Hartshorne's question on cofinite complexes

Let $\mathfrak{a}$ be a proper ideal of a commutative noetherian ring $R$ and $d$ a positive integer. We answer Hartshorne’s question on cofinite complexes completely in the cases $\mathrm{d}\mathrm{i}\mathrm{m}R=d$ or $\mathrm{d}\mathrm{i}\mathrm{m}R\mathrm{/}\mathfrak{a}=d-1$ or $\mathrm{a}\mathrm{r}\mathrm{a}(\mathfrak{a})=d-1$, show that if $d\le 2$ then an $R$-complex $X\in {\mathrm{D}}_{⊏}(R)$ is $\mathfrak{a}$-cofinite if and only if each homology module ${\mathrm{H}}_i(X)$ is $\mathfrak{a}$-cofinite; if $\mathfrak{a}$ is a perfect ideal and $R$ is regular local with $d\le 2$ then an $R$-complex $X\in \mathrm{D}(R)$ is $\mathfrak{a}$-cofinite if and only if ${\mathrm{H}}_i(X)$ is $\mathfrak{a}$-cofinite for every $i\in \mathbb{Z}$; if $d\ge 3$ then for an $R$-complex $X$ of $\mathfrak{a}$-cofinite $R$-modules, each ${\mathrm{H}}_i(X)$ is $\mathfrak{a}$-cofinite if and only if ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^j(R\mathrm{/}\mathfrak{a},\mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}{d}_i)$ are finitely generated for $j\le d-2$. We also study cofiniteness of local cohomology ${\mathrm{H}}_{\mathfrak{a}}^i(X)$ for an $R$-complex $X\in {\mathrm{D}}_{⊏}(R)$ in the above cases. The crucial step to achieve these is to recruit the technique of spectral sequences.