Are there nonzero morphisms between families of stable sheaves, with slopes decreasing?

Question

Consider over $\mathbb{C}$. Let $C$ be a smooth projective curve. Fix degrees $d_1,d_2$ and ranks $r_1,r_2$ such that $\mu_1=\frac{d_1}{r_1}>\mu_2=\frac{d_2}{r_2}$.

It is well known that if $E,F$ are stable sheaves with slopes $\mu(E)>\mu(F)$, then $Hom(E,F)=0$. I wonder if this still holds for families.

Let $S$ be a scheme. Let $\mathcal{E}_i$ be a flat $S$-family of stable sheaves of degree $d_i$ and rank $r_i$, for $i=1,2$. Is $Hom_{C\times S}(\mathcal{E}_1,\mathcal{E}_2)=0$?

Here is my try. Suppose $\varphi:\mathcal{E}_1\to\mathcal{E}_2$. Then immediately $\varphi$ pulls back to zero along any points $\mathrm{Spec}(\kappa(s))\to S$. However, this pointwise argument would not detect any nilpotent phenomena, and can not prove $\varphi=0$.

Answer 1

Let $R$ be a local ring of finite dimension over $\mathbb C$, and let $\mathcal{E}_1$ and $\mathcal{E}_2$ be flat $\operatorname{Spec} R$-families of stable sheaves.

Then $R$ admits a finite filtration as an $R$-module with each associated graded component isomorphic to the residue field. Hence $\mathcal{E}_1$ and $\mathcal{E}_2$ admit finite filtrations with each associated graded quotient isomorphic to their special fibers $E_1$ and $E_2$, which are stable sheaves of the same slope. Any nonzero map $\mathcal{E}_1 \to \mathcal{E}_2$ must induce a nontrivial map between one of the associated graded components, for general reasons about filtrations, which is impossible.

Since a map must vanish over finite-dimensional local rings, it must vanish over finitely-generated local rings, hence over all finitely-generated rings, and finally over all rings.