Proof. The sheaf $I_{\ell }(1)$ is the restriction of the analogous sheaf on $\mathbf {P}^{3}_{L}$, and the sheaf on $\mathbf {P}^{3}_{L}$ is generated by $T$ by the definition of the subscheme $\ell \subset \mathbf {P}^{3}_{L}$ (see § 2). We conclude that the same holds on $V$, and $T$ generating $I_{\ell }(1)$ is equivalent to base-point-freeness.

Proof. The point corresponding to $(a, b, 0, 0)$ is the zero locus of $b x_{1} - a x_{2}$, considered as a global section of ${\mathcal {O}}_{\mathbf {P}(T^{\vee })}(1)$. By construction, the preimage of this point under $\pi \colon \ell \to \mathbf {P}( T^{\vee })$ is the zero locus of $b x_{1} - a x_{2}$ considered as a global section of ${\mathcal {O}}_{\ell } \otimes I_{\ell }(1)$. We prove the lemma by identifying ${\mathcal {O}}_{\ell } \otimes I_{\ell }(1)$ with ${\mathcal {O}}_{\ell }(2)$ in such a way that $b x_{1} - a x_{2}$ is identified with the polynomial in (5).

Consider the line bundle $I_{\ell }(1)$. On $V$, we have $0 = x_{1} P_1 + x_{2} P_2$, so $x_{1} = - P_{2}/P_{1} \cdot x_{2}$, showing that $x_{2}$ generates $I_{\ell }(1)$ on $\{ P_1 \ne 0 \}$ and $x_{1}$ generates $I_{\ell }(1)$ on $\{ P_{2} \ne 0 \}$. We conclude that the analogue is true for ${\mathcal {O}}_{\ell } \otimes I_{\ell }(1)$, and the map sending $x_{2}$ to $-P_{1}(0, 0, x_{3}, x_{4})$ and $x_{1}$ to $P_{2}(0, 0, x_{3}, x_{4})$ defines an isomorphism ${\mathcal {O}}_{\ell } \otimes I_{\ell }(1) \cong {\mathcal {O}}_{\ell }(2)$ that sends $b x_{1} - a x_{2}$ to $a P_{1}(0, 0, x_{3}, x_{4}) + b P_{2}(0, 0, x_{3}, x_{4})$.

Proof. This is [Reference BeauvilleBea10, Theorem 4.2].

Proof. This is a special case of, for example, the main result of [Reference CazanaveCaz12].

Proof. Both the $\mathbf {A}^{1}$-degree and the class of the discriminant are unchanged by conjugation, so by Lemma 11, it is enough to prove the result when $i$ is the involution $i(z) = -\alpha /z$. In this case, the fixed subscheme is $\{ z^2+\alpha =0 \}$, which has discriminant $-4 \alpha$. We have that $\langle -4 \alpha \rangle = \langle -\alpha \rangle$, and the second class is $\langle -1 \rangle \cdot \deg ^{\mathbf {A}^{1}}(i)$ by Lemma 12.

Proof. By Corollary 13, it is enough to show that the right-hand side of (6) equals the class of the discriminant of the fixed locus of $i$. This fixed locus maps isomorphically onto the ramification locus of $\pi \colon \ell \to \mathbf {P}( T^{\vee })$, and we compute by directly computing the discriminant of the ramification locus using Lemma 10 as follows.

If we write $f = x_{1} P_{1} + x_{2} P_{2}$, then Lemma 10 implies that the ramification locus is the locus where the polynomial

(7)\begin{equation} a \cdot P_1(0, 0, x_{3}, x_{4})+ b \cdot P_2(0, 0, x_{3}, x_{4}) \end{equation}

in $x_{3}, x_{4}$ has a multiple root. The ramification locus is thus the zero locus of $\operatorname {Disc}_{x_{3}, x_{4}}( a \cdot P_{1}(0, 0, x_{3}, x_{4}) + b \cdot P_{2}(0, 0, x_{3}, x_{4}))$, the discriminant of (7) considered as a polynomial in $x_{3}$ and $x_{4}$. Consequently the discriminant of the ramification locus is $\operatorname {Disc}_{a, b}( \operatorname {Disc}_{x_{3}, x_{4}}( a \cdot P_{1}(0, 0, x_{3}, x_{4}) + b \cdot P_{2}(0, 0, x_{3}, x_{4}))) \in k^{\ast }/ (k^{\ast })^{2}$.

The right-hand side of (6) can also be described in terms of $P_{1}, P_{2}$. Differentiating $f = x_{3} P_{1} + x_{4} P_{2}$, we get $({\partial f}/{\partial e_1})\,|\,S = P_{1}(0, 0, x_{3}, x_{4})$ and $({\partial f}/{\partial e_2}) = P_{2}(0, 0, x_{3}, x_{4})$. We now complete the proof by computing explicitly. If $P_1(0, 0, x_{3}, x_{4}) = \sum a_{i} x_{3}^{i} x_{4}^{2-i} \text { and } P_{2}(0, 0, x_{3}, x_{4}) = \sum b_{i} x_{3}^{i} x_{4}^{2-i}$, then resultant computations show

\begin{align*} & \operatorname{Res}(P_1(0, 0, x_{3}, x_{4}), P_2(0, 0, x_{3}, x_{4}))\\ &\quad = a_1^2 b_0 b_2-a_2 a_1 b_0 b_1-a_0 a_1 b_1 b_2 +a_2^2 b_0^2+a_0 a_2 b_1^2 + a_0^2 b_2^2-2 a_0 a_2 b_0 b_2 \\ &\quad = \frac{1}{16} \operatorname{Disc}_{a,b}(\operatorname{Disc}_{x_{3},x_{4}} (a \cdot P_{1}(0, 0, x_{3}, x_{4})+ b \cdot P_{2}(0, 0, x_{3}, x_{4}))). \end{align*}

Proof. Let $X \to \operatorname {Spec} k$ be smooth of dimension $r \geq 1$ and let $p$ be a closed point of $X$ such that $k \subseteq k(p)$ is separable. We may assume that $X$ is affine. Let $p$ also denote the ideal corresponding to $p$. By [SGA1, II Corollaire 5.10 and 5.9], there are $x_1, \ldots , x_r$ in $\mathcal {O}_X$ such that $x_1, \ldots , x_r$ generate $p$. Since $k \subseteq k(p)$ is separable, there is $x \in \mathcal {O}_X$ which generates $k(p)$ as an extension of $k$ by the primitive element theorem. If ${\textrm {d}x}$ is zero in $\Omega _{X/k}^1 \otimes k(p)$, then $d (x + x_1), d x_2 ,\ldots , d x_r$ is a $k(p)$-basis of $\Omega _{X/k}^1 \otimes k(p)$, whence $X \to \operatorname {Spec} k [x + x_1, x_2, \ldots , x_r]$ is étale at $p$. Furthermore, $x + x_1 = x$ modulo $p$. It follows that the map $X \to \operatorname {Spec} k [x + x_1, x_2, \ldots , x_r]$ gives Nisnevich coordinates around $p$. If ${\textrm {d}x}$ is non-zero in $\Omega _{X/k}^1 \otimes k(p)$, then we may use ${\textrm {d}x}$ as the first basis element in a basis formed from the spanning set $\{ {\textrm {d}x}, dx_1, \ldots , dx_r \}$. The map to $r$-dimensional affine space over $k$ corresponding to this basis gives Nisnevich coordinates.

Proof. Nisnevich coordinates exist around closed points of smooth $r$-dimensional $k$-schemes when $k$ is infinite and $r\geq 1$ by [Reference KnusKnu91, Chapter 8, Proposition 3.2.1]. Combining with the previous lemma, we have the claimed existence of Nisnevich coordinates.

Proof. If $p$ is an isolated zero of $\sigma$, then $\mathcal {O}_{Z,p}$ is dimension zero. Since $p$ is a closed point, $\mathcal {O}_{Z,p}/p$ has dimension zero. Let $Z^0$ be an irreducible component of $Z$ containing $p$. Since $Z^0$ is finite type over a field, $\dim Z^0 = \dim \mathcal {O}_{Z,p} + \dim \mathcal {O}_{Z,p}/p = 0$. Thus $Z^0$ is an irreducible dimension zero scheme which is finite type over $k$ and is therefore a single point, which must be $p$. Thus we may take $U$ to be the complement of the other irreducible components of $Z$.

If $p$ is a closed point with a Zariski open neighborhood $U$ such that $U \cap Z= \{ p\}$, then $\dim Z^0 =0$. A zero-dimensional finite type $k$-algebra is finite.

Since a zero-dimensional Noetherian ring has finitely many points, if $\sigma$ has isolated zeros, then $Z$ has finitely many points. These points are closed because since $Z$ is closed, any specialization of a point of $Z$ is in $Z$, and since $Z$ is zero-dimensional, there are no such specializations.

If $Z$ consists of finitely many closed points, then $Z$ is a zero-dimensional finite type $k$-algebra, which is thus finite.

Proof. Since $\mathcal {O}_{Z,p}$ is finite, there exists an $m$ such that $p^m = 0$ in $\mathcal {O}_{Z,p}$. Since $\mathcal {O}_{Z,p}$ is a quotient of $\mathcal {O}_{X,p}$, it thus suffices to show the second assertion. Let $q\subset k[x_1, \ldots , x_r]$ be the prime ideal $q = \phi (p)$. By construction of $\phi$, the induced map $k[x_1, \ldots , x_r]/q \to \mathcal {O}_{X,p}/p$ on residue fields is an isomorphism. We claim by induction that the map $k[x_1, \ldots , x_r] \to \mathcal {O}_{X,p}/p^{m}$ is a surjection. Given an element $y$ of $\mathcal {O}_{X,p}/p^{m}$, by induction on $m$, we can find an element $y'$ of the image such that $y - y'$ is in $p^{m-1}$. We can therefore express $y-y'$ as $y - y' = \sum _i a_i b_i$ where $a_i$ is in $p^{m-2}$ and $b_i \in p$. Since $\phi$ is étale, $\phi$ induces an isomorphism on cotangent spaces. It follows that there exist $a_i' \in p^{m-2}$ and $b_i' \in p$ in the image, such that $a_i - a_i'$ is in $p^{m-1}$ and $b_i - b_i'$ is in $p^2$. Then $\sum _i a_i b_i = \sum _i a_i' b_i'$ in $\mathcal {O}_{X,p}/p^{m}$ and the latter is in the image, showing the lemma.

Proof. It suffices to show the lemma for $e=1$. Since $\langle f_1,\ldots , f_r \rangle \supseteq \langle f_1,\ldots , f_r \rangle + p^m \supseteq \langle f_1,\ldots , f_r \rangle + p^{m+1}$, we have that

\[ \langle g_1, \ldots, g_r \rangle \subseteq \langle f_1,\ldots, f_r \rangle . \]

In the other direction, the $g_i$ generate $\langle g_1,\ldots , g_r \rangle + p^m$ modulo $p^{m+1}$ because modulo $p^{m+1}$, the $g_i$ are equal to the $f_i$. Thus $p^m \subseteq \langle g_1,\ldots , g_r \rangle$, giving equality.

Proof. It suffices to show that the quotient map

(8)\begin{equation} a: k[x_1, \ldots, x_r]_q/\langle g_1, \ldots, g_r \rangle^e \to k[x_1, \ldots, x_r]_q/ (\phi^*)^{{-}1} (\langle f_1,\ldots, f_r \rangle^e) \end{equation}

is injective. Since $\phi$ is flat, the map of local rings $\phi ^*: k[x_1, \ldots , x_r]_q \to \mathcal {O}_{X,p}$ is faithfully flat. It thus suffices to show the injectivity of the map

\[ a\otimes_{k[x_1, \ldots, x_r]_q} \mathcal{O}_{X,p}: \mathcal{O}_{X,p}/\langle g_1, \ldots, g_r \rangle^e \to \mathcal{O}_{X,p}/ \mathcal{O}_{X,p} ((\phi^*)^{{-}1} (\langle f_1,\ldots, f_r \rangle)^e). \]

This map is injective by Lemma 25.

Proof. By Lemma 24, the map $k[x_1, \ldots , x_r]_{q} \to \mathcal {O}_{Z,p}$ is surjective. The kernel is $(\phi ^*)^{-1} \langle f_1,\ldots , f_r \rangle$, so the lemma follows by Lemma 26.

Proof. $\eta$ commutes with base change, as can be seen from the construction [Reference Scheja and StorchSS75, § 3], so we may assume that $k(p) = k$, and that $\phi (p)$ is the origin by translation. Let $g_1',\ldots ,g_r'$ be another choice of $g_1, \ldots , g_r$, and let $\eta '$ and $\eta$ denote the corresponding elements of $\operatorname {Hom}_k(\mathcal {O}_{Z,p},k)$. Since $\phi$ is a flat map of integral domains, $k[x_1,\ldots , x_r] \to \mathcal {O}_{X,p}$ is injective. By construction, $g_i' - g_i$ is in $p^{2m} \cap k[x_1,\ldots , x_r]_q$. Since $p^m \subseteq \langle f_1, \ldots , f_r \rangle$, it follows that $p^{2m}\subseteq \langle f_1, \ldots , f_r \rangle ^2$. Thus $p^{2m} \cap k[x_1,\ldots , x_r]_q \subseteq \langle f_1, \ldots , f_r \rangle ^2 \cap k[x_1,\ldots , x_r]_q$. By Lemma 26, it follows that $p^{2m} \cap k[x_1,\ldots , x_r]_q \subseteq \langle g_1, \ldots , g_r \rangle ^2$. Thus we may express $g_i' - g_i$ as a sum $g_i' - g_i = \sum _{j=1}^r \tilde {c}_{i,j} g_j$ with $\tilde {c}_{i,j}$ in $\langle g_1, \ldots , g_r \rangle$. Let $c_{i,j} = \tilde {c}_{i,j}$ for $i \neq j$ and let $c_{i,i} = 1 + \tilde {c}_{i,i}$. Then $g_i' = \sum _{j=1}^r c_{i,j} g_j$. Let $c$ in $\mathcal {O}_{Z,p}$ denote the image of $\det (c_{i, j})$. By [Reference Scheja and StorchSS79, Satz 1.1], for all $y$ in $\mathcal {O}_{Z,p}$ there is equality $\eta (y) = \eta ' (c y)$. Since $(c_{i, j})$ is congruent to the identity modulo $\langle g_1, \ldots , g_r \rangle$, and $\langle g_1, \ldots , g_r \rangle$ is in the kernel of $k[x_1, \ldots , x_r] \to \mathcal {O}_{Z,p}$, it follows that $c=1$.

Proof. The lemma is equivalent to the assertion that $\eta (y) = \eta ' ((r_U/r'_U)^2 y)$ for all $y$ in $\mathcal {O}_{Z,p}$.

Suppose first that $\phi ' = \phi$ and that $r_U'=r_U$. Let $(f_1, \ldots , f_r)$ and $(f_1', \ldots , f_r')$ denote $\psi (\sigma )$ and $\psi '(\sigma )$ expressed as $r$-tuples of regular functions on $U$. Then there is $M: U \to \operatorname {SL}_r$ such that

\[ f_j' = \sum_{i=1}^r M_{ji} f_i. \]

By Lemma 24, there exist $M'_{ij} \in k[x_1,\ldots , x_r]$ such that $M'_{ij} - M_{ij}$ is in $p^{2m}$, where as before $m$ is chosen so that $p^m = 0$ in $\mathcal {O}_{Z,p}$. By potentially shrinking $U$, we may assume that $U$ is affine and $M'_{ij} - M_{ij} \subseteq p^{2m} \mathcal {O}(U)$. Define $g_j'$ in $k[x_1, \ldots , x_r]$ by

\[ g_j' = \sum_{i=1}^r M_{ji}' g_i. \]

By construction $f_j' - g_j'$ is in $p^{2m}$, and we may therefore use $g_1',\ldots , g_r'$ to compute $\eta '.$ Because $\det M = 1$, the difference $\det M' -1$ is in $p^{2m}$, which is therefore $0$ in $\mathcal {O}_{Z,p}$. Thus $\eta = \eta '$, as claimed.

Now suppose that $\phi ' = \phi$, and $\psi ' = A \psi$ where $A$ in $\operatorname {GL}_r \mathcal {O}_U$ is the matrix restricting to the identity on $\mathcal {O}_U^{r-1}$ and multiplying the last coordinate by $\alpha ^2$ for $\alpha$ in $\Gamma (U, \mathcal {O}^*)$. As before, let $(f_1, \ldots , f_r)$ and $(f_1', \ldots , f_r')$ denote $\psi (\sigma )$ and $\psi '(\sigma )$ expressed as $r$-tuples of regular functions on $U$, so $( f_1', \ldots , f_r') = (\alpha ^2 f_1, \ldots , f_r)$. Then $\eta ' (\alpha ^2 y) = \eta (y)$ for all $y$ in $\mathcal {O}_{Z,p}$ by [Reference Scheja and StorchSS79, Satz 1.1]. Furthermore, the distinguished basis of $\wedge ^r E\vert _U$ via $\psi$ is $\alpha ^2$ times the distinguished basis via $\psi '$ by construction. Since $\phi =\phi '$, it follows that $\alpha ^2 (r'_U)^2 = r_U^2$, showing the claim.

Combining the previous two paragraphs, we see that the lemma holds when $\phi '=\phi$.

Suppose given $\phi , \phi ': U \to \operatorname {Spec} k[x_1, \ldots x_r]$ Nisnevich coordinates around $p$. By Lemma 24, $\phi$ and $\phi '$ induce surjections $\operatorname {Spec} k[x_1, \ldots x_r] \to \mathcal {O}_{X,p}/ p^{N}$ for any chosen positive integer $N$. We may therefore choose a map $\varphi : \operatorname {Spec} k[x_1, \ldots x_r] \to \operatorname {Spec} k[x_1, \ldots x_r]$ fitting into the following commutative diagram.

Let $\tilde {\phi } = \varphi \circ \phi '$. It follows that $\tilde {\phi }: U \to \operatorname {Spec} k[x_1, \ldots x_r]$ determines Nisnevich coordinates around $p$, and $\tilde {\phi }^* x_i - \phi ^* x_i$ is contained in $p^{N}$ for $i=1, \ldots r.$ The coordinates $\tilde {\phi }$ and $\phi$ determine trivializations $t_{\tilde {\phi }}, t_{\phi }: TX \vert _U \stackrel {\cong }{\longrightarrow } \mathcal {O}_U^r$. Let $A = t_{\tilde {\phi }} \circ t_{\phi }^{-1}$. By choosing $N$ sufficiently large, we may assume that $A \in \operatorname {GL}_r \mathcal {O}_{U}$ is congruent to the identity mod $p^{2m}$. Let $\psi : E\vert _U \to \mathcal {O}_U^r$ be a trivialization of $E \vert _U$ compatible with $\phi$. Define $\tilde {\psi }$ by $\tilde {\psi }=A \psi$. Then $\tilde {\psi }$ is a trivialization of $E \vert _U$ compatible with $\tilde {\phi }$ such that $\tilde {r}_U = r_U,$ where $\tilde {r}_U$ is defined by Definition 21 for $(\tilde {\phi }, \tilde {\psi })$. Let $(f_1, \ldots , f_r)$ and $(\tilde {f}_1, \ldots , \tilde {f}_r)$ denote $\psi (\sigma )$ and $\tilde {\psi }(\sigma )$ expressed as $r$-tuples of regular functions on $U$, so in particular $(\tilde {f}_1, \ldots , \tilde {f}_r) = A (f_1, \ldots , f_r)$. Choose $g_1, \ldots , g_r$ in $k[x_1, \ldots , x_r]$ such that $g_i - f_i$ is in $p^{2m}$. By Lemma 24, we may choose $\tilde {A} \in \operatorname {GL}_r k[x_1, \ldots ,x_r]$ such that $\phi ^*\tilde {A} - A$ is congruent to the identity mod $p^{2m}$. Define $(\tilde {g}_1, \ldots , \tilde {g}_r)$ by the matrix equation $(\tilde {g}_1, \ldots , \tilde {g}_r) = \tilde {A}(g_1, \ldots , g_r)$. Then $\tilde {\phi '} \tilde {g}_i - \tilde {f}_i$ is in $p^{2m}$. By construction (Lemma 27), we obtain presentations $\mathcal {O}_{Z,p} \cong k[x_1, \ldots , x_r]_{q}/\langle g_1, \ldots , g_r \rangle$ and $\mathcal {O}_{Z,p} \cong k[x_1, \ldots , x_r]_{q}/\langle \tilde {g}_1, \ldots , \tilde {g}_r \rangle$ of $\mathcal {O}_{Z,p}$. Furthermore, mapping $x_i$ to $x_i$ for $i=1, \ldots , r$ determines a commutative diagram

because $p^m = 0$ in $\mathcal {O}_{Z,p}$. Let $\tilde {\eta }: \mathcal {O}_{Z,p} \to k$ denote the homomorphism corresponding to the presentation $\mathcal {O}_{Z,p} \cong k[x_1, \ldots , x_r]_{q}/\langle \tilde {g}_1, \ldots , \tilde {g}_r \rangle$ as in [Reference Scheja and StorchSS75, p. 182]. By [Reference Scheja and StorchSS79, Satz 1.1] there is the equality $\eta = \tilde {\eta }$.

It follows that we may replace $\phi$ by $\tilde {\phi }$ and assume that $\phi = \varphi \circ \phi '$. $\eta$ commutes with base change, so we may assume that $k(p) = k$, and that $\phi (p)$ is the origin by translation. We may likewise assume that $\phi '(p)$ is the origin, and therefore that $\varphi$ takes the origin to the origin. Let $\psi ': E\vert _U \to \mathcal {O}_U^r$ denote a trivialization of $E\vert _U$ compatible with $\phi '$. Since $\varphi$ is étale on a neighborhood of $0$, the Jacobian of $\varphi$ defines a map $T\varphi$ from this neighborhood to $\operatorname {GL}_r$. By possibly shrinking $U$, we therefore have a map $T\varphi \circ \phi ': U \to \operatorname {GL}_r$. Using the canonical action of $\operatorname {GL}_r$ on the free sheaf of rank $r$, we obtain a second trivialization of $E\vert _U$ given by $\psi =(T\varphi \circ \phi ') \psi ': E\vert _U \to \mathcal {O}_U^r$, which is compatible with $\phi$, and $r_U = r_U'$. Let $(f_1', \ldots , f_r') \in \mathcal {O}_U^r$ denote the coordinate projections of $\sigma$ under the trivialization $\psi '$. It follows that the coordinate projections $(f_1, \ldots , f_r) \in \mathcal {O}_U^r$ of $\sigma$ under the trivialization $\psi$ are $T\varphi \circ \phi (f_1', \ldots , f_r')$. Choose $g_i$ in $k[x_1, \ldots , x_r]$ for $i=1, \ldots , r$ such that $(\phi )^* g_i - f_i$ is in $p^{2m}$. Then we may define $g_i'$ by $(g_1', \ldots , g_r')=(T\varphi )^{-1}(\varphi ^* g_1, \ldots , \varphi ^* g_r)$ and have that $(\phi ')^* g_i' - f'_i$ is in $p^{2m}$ as required. By [Reference Scheja and StorchSS79, Satz 1.5 and 1.1], it follows that $\eta = \eta '$. We have thus reduced to the case where $\phi = \phi '$, completing the proof.

Proof. By Remark 21, Nisnevich coordinates exist around $p$. By Lemma 27 and the construction of Scheja and Storch [Reference Scheja and StorchSS75, § 3] discussed immediately after, $\operatorname {ind}_p \sigma$ exists. The independence of the choice of $g_1, \ldots , g_r$ follows from Lemma 28. The independence of the choice of Nisnevich coordinates and compatible trivialization of $E \vert _U$ follows from Lemma 29.

Proof. Let $k(p) \subseteq L$ be an extension such that $k \subseteq L$ is Galois. Let $\beta$ denote the bilinear form representing $\operatorname {ind}_p \sigma$ and $\beta _{k(p)}$ denote the bilinear form representing $\operatorname {ind}_{p} \sigma _{k(p)}$ as in Definition 30. We identify the bilinear forms $L \otimes \beta$ and $L \otimes \operatorname {Tr}_{k(p)/k}\beta _{k(p)}$ together with the associated descent data.

The bilinear form $L \otimes \beta$ has underlying vector space $L \otimes \mathcal {O}_{Z, p}$. The bilinear form $\operatorname {Tr}_{k(p)/k} \beta _{k(p)}$ has underlying vector space given by $\mathcal {O}_{Z_{k(p)}, p}$, so $L \otimes \operatorname {Tr}_{k(p)/k} \beta _{k(p)}$ has underlying vector space given by $L \otimes _k \mathcal {O}_{Z_{k(p)}, p}$.

For each coset $g \operatorname {Gal}_{L/k(p)}$ of $\operatorname {Gal}_{L/k}$, there is a point $g p$ in $X (L)$, and a corresponding local ring $\mathcal {O}_{Z_{L}, g p}$. The ring $L \otimes \mathcal {O}_{Z,p}$ decomposes into idempotents corresponding to the $g p$, giving rise to an isomorphism

(10)\begin{equation} L \otimes \mathcal{O}_{Z,p}\cong \prod_{g \in \operatorname{Gal}_{L/k}/ \operatorname{Gal}_{L/k(p)}} \mathcal{O}_{Z_{L}, g p}. \end{equation}

It follows from the construction on [Reference Scheja and StorchSS75, p. 182] that the restriction of the map $L \otimes \eta _{k,p} : L \otimes \mathcal {O}_{Z,p} \to L$ to an idempotent $\mathcal {O}_{Z_{L}, g p}$ is the corresponding $\eta _{L, g p}$.

The map $g: \mathcal {O}_{Z_{k(p)}, p} \to \mathcal {O}_{g Z_{k(p)}, g p}$ determines a quotient map

\[ L \otimes_k \mathcal{O}_{Z_{k(p)}, p} \to L \otimes_{g k(p)} \mathcal{O}_{Z_{g k(p)},g p}. \]

These maps determine an isomorphism

\[ L \otimes_k \mathcal{O}_{Z_{k(p)}, p} \cong \prod_{g \in \operatorname{Gal}_{L/k}/ \operatorname{Gal}_{L/k(p)}} L \otimes_{g k(p)} \mathcal{O}_{Z_{g k(p)}, g p}. \]

We have $L \otimes _{g k(p)} \mathcal {O}_{Z_{g k(p)}, g p} \cong \mathcal {O}_{Z_{L}, g p}$. We therefore have constructed a $k$-linear isomorphism

(11)\begin{equation} L \otimes_k\mathcal{O}_{Z_{k(p)}, p} \cong \prod_{g \in \operatorname{Gal}_{L/k}/ \operatorname{Gal}_{L/k(p)}} \mathcal{O}_{Z_{L}, g p}. \end{equation}

By the functoriality of $\eta$, the pullback of $\eta _{L, g p}$ by $g : \mathcal {O}_{Z_{L}, p} \to \mathcal {O}_{Z_{L}, g p}$ is $\eta _{L,p}$. By the proof of [Reference LamLam05, VII Theorem 6.1], it follows that the isomorphism (11) takes $L \otimes \operatorname {Tr}_{k(p)/k} \beta _{k(p)}$ from the left-hand side to the orthogonal direct sum $\oplus _{g \in \operatorname {Gal}_{L/k}/ \operatorname {Gal}_{L/k(p)}} \beta _{L, g p}$.

Combining with (10) and (11), we have an isomorphism

\[ L \otimes \mathcal{O}_{Z,p} \cong L \otimes_k \mathcal{O}_{Z_{k(p)}, p} \]

taking $L \otimes \beta$ on the left to $L \otimes \operatorname {Tr}_{k(p)/k}\beta _{k(p)}$ on the right.

There are canonical $\operatorname {Gal}(L/k)$ actions on

\[ L \otimes \mathcal {O}_{Z,p},\quad L \otimes _k \mathcal {O}_{Z_{k(p)}}, p,\quad \text{and}\quad \prod _{g \in \operatorname {Gal}_{L/k}/ \operatorname {Gal}_{L/k(p)}} \mathcal{O}_{Z_{L}, g p}.\]

Unwinding definitions shows that the isomorphisms (10) and (11) are equivariant, identifying the appropriate descent data.

Proof. Let $L \to X$ and $L^{\otimes 2} \cong \operatorname {Hom} (\wedge ^r {\mathcal {T}}(X), \wedge ^r E)$ be the relative orientation of $E$. Let $\mathcal {X} = X \times \mathbf {A}_{k}^1$, and let $\mathcal {L}$ be the pullback of $L$ by the projection $\mathcal {X} \to X$. The canonical isomorphism $\wedge ^{r+1} {\mathcal {T}}(X \times \mathbf {A}_{k}^1) \cong \wedge ^r {\mathcal {T}}(X)$ and the isomorphism $L^{\otimes 2} \cong \operatorname {Hom} (\wedge ^r {\mathcal {T}}(X), \wedge ^r E)$ give rise to a canonical isomorphism $\mathcal {L}^{\otimes 2} \cong \operatorname {Hom} (\wedge ^{r+1} {\mathcal {T}}(\mathcal {X}), \wedge ^r \mathcal {E}),$ which is the relative orientation of $\mathcal {E}$.

Let $\mathcal {Z} \hookrightarrow \mathcal {X}$ be the closed immersion defined by $s=0$. Since $X$ is proper over $\operatorname {Spec} k$, it follows that $\mathcal {X} \to \mathbf {A}_{k}^1$ is proper, whence $p: \mathcal {Z} \to \mathbf {A}_{k}^1$ is proper. For any closed point $t$ of $\mathbf {A}_{k}^1$, the fiber $\mathcal {Z}_t \to \mathbf {A}_{k(t)}^1$ is the zero locus of the section $s_t$, which is finite by hypotheses. Thus $\mathcal {Z} \to \mathbf {A}_{k}^1$ has finite fibers and is therefore finite because it was also seen to be proper. We construct $\beta$ on the finite $\mathcal {O}(\mathbf {A}_{k}^1)$-module $p_* \mathcal {L}$.

Let $z$ be a closed point of $\mathcal {Z}$ and let $t$ be its image in $\mathbf {A}_{k}^1 = \operatorname {Spec} k[\tau ]$. By Remark 20, we may choose Nisnevich coordinates around $z$ in $\mathcal {X}_t$. Therefore we have functions $\bar {x}_1, \ldots , \bar {x}_r$ in $\mathcal {O}_{\mathcal {X}_t}$ such that the $\bar {x}_i$ generate $k(z)$ over $k(t)$ and $d \bar {x}_1, \ldots , d\bar {x}_r$ are a basis of $\Omega _{\mathcal {X}_t/k(t)}$ at $z$. Let $x_i$ be an element of $\mathcal {O}_{\mathcal {X}}$ lifting $\bar {x}_i$. It follows that $d x_1, \ldots , dx_r$ form a basis of the fiber of $\Omega _{\mathcal {X}/\mathbf {A}_{k}^1}$ at $z$. Furthermore, $\tau , x_1, \ldots , x_r$ generate $k(z)$ over $k$. Thus $\phi : U \to \operatorname {Spec} k[\tau , x_1, \ldots , x_r]$ define Nisnevich coordinates around $z$.

It follows from Lemma 24 that $\mathcal {O}_{\mathcal {Z},z}$ is generated as a $k[\tau ]_t$-algebra by $x_1, \ldots , x_r$.

Let $I \subset k[\tau ]_t[x_1, \ldots , x_r]$ denote the kernel of the surjection $k[\tau ]_t[x_1, \ldots , x_r] \to \mathcal {O}_{\mathcal {Z},z}$. Since the tensor product is right exact, the sequence

\[ k(t) \otimes_{k[\tau]_t} I \to k(t)[x_1, \ldots, x_r] \to \mathcal{O}_{\mathcal{Z}_t,z} \]

is exact. Since $k[\tau ]_t \to k(t)$ is surjective, so is $I \to k(t) \otimes _{k[\tau ]_t} I$. Thus we may choose $g_1, \ldots , g_r$ in $I$ lifting elements of $k(t)[x_1, \ldots , x_r]$ as in the construction of $\operatorname {ind}_{z} s_t$.

Let $q\subset k[\tau ][x_1, \ldots , x_r]$ be the prime ideal determined by $\phi (z)$. By Nakayama's lemma and Lemma 27, $g_1, \ldots , g_r$ generate the kernel of $k[\tau ][x_1, \ldots , x_r]_{q} \to \mathcal {O}_{\mathcal {Z},z}.$ Thus

\[ \mathcal{O}_{\mathcal{Z},z} \cong k[\tau][x_1, \ldots, x_r]_{q}/\langle g_1, \ldots, g_r \rangle, \]

expressing $k[\tau ]_t \to \mathcal {O}_{\mathcal {Z},z}$ as a relative finite complete intersection. The morphism $k[\tau ]_t \to \mathcal {O}_{\mathcal {Z},z}$ is furthermore flat by [Sta17, Lemma 10.98.3 Tag 00MD]. Repeating this process for each $z$ in $p^{-1}(p(z))$, we have expressed $\mathcal {O}_{{\mathbf {A}}^1,t} \to \mathcal {O}_{\mathcal {Z},z} \otimes _{\mathcal {O}_{\mathbf {A}^1}} \mathcal {O}_{\mathbf {A}^1,t}$ as a relative finite complete intersection. It follows that there is an open affine neighborhood $W$ of $t$ such that $p^{-1} W \to W$ is a flat relative finite complete intersection.

Let $\eta _W$ denote the element in $\operatorname {Hom}_{\mathcal {O}_W}( \mathcal {O}_{p^{-1}W} ,\mathcal {O}_W )$ corresponding to $1$ under the canonical isomorphism

\[ \operatorname{Hom}_{\mathcal{O}_W} (\mathcal{O}_{p^{{-}1}W}, {\mathcal{O}_W}) \cong \mathcal{O}_{p^{{-}1}W} \]

of $\mathcal {O}_{p^{-1}W}$-modules of [Reference Scheja and StorchSS75, § 3]. Let $\beta _{W,t}$ denote the non-degenerate bilinear form

\begin{gather*} \beta_{W,t}(x,y)= \eta_{W,t}(xy),\\ \mathcal{O}_{\mathcal{Z}} (p^{{-}1}(W)) \otimes \mathcal{O}_{\mathcal{Z}} (p^{{-}1}(W)) \to \mathcal{O}_W, \end{gather*}

specializing at $t$ to $e(\mathcal {E}_t, s_t)$.

Let $\mathcal {W}$ denote the set of those neighborhoods $W$. By Lemma 29, we may define elements $r_W$ in $p_* \mathcal {L} (W)$ for each $W$ in $\mathcal {W}$ such that for all $W$,$W'$ in $\mathcal {W}$, we have that

\[ \eta_W (y) = \eta_{W'} ((r_W/r_{W'})^2 y) \]

for all $y$ in $\mathcal {O}_{\mathcal {Z}} (p^{-1}(W \cap W'))$. The $r_W$ therefore define a descent datum on the $\beta _{W,t}$, which defines the bilinear form $\beta$ as claimed.

Proof. It suffices to prove the first claim. For $k \subseteq L$ a field extension of finite odd dimension, tensoring with $L$ is an injective map $\mathrm {GW}(k) \to \mathrm {GW}(L)$. It follows that we may assume that $\sigma$ and $\sigma '$ can be connected by sections with isolated zeros (over $k$). Since $\sigma$ and $\sigma '$ can be connected by sections with isolated zeros, it suffices to show that for a section $s$ of $\mathcal {E}$ such that $s_t$ has isolated zeros for all closed points $t$, then

\[ e(E, s_t) = e(E, s_{t'}) \]

for $k$-rational points $t$ and $t'$ of $\mathbf {A}_{k}^1$. By Lemma 36, there is a finite $\mathcal {O}(\mathbf {A}_{k}^1)$-module equipped with a non-degenerate symmetric bilinear form $\beta$ such that $\beta _t = e(E, s_t)$ and $\beta _{t'} = e(E, s_{t'})$. It therefore suffices to show that $\beta _t = \beta _{t'}$ in $\mathrm {GW}(k)$, which is true by the Serre problem for bilinear forms or Harder's theorem, for instance the version in [Reference Kass and WickelgrenKW19, Lemma 31].