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Let $X$ be something. (smooth and projective variety over C are my assumptions) The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, where $\alpha$ ranges over the possible chern characters of the sheaves. (one could equivalently take hilbert polynomials)

What is nice about $M$ is that the stack parameterising coherent sheaves on $X$ with support of dimension $\leq r$ can be written as a union of $M_\alpha$. This implies that $M_{\leq r}$ is an open substack of $M$.

If now we take $D$ to be the stack parameterising complexes on $X$, that is objects of the bounded derived category of coherent sheaves, then the situation becomes more complicated. (see the paper by Lieblich for details) While I still believe that $D$ splits as a disjoint union according to the chern character of the complex*, the deal with supports is a whole other story. Usually, one defines the support of a complex to be the union of the supports of its cohomology sheaves. For example, if one takes $X = P^1$, and considers $E = O \oplus O(-1)[1]$, then the support of $E$ is certainly one-dimensional, but its chern character does not see this.

So my question is: is the subprestack $D_{\leq r}$ of $D$ parameterising complexes with support in dimension at most $r$ a substack of $D$? (in other words, does it satisfy descent?) Is it a locally closed substack?

I'm happy to make further assumptions. For example we could take $A$ to be a tilt via a torsion pair in $Coh(X)$, and we consider the stack parameterising objects in $A$.


*Is it written down anywhere that the chern character of a (relatively perfect) family of complexes is locally constant over the base?

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