Sheaves of t-structures and valuative criteria for stable complexes

In this work more questions arise than answers given, for which of course we do not apologize. The core of this paper is concerned with the construction of a ``constant’’ t-structure on the bounded derived category of coherent sheaves $D(X\times S)$, with $X$ and $S$ smooth varieties, given a nondegenerate t-structure on $D(X)$ with noetherian heart. While we believe this construction and the methods involved should be generally useful in studying derived categories, we present here but one application: we prove a valuative criterion for separation and properness for the collection P(1) of stable objects of phase 1 under a numerical, locally finite and noetherian Bridgeland-Douglas stability condition $(Z,\text{cP})$ on $D(X)$, where $X$ is a smooth projective variety. As an immediate result, the number of long-exact sequences in the first named author’s output is no longer an embarrassment.