# Deformations and moduli of semistable sheaves in mixed characteristic

Let $X$ be a projective scheme over an algebraically closed field $k$. There is the coarse moduli space $M_X$ parametrizing semistable sheaves on $X$ with fixed reduced Hilbert polynomial $p$. Now, the deformation theory of $M_X$ at a stable sheaf $E$ is (in some sense) understood:

Let $Def_E$ be the deformation functor, that is, it assigns to each local Artinian $k$-algebra $A$ with residue field $k$, the isomorphism classes of sheaves $\mathcal E$ on $X \otimes_k A$, flat over $A$ and with $\mathcal E \otimes_A k \cong E$. Then, $Def_E$ is pro-represented by the completion of the local ring $\mathcal O_{M_X, E}$.

Now, the moduli space $M_X$ also exists in mixed characteristic, e.g. for $X$ a smooth projective surface over a discrete valuation ring $R$ of characteristic $(0,p > 0)$. If $E$ is a stable sheaf on the special fiber $X_\kappa$ of $X$ ($\kappa$ - residue field of $R$), then I think the deformation functor to consider here should live on the category of Artinian rings $A$ with residue field $\kappa$, which are not necessarily $\kappa$-algebras. For example, if one wants to consider $Def_E(R_n)$ where $R_n = R/\mathfrak m_R^n$ is the truncated ring $R$ for some $n \geq 2$.

Is there a similar description of $Def_E$ in mixed characteristic (or in this particular situation)?

My motivation to ask this (rather vague, sorry for that) question comes from the question: when two semistable families (or deformations) $\mathcal E, \mathcal E'$ on $X \otimes R_n$ define the same point in $M_X(R_n)$? In the situation, where $X$ is defined over a field, the answer (if we replace $R_n$ for example with $k[t]/t^n$) is given by the above description of $Def_E$: they are just isomorphic.

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