Perfect dg modules have finite total Ext

For representables $h_a=\mathcal{T}(-,a)$ and $h_b=\mathcal{T}(-,b)$, evaluation at $1_a$ identifies the dg Hom complex $\underset{}{\mathrm{Hom}}(h_a,h_b)$ with $\mathcal{T}(a,b)$. Indeed, a module map $\varphi :h_a\to h_b$ is determined by ${\varphi }_a(1_a)$, and for any $\gamma :c\to a$ naturality forces ${\varphi }_c(\gamma )={\varphi }_a(1_a)\circ \gamma$; conversely composition by any homogeneous element of $\mathcal{T}(a,b)$ defines such a map and respects the differential. The standard projective presentation in [F1] computes derived Hom by this dg Hom complex, so ${⨁}_i{\mathrm{Ext}}^i(h_a,h_b)$ is finite-dimensional.

Fix $x=h_a$. Shifts and finite sums of the second argument preserve finite total Ext. If $y_1\to y_2\to y_3\to y_1[1]$ is a triangle, applying ${\mathrm{Hom}}_{\mathcal{D}(\mathcal{T})}(h_a,-[i])$ gives an exact sequence; each ${\mathrm{Ext}}^i(h_a,y_2)$ is a subquotient of the neighboring finite groups, and outside their finite degree range it is zero. Direct summands give subgroups. Induction through the finite construction of a perfect $y$ proves the claim for $(h_a,y)$.

For fixed perfect $y$, the contravariant long exact Hom sequence gives the same closure under shifts, finite sums, cones, and retracts in the first argument. Induction through the finite construction of $x$ extends step 2.1 to every pair of perfect modules. The finite support in degrees is preserved at each finite construction step, proving the statement without a choice of resolutions or simultaneous representatives.