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 $\text {Spec}(R)$ 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 an affine Noetherian scheme $\mathbb {X}$, we get the following: 1) if the sequence $(\mathcal {U}[-n]\cap \mathcal {D}^{\leq 0}(\mathbb {X}))_{n\in \mathbb {N}}$ is stationary, 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-structure 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 $j>m$.