This question comes when I try the valuative criterion on properness of the moduli space of stable sheaves. Let $X$ be a projective scheme over $\Bbbk$ with an ample line bundle $\mathcal L$. Let $P(t)$ be a Hilbert polynomial. I know the following fact:
the moduli scheme of stable sheaves is the étale sheafification of the moduli functor of stable sheaves $$(Sch/\Bbbk)^{opp}\to Sets,\quad T\mapsto \Big\{\begin{matrix}\textrm{flat }T\textrm{-families of }\mathcal O_X\textrm{-modules}\\\textrm{stable at closed points, with Hilbert polynomial }P(t)\end{matrix}\Big\}\Big/\sim; $$
In a few cases, the moduli scheme of stable sheaves is proper over $\Bbbk$, (when semistability coincides with stability). Since the moduli functor above determines the moduli scheme and the condition that semistability coincides with stability do not depend on the GIT construction, I assume we can show properness from them directly.
The fact that the moduli scheme is an étale sheafification asks us to consider étale coverings of $\operatorname{Spec}(A)$ for $A$ a discrete valuation ring.
My questions:
- What are étale coverings of $\operatorname{Spec}(A)$ for a DVR $A$?
- Are there literatures proving properness of moduli schemes of stable sheaves using the above method? If there is no answer to 1, what special properties of this moduli problem played a role?
