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 eacch 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.

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.

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 eacch 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.

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 eacch 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.