Inherited bounded t structure on supported perfect complexes

Kernels and cokernels of maps of coherent sheaves are coherent by [F1]. Their stalks outside $\mathrm{\mid }T\mathrm{\mid }$ are zero if the original sheaves have zero stalks there. The same holds for images and extensions by exactness of stalks. Hence ${\mathcal{A}}_T$ is an abelian full subcategory, closed under the indicated exact operations. Every sheaf of this category placed in degree zero has bounded coherent cohomology, so is perfect by regularity in [F1] and belongs to ${\mathcal{C}}_T$ by [F2].

For $E\in {\mathcal{C}}_T$, bounded coherent cohomology follows from [F1]. Each ambient truncation retains only a subset of those cohomology sheaves; it therefore remains bounded coherent, perfect by [F1], and supported on $\mathrm{\mid }T\mathrm{\mid }$. Thus the ambient t-structure restricts: its orthogonality and truncation sequences restrict to this full subcategory. It is bounded because each $E$ has finitely many nonzero cohomology sheaves. Its heart consists exactly of the objects $M[0]$ with $M\in {\mathcal{A}}_T$.

For a short exact sequence in ${\mathcal{A}}_T$, the mapping cone of $M^{\mathrm{\prime }}[0]\to M[0]$ is quasi-isomorphic to $M^{\mathrm{\prime }\mathrm{\prime }}[0]$ by its cohomology sequence. Mapping cones give actual cofibers in the supplied enhancement, so the sequence is a cofiber sequence there. No global locally free presentation was used.