Embedding compatibility of smooth-projective Gysin traces

For an embedding $i$ of codimension $c=N-n$, combine [F1] in degree $c+n=N$ for $E={\mathcal{O}}_X$ with [F2] in degree $N$. This gives a perfect pairing $H^0(X,{\mathcal{O}}_X)\times H^n(X,{\omega }_X)⟶k,$ whose value at $(1,\eta )$ defines $t_i(\eta )$. The same construction defines $t_j$. Naturality of the Ext collapse and of Yoneda evaluation shows that the pairing is $(f,\eta )\mapsto t_i(f\eta )$; in particular $H^n(X,{\omega }_X)$ has dimension ${\dim }_k{H}^0(X,{\mathcal{O}}_X)$. This uses only the already proved projective-space duality and collapse; no smooth-projective duality is assumed.

First suppose $k$ is algebraically closed. By [F4], the algebra $A=H^0(X,{\mathcal{O}}_X)$ is finite-dimensional over $k$. It is reduced: if $f^m=0$ as a global section, then every germ $f_x$ is nilpotent, so $f_x=0$ because smooth $X$ is reduced, and hence $f=0$. A finite reduced commutative $k$-algebra over algebraically closed $k$ is a product $k^s$: it is Artinian, its distinct maximal ideals have zero intersection, and the Chinese remainder theorem gives the product of their finite field quotients, each equal to $k$. The primitive idempotents cut out the nonempty open-and-closed connected components $X_1,\dots ,X_s$. Choose a rational point $x_a\in X_a$ for each component; it exists because a nonempty finite-type $k$-scheme has a closed point, and a closed point has residue field $k$ here. Evaluations ${\mathrm{ev}}_{x_a}:A\to k$ are the coordinate projections and form a basis of $A^{\vee }$. If $X=\mathrm{\varnothing }$, then $A=0$ and both traces are zero, so the conclusion is immediate.

For each chosen $x_a$ let ${ϵ}_{x_a}$ be the normalized class in ${\mathrm{Ext}}_X^n(k_{x_a},{\omega }_X)$ of [F3]. The quotient ${\mathcal{O}}_X\to k_{x_a}$ induces a class ${\eta }_{x_a}\in {\mathrm{Ext}}_X^n({\mathcal{O}}_X,{\omega }_X)=H^n(X,{\omega }_X)$. Naturality of Yoneda composition and [F3] give $t_i(f{\eta }_{x_a})=f(x_a),t_j(f{\eta }_{x_a})=f(x_a)(f\in A)\mathrm{.}$ In particular $t_i({\eta }_{x_a})=t_j({\eta }_{x_a})=1$. The perfect pairing of 1.1 sends ${\eta }_{x_a}$ to ${\mathrm{ev}}_{x_a}\in A^{\vee }$, so the $s$ classes ${\eta }_{x_a}$ form a basis of $H^n(X,{\omega }_X)$ by 2.1. The two linear forms agree on this basis, hence $t_i=t_j$ when $k$ is algebraically closed. This point-basis argument avoids presuming duality for arbitrary coherent sheaves.

To extend the algebraically closed comparison of step 3.1 to a general field $k$, choose an algebraic closure $K$ and use [F4] to identify $H^n(X,{\omega }_X){\otimes }_{k}K$ with $H^n(X_K,{\omega }_{X_K})$. Let $\mathcal{I}$ be the coherent ideal of $i(X)$ in ${\mathbb{P}}_k^N$. Resolve $\mathcal{I}$ by [F7] and splice its finite twisted locally free resolution with ${\mathcal{O}}_{{\mathbb{P}}^N}\twoheadrightarrow i_{\ast }{\mathcal{O}}_X$. This gives a finite locally free resolution $P^{\bullet }\to i_{\ast }{\mathcal{O}}_X$ with $P^0={\mathcal{O}}_{{\mathbb{P}}^N}$. On the finite standard affine cover form the bicomplex $C^{p,q}={\check{C}}^q(\mathcal{H}om(P^{-p},{\omega }_{{\mathbb{P}}^N}))$, with internal Hom degree $p$ first, Čech degree $q$ second, total differential $D=d_{\mathcal{H}om}+(-1{)}^p{\delta }_{\check{C}}$, and the ordered Alexander–Whitney Čech product. This is the Koszul/Hom-first convention used in the rational-point normalization [F3] and in the local comparison below.

Every Hom term in the bicomplex of step 4.1 is a finite sum of twists, and its higher cohomology on each affine intersection vanishes by [F7]. Comparing with an injective resolution of ${\omega }_{{\mathbb{P}}^N}$ therefore identifies $H^m(\mathrm{Tot}C)$ with ${\mathrm{Ext}}_{{\mathbb{P}}^N}^m(i_{\ast }{\mathcal{O}}_X,{\omega }_{{\mathbb{P}}^N})$. Filter by Čech degree and first take internal Hom cohomology on each affine intersection; this gives local sheaf Ext, and the next page takes its sheaf cohomology, yielding the raw Hom-first local-to-global Ext edge $e_m$ of [F1]. To identify an intrinsic class with this global Ext group, apply the normalized inverse $D_j^{-1}$ of [F1], which inserts ${\sigma }_c=(-1{)}^{c(c+1)\mathrm{/}2}$ exactly once after the raw edge and Hodge identification. The chain map ${\mathcal{O}}_{{\mathbb{P}}^N}\to P^{\bullet }$ equal to the identity in degree zero makes precomposition a map of these bicomplexes to ${\check{C}}^{\bullet }({\omega }_{{\mathbb{P}}^N})$, representing the Gysin map used in step 1.1.

Tensor the finite locally free resolution $P^{\bullet }$ of step 4.1 with $K$; because $K\mathrm{/}k$ is flat, it remains exact and resolves $i_{K\ast }{\mathcal{O}}_{X_K}$. On every standard affine intersection, sections of a twisting bundle and all maps of the finite bicomplex in step 5.1 commute termwise with $k\to K$. Flatness carries its cohomology, filtration and raw edge maps to those for $i_K$, and preserves the fixed scalar ${\sigma }_c$ in the normalized collapse. By [F9], ${\mathrm{\Omega }}_{X_K\mathrm{/}K}^1\cong {\mathrm{\Omega }}_{X\mathrm{/}k}^1{\otimes }_{k}K$ on affine charts; since $X$ is smooth of pure dimension $n$, taking ${\bigwedge }^n$ identifies ${\omega }_{X_K}\cong {\omega }_X{\otimes }_{k}K$, so the cohomology comparison of [F4] has the required coefficient. On a local regular-sequence chart the degree-$c$ sheaf-Ext identification is the dual Koszul determinant map of [F1]; its fixed integer signs, generator matrices and determinant adjunction commute with extension of scalars. The Laurent coefficient trace sends the same ordered monomial to $1$ over $K$. Thus the whole embedding trace $t_i$, and similarly $t_j$, commutes with $k\to K$, beyond the cohomology comparison of [F4]. By step 3.1 their extensions to $K$ agree, so faithful flatness gives $t_i=t_j$ over $k$.

Let $E$ be finite locally free. Choose an injective ${\mathcal{O}}_{{\mathbb{P}}^N}$-module resolution ${\omega }_{{\mathbb{P}}^N}\to I^{\bullet }$ as in the proof of [F1]. For every ambient open $V$ and $e\in (i_{\ast }E)(V)$, multiplication by $e$ is the canonical ${\mathcal{O}}_V$-linear map $u_e:i_{\ast }{\mathcal{O}}_X{\mathrm{\mid }}_V\to i_{\ast }E{\mathrm{\mid }}_V$. Precomposition defines, without a frame or lift, a restriction-compatible map of sheaf complexes $i_{\ast }E\otimes \mathcal{H}om(i_{\ast }E,I^{\bullet })\to \mathcal{H}om(i_{\ast }{\mathcal{O}}_X,I^{\bullet })$, $e\otimes \varphi \mapsto \varphi \circ u_e$. The extension-by-zero/injectivity argument in [F1, proof 2.1(ii)] makes every sheaf $\mathcal{H}om(i_{\ast }E,I^p)$ and $\mathcal{H}om(i_{\ast }{\mathcal{O}}_X,I^p)$ flasque; their ordered Čech bicomplexes on the common finite affine cover therefore compute global Ext, while the ordered Čech complex of the quasi-coherent $i_{\ast }E$ computes $H^q(X,E)$ by [F7]. Apply the Alexander–Whitney cup to this strict global map: for a Čech $q$-cochain $a$ and a Čech $(n-q)$-cochain $b$ of internal Hom degree $c$, the total tensor convention of 4.1 evaluates their product as $(-1{)}^{cq}b\circ u_a$ on the ordered intersection. The injective-resolution lane of the shifted derived composition in [F8] uses this same total tensor rule, so the strict map computes the ambient Yoneda product in [F2]; it is restriction-compatible and requires no coherent choices of lifted frames on triple overlaps. For the Hom-first differential of 4.1, the internal-Hom-degree $c$ row has horizontal differential $(-1{)}^c{\delta }_{\check{C}}$. Its comparison with ordinary Čech cohomology in degree $j$ therefore multiplies a row cocycle by $T_j=(-1{)}^{cj}$. For an Ext class with row Čech degree $r=n-q$, the two routes through the product square differ in edge conversion by $T_{q+r}\mathrm{/}{T}_r=(-1{)}^{cq}$, exactly the graded interchange factor in the strict total evaluation; their product is $1$. This calculates the arbitrary-degree sign rather than importing a projective-resolution cochain convention. The strict map acts on the entire Čech–Hom total complexes, including every correction component of an Ext cocycle. It respects the filtration by Čech degree; taking internal Hom cohomology first as in 5.1 leaves only the sheaf-Ext row $c$ for both targets by [F1]. The induced product on this one row determines the product on the abutments, without choosing a pure-degree-$c$ representative. To identify its internal-Hom-degree $c$ local sheaf-Ext row, take an ambient affine chart $U=\mathrm{Spec}A$ with regular ideal $I=(f_1,\dots ,f_c)$, choose a frame $E{\mathrm{\mid }}_{X\cap U}\cong (A\mathrm{/}I{)}^r$, and put $F_U=A^r$ as its ambient free lift. The finite free complexes $K(f;A)$ and $K(f;A){\otimes }_A{F}_U$ resolve $i_{\ast }{\mathcal{O}}_X{\mathrm{\mid }}_U$ and $i_{\ast }E{\mathrm{\mid }}_U$; on their top Hom cochains precomposition by a lift $\tilde{e}\in F_U$ is ordinary evaluation. If two lifts differ by $\sum f_i{e}_i$, exterior multiplication with the $i$th Koszul basis vector gives a chain homotopy for multiplication by $f_i$, so the induced local Ext map is independent of lifts. The Hodge determinant identification and adjunction of [F1] carry it to contraction $E\otimes E^{\vee }\otimes {\omega }_X\to {\omega }_X$. In the Čech–Hom comparison, moving the local Hom degree $c$ past the Čech degree $q$ gives precisely the $(-1{)}^{cq}$ already displayed; hence the local contraction square and the strict global evaluation square agree with the intrinsic Čech cup of [F5] after the natural comparison of [F8]. The normalized collapse for $E$ and for ${\mathcal{O}}_X$ is $D={\sigma }_c$ times the raw edge in both cases [F1, step 5.1]; the same fixed factor occurs once on each route of this square, so no second sign is applied. Thus for every $q$ the ambient Yoneda pairing is the intrinsic cup/contraction pairing followed by $t_i$. Replacing $i$ by $j$ changes only that final trace, equal by 6.1. The strict map commutes with restriction to local charts and is natural under bundle maps on the fixed $X$ and isomorphisms of the embedded data; field-extension compatibility is 6.1. AC is inherited through the cited injective and cohomology suppliers.