Constant families of t-structures on derived categories of coherent sheaves

TL;DR

This paper extends the construction of constant t-structures on derived categories of coherent sheaves to more general, possibly non-smooth, schemes, and explores their invariance properties under autoequivalences and local conditions.

Contribution

It generalizes the construction of constant t-structures to non-Noetherian and non-smooth schemes using unbounded derived categories, and studies their invariance and classification.

Findings

Every bounded nondegenerate t-structure with Noetherian heart is autoequivalence-invariant.

On smooth schemes, local t-structures are precisely the perverse t-structures.

The construction applies without smoothness or quasiprojectivity assumptions.

Abstract

We generalize the construction given in math.AG/0309435 of a "constant" t-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a t-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity assumptions on $X$ and $S$ and work with t-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced t-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of \cite{AJS}. As an application of the "constant" t-structures techniques we prove that every bounded nondegenerate t-structure on $D(X)$ with Noetherian heart is invariant under the action of a connected group of autoequivalences of $D(X)$. Also, we show that if $X$ is smooth then the only local t-structures on $D(X)$, i.e., those for which there exist compatible t-structures on $D(U)$ for all open $U\subset X$, are the perverse t-structures considered in math.AG/0005152.