Hearts of t-structures in the derived category of a commutative Noetherian ring

Abstract

Let $R$ be a commutative Noetherian ring and let $\mathcal{D}(R)$ be its (unbounded) derived category. We show that all compactly generated t-structures in $\mathcal{D}(R)$ associated to a left bounded filtration by supports of Spec$(R)$ have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in $\mathcal{D}(R)$ whose heart is a module category. As geometric consequences for a compactly generated t-structure $(\mathcal{U},{\mathcal{U}}^{\perp }[1])$ in the derived category $\mathcal{D}(\mathbb{X})$ of a Noetherian scheme $\mathbb{X}$, we get the following: 1) If the sequence $(\mathcal{U}[-n]\cap {\mathcal{D}}^{\le 0}(\mathbb{X}){)}_{n\in \mathbb{N}}$ is stationnary, then the heart $\mathcal{H}$ is a Grothendieck category; 2) If $\mathcal{H}$ is a module category, then $\mathcal{H}$ is always equivalent to $\text{Qcoh}(\mathbb{Y})$, for some affine subscheme $\mathbb{Y}\subseteq \mathbb{X}$; 3) If $\mathbb{X}$ is connected, then: a) when ${\bigcap }_{k\in \mathbb{Z}}\mathcal{U}[k]=0$, the heart $\mathcal{H}$ is a module category if, and only if, the given t-structure is a translation of the canonical t-estructure in $\mathcal{D}(\mathbb{X})$; b) when $\mathbb{X}$ is irreducible, the heart $\mathcal{H}$ is a module category if, and only if, there are an affine subscheme $\mathbb{Y}\subseteq \mathbb{X}$ and an integer $m$ such that $\mathcal{U}$ consists of the complexes $U\in \mathcal{D}(\mathbb{X})$ such that the support of $H^j(U)$ is in $\mathbb{X}\setminus \mathbb{Y}$, for all $j>m$.

arXiv commentThis version is a modification of the previous one following the suggestions of the referee