Rational-point Koszul residue normalization for a smooth projective embedding

A projective linear coordinate change moves $x$ to $[1:0:\cdots :0]$. On $D_{+}(x_0)={\mathbb{A}}_k^N$ put $z_a=x_a\mathrm{/}{x}_0$ for $1\le a\le N$. In the local ring at $x$ the ordered sequence $z=(z_1,\dots ,z_N)$ is regular, and ${\mathrm{\Omega }}_z=d{z}_1\wedge \cdots \wedge d{z}_N$ is the corresponding generator of ${\omega }_{{\mathbb{P}}^N}$ on this chart. Give $K(z)$ the differential $d(e_{i_1}\wedge \cdots \wedge e_{i_p})={\sum }_{j=1}^p(-1{)}^{j-1}{z}_{i_j}{e}_{i_1}\wedge \cdots \widehat{e_{i_j}}\cdots \wedge e_{i_p}\mathrm{.}$ Write $a$ for the raw dual top cochain $e_1\wedge \cdots \wedge e_N\mapsto {\mathrm{\Omega }}_z$. We shall prove that its trace is $(-1{)}^N$; the normalized ambient point class is therefore represented by $(-1{)}^{N}a$. The same convention with $n$ parameters gives the intrinsic class in the statement.

Work locally near $x$ and choose the regular equations $f_1,\dots ,f_c$ of [F1]. Lift the parameters $t_1,\dots ,t_n$ from ${\mathcal{O}}_{X,x}$ to the regular local ring ${\mathcal{O}}_{{\mathbb{P}}^N,x}$. Since the conormal sequence is exact and $x$ is smooth, the ordered sequence $(f_1,\dots ,f_c,t_1,\dots ,t_n)$ is a regular system of parameters of the ambient local ring. The Koszul concatenation map of [F2] identifies $K(f)\otimes K(t)$ with $K(f,t)$ and sends the ordered top tensor to the ordered top wedge. The determinant adjunction of [F1] uses this same conormal-first order: $d{f}_1\wedge \cdots \wedge d{f}_c\wedge d{t}_1\wedge \cdots \wedge d{t}_n$ corresponds to $d{t}_1\wedge \cdots \wedge d{t}_n$.

Use the ordered affine cover $U_j=D_{+}(x_j)$ and the homogeneous Koszul resolution on $x_1,\dots ,x_N$; on $U_0$ it is $K(z)$ of 1.1. Put the dual Koszul degree first, so that for a cochain of Koszul degree $p$ and Čech degree $q$ the mixed total differential is $D=h+(-1{)}^p\delta$, where $h$ is precomposition with the Koszul differential and $\delta$ is the ordered Čech differential. Start with $a$ on $U_0$ and zero on the other $U_j$. For $J=\{i_1<\cdots <i_q\}\subseteq \{1,\dots ,N\}$, its correction on $U_{0,i_1,\dots ,i_q}$ is supported on the Koszul wedge complementary to $J$ and has the form $a_q=A_q\frac{{\iota }_{i_q}\cdots {\iota }_{i_1}a}{z_{i_1}\cdots z_{i_q}},A_0=1,$ where ${\iota }_i$ inserts $e_i$ into the argument of the alternating dual cochain. The Koszul deletion formula gives $h{\iota }_{J}a={\sum }_j(-1{)}^{q-j}{z}_{i_j}{\iota }_{J\setminus i_j}a$; comparing this with the face of $\delta a_{q-1}$ and the factor $(-1{)}^{N-q+1}$ in $D$ gives $A_q=(-1{)}^N{A}_{q-1}$. Thus $A_N=(-1{)}^{N^2}=(-1{)}^N$. The last term is $(-1{)}^N{\mathrm{\Omega }}_z\mathrm{/}(z_1\cdots z_N)$, corresponding under the Euler trivialization of ${\omega }_{{\mathbb{P}}^N}=\mathcal{O}(-N-1)$ to $(-1{)}^N{x}_0^{-1}\cdots x_N^{-1}$. By [F3] the raw class has trace $(-1{)}^N$, and the normalized class $(-1{)}^{N}a$ has trace $1$ under Yoneda evaluation at $1\in H^0(k_x)$. For $N=0$ the empty Koszul complex gives $k\stackrel{\mathrm{i}\mathrm{d}}{\to }k$.

The normalized regular-immersion purity map multiplies the determinant-to-sheaf-Ext identification of [F2] by ${\sigma }_c$. Its inverse Hodge map contributes $s_c=(-1{)}^{c(c-1)\mathrm{/}2}$, so the determinant frame corresponds to $s_c{\sigma }_c=(-1{)}^c$ times the raw top cochain for $K(f)$. This sign comparison is made on the affine local ring with its finite-free Koszul resolution, then carried to sheaf Ext by the local comparison in [F2]; it does not require global projectives among sheaves. The intrinsic point class of the statement contributes $(-1{)}^n$ times its raw top cochain. Ordinary Yoneda composition of raw ordered Koszul classes concatenates with coefficient $+1$: the graded tensor–Hom interchange contributes $(-1{)}^{cn}$, while the resolution-to-derived comparison contributes the same factor because ${\sigma }_{c+n}={\sigma }_c{\sigma }_n(-1{)}^{cn}$. The factors cancel. Hence the composite is represented on $K(f,t)$ by $(-1{)}^{c+n}$ times its raw top cochain, exactly the ambient normalization of 1.1–2.1. The cases $c=0$ and $n=0$ use empty Koszul factors and satisfy the same identities.

Compare $(f,t)$ to $z$. Their images in the cotangent space at $x$ are two bases, so their Jacobian matrix $J$ has determinant in $k^{\times }$. The Koszul generator matrix changes the ordered top Koszul basis by $\det J$ and the dual top cochain by $(\det J{)}^{-1}$, whereas the ambient differential form changes by $\det J$. The two factors cancel in the Ext class, and the normalization factor $(-1{)}^N$ is unchanged. The same calculation applies to changes of $f$, of the parameters $t$, and of projective coordinates, so the class constructed from $(f,t)$ equals the normalized class of 1.1 and has trace $1$ by 2.1. All matrix, wedge, Koszul and Laurent formulas commute with a field extension $k\to k^{\mathrm{\prime }}$, and the coefficient $1$ remains $1$. AC is inherited through the Ext and cohomology suppliers and supplies DC for the derived-Ext comparison in [F2], exactly as recorded in [F4].