Degree of isogenies between (semi-)abelian schemes

Question

Let $S$ be a connected noetherian normal scheme of dimension 0 or 1 (i.e. $S$ is a connected Dedekind scheme).

Let $f:G\to G'$ be a morphism of semi-abelian schemes over $S$. In their book on Neron models, Bosch-Lutkebohmert-Raynaud define that $f$ is an isogeny if for all points $s\in S$ the base changed morphism $f_s:G_s\to G'_s$ is an isogeny (i.e. $f_s$ is finite and surjective on identity components).

Is $\deg(f_s)=\deg(f_{s'})$ for all $s,s'\in S$?

If this is the case, can one then define the degree of $f$ just by $\deg(f_s)$ for $s$ the generic point of $S$?

If the above questions have negative answers, can one answer them affirmatively for abelian schemes?

Answer 1

I am going to assume that $S$ is Noetherian. By limit arguments, one should always be able to reduce to that case. Also I am going to assume that your semi-Abelian schemes are smooth over $S$; I believe this is part of the usual definition.

In fact, $f$ should be flat, which implies constancy of the degree of $f_s$. The point is that both $G$ and $G'$ are smooth over $S$. Thus the fiber product, $G\times_S G'$, is flat over $G'$ of relative dimension $g=\text{dim}(G)$. Also $G'$ is smooth over $S$ of some relative dimension $g'$, so that the diagonal morphism, $$\Delta_{G'}:G'\to G'\times_S G',$$ is a regular embedding of codimension $g$. Thus, the base change graph morphism, $$\Gamma_f:G\to G\times_S G',$$ is also a regular embedding of codimension $g$.

Finally, for a flat morphism $X\to Y$ of relative dimension $d$ and for a regular embedding $Z\hookrightarrow X$ of codimension $e$, the flat locus of $Z\to Y$ contains every point of $Z$ where the fiber has dimension $d-e$. This follows from the local flatness criterion (use the Koszul complex of the regular sequence to resolve the ideal sheaf, and then use the codimension criterion for exactness of a Koszul complex to check exactness of the basechange of the Koszul complex to each fiber). Apply this with $Y=G'$, $X=G\times_S G'$ and $Z$ is the image of $\Gamma_f$.