Adjunction for a smooth closed subvariety

The conormal sequence and its ranks. By [F1] the sequence $0⟶\mathcal{A}⟶\mathcal{B}⟶\mathcal{C}⟶0,\mathcal{A}=\mathcal{I}\mathrm{/}{\mathcal{I}}^2,\mathcal{B}=i^{\ast }{\mathrm{\Omega }}_{{\mathbb{P}}^N\mathrm{/}k}^1,\mathcal{C}={\mathrm{\Omega }}_{X\mathrm{/}k}^1,$ is an exact sequence of finite locally free ${\mathcal{O}}_X$-modules of ranks $c$, $N$ and $n$. In particular every point of $X$ has an affine open neighbourhood $U=\mathrm{Spec}A$ over which all three restrictions $\mathcal{A}{\mathrm{\mid }}_U,\mathcal{B}{\mathrm{\mid }}_U,\mathcal{C}{\mathrm{\mid }}_U$ are free $A$-modules of ranks $c,N,n$: a finite intersection of trivialising opens for the three locally free modules, shrunk to an affine open.

Top exterior powers. By [F2] the dualizing line bundles are ${\omega }_X={\bigwedge }^n\mathcal{C},{\omega }_{{\mathbb{P}}^N}={\bigwedge }^N{\mathrm{\Omega }}_{{\mathbb{P}}^N\mathrm{/}k}^1,$ and forming the top exterior power commutes with pullback along $i$ for a locally free module of finite rank: restricting to a chart on which ${\mathrm{\Omega }}_{{\mathbb{P}}^N\mathrm{/}k}^1$ is free and $i$ is given by a ring map, the pullback of a free module is free and the map on top exterior powers of the pulled-back basis is the pullback of the corresponding wedge, so the identifications are compatible on overlaps and glue. Hence $i^{\ast }{\omega }_{{\mathbb{P}}^N}\cong {\bigwedge }^N\mathcal{B},{\omega }_X={\bigwedge }^n\mathcal{C}\mathrm{.}$

The normal bundle. By [F1] and the definition of the normal bundle, $\mathcal{N}={\mathcal{A}}^{\vee }$ is finite locally free of rank $c$, so by [F3] applied to $\mathcal{E}=\mathcal{A}$ there is a canonical isomorphism $\det \mathcal{N}={\bigwedge }^c({\mathcal{A}}^{\vee })\cong ({\bigwedge }^c\mathcal{A}{)}^{\vee }=(\det \mathcal{A}{)}^{\vee }\mathrm{.}$

The determinant of the conormal sequence. We construct a canonical isomorphism $\mathrm{\Phi }:{\bigwedge }^c\mathcal{A}{\otimes }_{{\mathcal{O}}_X}{\bigwedge }^n\mathcal{C}⟶{\bigwedge }^N\mathcal{B}\mathrm{.}$ On an affine chart $U=\mathrm{Spec}A$ as in step 1.1 choose a splitting $s:\mathcal{C}{\mathrm{\mid }}_U\to \mathcal{B}{\mathrm{\mid }}_U$ of $\mathcal{B}{\mathrm{\mid }}_U\to \mathcal{C}{\mathrm{\mid }}_U$, which exists because $\mathcal{C}{\mathrm{\mid }}_U$ is free, hence projective. For $\alpha \in {\bigwedge }_A^c\mathcal{A}(U)$ and a decomposable $\gamma =c_1\wedge \cdots \wedge c_n\in {\bigwedge }_A^n\mathcal{C}(U)$ put ${\mathrm{\Phi }}_U(\alpha \otimes \gamma )=\alpha \wedge s(c_1)\wedge \cdots \wedge s(c_n)\in {\bigwedge }_A^N\mathcal{B}(U),$ extended to all of ${\bigwedge }^c\mathcal{A}(U)\otimes {\bigwedge }^n\mathcal{C}(U)$ by linearity. This is well defined: for fixed $\alpha$ the assignment $(c_1,\dots ,c_n)\mapsto \alpha \wedge s(c_1)\wedge \cdots \wedge s(c_n)$ is alternating $A$-multilinear, so it factors through ${\bigwedge }_A^n\mathcal{C}(U)$ by the universal property of exterior powers, and for fixed $\gamma$ the assignment is alternating $A$-multilinear in the $\alpha$-variables.

Independence of the splitting. Let $s^{\mathrm{\prime }}$ be another splitting. For each $j$ one has $s^{\mathrm{\prime }}(c_j)-s(c_j)\in \mathcal{A}{\mathrm{\mid }}_U$ because both map to $c_j$ in $\mathcal{C}{\mathrm{\mid }}_U$. Expanding the product $s^{\mathrm{\prime }}(c_1)\wedge \cdots \wedge s^{\mathrm{\prime }}(c_n)$ by multilinearity, every term in which at least one factor $s^{\mathrm{\prime }}(c_j)-s(c_j)\in \mathcal{A}$ occurs is a wedge product in which $c+1$ elements of the rank-$c$ free module $\mathcal{A}{\mathrm{\mid }}_U$ occur ($\alpha$ contributes $c$ of them), hence vanishes; only the term $s(c_1)\wedge \cdots \wedge s(c_n)$ survives. Therefore ${\mathrm{\Phi }}_U$ does not depend on the chosen splitting. It also does not depend on the chart: restrictions of splittings are splittings, and the construction is compatible with restriction, so the maps ${\mathrm{\Phi }}_U$ for the members of a trivialising affine cover agree on overlaps and glue to a global morphism $\mathrm{\Phi }$ of ${\mathcal{O}}_X$-modules, without any choice of splitting.

$\mathrm{\Phi }$ is an isomorphism. It suffices to check this on the members of the cover, where we may choose a splitting and bases $a_1,\dots ,a_c$ of $\mathcal{A}{\mathrm{\mid }}_U$ and ${\bar{c}}_1,\dots ,{\bar{c}}_n$ of $\mathcal{C}{\mathrm{\mid }}_U$; then $a_1,\dots ,a_c,s({\bar{c}}_1),\dots ,s({\bar{c}}_n)$ is a basis of the free module $\mathcal{B}{\mathrm{\mid }}_U$ (the sequence is split exact). For every subset $J=\{j_1<\cdots <j_n\}$ the element $a_1\wedge \cdots \wedge a_c\otimes {\bar{c}}_{j_1}\wedge \cdots \wedge {\bar{c}}_{j_n}$ is mapped to the corresponding determinant basis element of the complement, up to the sign of the shuffle; these elements form a basis of ${\bigwedge }_A^c\mathcal{A}(U){\otimes }_A{\bigwedge }_A^n\mathcal{C}(U)$ (tensor of two free modules with the displayed bases), so ${\mathrm{\Phi }}_U$ is an isomorphism. Hence $\mathrm{\Phi }$ is an isomorphism of finite locally free modules everywhere. Inverting it and using [F3] to dualise the rank-one factor ${\bigwedge }^c\mathcal{A}$ gives a canonical isomorphism ${\bigwedge }^n\mathcal{C}\cong ({\bigwedge }^c\mathcal{A}{)}^{\vee }{\otimes }_{{\mathcal{O}}_X}{\bigwedge }^N\mathcal{B},\text{that is}{\omega }_X\cong (\det \mathcal{A}{)}^{\vee }\otimes i^{\ast }{\omega }_{{\mathbb{P}}^N}\mathrm{.}$

Conclusion. Combining step 4.1 and step 1.3 gives a canonical isomorphism ${\omega }_X\cong i^{\ast }{\omega }_{{\mathbb{P}}^N}{\otimes }_{{\mathcal{O}}_X}\det {\mathcal{N}}_{X\mathrm{/}{\mathbb{P}}^N}\mathrm{.}$ As a consistency check, for a linear subspace $i:{\mathbb{P}}_k^n\hookrightarrow {\mathbb{P}}_k^N$ one has $\mathcal{I}\mathrm{/}{\mathcal{I}}^2\cong {\mathcal{O}}_{{\mathbb{P}}^n}(-1{)}^{\oplus c}$ and $\det \mathcal{N}\cong {\mathcal{O}}_{{\mathbb{P}}^n}(c)$, so the right hand side is $\mathcal{O}(-N-1)\otimes \mathcal{O}(c)=\mathcal{O}(-n-1)={\omega }_{{\mathbb{P}}^n}$, matching the projective-space model of [F2]; the same value is the one fixed by Serre duality for twisting sheaves on projective space for the trace normalisation. The Axiom of Choice [A1] is assumed in the statement and is inherited through the conormal-sequence supplier [F1] and the dualizing-bundle definition [F2]; the determinant construction above chooses only finitely many splittings on the members of a fixed finite trivialising cover, hence adds no further choice. The conormal-sequence supplier is used at steps 1.1 and 1.3 for the exact sequence and ranks, while the dualizing definition is used at step 1.2 for the top-exterior identification.