Ideal-power dévissage and compatibility with coherent pushforward

Both categories are essentially small: in the supported perfect-category realization used by [F3], ${\mathcal{A}}_T$ is the heart of the essentially small perfect category, and $\mathrm{C}\mathrm{o}\mathrm{h}(T)$ is equivalent to its full subcategory ${\mathcal{B}}_T$ by [F1]. Choose equivalent small presentations under [A1]. The exactness, subobject/quotient closure and finite-filtration hypotheses of [F2] are supplied by [F1], and essential smallness was just checked. Thus exact inclusion ${\mathcal{B}}_T\hookrightarrow {\mathcal{A}}_T$ induces an isomorphism. Precompose with the exact equivalence $i_{\ast }$ of [F1]. This is precisely $d$, and its inverse is the displayed quotient sum by [F2]. The theorem’s independence statement also shows that any exponent which annihilates $M$ gives the same answer.

On a generator $[F]\in G_0(T)$, $d$ sends it to $[i_{\ast }F]$ in $G_0({\mathcal{A}}_T)$. Forgetting support sends this to the same coherent sheaf class in $G_0(U)$, which is the definition of closed coherent pushforward. Since the groups are generated by these classes, this proves the first square. By [F3], their images in the supported and global spectral groups are exactly the object classes of $i_{\ast }F[0]$ and the actual inclusion sends one to the other. Thus the square after the regular perfect comparisons also commutes.