Local-to-global Ext collapse for a regular immersion

The functors and their derived objects. Let $F=\mathcal{H}o{m}_{{\mathcal{O}}_{{\mathbb{P}}^N}}(i_{\ast }E,-)$ and $G=\mathrm{\Gamma }({\mathbb{P}}^N,-)$. By [F1] the functor $F$ is additive and left exact, and by [F10] the global-sections functor $G$ is additive and left exact; the composite $GF$ sends an ${\mathcal{O}}_{{\mathbb{P}}^N}$-module $\mathcal{M}$ to ${\mathrm{Hom}}_{{\mathcal{O}}_{{\mathbb{P}}^N}}(i_{\ast }E,\mathcal{M})$, because global sections of the internal Hom are the Hom group. With the supplied injective resolution datum of [F6] in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_{{\mathbb{P}}^N})$, the definitions of [F2] identify $R^{q}F(\mathcal{M})=\mathcal{E}x{t}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^q(i_{\ast }E,\mathcal{M})$ and $R^q(GF)({\omega }_{{\mathbb{P}}^N})={\mathrm{Ext}}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^q(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})$ for every $q\ge 0$.

Extension by zero for $\mathcal{O}$-modules and its adjunction. Let $o:W\hookrightarrow T$ be an open immersion of ringed spaces and let $\mathcal{G}$ be an ${\mathcal{O}}_W$-module. Define $o_{!}\mathcal{G}$ by the formula of [F7] applied to the underlying abelian sheaf, that is $(o_{!}\mathcal{G})(V)=\{s\in \mathcal{G}(V\cap W):\mathrm{Supp}(s)\text{ is closed in }V\}$ for open $V\subseteq T$, with the ${\mathcal{O}}_T(V)$-module structure induced by the ring map ${\mathcal{O}}_T(V)\to {\mathcal{O}}_W(V\cap W)$. The support condition is stable under multiplication by functions and compatible with restrictions, so $o_{!}\mathcal{G}$ is a sheaf of ${\mathcal{O}}_T$-modules whose underlying abelian sheaf is exactly the extension by zero of the underlying abelian sheaf of $\mathcal{G}$. Consequently $o_{!}$ is exact on $\mathcal{O}$-modules: the forgetful functor from ${\mathcal{O}}_T$-modules to abelian sheaves preserves kernels and cokernels, so a short exact sequence of $\mathcal{O}$-modules has a short exact underlying sequence of abelian sheaves, exact by [F7]. The transposition of [F7] preserves $\mathcal{O}$-linearity in both directions: it sends an ${\mathcal{O}}_T$-linear morphism to the family of its components over opens inside $W$, which are ${\mathcal{O}}_W$-linear, and it sends an ${\mathcal{O}}_W$-linear morphism $\mathrm{\Psi }$ to the morphism whose section over an open $V\subseteq T$ is the gluing of $\mathrm{\Psi }(s{\mathrm{\mid }}_{V\cap W})$ with the zero sections near $V\setminus W$, which is ${\mathcal{O}}_T(V)$-linear because on $V\cap W$ it is the ${\mathcal{O}}_W(V\cap W)$-linear map $\mathrm{\Psi }$ and near $V\setminus W$ both sides vanish. Hence there is a natural bijection ${\mathrm{Hom}}_{{\mathcal{O}}_T}(o_{!}\mathcal{G},\mathcal{H})\cong {\mathrm{Hom}}_{{\mathcal{O}}_W}(\mathcal{G},\mathcal{H}{\mathrm{\mid }}_W)$. Finally, the transpose of the identity of $\mathcal{G}=\mathcal{H}{\mathrm{\mid }}_W$ is the ${\mathcal{O}}_T$-linear map $u_{\mathcal{H}}:o_{!}(\mathcal{H}{\mathrm{\mid }}_W)\to \mathcal{H}$ that glues a section $s\in \mathcal{H}(V\cap W)$ with closed support in $V$ to the zero sections on a cover of $V$ by neighbourhoods of the points of $V\setminus W$; its section maps are injective because a section of $\mathcal{H}$ over $V$ restricts to its given values on $V\cap W$, so $u_{\mathcal{H}}$ is a monomorphism.

Calculate the local comparison signs. Regard a homological Koszul resolution $K_p$ as $K^{-p}$, with differential $\mathrm{\partial }$. The classical dual differential in degree $p$ is $h(\varphi )=\varphi \mathrm{\partial }$, whereas the cochain Hom differential into a module in degree zero is $(-1{)}^{p+1}\varphi \mathrm{\partial }$. With ${\sigma }_0=1$, the recurrence ${\sigma }_{p+1}=(-1{)}^{p+1}{\sigma }_p$ gives ${\sigma }_p=(-1{)}^{p(p+1)\mathrm{/}2}$. This also fixes the injective comparison: in bidegree $(p,q)$, the classical mixed complex $\mathrm{Hom}(K_p,I^q)$ has total differential $h+(-1{)}^{p}v$, while the cochain Hom complex has $v+(-1{)}^{p+q+1}h$. Multiplication by ${\sigma }_p(-1{)}^{pq}$ intertwines both differentials, equals ${\sigma }_p$ on the projective edge, and equals $1$ on the injective edge. Thus [F11] carries a raw degree-$c$ Koszul cochain to ${\sigma }_c$ times its cochain-derived representative. For the Hodge identification, let ${\mathrm{\Theta }}^p(\varphi )$ be characterized by $\varphi (x)=\lambda (x\wedge z)$ in the determinant pairing. The Koszul Leibniz rule on $x\wedge z=0$ gives ${\mathrm{\Theta }}^{p+1}(\varphi \mathrm{\partial })=(-1{)}^p\mathrm{\partial }{\mathrm{\Theta }}^p(\varphi )$. Hence $s_p{\mathrm{\Theta }}^p$ with $s_p=(-1{)}^{p(p-1)\mathrm{/}2}$ is a chain map, and in degree $c$ it sends the raw top cochain to $s_c$ times its determinant frame. This is the local map constructed in the proof of [F3], and its scalar depends only on $c$, so it survives restriction and changes of generators.

Injective modules restrict to injective modules, and internal Hom into an injective is flasque. (i) Let $I$ be an injective ${\mathcal{O}}_T$-module and let $W\subseteq T$ be open. Then $I{\mathrm{\mid }}_W$ is injective in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_W)$: given a monomorphism $\alpha :\mathcal{A}\rightarrowtail \mathcal{B}$ of ${\mathcal{O}}_W$-modules and a morphism $\beta :\mathcal{A}\to I{\mathrm{\mid }}_W$, step 1.2 transposes $\beta$ into a morphism ${\beta }^{\mathrm{♯}}:o_{!}\mathcal{A}\to I$, the morphism $o_{!}\alpha$ is a monomorphism because $o_{!}$ is exact, injectivity of $I$ extends ${\beta }^{\mathrm{♯}}$ over $o_{!}\alpha$ to $o_{!}\mathcal{B}\to I$, and transposing back gives $\mathcal{B}\to I{\mathrm{\mid }}_W$ whose composite with $\alpha$ is $\beta$ by the functoriality of the transposition. (ii) Let $\mathcal{F}$ be an ${\mathcal{O}}_T$-module and let $U\subseteq V\subseteq T$ be open. A section $\varphi \in {\mathrm{Hom}}_{{\mathcal{O}}_U}(\mathcal{F}{\mathrm{\mid }}_U,I{\mathrm{\mid }}_U)$ transposes by step 1.2, applied to the open immersion $m:U\hookrightarrow V$, to a morphism $m_{!}(\mathcal{F}{\mathrm{\mid }}_U)\to I{\mathrm{\mid }}_V$. The monomorphism used to extend it by injectivity of $I{\mathrm{\mid }}_V$ is $u_{\mathcal{F}{\mathrm{\mid }}_V}:m_{!}(\mathcal{F}{\mathrm{\mid }}_U)\rightarrowtail \mathcal{F}{\mathrm{\mid }}_V$ from step 1.2. The extension is a morphism $\tilde{\varphi }:\mathcal{F}{\mathrm{\mid }}_V\to I{\mathrm{\mid }}_V$; restricting it to opens inside $U$ recovers $\varphi$, because there $m_{!}$ and $u_{\mathcal{F}{\mathrm{\mid }}_V}$ are the identity. Hence every section of the abelian sheaf underlying $\mathcal{H}o{m}_{{\mathcal{O}}_T}(\mathcal{F},I)$ over $U$ extends to $V$, so that abelian sheaf is flasque. (iii) Taking $\mathcal{F}={\mathcal{O}}_T$ in (ii), and using that the canonical evaluation $\mathcal{H}o{m}_{{\mathcal{O}}_T}({\mathcal{O}}_T,I)\to I$, which sends a morphism to its value at the section $1$, is an isomorphism over every open, an injective ${\mathcal{O}}_T$-module is flasque as an abelian sheaf.

Derived global sections agree with sheaf cohomology. On $T={\mathbb{P}}^N$ and for every ${\mathcal{O}}_T$-module $\mathcal{M}$ one has $R^{q}G(\mathcal{M})\cong H^q(T,\mathcal{M})$ for all $q\ge 0$. Indeed, take the injective resolution $\mathcal{M}\to J^{\bullet }$ supplied by [F6]; each $J^p$ is flasque by step 2.1(iii), hence $H^q(T,J^p)=0$ for every $q>0$ by [F8], so the underlying abelian complex is a $\mathrm{\Gamma }(T,-)$-acyclic resolution of the underlying abelian sheaf of $\mathcal{M}$. With the Axiom of Dependent Choice, which holds by [F6], and with the class of all abelian sheaves on $T$, on which the supplied datum of [F6] is defined, the acyclic-resolution theorem [F9] gives $H^q(T,\mathcal{M})\cong H^q(\mathrm{\Gamma }(T,J^{\bullet }))$; the right hand side is $R^{q}G(\mathcal{M})$ computed from the same resolution, by [F10].

The acyclicity hypothesis of the spectral sequence. Let $I$ be an injective ${\mathcal{O}}_{{\mathbb{P}}^N}$-module. By step 2.1(ii) the abelian sheaf underlying $\mathcal{H}o{m}_{{\mathcal{O}}_{{\mathbb{P}}^N}}(i_{\ast }E,I)$ is flasque, so $H^q({\mathbb{P}}^N,\mathcal{H}o{m}_{{\mathcal{O}}_{{\mathbb{P}}^N}}(i_{\ast }E,I))=0$ for every $q>0$ by [F8], and step 3.1 identifies these groups with $R^{q}G(\mathcal{H}o{m}_{{\mathcal{O}}_{{\mathbb{P}}^N}}(i_{\ast }E,I))$. Hence $F$ carries injective objects to $G$-acyclic objects in the sense of the hypothesis of [F4].

The local-to-global spectral sequence. Apply [F4] to the pair of additive left exact functors $(F,G)$ of step 1.1: both source and target categories have enough injectives by [F6], the injective and Cartan-Eilenberg resolutions and comparison data are supplied under the Axiom of Choice [A1], and step 4.1 verifies the acyclicity hypothesis. The resulting natural first-quadrant spectral sequence is $E_2^{p,q}=R^{p}G(R^{q}F({\omega }_{{\mathbb{P}}^N}))⟹R^{p+q}(GF)({\omega }_{{\mathbb{P}}^N}),$ with differentials of bidegree $(r,1-r)$ and a finite decreasing filtration of the abutment. By steps 1.1 and 3.1 its $E_2$-page and abutment are $E_2^{p,q}=H^p({\mathbb{P}}^N,\mathcal{E}x{t}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^q(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})),E_{\mathrm{\infty }}^{p,q}\Rightarrow {\mathrm{Ext}}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^{p+q}(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})\mathrm{.}$

Degeneration. By [F3] the sheaf $\mathcal{E}x{t}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^q(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})$ vanishes unless $q=c$, where it is $i_{\ast }(E^{\vee }{\otimes }_{{\mathcal{O}}_X}{\omega }_X)$. Thus $E_2^{p,q}=0$ unless $q=c$. Each differential has bidegree $(r,1-r)$ for $r\ge 2$, changing the second index, so no differential can meet the single nonzero row and $E_2=E_{\mathrm{\infty }}$. In total degree $m\ge c$ the abutment filtration has just the graded piece $(m-c,c)$; its one-row edge $e_m$ is an isomorphism from ${\mathrm{Ext}}_{{\mathcal{O}}_{{\mathbb{P}}^N}}^m(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})$ to $H^{m-c}({\mathbb{P}}^N,\mathcal{E}x{t}^c(i_{\ast }E,{\omega }_{{\mathbb{P}}^N}))$. For $m<c$ every piece vanishes, hence the Ext group vanishes.

Normalize the edge orientation. Let $h:\mathcal{E}x{t}^c(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})\stackrel{\sim }{\to }{i}_{\ast }(E^{\vee }\otimes {\omega }_X)$ be the local Koszul/Hodge identification of [F3]. The ordered top Koszul cochain is carried by the Hodge chain map calculated in step 1.3 to $s_c=(-1{)}^{c(c-1)\mathrm{/}2}$ times its determinant frame. On each affine Koszul chart the classical-projective to derived/injective Ext comparison calculated in step 1.3 uses the factor ${\sigma }_c=(-1{)}^{c(c+1)\mathrm{/}2}$ in degree $c$; we make no global-projective-resolution claim for $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_{{\mathbb{P}}^N})$. Therefore we define the normalized collapse in total degree $c+j$ by $D_j:={\sigma }_c{H}^j(h)\circ e_{c+j}$, inserting this factor once rather than assuming it is implicit in the Grothendieck edge. Equivalently, its inverse sends the ordered determinant frame to ${\sigma }_c{s}_c=(-1{)}^c$ times the raw ordered top Koszul cochain. This convention is independent of $j$, commutes with restriction and changes of regular generators because both signs depend only on $c$, and is the one used for Gysin/Yoneda composition. Since ${\sigma }_c$ is a unit, $D_j$ is still a natural isomorphism.

Identification of the cohomology. The ${\mathcal{O}}_X$-module $E^{\vee }{\otimes }_{{\mathcal{O}}_X}{\omega }_X$ is finite locally free, hence quasi-coherent, so [F5] applied to the closed immersion $i$ gives a canonical isomorphism $H^j({\mathbb{P}}^N,i_{\ast }(E^{\vee }{\otimes }_{{\mathcal{O}}_X}{\omega }_X))\cong H^j(X,E^{\vee }{\otimes }_{{\mathcal{O}}_X}{\omega }_X)$. Composing the normalized map $D_j$ of step 7.1 with this pushforward comparison proves the isomorphism of the statement.

Naturality, boundary cases and the Axiom of Choice. A morphism $u:E\to E^{\mathrm{\prime }}$ of finite locally free ${\mathcal{O}}_X$-modules induces a morphism of functors $\mathcal{H}om(i_{\ast }{E}^{\mathrm{\prime }},-)\to \mathcal{H}om(i_{\ast }E,-)$ and hence, by the naturality assertions of [F4], a morphism of the spectral sequences of step 5.1 compatible with the abutments and their filtrations; the degeneration of step 6.1 and the fixed sign of step 7.1 are natural in these data, the identification of the row $q=c$ is the natural-in-$E$ isomorphism of [F3], and the isomorphism of [F5] is natural in the sheaf argument, so the isomorphism of step 8.1 is natural in $E$. The boundary cases are consistent: for $X=\mathrm{\varnothing }$ or $E=0$ both sides vanish; for $c=0$, [F3] reads $\mathcal{E}x{t}^0(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})\cong i_{\ast }(E^{\vee }\otimes {\omega }_X)$ and $\mathcal{E}x{t}^q=0$ for $q\mathrm{\ne }0$, so steps 6.1–8.1 give ${\mathrm{Ext}}^j(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})\cong H^j(X,E^{\vee }\otimes {\omega }_X)$ directly; for $j=0$ the statement reads ${\mathrm{Ext}}^c(i_{\ast }E,{\omega }_{{\mathbb{P}}^N})\cong H^0(X,E^{\vee }{\otimes }_{{\mathcal{O}}_X}{\omega }_X)$; for total degree below $c$, the Ext group vanishes by step 6.1. The Axiom of Choice [A1] is assumed in the statement and is used exactly through the injective-resolution data of [F6] for modules and for abelian sheaves, through the Dependent Choice instance of [F6] in step 3.1, and through the resolution and comparison data of [F4] used in step 5.1. This proves the lemma.