Relative regularity, generation, and arbitrary base change

Work on an affine open of $T$ and make the complex in [F1] finite free locally. At any point its residue-field complex has zero positive cohomology by regularity. At the highest nonzero positive degree, exactness modulo the maximal ideal makes the incoming differential surjective; an invertible maximal minor splits off that final term together with an equal direct summand of the preceding term as a contractible pair. Repeat downwards. The remaining complex is a finite free module in degree zero. Every splitting survives arbitrary tensoring, so this description computes $H^0$ after every algebra change and gives zero higher cohomology. The descriptions agree through the canonical cohomology comparison and glue.

The cokernel $C$ of evaluation is of finite type. Formation of $H^0$ commutes with residue-field extension by step 1.1, and each fibre evaluation is onto by [F1]. At a stalk above $t$, therefore $C_x\mathrm{/}{\mathfrak{m}}_t{C}_x=0$. Since ${\mathfrak{m}}_t{\mathcal{O}}_x$ lies in the maximal ideal of the local ring ${\mathcal{O}}_x$, Nakayama gives $C_x=0$. Evaluation is onto. The rank equals $h^0(F_t(r))=\chi (F_t(r))$ since all higher cohomology vanishes.

The kernel $K$ is base-flat: tensor the exact sequence by any base module; the Tor sequence and flatness of $E,F$ show injectivity at the left and preservation of exactness, which is precisely flatness of $K$. Apply steps 1.1–2.1 to each sheaf and take the long exact direct-image sequence; $R^1{\pi }_{\ast }K(r)=0$ gives the stated short exact sequence. The locally free quotient makes it split locally, hence every base change preserves it. The canonical cohomology comparisons identify the pulled-back sequence with that of the pulled-back sheaves.