# The locus where a sheaf is supported in a certain dimension

I am trying to understand a particular case of this question.

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 $r$.

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.

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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 $T$.
As the fiber dimension doesn't change by base change, if there is a scheme which represents your functor, it will be reduced with underlying space equal to the set of $t\in T$ such that $\dim Z_t\le r$. If the latter is locally closed, then endow it with the structure of a reduced subscheme of $T$ and you get a scheme representing your functor. Otherwise I don't know, probably your functor is then not representable. – Qing Liu Feb 25 '13 at 12:13
sorry for the barrage of comments. Your reference got me reading the wonderful book by Goertz-Wedhorn again, and Corollary 14.113 applied to Z = supp E --> T, seems to suggest that the locus of where the fibre $Z_t$ is of dimension $\geq r+1$ is closed. My foggy brain right now concludes that then the locus I'm interested in is in fact open. – Jacob Bell Feb 25 '13 at 14:51
Yes because in your special case $X\to T$ is proper. As for the reducedness (in the general case), apply your functor to all $U$ affine open in the (moduli scheme)$_{\mathrm{red}}$. – Qing Liu Feb 25 '13 at 17:20