Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension

A cover by charts with Koszul resolutions. By [F1] the conormal sheaf $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ is locally free of rank $c$, so for every $y\in X$ the minimal number of generators of ${\mathcal{I}}_y$ is $c$ by Nakayama; hence two $c$-element systems of generators of ${\mathcal{I}}_y$ differ by an invertible matrix over ${\mathcal{O}}_{{\mathbb{P}}^N,y}$, and by [F1] one of them, the system $\mathbf{g}$ of that item, is a regular sequence at $y$. Fix a finite affine open cover of $X$ by charts $U=\mathrm{Spec}A\subseteq {\mathbb{P}}^N$ on which $\mathcal{I}{\mathrm{\mid }}_U=(f_1,\dots ,f_c)$ and $E{\mathrm{\mid }}_{X\cap U}\cong {\mathcal{O}}_{X\cap U}^{\oplus r}$; near each $y\in X$ first shrink an ambient affine neighbourhood until the conormal generators extend and the frame of $E$ persists on $X\cap U$, then take a finite subcover by quasi-compactness of $X$.

Hodge-star duality for the dual of a Koszul complex. Let $F=A^c$ have basis $e_1,\dots ,e_c$ with image $\mathbf{f}$ in $A$, and let $N$ be an $A$-module. For $0\le p\le c$ define ${\mathrm{\Theta }}^p:{\mathrm{Hom}}_A({\mathrm{\Lambda }}^{p}F,N)⟶{\mathrm{\Lambda }}^{c-p}F{\otimes }_A{\mathrm{\Lambda }}^c{F}^{\vee }{\otimes }_{A}N$ by the canonical perfect pairing ${\mathrm{\Lambda }}^{p}F{\otimes }_A{\mathrm{\Lambda }}^{c-p}F\to \det F$: if ${\mathrm{\Theta }}^p(\varphi )={\sum }_j{z}_j\otimes {\lambda }_j\otimes n_j$, then $\varphi (x)={\sum }_j{\lambda }_j(x\wedge z_j)n_j$ for every $x\in {\mathrm{\Lambda }}^{p}F$. By [F4] the wedge monomials form bases on both sides, so ${\mathrm{\Theta }}^p$ is an isomorphism. To check the differential, take $x\in {\mathrm{\Lambda }}^{p+1}F$ and $z\in {\mathrm{\Lambda }}^{c-p}F$. Since $x\wedge z=0$ in degree $c+1$, the Koszul differential's signed Leibniz rule, obtained from its coordinate formula in [F4], gives $0=\mathrm{\partial }(x\wedge z)=\mathrm{\partial }x\wedge z+(-1{)}^{p+1}x\wedge \mathrm{\partial }z$. Thus $\lambda (\mathrm{\partial }x\wedge z)=(-1{)}^p\lambda (x\wedge \mathrm{\partial }z)$, and the raw maps satisfy ${\mathrm{\Theta }}^{p+1}(\varphi \circ \mathrm{\partial })=(-1{)}^p\mathrm{\partial }{\mathrm{\Theta }}^p(\varphi )$. Set $s_p=(-1{)}^{p(p-1)\mathrm{/}2}$; then $s_{p+1}(-1{)}^p=s_p$, so the maps $s_p{\mathrm{\Theta }}^p$ commute with the differentials and give an isomorphism of complexes ${\mathrm{Hom}}_A(K(\mathbf{f};A{)}_{\bullet },N)\cong K(\mathbf{f};{\mathrm{\Lambda }}^c{F}^{\vee }{\otimes }_{A}N{)}_{c-\bullet }\mathrm{.}$ This also covers $c=0$, when both complexes have one term.

Extension by zero for modules. Give the abelian sheaf $j_{!}M$ of [F10] the ${\mathcal{O}}_Y$-action induced by restriction of functions to $U$: multiplication preserves sections with support closed in the ambient open. This defines $j_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}M$, with stalks $M_y$ on $U$ and zero off $U$, so it is exact by [F10]. The abelian adjunction restricts to module morphisms: the adjoint of an ${\mathcal{O}}_U$-linear map is ${\mathcal{O}}_Y$-linear on stalks in $U$, while outside $U$ its source stalk is zero. Thus $j_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}$ is left adjoint to module restriction. Given an injective $I$ upstairs and a monomorphism downstairs, exact $j_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}$ carries it to a monomorphism; the adjunction and injectivity solve the corresponding extension problem. Therefore $I{\mathrm{\mid }}_U$ is injective.

The Koszul complex on each chart is a resolution. Fix such a chart and let $\mathbf{f}=(f_1,\dots ,f_c)$. At a point $y\in X\cap U$ the systems $\mathbf{f}$ and $\mathbf{g}$ generate ${\mathcal{I}}_y$ and are minimal, so by step 1.1 the generator-matrix chain map of [F5] is an isomorphism $K(\mathbf{f};A_{{\mathfrak{p}}_y})\cong K(\mathbf{g};A_{{\mathfrak{p}}_y})$ and the right hand side is acyclic in positive degrees by [F5] since $\mathbf{g}$ is regular at $y$. Here $A_{{\mathfrak{p}}_y}\mathrm{/}(\mathbf{f})\mathrm{\ne }0$, so the local converse in [F5] also makes $\mathbf{f}$ a regular sequence at every point of $X\cap U$, as asserted in the Statement. At a point $y\in U\setminus X$ some $f_i$ is a unit, and by [F4] the complex $K(\mathbf{f};A_{{\mathfrak{p}}_y})$ is the tensor product of the contractible two-term complex of that unit with the Koszul complex of the remaining elements, hence is contractible and thus acyclic in positive degrees. The positive homology modules of the complex of finite free $A$-modules $K(\mathbf{f};A)$ are finitely generated, and a finitely generated module over the Noetherian ring $A$ all of whose localisations are zero is zero; hence $H_i(K(\mathbf{f};A))=0$ for $i>0$ and $H_0(K(\mathbf{f};A))=A\mathrm{/}(\mathbf{f})$ by [F5]. Thus $K(\mathbf{f};A)$ is a finite free resolution of $A\mathrm{/}(\mathbf{f})$, and its associated sheaf complex on $U$ resolves $i_{\ast }{\mathcal{O}}_X{\mathrm{\mid }}_U$; after the chosen frame of $E{\mathrm{\mid }}_{X\cap U}$, its $r$-fold direct sum resolves $i_{\ast }E{\mathrm{\mid }}_U$.

The local computation of sheaf Ext. Fix a chart $U$ as in step 2.1 and write ${\mathcal{K}}_p=\widetilde{K(\mathbf{f};A{)}_p^{\oplus r}}$ for the associated finite free ${\mathcal{O}}_U$-module. Choose an ${\mathcal{O}}_U$-injective resolution ${\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_U\to I^{\bullet }$, which exists by [F6], and form the first-quadrant double cochain complex $C^{p,q}=\mathcal{H}o{m}_{{\mathcal{O}}_U}({\mathcal{K}}_p,I^q),p,q\ge 0,$ whose horizontal differential is induced by ${\mathcal{K}}_{p+1}\to {\mathcal{K}}_p$ and whose vertical differential is induced by $I^q\to I^{q+1}$; every diagonal is finite because ${\mathcal{K}}_p=0$ for $p>c$. For fixed $q$ the augmented row $0\to \mathcal{H}om(i_{\ast }E{\mathrm{\mid }}_U,I^q)\to \mathcal{H}om({\mathcal{K}}_0,I^q)\to \cdots$ is exact as a sequence of sheaves: on each smaller open $V\subseteq U$, step 1.3 makes $I^q{\mathrm{\mid }}_V$ injective, so ${\mathrm{Hom}}_{{\mathcal{O}}_V}(-,I^q{\mathrm{\mid }}_V)$ sends the restricted resolution to an exact sequence; for fixed $p$ the augmented column $0\to \mathcal{H}om({\mathcal{K}}_p,{\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_U)\to \mathcal{H}om({\mathcal{K}}_p,I^0)\to \cdots$ is exact because ${\mathcal{K}}_p$ is finite free, so that $\mathcal{H}om({\mathcal{K}}_p,-)$ is exact. The two assembly lemmas [F7] (applied in the abelian category of ${\mathcal{O}}_U$-modules, whose arguments use only these two exactness statements) then make both edge complexes quasi-isomorphic to the total complex, so that $H^q(\mathcal{H}o{m}_{{\mathcal{O}}_U}({\mathcal{K}}_{\bullet },{\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_U))\cong H^q(\mathcal{H}o{m}_{{\mathcal{O}}_U}(i_{\ast }E{\mathrm{\mid }}_U,I^{\bullet }))=\mathcal{E}x{t}_{{\mathcal{O}}_U}^q(i_{\ast }E{\mathrm{\mid }}_U,{\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_U),$ the last term being $\mathcal{E}x{t}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^q(i_{\ast }E,{\omega }_{{\mathbb{P}}^N}){\mathrm{\mid }}_U$ because restriction to the open $U$ is exact and commutes with $\mathcal{H}om$ and, by step 1.3, preserves injectives: if $j_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}$ is exact and left adjoint to restriction, every extension problem for the restricted injective adjoints to an extension problem upstairs. Thus the restricted injective resolution computes the same sheaf Ext. This proves the local computation asserted in [F3] and in the statement.

Concentration in the top degree. If $H_i(K(\mathbf{f};N))=0$ for all $i>0$ then step 1.2 gives $H^q({\mathrm{Hom}}_A(K(\mathbf{f};A{)}_{\bullet },N))\cong H_{c-q}(K(\mathbf{f};{\mathrm{\Lambda }}^c{F}^{\vee }{\otimes }_{A}N)),$ which vanishes for $q\mathrm{\ne }c$ and equals ${\mathrm{\Lambda }}^c{F}^{\vee }{\otimes }_{A}N\mathrm{/}(\mathbf{f})N$ for $q=c$, by the description of $H_0$ in [F5]. Take $N={\omega }_{{\mathbb{P}}^N}(U{)}^{\oplus r}$. By [F2] the sheaf ${\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_U$ is invertible; [F9] identifies its sections with an $A$-module whose prime localisations are free of rank one, hence $N$ is flat. Thus $K(\mathbf{f};N)=K(\mathbf{f};A){\otimes }_{A}N$ is acyclic in positive degrees by the finite-free resolution of step 2.1 and flatness, with $H_0=N\mathrm{/}(\mathbf{f})N$. Since every ${\mathcal{K}}_p$ in step 3.1 is the associated sheaf of the finite free module $K(\mathbf{f};A{)}_p^{\oplus r}$, affine quasi-coherent equivalence [F9] identifies $\mathcal{H}o{m}_{{\mathcal{O}}_U}({\mathcal{K}}_p,{\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_U)$ with the associated sheaf of ${\mathrm{Hom}}_A(K(\mathbf{f};A{)}_p^{\oplus r},{\omega }_{{\mathbb{P}}^N}(U))$. Combining with step 3.1, the sheaf $\mathcal{E}x{t}^q(i_{\ast }E,{\omega }_{{\mathbb{P}}^N}){\mathrm{\mid }}_U$ vanishes for $q\mathrm{\ne }c$, while for $q=c$ it is canonically the associated sheaf of ${\mathrm{\Lambda }}^c{F}^{\vee }{\otimes }_A{\omega }_{{\mathbb{P}}^N}(U){\otimes }_A(A\mathrm{/}(\mathbf{f}){)}^{\oplus r}\mathrm{.}$ This module is killed by $(\mathbf{f})=\mathcal{I}(U)$, so its associated sheaf is the pushforward from $X\cap U$.

Identification with $E^{\vee }\otimes {\omega }_X$. The assignment $e_i\mapsto f_i<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\mathcal{I}}^2$ is a surjection of ${\mathcal{O}}_{X\cap U}$-modules $F{\otimes }_A{\mathcal{O}}_{X\cap U}\to \mathcal{I}\mathrm{/}{\mathcal{I}}^2$ between finite locally free modules of the same rank $c$, hence an isomorphism by [F8] and [F1]; taking $c$-th exterior powers and dualising gives a canonical isomorphism ${\mathrm{\Lambda }}^c{F}^{\vee }{\otimes }_A{\mathcal{O}}_{X\cap U}\cong \det (\mathcal{I}\mathrm{/}{\mathcal{I}}^2{)}^{\vee }{\mathrm{\mid }}_{X\cap U}$. Substituting this into step 4.1, and using $i^{\ast }{\omega }_{{\mathbb{P}}^N}={\omega }_{{\mathbb{P}}^N}{\mathrm{\mid }}_{X\cap U}$ and $(A\mathrm{/}(\mathbf{f}){)}^{\oplus r}$ being the local frame of $E^{\vee }{\mathrm{\mid }}_{X\cap U}$, gives a canonical isomorphism $\mathcal{E}x{t}^c(i_{\ast }E,{\omega }_{{\mathbb{P}}^N}){\mid }_U\cong i_{\ast }(E^{\vee }\otimes i^{\ast }{\omega }_{{\mathbb{P}}^N}\otimes \det (\mathcal{I}\mathrm{/}{\mathcal{I}}^2{)}^{\vee }){\mid }_U,$ and by the adjunction formula [F2] the right hand side is $i_{\ast }(E^{\vee }\otimes {\omega }_X){\mathrm{\mid }}_U$.

Gluing and vanishing in the remaining degrees. Near each point of $X$ in the overlap, write the second regular-generator tuple as ${\mathbf{f}}^{\mathrm{\prime }}=B\mathbf{f}$. Both tuples give bases of $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$, so $\bar{B}$ is invertible modulo $\mathcal{I}$; its determinant is a unit after shrinking an ambient neighbourhood of that point. On this smaller neighbourhood the generator-matrix chain map of [F5] is an isomorphism. In top degree it acts by $\det B$, while dualizing the Koszul complex acts by the corresponding dual determinant; the identification in step 5.1 uses exactly the induced change of the conormal basis. Different lifts $B$ of the same conormal change have the same determinant modulo $\mathcal{I}$ and thus induce the same map on the top Ext module, which is killed by $\mathcal{I}$. A change of the chosen frame of $E$ similarly acts on the dual Koszul complex by the dual transition matrix and agrees with the transition of $E^{\vee }$. Hence the local isomorphisms of steps 4.1–5.1 agree after shrinking around every point of an overlap and therefore agree on the overlap itself; both sheaves vanish off $X$, so they glue to a global isomorphism $\mathcal{E}x{t}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^c(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})\cong i_{\ast }\left(E^{\vee }{\otimes }_{{\mathcal{O}}_X}{\omega }_X\right)\mathrm{.}$ The same local computation gives $\mathcal{E}x{t}^q(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})=0$ for $q\mathrm{\ne }c$, on a cover of $X$ and on its open complement.

Naturality, degenerate cases and the Axiom of Choice. An ${\mathcal{O}}_X$-linear map $u:E\to E^{\mathrm{\prime }}$ between finite locally free modules induces a map from the Koszul resolution for $E$ to that for $E^{\mathrm{\prime }}$, and hence, after applying $\mathcal{H}om(-,{\omega }_{{\mathbb{P}}^N})$, a map in the reverse direction between the complexes of steps 3.1, 4.1 and 5.1; the Hodge-star isomorphism, the identification of ${\mathrm{\Lambda }}^c{F}^{\vee }$ with $\det (\mathcal{I}\mathrm{/}{\mathcal{I}}^2{)}^{\vee }$ and the gluing of step 6.1 are natural, so in degree $c$ the resulting map from $\mathcal{E}x{t}^c(i_{\ast }{E}^{\mathrm{\prime }},{\omega }_{{\mathbb{P}}^N})$ to $\mathcal{E}x{t}^c(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})$ is $i_{\ast }(u^{\vee }\otimes {\mathrm{id}}_{{\omega }_X})$, which is the asserted contravariant naturality in $E$. If $X=\mathrm{\varnothing }$, then $i_{\ast }E=0$ and both sides of the claimed isomorphism are zero. For nonempty $X$ and $c=0$, the closed immersion identifies $X$ with ${\mathbb{P}}^N$ because $X$ is reduced and has full-dimensional closed support in the irreducible projective space; the empty Koszul complex is ${\mathcal{O}}_U$ in degree zero by [F4], $\det \mathcal{N}$ is trivial, ${\omega }_X={\omega }_{{\mathbb{P}}^N}$, and the conclusion reads $\mathcal{E}x{t}^0(E,{\omega }_{{\mathbb{P}}^N})=\mathcal{H}om(E,{\omega }_{{\mathbb{P}}^N})=E^{\vee }\otimes {\omega }_{{\mathbb{P}}^N}$. Finally, the Axiom of Choice [A1] is assumed in the statement and is consumed exactly through the conormal supplier [F1], the enough-injectives theorem [F6] that supplies the injective resolution in the definition of sheaf Ext, and the Koszul-regularity input [F5]; the argument itself selects only finitely many charts, regular systems and frames on a fixed finite cover, and no further choice is made. This proves the statement.