  In this note we look at the freeness for complex affine hypersurfaces. If $X
\subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated
projective hypersurface, obtained by taking the closure of $X$ in
$\mathbb{P}^n$, then we relate first the Jacobian syzygies of $D$ and those of
$X$. Then we introduce two types of freeness for an affine hypersurface $X$,
and prove various relations between them and the freeness of the projective
hypersurface $D$. We write down a proof of the folklore result saying that an
affine hypersurface is free if and only if all of its singularities are free,
in the sense of K. Saito's definition in the local setting. In particular,
smooth affine hypersurfaces and affine plane curves are always free. Some other
results, involving global Tjurina numbers and minimal degrees of non trivial
syzygies are also explored.
