operatorname{SL}\_n(\mathbb Z/2\mathbb Z)$ is generated by transvections, and these obviously lift to $\operatorname{GL}\_n(\mathbb Z)$.


