Shifted smooth complex pullback is exact for the middle perverse t structure

Preservation of the upper aisle. Choose the finite compatible smooth target stratification of [F1] adapted to $A$, with the compatible refinements of [F3] available on affine charts, and work with the connected components $T$ of the full inverse images $a^{-1}(S)$. Smooth base change makes each nonempty such component smooth of dimension ${\dim }_{\mathbb{C}}S+d$, and smooth local charts make this the product pullback stratification. It has the frontier and Whitney conditions locally because the target stratification does and a smooth chart adds a ball factor; finitely many algebraic components give a finite global stratification. This choice matters: a further arbitrary source refinement need not have the full relative dimension. Write $e={\dim }_{\mathbb{C}}S$ and $a_T:T\to S$. Composition of inverse images gives $i_T^{\ast }{a}^{\ast }A=a_T^{\ast }{i}_S^{\ast }A$, and exactness gives $H^q(i_T^{\ast }{a}^{\ast }A[d])=a_T^{\ast }{H}^{q+d}(i_S^{\ast }A)$. For $A$ in the upper aisle this vanishes whenever $q+d>-e$, or $q>-(e+d)$, precisely the source support inequality. Independence of compatible refinement [F1] proves upper-aisle preservation on $D_c^b$.

Preservation of the lower aisle. The ordinary six-operation construction [F2] gives $D_X{a}^{\ast }\cong a^{!}{D}_Y$ and the complex-orientation normalization $a^{!}\cong a^{\ast }[2d]$, where the relative dimension is $d$. Consequently $D_X(a^{\ast }A[d])\cong a^{\ast }{D}_{Y}A[d]$. The ordinary middle-perverse core of [F1] identifies the lower aisle as the duality image of the upper aisle, and both the duality exchange of [F1] and the pullback comparison of [F2] are used here, so the first step proves preservation of the lower aisle as well. The orientation unit is nonzero over every field, exact finite-dimensional duality is field-general, and $[2d]$ is a cohomological shift rather than a scalar inversion; thus the support-and-duality argument applies to arbitrary $k$.

Truncation, cohomology and reflection. Applying the t-exact functor $a^{\ast }[d]$ of steps 1.1 and 2.1 to a truncation triangle produces the unique target truncation triangle, proving that it commutes with perverse truncation and perverse cohomology. If $a$ is surjective, inverse image is stalkwise conservative and reflects ordinary cohomological vanishings; applying the same dimension calculation as in step 1.1 to the support and dual support inequalities reflects both perverse inequalities. The dimension count is the same over every field, and no characteristic hypothesis enters, so the stated conservativity and reflection hold for arbitrary $k$.