I am trying to understand a particular case of this [question](https://mathoverflow.net/questions/122655/what-is-the-structure-of-the-stack-of-complexes-supported-in-dimension-less-than).


Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base change of a smooth and projective variety over C. In particular $X \to T$ is flat and proper.


Let $E$ be a coherent sheaf on $X$, which I do *not* assume to be flat over $T$ (and this is where the problems come in). Fix a number $r$.
If $t \in T$ is a (geometric) point of $T$, let $E\_t$ be the pullback to the fibre $X\_t$.
I would like to understand what the locus of points in $T$ such that $E\_t$ is supported in dimension at most $r$.


In other words, I want to characterise the subfunctor of $T$, taking an affine $U$ to the set of morphisms $U \stackrel{g}{\to} T$, such that the pullback sheaf $g^\*E$ is fiberwise supported in dimension at most $r$.




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The condition can be phrased with chern characters (or hilbert polynomials I guess), as if $r$ corresponds to codimension $k$ in $X\_t$ then I'm looking at the locus where $E\_t$ has chern character with zeros in degrees lower than $k$.
If $E$ were flat over $T$ then this locus would be open in $T$.


I suspect this condition for general $E$ should be closed or locally closed, but I can't seem to work it out. One thought would be to use the flattening stratification for $E$ and then impose the condition on chern characters, but that isn't really what we want.
The other thought would be to take some locus where $E$ itself has support of dimension at most $r + \dim T$. 
While it is true that, for any $d$, one can split $E$ in $0 \to E\_{\leq d} \to E \to F \to 0$, with $E$ support in dimension at most $d$, I'm not sure taking, say, the schematic image of the (schematic) support of $E\_{\leq r + \dim T}$ is the right thing.



The locus where a sheaf is supported in a certain dimension
Maybe I misunderstand something, but by Nakayama, the support of $E\_t$ is just the support of $E$ intersected with $X\_t$, and the support of $E$ is closed in $X$. Let $Z$ be the suppor of $E$. By Chevalley's semi-continuity theorem (EGA, IV.13.1.3), the set of $x\in X$ such that $\dim\_x Z\_t\le r$ ($x\mapsto t$) is open, so the set of $t\in T$ such that $\dim Z\_t\le r$ is constructible in $T$. 


This is true for any morphism of finite type $X\to T$ over a noetherian scheme $T$. 


