Relative rank-one classification of the singular differential kernel

Morita untwisting identifies $E$ with $L\otimes D$. Its evaluation map $ϵ:E\to L$ sends $\mathrm{\ell }\otimes P$ to $\mathrm{\ell }P(1)$. In an affine coordinate $x$, the unique normal form $P={\sum }_{j\ge 0}{a}_j{\mathrm{\partial }}_x^j$ shows that $K=\ker ϵ$ is locally $L\otimes D{\mathrm{\partial }}_x$. Its order-one part is canonically $L\otimes T_Z$: a vector field intrinsically annihilates $1$, so this identification does not choose a connection or coordinate. PBW shows $K=D_0{K}_{\le 1}$. By [F3] the two global sections of $K_{\le 1}={\mathcal{O}}_Z(1)$ generate $K$. Conversely every global section of $E$ maps to zero because $\mathrm{\Gamma }L=0$. Thus $D_0\mathrm{\Gamma }E=K$, proving the universal quotient.

Let $q:Z\times U\to Z$ be the other projection. It is affine, so $q_{\ast }Q$ is quasi-coherent, carries the relative $D_0$-action, and has no positive higher direct images by [F4]. Leray and singular vanishing [F2] give $H^i(Z\times U,Q)=0$ for $i>0$. Apply the same argument over every affine open subset of $U$; the sheafification of these cohomology groups is $R^i{r}_{\ast }Q$, proving its vanishing. In particular $r_{\ast }$ is exact on quasi-coherent product differential modules.

Put $T=L^{-1}\otimes Q$. By the regular equivalence in [F2], $T$ is generated over $D$ by its global sections, so $E^{(I)}\twoheadrightarrow Q$ for some set $I$. Since $\mathrm{\Gamma }Q=0$, each summand kills $\mathrm{\Gamma }E$ and hence $D_0\mathrm{\Gamma }E$. Step 1.1 gives $L^{(I)}\twoheadrightarrow Q$, or ${\mathcal{O}}_Z^{(I)}\twoheadrightarrow T$. The kernel $H$ is a quasi-coherent $D$-module. Exactness of $\mathrm{\Gamma }$ embeds $\mathrm{\Gamma }H$ into ${\mathbb{C}}^{(I)}$; its ${\mathfrak{s}\mathfrak{l}}_2$-action is trivial. The natural map ${\mathcal{O}}_Z{\otimes }_{\mathbb{C}}\mathrm{\Gamma }H\to H$ sends a constant horizontal section to that same section, and is $D$-linear because $H\subseteq {\mathcal{O}}_Z^{(I)}$. Its map on global sections is the identity of $\mathrm{\Gamma }H$. Exactness and regular conservativity in [F2] kill its kernel and cokernel, so it is an isomorphism. Thus $H={\mathcal{O}}_Z\otimes \mathrm{\Gamma }H$ and $T={\mathcal{O}}_Z\otimes V$, with $V=\mathrm{\Gamma }T$. Untwisting gives the claimed canonical evaluation isomorphism and trivial action. The converse follows from $\mathrm{\Gamma }L=0$.

For any central base ring $R$, push a module on $Z_R=Z\times \mathrm{Spec}R$ along the affine map to $Z$ and apply step 2.1. Multiplication by $R$ commutes with all vertical operators, so its natural evaluation is $R$-linear and reconstructs the sheaf as $L_R{\otimes }_{R}N$. Conversely this form has zero direct image by the Čech complex for $L$ tensored over $\mathbb{C}$ with $N$. Tensoring this form over $R$ with $R^{\mathrm{\prime }}$ proves the asserted scalar-extension compatibility without assuming flatness. If $R$ is noetherian, finite relative differential generators involve finitely many coefficients in $N$; these coefficients generate $N$ since vertical operators create no new coefficients. Conversely finitely generated $N$ gives a finite relative differential generating set, and PBW over the noetherian ring $R$ makes the relative differential ring left noetherian, proving coherence.

Suppose $r_{\ast }Q=0$. Then $\mathrm{\Gamma }(Z,q_{\ast }Q)=\mathrm{\Gamma }(Z\times U,Q)=0$. Step 2.1 applied to $q_{\ast }Q$ yields its canonical form $L\otimes V$. Multiplication by $\mathrm{\Gamma }(U,{\mathcal{O}}_U)$ and every operator in $\mathrm{\Gamma }(U,A)$ commute with the relative $D_0$ action, so naturality of step 2.1 makes $V$ a module over that base differential ring. Affine quasi-coherent sheafification identifies its sheaf with $N=r_{\ast }(L^{-1}\otimes Q)$ and the obtained isomorphism with $Q=L⊠N$. No tensor product is taken over a noncommutative ring: the exterior product is over $\mathbb{C}$ with the two commuting actions. Conversely $r_{\ast }(L⊠N)=0$ follows by the two-affine-open Čech calculation for $L$ tensored over $\mathbb{C}$ with $N$.

The $A$-module $N$ of step 3.2 is coherent. On each of the two affine fiber charts choose finitely many product-differential generators of $Q$. This is possible for a coherent differential module on an affine smooth chart: its quasi-coherent underlying module is determined by sections; local finite generators on a finite principal-open cover become global generators after clearing denominators, and differential operators localize by the Leibniz rule. Trivialize $L$ there and express each chosen generator as a finite sum of polynomials in the fiber coordinate times sections of $N$. Let $N_0$ be the base differential submodule generated by the finitely many coefficients from both charts. Product differential operators preserve $L⊠N_0$, so the chosen generators imply $L⊠N_0=L⊠N$. Evaluation at any fiber point gives $N_0=N$, proving finite generation. A TDO on a smooth affine chart is left noetherian: PBW gives the noetherian graded algebra $\mathrm{Sym}{T}_U$; lift finitely many graded generators and induct on order to generate any left ideal. Thus finite generation implies coherence.

Finally the map $p^{†}{p}_{\ast }(L^{-1}\otimes Q)\to L^{-1}\otimes Q$ is the intrinsic differential evaluation map. On each product chart it is the isomorphism of step 3.2. A change of torsor section acts on the fiber coordinate by its transition, and the chain rule intertwines relative differentiation; the base differential action transforms by the given TDO cocycle. Evaluation therefore agrees on overlaps and glues. Moreover $p_{\ast }{p}^{†}N=N$, since $p_{\ast }\mathcal{O}=\mathcal{O}$ on a ${\mathbb{P}}^1$-bundle, so $N$ is unique up to its canonical isomorphism. The argument uses choice only for the permitted bases and generating sets in [F2] and the inherited suppliers.