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Let $R$ be a discrete valuation ring, $\{B_i\}_{i\in I}$ be an inductive system of $R$-algebras of finite type and $B$ the direct limit of the inductive system. Let $X$ be a regular, projective scheme, flat over $\mathrm{Spec}(R)$. Fix a polarization $H$ on $X$, under which there is a closed immersion $X \hookrightarrow \mathbb{P}^n_R$ for some integer $n$.

Denote by $H_i$ the pull-back of $H$ under the natural morphism $X \times_R \mathrm{Spec}(B_i) \to X$. Let $\mathcal{F}$ be a coherent sheaf on $X \times_R \mathrm{Spec}(B)$ and $\{\mathcal{F}_i\}_{i \in I}$ be a sequence of coherent sheaves on $X \times_R \mathrm{Spec}(B_i)$ for $i \in I$ satisfying:

1) For each $i \in I$, $\mathcal{F}_i$ is semi-stable under the polarization $H_i$

2) The pullback of $\mathcal{F}_i$ under the natural morphism $X \times_R \mathrm{Spec}(B) \to X \times_R \mathrm{Spec}(B_i)$, arising from the natural morphism $B_i \to B$, is isomorphic to $\mathcal{F}$.

The question is: Is $\mathcal{F}$ semi-stable under the polarization $H$?

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    $\begingroup$ This question appears to be written backwards. Surely you meant to ask if $F_i$ is semistable with respect to $H_i$ for large $i$ when $F$ is semistable with respect to $H$. The opposite direction in which the question is written now is a special case of asking if semistability is preserved under base change, which is trivial. If you gave motivation for the question then it would be more apparent what you intended to ask. For what I am predicting you meant to ask, think through the more "classical" analogue for semistability of a proper flat family of curves (hint: constructibility!). $\endgroup$
    – grghxy
    Commented Sep 27, 2015 at 16:43
  • $\begingroup$ @grghxy Thank you. Could you give a reference for semi-stability preserved under base change? $\endgroup$
    – Ron
    Commented Sep 27, 2015 at 16:53
  • $\begingroup$ Preservation by base change is a tautology by the definition of "semistable sheaf" in the relative setting. What definition are you using? There is no reference or argument to give; it is baked into the definition. The real content is that fibral semistability cuts out a reasonable locus in the base for an $S$-flat coherent sheaf on a smooth proper $S$-scheme (and perhaps in your setup you meant to assume $X$ is $R$-smooth, not merely regular, the latter a much weaker condition): it is an open subset of the base; see Theorem 2.3.1 in the book "The geometry of moduli spaces of sheaves". $\endgroup$
    – grghxy
    Commented Sep 27, 2015 at 20:30
  • $\begingroup$ @grghxy I definitely do not want to assume smoothness. I am not using a relative version of semistability. There is an absolute definition of semi-stability for coherent sheaves on projective schemes (mind not varieties) in Definition $1.2.4$ page $11$ of "The geometry of moduli spaces of sheaves" Using this definition, I do not get the tautology that you mention. $\endgroup$
    – Ron
    Commented Sep 27, 2015 at 20:41
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    $\begingroup$ In section 1.2 the projective scheme is over a field, so you have not said what definition you are using when the base is not a field. Corollary 1.3.8 shows that the notion over a field is insensitive to ground field extension (in both directions). In general, by definition a coherent sheaf on a projective scheme $X$ over a noetherian base $S$ is "semistable" when it is $S$-flat and semistable on every fiber (equivalent to work with geometric fibers, by Cor. 1.3.8). By Cor. 1.3.8 this notion is preserved by base change to another noetherian scheme. What is unclear? $\endgroup$
    – grghxy
    Commented Sep 27, 2015 at 22:55

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