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 $. References