Koszul resolutions and restriction to flat fibres

Koszul exactness. At a point of $Z(s)$ choose the frame in the hypothesis and let $K(a_1,\dots ,a_d;M)$ be the Koszul complex. For $d=0$ it is $M$ in degree zero. For $d>0$, by [L1] it is the mapping cone of multiplication by $a_d$ from $K(a_1,\dots ,a_{d-1};M)$ to itself, so its long exact homology sequence identifies $H_i(K(a_1,\dots ,a_d))$ with the kernel and cokernel of the map induced by $a_d$ on $H_{\ast }(K(a_1,\dots ,a_{d-1}))$; by induction $a_d$ acts on the only nonzero homology $H_0=M\mathrm{/}(a_1,\dots ,a_{d-1})M$ as a nonzerodivisor, so the only nonzero homology of the full complex is $H_0=M\mathrm{/}(a_1,\dots ,a_d)M$ and the complex resolves that quotient. At a point outside $Z(s)$ a component $a_j$ is a unit; if $e_j$ is the corresponding exterior generator, the homotopy $h(v)=a_j^{-1}{e}_j\wedge v$ satisfies $\mathrm{\partial }h+h\mathrm{\partial }=\mathrm{id}$ by the contraction differential, so the complex is exact there and resolves the zero stalk. The construction is canonical under change of frame, because exterior powers and contraction by $s$ are, so the local complexes glue to the global resolution ${\bigwedge }^d{Q}^{\vee }\to \cdots \to Q^{\vee }\to \mathcal{O}\to {\mathcal{O}}_{Z(s)}\to 0$; tensoring with a vector bundle $W$ preserves exactness and produces the resolution of $W{\mathrm{\mid }}_{Z(s)}$ twisted by the Koszul terms.

Restriction to a flat fibre. Let $0\to H_n\to \cdots \to H_0\to H\to 0$ be a finite locally free resolution and let $t$ be a Cartier equation acting injectively on $H$; since the $H_i$ are locally free, $t$ acts injectively on each of them. Tensoring with the two-term resolution $0\to \mathcal{O}\stackrel{t}{\to }\mathcal{O}\to \mathcal{O}\mathrm{/}(t)\to 0$ of the principal quotient and computing Tor, the identity ${\mathrm{Tor}}_1(H,\mathcal{O}\mathrm{/}(t))=\ker (t:H\to H)=0$ together with the snake lemma applied to the short exact sequences of complexes shows that ${\mathrm{Tor}}_i(H,\mathcal{O}\mathrm{/}(t))=0$ for all $i>0$, and the complex $H_{\bullet }\mathrm{/}t{H}_{\bullet }$ is exact with $H_0(H_{\bullet }\mathrm{/}t{H}_{\bullet })=H\mathrm{/}tH$. Hence the restricted complex $H_{\bullet }{\mathrm{\mid }}_{t=0}$ remains exact and resolves $H\mathrm{/}tH$; the essential extra hypothesis is injectivity on $H$: injectivity on the locally free terms alone does not suffice.

The deformation blowup. By [L3], $M$ is smooth and quasi-projective over the field, so every coherent sheaf on $M$ has a finite locally free resolution by [L2]; the strict transform $F$ is isomorphic to $X\times {\mathbb{P}}^1$ and meets the fibre at infinity in the section $[0:1]$ of $P=\mathbb{P}(N\oplus 1)$, while it is disjoint from $B={\mathrm{Bl}}_{X}Y$, since $B$ meets $P$ in $\mathbb{P}(N)$ and the strict transform of $X\times {\mathbb{P}}^1$ lies over $X\times \{\mathrm{\infty }\}$ inside the exceptional component. On the chart where the infinity fibre is Cartier with equation $t$, the ordinary fibre $Y$ has equation $t-1$, and both act injectively on the coherent sheaf $F_{\ast }{\mathrm{pr}}_X^{\ast }E$, since on its support $F\cong X\times {\mathbb{P}}^1$ they are parameter equations on the second factor and ${\mathrm{pr}}_X^{\ast }E$ is locally free there. This sheaf need not be locally free on $M$. Near its intersection with $P$ the component $B$ is absent, so the Cartier equation of $P$ agrees with that of the infinity fibre; step 1.2 therefore restricts a finite locally free resolution of $F_{\ast }{\mathrm{pr}}_X^{\ast }E$ exactly to the ordinary fibre and to $P$, where the restricted complex resolves the restrictions of the sheaf. Near $B$ the sheaf is zero, and a bounded exact complex of vector bundles resolving the zero sheaf is split exact by induction on its length, starting from the last cokernel; hence the restriction of the resolution to $B$ is an exact complex, as asserted.