Given $n$ a non-zero integer, it is known that multiplication by $n$ on an abelian variety (defined over any field $k$) is an isogeny. The proof of this fact uses the existence of an ample symmetric divisor on these varieties, which are projective.
Is this statement also true in general for abelian schemes, which may not be projective ?
I know that it is true for elliptic curves (as schemes), as it is proved in Katz and Mazur's book. However, the proof also makes use of the projectivity of such curves and their concrete description in terms of a Weierstraß equation.
For reference, an abelian scheme $X$ over a base scheme $S$ is a smooth proper $S$-group scheme with geometrically connected fibers. A homomorphism $f:X\rightarrow Y$ (as $S$-group schemes) of abelian schemes is an isogeny if it is surjective with a finite kernel. By "finite", we mean that the kernel is an $S$-group scheme which is locally free of finite rank over $S$. When the base is noetherian, this is just a finite flat group scheme over $S$.