2.1 Coherent-duality Euler Number

Let $f:X \to \operatorname {Spec} k$ be a smooth, proper k-scheme of dimension n, and let V be a rank n vector bundle, relatively oriented by the line bundle $\mathcal L$ on X and the isomorphism $\rho : \det V \otimes \omega _{X/k} \to \mathcal L^{\otimes 2}$ . Let $\sigma : X \to V$ be a section, and let $K(\sigma )^\bullet $ denote the Koszul complex

$$\begin{align*}0 \to \wedge^n V^* \to \wedge^{n-1} V^* \to \cdots \to V^* \to \mathcal{O} \to 0, \end{align*}$$

with $\mathcal {O}$ in degree $0$ and differential of degree $+1$ given by

$$ \begin{align*}d\left(v_1 \wedge v_2 \wedge \cdots \wedge v_j\right) = \sum_{i=1}^j (-1)^{i-1} v_i(\sigma) v_1 \wedge \cdots \wedge v_{i-1} \wedge v_{i+1} \wedge \cdots \wedge v_j.\end{align*} $$

This choice of $K(\sigma )^{\bullet }$ is $\operatorname {Hom}_{\mathcal {O}}(-,\mathcal {O})$ applied to the Koszul complex of [Reference Eisenbud27, 17.2]. $K(\sigma )^{\bullet }$ carries a canonical multiplication

(2) $$ \begin{align}m: K(\sigma)^{\bullet} \otimes K(\sigma)^{\bullet} \to K(\sigma)^{\bullet}\end{align} $$

defined in degree $-p$ by $m = \oplus _{i+j = p} 1_{\wedge ^i V^*} \wedge 1_{\wedge ^j V^*}$ . Composing m with the projection $p: K(\sigma )^{\bullet } \to \det {V}^*[n]$ defines a nondegenerate bilinear form

$$ \begin{align*}\beta_{\left(V,\sigma\right)}: K(V, \sigma) \otimes K(V, \sigma) \to \det{V}^*[n],\end{align*} $$

$$ \begin{align*}\beta_{\left(V,\sigma\right)} =p m .\end{align*} $$

Tensoring $\beta _{\left (V,\sigma \right )}$ by $\mathcal L^{\otimes 2}$ and reordering the tensor factors of the domain, we obtain a nondegenerate symmetric bilinear form on $K(V, \sigma ) \otimes \mathcal L$ valued in $\left (\det {V}^* \otimes \mathcal L^{\otimes 2}\right )[n]$ . The orientation $\rho $ determines an isomorphism $\left (\det {V}^* \otimes \mathcal L^{\otimes 2}\right )[n] \to \omega _{X/k}[n]$ . Composing $\beta _{\left (V,\sigma \right )} \otimes \mathcal L^{\otimes 2}$ with this isomorphism produces a nondegenerate bilinear form

$$ \begin{align*}\beta_{\left(V,\sigma, \rho\right)}: (K(V, \sigma) \otimes \mathcal L) \otimes (K(V, \sigma) \otimes \mathcal L) \to \omega_{X/k}[n].\end{align*} $$

Let $D(X)$ denote the derived category of quasi-coherent $\mathcal {O}_X$ -modules. Serre duality determines an isomorphism $R f_* \omega _{X/k}[n] \cong \mathcal {O}_k$ [Reference Hartshorne41, III Corollary 7.2 and Theorem 7.6]. Since $Rf_*$ is lax symmetric monoidal (being right adjoint to a symmetric monoidal functor), we obtain a symmetric morphism

$$\begin{align*}Rf_*\beta_{\left(V,\sigma,\rho\right)}: [Rf_*(K(V, \sigma) \otimes \mathcal L)]^{\otimes 2} \to Rf_* \omega_{X/k}[n] \simeq \mathcal O_k \end{align*}$$

in $D(k)$ , which is nondegenerate by Serre duality.

The derived category $D(k)$ is equivalent to the category of graded k-vector spaces, by taking cohomology.Footnote ³ If V is a (nondegenerate) symmetric bilinear form in graded k-vector spaces, denote by $V^{(n)} = V_n \oplus V_{-n}$ (for $n \ne 0$ ) and $V^{(0)} = V_0$ the indicated subspaces; observe that they also carry (nondegenerate) symmetric bilinear forms.

Definition 2.1. For a relatively oriented rank n vector bundle $V \to X$ with section $\sigma $ and orientation $\rho $ , over a smooth and proper variety $f: X \to k$ of dimension n, the Grothendieck–Serre-duality Euler number with respect to $\sigma $ is

$$\begin{align*}n^{\mathrm{GS}}(V, \sigma, \rho) = \sum_{i \ge 0} (-1)^i \left[\left(Rf_*\beta_{\left(V,\sigma,\rho\right)}\right)^{(i)}\right] \in \mathrm{GW}(k). \end{align*}$$

Remark 2.2. In order not to clutter notation unnecessarily, we also write Definition 2.1 as

$$\begin{align*}n^{\mathrm{GS}}(V, \sigma, \rho) = \sum_i (-1)^i \left[\left(Rf_*\beta_{\left(V,\sigma,\rho\right)}\right)_i\right]. \end{align*}$$

We shall commit to this kind of abuse of notation from now on.

Recall that $n^{\mathrm {GS}}(V, \rho ) \in \mathrm {GW}(k)$ was defined in the introduction in terms of the symmetric bilinear form on $\bigoplus _{i,j} H^i\left (X, \Lambda ^j V^* \otimes \mathcal L\right )$ .

To prove Proposition 2.3, we use the hypercohomology spectral sequence $E^{i,j}_r(K^{\bullet })$ associated to a complex $K^{\bullet }$ of locally free sheaves on X:

$$ \begin{align*}E^{i,j}_1(K^{\bullet}) : = \mathrm{H}^j\left(X, K^i\right) \Rightarrow R^{i+j} f_* K^{\bullet}.\end{align*} $$

Let $F_i$ denote the resulting filtration on $R^{*} f_* K^{\bullet }$ , such that

$$ \begin{align*}\cdots \supseteq F_i = \mathrm{Im}\left(\mathrm{H}^*\left(X, K^{\bullet \geq i}\right) \to \mathrm{H}^*(X, K^{\bullet})\right) \supseteq F_{i+1} \supseteq \cdots.\end{align*} $$

Given a perfect symmetric pairing of chain complexes $\beta : K^{\bullet } \otimes K^{\bullet } \to \omega _{X/k} [n]$ , the cup product induces pairings

$$\begin{align*}\beta': R^*f_* K^{\bullet} \otimes R^*f_* K^{\bullet} \to R^*f_* \omega_{X/k} [n] \to k \end{align*}$$

and

$$\begin{align*}\beta_1: E^{*,*}_1(K^{\bullet}) \otimes E^{*,*}_1(K^{\bullet}) \to k.\end{align*}$$

The following properties hold:

1. (1) Placing the k in the codomain of $\beta _1$ in bidegree $(-n,n)$ , $\beta _1$ is a map of bigraded vector spaces and satisfies the Leibniz rule with respect to $d_1$ . It thus induces $\beta _2: E^{*,*}_2(K^{\bullet }) \otimes E^{*,*}_2(K^{\bullet }) \to k$ . Then $\beta _2$ satisfies the Leibnitz rule with respect to $d_2$ and hence induces $\beta _3$ , and so on.

2. (2) All the pairings $\beta _i$ are perfect.

3. (3) The pairing $\beta '$ is compatible with the filtration in the sense that $\beta '(F_i,F_k) = 0$ if $i+k>-n$ .

4. (4) It follows that $\beta '$ induces a pairing on $\mathrm {gr}_{\bullet } R^*f_* K^{\bullet } $ . Under the isomorphism $\mathrm {gr}_{\bullet } \simeq E_\infty $ , it coincides with $\beta _\infty $ .

5. (5) $\beta '$ is perfect in the filtered sense: the induced pairing $F_i \otimes R^*f_* K^{\bullet }/F_{-n-i+1} \to k$ is perfect. (In particular, the pairing $\beta '$ is perfect.)

Remark 2.4. We do not know a reference for these facts, and proving them would take us too far afield. The main idea is that we have a sequence of duality-preserving functors

$$\begin{align*}C^{\mathrm{perf}}(X) \xrightarrow{\sigma_{\bullet}} D(X)^{\mathrm{fil}} \xrightarrow{\pi_*} D(k)^{\mathrm{fil}}. \end{align*}$$

Here $C^{\mathrm {perf}}(X)$ denotes the category of bounded chain complexes of vector bundles, $D(X)^{\mathrm {fil}}$ is the filtered derived category [Reference Gwilliam and Pavlov39], and $\sigma _{\bullet }$ is the ‘stupid truncation’ functor (composed with forgetting to the filtered derived category). The first duality is with respect to $\underline {\operatorname {Hom}}(\mathord -, \omega [n])$ , the second with respect to $\underline {\operatorname {Hom}}(\mathord -, \sigma _{\bullet }(\omega [n])) = \underline {\operatorname {Hom}}(\mathord -, \omega [n](-n))$ , and the third with respect to $\underline {\operatorname {Hom}}(\mathord -, k[0](-n))$ . There are further duality-preserving functors

$$\begin{align*}(\mathord-)^{\mathrm{gr}}: D(k)^{\mathrm{fil}} \to D(k)^{\mathrm{gr}} \quad \text{and} \quad U: D(k)^{\mathrm{fil}} \to D(k), \end{align*}$$

where $D(X)^{\mathrm {gr}} = \mathrm {Fun}(\mathbb {Z}, D(X))$ , with $\mathbb {Z}$ viewed as a discrete category. Hence any perfect pairing $C \otimes C \to k[0](-n) \in D(k)^{\mathrm {fil}}$ induces a perfect pairing on $H_*C^{\mathrm {gr}} \otimes H_*C^{\mathrm {gr}} \to k(-n,n)$ , satisfying property (1), and a pairing $H_*UC \otimes H_* UC \to k$ , satisfying properties (3) and (5). Moreover there is a spectral sequence $E_1 = H_*C^{\mathrm {gr}} \Rightarrow H_* UC$ , satisfying properties (1) and (4). Property (2) is obtained from the fact that passage to homology is a duality-preserving functor.

We apply this to $K^{\bullet } \in C^{\mathrm {perf}}(X)$ ; then $\mathrm {gr}_i \sigma _{\bullet } K^{\bullet } = K^i[i]$ and hence $\mathrm {gr}_i(\pi _*\sigma _{\bullet } K^{\bullet }) = \pi _* K^i[i]$ .

Lemma 2.5. Let X be a graded k-vector space with a finite decreasing filtration

$$ \begin{align*}X \supset \cdots \supset X_{\bullet} \supset X_{\bullet +1} \supset \cdots.\end{align*} $$

Suppose that $X \otimes X \to k$ is a perfect symmetric bilinear pairing, which is compatible with the filtration in the sense of properties (3) and (5). Let $X^i$ denote the ith graded subspace of X and $X_{\bullet }^i$ denote the ith graded subspace of $X_{\bullet }$ . Then in $\mathrm {GW}(k)$ , there is an equality

$$\begin{align*}\sum_i (-1)^i \left[X^i\right] = \sum_i (-1)^i \left[\mathrm{gr}_{\bullet} X^i\right].\end{align*}$$

Proof. Note that property (5) implies that the pairing $\mathrm {gr}_{\bullet } X$ is nondegenerate, so the statement makes sense (recall Remark 2.2). On any graded symmetric bilinear form, the degree i and $-i$ parts for $i\ne 0$ assemble into a metabolic space, with Grothendieck–Witt class determined by the rank (see Lemma B.2). It is clear that the ranks on both sides of our equation are the same; hence it suffices to prove the lemma in the case where $X^i=0$ for $i \ne 0$ . We may thus ignore the gradings.

Let N be maximal with the property that $X_N \ne 0$ . We have a perfect pairing

$$\begin{align*}X_{N+1} \otimes X/X_{-n-N} \to k. \end{align*}$$

Since $X_{N+1}=0$ , we deduce that $X_{-n-N} = X$ and hence $X_j = X$ for all $j \le -n-N$ . If $-n-N \ge N$ , then $X = X_N(N)$ and there is nothing to prove; hence assume the opposite.

We have the perfect pairing

$$\begin{align*}X_N/X_{N+1} \otimes X_{-n-N}/X_{-n-N+1} \simeq X_N \otimes X/X_{-n-N+1} \to k. \end{align*}$$

Pick a sequence of subspaces $X \supset X^{\prime }_{-n-N+1} \supset \dots \supset X^{\prime }_{N-1}$ such that $X^{\prime }_i \subset X_i$ and the canonical projection $X^{\prime }_i \to X_i/X_{N}$ is an isomorphism. Extend the filtration $X'$ by $0$ on the left and constantly on the right. By construction, $X^{\prime \mathrm {gr}}_i = X^{\mathrm {gr}}_i$ for $i \ne N,-n-N$ , and the pairing on $X' \subset X$ is perfect in the filtered sense. By [Reference Milnor and Husemoller59, Lemma I.3.1], we have $X = X'\oplus (X')^\perp $ . By induction on N, we have $[X'] = [\mathrm {gr}_{\bullet } X']$ . It thus suffices to show that $\left [(X')^\perp \right ] = [\mathrm {gr}_N X \oplus \mathrm {gr}_{-n-N} X]$ . This holds because both sides are metabolic of the same rank: $X_{-n-N}$ is an isotropic subspace of half rank on either side (see again Lemma B.2).

Lemma 2.6. Let $E^{\bullet }$ be a chain complex with a nondegenerate, symmetric bilinear form $E^{\bullet } \otimes E^{\bullet } \to k[0]$ . Then

$$\begin{align*}\sum_i (-1)^i \left[H^i(E)\right] = \sum_i \left[E^i\right] \in \mathrm{GW}(k). \end{align*}$$

Proof. Since passing to homology is a duality-preserving functor, the statement makes sense. Both sides have the same rank, so it suffices to prove equality in $\mathrm {W}(k)$ (see Lemma B.2). We have a perfect pairing $C^i \otimes C^{-i} \to k$ , and similarly for homology. Both are metabolic unless $i = 0$ . We can choose a splitting

$$\begin{align*}C^{0} = H \oplus C', \end{align*}$$

where $H \subset ker\left (C^{0} \to C^{1}\right )$ maps isomorphically to $H^{0}(C)$ . The restriction of the pairing on $C^{0}$ to H is perfect by construction, and hence $C^{0} = H \oplus H^\perp $ . It suffices to show that $H^\perp $ is metabolic. Compatibility of the pairing with the differential shows that $d\left (C^{-1}\right ) \subset C^{0}$ is an isotropic subspace. Self-duality shows that

$$\begin{align*}\mathrm{Im}\left(d: C^{-1} \to C^{0}\right) \simeq \mathrm{Im}\left(d^\vee: \left(C^{1}\right)^\vee \to \left(C^{0}\right)^\vee\right) \simeq \mathrm{Im}\left(d: C^{0} \to C^{1}\right)^\vee, \end{align*}$$

which implies that $d\left (C^{-1}\right ) \subset H^\perp $ is of half rank. This concludes the proof.

Proof of Proposition 2.3. Let $K^{\bullet } =K(V, \sigma )^{\bullet } \otimes \mathcal L $ . We compute

$$ \begin{align*} & n^{\mathrm{GS}}(V, \rho) \stackrel{\text{def.}}{=} \sum (-1)^{i+j} \left[E^{i,j}_1(K^{\bullet})\right] \\ & \qquad \quad \stackrel{\text{Lemma~2.6}}{=} \sum (-1)^{i+j} \left[E^{i,j}_{\infty}(K^{\bullet})\right] \\ & \qquad \quad \stackrel{\text{Lemma~2.5}}{=} \sum_i (-1)^i \left[R^if_* K^{\bullet}\right] \\ & \qquad \qquad \ \stackrel{\text{def.}}{=} n^{\mathrm{GS}}(V, \sigma, \rho). \end{align*} $$

This is the desired result.

2.2 Local indices for $n^{\mathrm {GS}}(V, \sigma, \rho )$

Suppose that $\sigma $ is a section with only isolated zeros. Let i denote the closed immersion $i: Z = Z(\sigma ) \hookrightarrow X$ given by the zero locus of $\sigma $ . We express $n^{\mathrm {GS}}(V, \sigma , \rho )$ as a sum over the points z of Z of a local index at z. To do this, we use a push-forward in a suitable context and show that $\beta _{\left (V,\sigma \right )}$ is a push-forward from Z.

For a line bundle $\mathcal L$ on a scheme X, denote by $\mathrm {BL}_{\mathrm {naive}}(D(X), \mathcal L[n])$ the set of isomorphism classes of nondegenerate symmetric bilinear forms on the derived category of perfect complexes on X, with respect to the duality $\underline {\operatorname {Hom}}(\mathord -, \mathcal L[n])$ . For a proper, lci map $f: X' \to X$ , coherent duality supplies us with a trace map $\eta _{f,\mathcal L}: f_* f^!(\mathcal L)\to \mathcal L$ . We can use this [Reference Calmès and Hornbostel21, Theorem 4.2.9] to build a push-forward

$$ \begin{gather*} f_*: \mathrm{BL}_{\mathrm{naive}}\left(D(X'), f^!\mathcal L\right) \to \mathrm{BL}_{\mathrm{naive}}(D(X), \mathcal L), \\ \left[E \otimes E \xrightarrow{\phi} f^! \mathcal L\right] \mapsto \left[f_* E \otimes f_*E \to f_*(E \otimes E) \xrightarrow{f_* \phi} f_*\left(f^! \mathcal L\right) \xrightarrow{\eta_{f,\mathcal L}} \mathcal L\right]. \end{gather*} $$

Remark 2.8. There is a canonical weak equivalence $f^!\mathcal L \simeq f^! \mathcal O_X \otimes f^* \mathcal L$ , and $\eta _{f,\mathcal L}$ is given by the composition

$$ \begin{align*}f_*\left(f^!\mathcal L\right) \simeq f_*\left( f^! \mathcal O_X \otimes f^* \mathcal L \right) \simeq f_* f^! \mathcal O_X \otimes \mathcal L \xrightarrow{\eta_f \otimes \operatorname{id}_{\mathcal L}} \mathcal L,\end{align*} $$

where $\eta _f = \eta _{f, \mathcal O_S}$ [73, Lemma 47.17.8].

Example 2.9. Consider the case of a relatively oriented vector bundle V on a smooth, proper variety $f: X \to \operatorname {Spec}(k)$ . Note that elements of $\mathrm {BL}^{\mathrm {naive}}(k)$ are just isomorphism classes of symmetric bilinear forms on graded vector spaces. The orientation supplies us with an equivalence

$$\begin{align*}f^!(\mathcal O_k) \simeq \omega_{X/k}[n] \simeq \det V^* [n] \otimes \mathcal L^{\otimes 2}. \end{align*}$$

Under the induced push-forward map we have

$$\begin{align*}f_* \left[\beta_{\left(V,\sigma,\rho\right)}\right] = n^{\mathrm{GS}}(V, \sigma, \rho) \in \mathrm{BL}^{\mathrm{naive}}(k), \end{align*}$$

where $\beta _{\left (V,\sigma ,\rho \right )} \in \mathrm {BL}^{\mathrm {naive}}\left (X, \det V^* [n] \otimes \mathcal L^{\otimes 2}\right )$ is the form on $K(V, \sigma ) \otimes \mathcal L$ defined in §2.1.

Let $1_Z$ denote the element of $\mathrm {BL}_{\mathrm {naive}}(D(Z), \mathcal O_Z[0])$ represented by $\mathcal {O}_Z \otimes \mathcal {O}_Z \to \mathcal {O}_Z$ .

Proposition 2.11. Let X be a scheme, V a vector bundle, and $\sigma \in \Gamma (X, V)$ a section locally given by a regular sequence. Write $i: Z = Z(\sigma ) \hookrightarrow X$ for the inclusion of the zero scheme. Proposition B.1 yields a canonical equivalence $i^!\det (V^*)[n] \simeq \mathcal O_Z[0]$ , where n is the rank of V; under the induced map

$$\begin{align*}i_*: \mathrm{BL}_{\mathrm{naive}}(D(Z), \mathcal O_Z[0]) \to \mathrm{BL}_{\mathrm{naive}}(D(X), \det(V^*)[n]), \end{align*}$$

we have $i_*(1_Z) = \beta _{\left (V,\sigma \right )}$ , where

$$\begin{align*}\beta_{\left(V,\sigma\right)}: K(V, \sigma) \otimes K(V, \sigma) \to \det(V^*)[n] \end{align*}$$

is the canonical pairing on the Koszul complex as in §2.1.

Proof. Because $\sigma $ locally corresponds to a regular sequence, the canonical map $r:K(V, \sigma )^{\bullet } \to i_*\mathcal {O}_Z$ is an equivalence in $D(X)$ . The canonical projection $i_* \mathcal O_Z \simeq K(V, \sigma ) \to \det (V^*)[n]$ induces by adjunction a map $\mathcal O_Z \to i^!\det (V^*)[n]$ . We claim that this is the equivalence of Proposition B.1. The proof of that proposition shows that the problem is local on Z, so we may assume that V is trivial. Then this map is precisely the isomorphism constructed in [Reference Hartshorne40, Proposition III.7.2 and preceeding pages], which is also the isomorphism used in the proof of Proposition B.1.

Now we prove that $i_*(1_Z) = \beta _{\left (V,\sigma \right )}$ . Consider the following diagram:

The map $m_K: K(V, \sigma ) \otimes K(V, \sigma ) \to K(V, \sigma )$ is the canonical multiplication (see §2.1, property (2)) and $m_Z: i_*\mathcal O_Z \otimes ^L i_* \mathcal O_Z \to i_* \mathcal O_Z \otimes i_* \mathcal O_Z \to i_* \mathcal O_Z$ is equivalently given by either multiplication in $\mathcal O_Z$ or the lax monoidal witness transformation of $i_*$ . The former interpretation shows that the left-hand square commutes. The pairing $i_*(1)$ is given by the composite from the top right corner to the bottom right corner. To prove the claim, it suffices to show that the bottom-row composite $K(V, \sigma ) \to \det (V^*)[n]$ is the canonical projection. This follows by adjunction from our choice of equivalence $\mathcal O_Z \simeq i^! \det (V^*)[n]$ .

This concludes the proof.

Proposition 2.11 is an example of a more general phenomenon given in Meta-Theorem 3.9.

Lemma 2.12 [Reference Calmès and Hornbostel21]

Let $g: Z \to Y$ and $f: Y \to X$ be proper maps.Footnote ⁴ Given equivalences $f^! \mathcal L \simeq \mathcal M [n]$ and $g^! \mathcal M [n] \simeq \mathcal N$ , the canonical equivalence $(f g)^! \simeq g^! f^!$ produces a weak equivalence $(f g)^! \mathcal L \simeq \mathcal N$ , and consequently push-forward maps

$$\begin{align*}g_*: \mathrm{BL}_{\mathrm{naive}}(D(Z), \mathcal N) \to \mathrm{BL}_{\mathrm{naive}}(D(Y), \mathcal M [n]), \end{align*}$$

$$\begin{align*}f_*: \mathrm{BL}_{\mathrm{naive}}(D(Y), \mathcal M [n]) \to \mathrm{BL}_{\mathrm{naive}}(D(X), \mathcal L), \end{align*}$$

$$\begin{align*}(f g)_*: \mathrm{BL}_{\mathrm{naive}}(D(Z), \mathcal N) \to \mathrm{BL}_{\mathrm{naive}}(D(X), \mathcal L). \end{align*}$$

There is a canonical equivalence $(f g)_* \simeq f_* g_*$ .

Now we get back to our Euler numbers. Let $X/k$ be smooth and proper, V a relatively oriented vector bundle, and $\sigma $ a section of V with only isolated zeros. Write $i: Z = Z(\sigma ) \hookrightarrow X$ for the inclusion of the zero scheme of $\sigma $ . Let $\varpi : Z \to \operatorname {Spec} k$ and $f: X \to \operatorname {Spec} k$ denote the structure maps, so that $\varpi = f i $ .

The weak equivalence $i^!\det (V^*)[n] \simeq \mathcal O_Z[0]$ of Proposition 2.11, with Remark 2.8, produces a weak equivalence $i^! \left (\det V^*[n] \otimes \mathcal L^{\otimes 2}\right ) \cong i^* \mathcal L^{\otimes 2}$ . The orientation $\rho $ gives an isomorphism $\det V^*[n] \otimes \mathcal L^{\otimes 2}\cong \omega _{X/k}[n]$ . Combining, we have a chosen weak equivalence

$$\begin{align*}i^! (\omega_{X/k}[n]) \cong i^* \mathcal L^{\otimes 2}. \end{align*}$$

Since also $f^! \mathcal {O}_k \simeq \omega _{X/k} [n]$ (see, e.g., Proposition B.1), we therefore obtain a canonical equivalence

$$\begin{align*}\varpi^! \mathcal{O}_k \simeq i^* \mathcal L^{\otimes 2}. \end{align*}$$

We use this equivalence to define

$$\begin{align*}\varpi_*: \mathrm{BL}^{\mathrm{naive}}\left(Z, i^* \mathcal L^{\otimes 2}\right) \to \mathrm{BL}^{\mathrm{naive}}(k). \end{align*}$$

Corollary 2.13. With this notation, we have

$$\begin{align*}n^{\mathrm{GS}}(V, \sigma, \rho) =\varpi_* \left(i^* \mathcal L \otimes i^* \mathcal L \to i^* \mathcal L^{\otimes 2}\right). \end{align*}$$

Suppose that $\sigma $ has isolated zeros, or in other words that the support of $\sigma $ is a disjoint union of points. Then $n^{\mathrm {GS}}(V, \sigma , \rho )$ can be expressed as a sum of local contributions. Namely, for each point z of Z, let $i_z: Z_z \hookrightarrow X$ denote the chosen immersion coming from the connected component of Z given by z. Let $\varpi _z: Z_z \to \operatorname {Spec} k$ denote the structure map. Then

$$ \begin{align*}n^{\mathrm{GS}}(V, \sigma, \rho) = \sum_{z \in Z}\varpi_{z*} \left(i_z^* \mathcal L \otimes i_z^* \mathcal L \to i_z^* \mathcal L^{\otimes 2}\right) .\end{align*} $$

In light of this we propose the following:

Definition 2.14. For a relatively oriented vector bundle with a section as described, and $z \in Z(\sigma )$ , we define

$$\begin{align*}\mathrm{ind}_z(\sigma) = \mathrm{ind}_z(V, \sigma, \rho) = \varpi_{z*}\left(i_z^* \mathcal L \otimes i_z^* \mathcal L \to i_z^* \mathcal L^{\otimes 2}\right) \in \mathrm{BL}^{\mathrm{naive}}(k). \end{align*}$$

The previous formula then reads

(3) $$ \begin{align} n^{\mathrm{GS}}(V, \sigma, \rho) = \sum_{z \in Z} \mathrm{ind}_z(V, \sigma, \rho). \end{align} $$

In the next two subsections, we compute the local contributions $\mathrm {ind}_z(\sigma )$ as an explicit bilinear form constructed by Scheja and Storch [Reference Scheja and Storch67], appearing in the Eisenbud–Levine–Khimshiashvili signature theorem [Reference Eisenbud and Levine29] and used as the local index of the Euler class constructed in [Reference Kass and Wickelgren49, Section 4].

2.3 Scheja–Storch and coherent duality

Let S be a scheme, $\pi : X \to S$ a smooth scheme of relative dimension n, and $Z \subset X$ closed with $\varpi : Z \to S$ finite. Suppose the following data:

1. (1) sections $T_1, \dots , T_n \in \mathcal O(X)$ such that $T_i \otimes 1 - 1 \otimes T_i$ generate the ideal of $X \subset X \times _S X$ ;

2. (2) sections $f_1, \dots , f_n \in \mathcal O(X)$ such that $Z = Z(f_1, \dots , f_n)$ .

Choose $a_{ij} \in \mathcal O(X \times _S X)$ such that

$$\begin{align*}f_i\otimes 1 - 1 \otimes f_i = \sum_j a_{ij}\left(T_j \otimes 1 - 1 \otimes T_j\right). \end{align*}$$

Let $\Delta \in \mathcal O(Z \times _S Z)$ be the image of the determinant of $a_{ij}$ . Since $\varpi $ is finite locally free, $\Delta $ determines an element $\tilde \Delta $ of

$$\begin{align*}\operatorname{Hom}_{\mathcal O_S}\left(\mathcal O_S, (\varpi \times_S \varpi)_* \mathcal O_{Z \times_S Z}\right) \simeq \operatorname{Hom}_{\mathcal O_S}(\mathcal O_S, \varpi_* \mathcal O_Z \otimes \varpi_* \mathcal O_Z) \simeq \operatorname{Hom}_{\mathcal O_S}((\varpi_* \mathcal O_Z)^*, \varpi_* \mathcal O_Z). \end{align*}$$

Remark 2.16. We can make $\tilde \Delta $ explicit: if $\Delta = \sum _i b_i \otimes b_i^{\prime }$ , then

$$\begin{align*}\tilde\Delta(\alpha) = \sum_i \alpha(b_i)b_i^{\prime}. \end{align*}$$

Remark 2.17. By construction, the pullback of $\Delta $ along the diagonal $\delta : Z \to Z \times _S Z$ is the determinant of the differentiation map $C_{Z/X} \to \Omega _X\rvert _Z$ with respect to the canonical bases. In other words,this is the Jacobian:

$$\begin{align*}\delta^*(\Delta) = \mathrm{Jac} F := \det \left(\frac{\partial f_i}{\partial T_j}\right)_{i,j=1}^n. \end{align*}$$

Theorem 2.18. Under the foregoing assumptions, the map

$$\begin{align*}\tilde\Delta: (\varpi_* \mathcal O_Z)^* \to \varpi_* \mathcal O_Z \end{align*}$$

is a symmetric isomorphism and hence determines a symmetric bilinear structure on $\varpi _* \mathcal O_Z$ . This is the same structure as $\varpi _*(1)$ –that is,

$$\begin{align*}\varpi_*(\mathcal O_Z) \otimes \varpi_*(\mathcal O_Z) \to \varpi_*(\mathcal O_Z) \simeq \varpi_*\left(\varpi^! \mathcal O_S\right) \xrightarrow{\eta_\varpi} \mathcal O_S. \end{align*}$$

Here the isomorphism $\varpi ^! \mathcal O_Z \simeq \mathcal O_Z$ arises from

$$\begin{align*}\varpi^!(\mathcal O_Z) \simeq \widetilde\det L_\varpi \simeq \omega_{Z/X} \otimes \omega_{X/S} \simeq \mathcal O, \end{align*}$$

with the first isomorphism given by Proposition B.1 and the third given by the sections $(T_i)$ and $(f_i)$ .

We begin with some preliminary observations before delving into the proof. The problem is local on S, so we may assume that $S = \operatorname {Spec}(A)$ ; then $Z = \operatorname {Spec}(B)$ . Since $\varpi $ is finite, there is a canonical isomorphism [Reference Hartshorne40, III §8 Theorem 8.7 (3), or Ideal Theorem (3) p. 6]

$$\begin{align*}\varpi^! \simeq \underline{\operatorname{Hom}}_A(B, \mathord-): D(A) \to D(B). \end{align*}$$

In particular,

$$\begin{align*}\varpi^!(A) \simeq \underline{\operatorname{Hom}}_A(B, A) \end{align*}$$

and the trace map takes the form [Reference Hartshorne40, Ideal theorem 3) pg 7]

$$\begin{align*}\varpi_* \varpi^! A \simeq \operatorname{Hom}_A(B, A) \to A, \eta \mapsto \eta(1). \end{align*}$$

Proof of Theorem 2.18. The isomorphisms

$$\begin{align*}B^* = \underline{\operatorname{Hom}}_A(B,A) \simeq \varpi^!(A) \simeq B \end{align*}$$

determine an element $\Delta ^{\prime } \in \operatorname {Hom}_A(B^*, B)$ . The theorem is equivalent to showing that $\tilde \Delta = \Delta ^{\prime }$ .

We thus need to make explicit the isomorphism

$$\begin{align*}\underline{\operatorname{Hom}}_A(B, A) \simeq \varpi^!(\mathcal O_A) \simeq i^! \pi^! (\mathcal O_A) \simeq \omega_{Z/X} \otimes \omega_{X} \simeq \mathcal O. \end{align*}$$

Tracing through the definitions (including the proof of [Reference Hartshorne40, III Proposition 8.2]), one finds that this isomorphism arises by computing

$$\begin{align*}\operatorname{Ext}^n_X(B, \mathcal O_X) \end{align*}$$

in two ways. One the one hand, the kernel of the surjection $\mathcal O_X \to B$ is generated by $f_1, \dots , f_n$ , which is a regular sequence by Remark 2.15; let $K_A(f)^{\bullet }$ denote the corresponding Koszul complex. On the other hand, we can consider the embedding $Z \hookrightarrow X \times Z$ ; its ideal is generated by the strongly regular sequence $T_i - t_i$ , where $t_i$ is the image of $T_i$ in B. We thus obtain a resolution $K_B(T-t)^{\bullet } \to B$ over $X \times Z$ . Since $p: X \times Z \to X$ is finite, $p_*K_B(T-t)^{\bullet } \to p_*B=B$ is still a resolution. We shall conflate $K_B(T-t)$ and $p_* K_B(T-t)$ notationally. We can thus compute

$$\begin{align*}\operatorname{Ext}^n_{X}(B, \mathcal O_X) \simeq coker\left(\operatorname{Hom}_{X}\left(K_B(T-t)^{n-1}, \mathcal O_X\right) \to \operatorname{Hom}_{X}(K_B(T-t)^{n}, \mathcal O_X)\right). \end{align*}$$

Since $\operatorname {Hom}_{X}(B \otimes \mathcal O_X, \mathcal O_X) \simeq \operatorname {Hom}_A(B, \mathcal O_X) \simeq \operatorname {Hom}_A(B, A) \otimes _A \mathcal O_X,$ there is a natural map $\xi : \operatorname {Hom}_A(B, A) \to \operatorname {Hom}_{X}(K_B(T-t)^{n}, \mathcal O_X)$ (sending $\alpha $ to $\alpha \otimes 1$ ). One checks that this induces $\operatorname {Hom}_A(B, A) \simeq coker(\dots ) \simeq \operatorname {Ext}^n_{X}(B, \mathcal O_X)$ .

We can write down a map of resolutions $\zeta : K_A(f) \to K_B(T-t)$ as follows: The kernel of $B \otimes \mathcal O_X \to B$ is by construction generated by $\{T_i - t_i\}_i$ , but it also contains $f_i$ . Note that $f_i = \sum _j \bar a_{ij}\left (T_j - t_j\right )$ , where we write $\bar a_{ij}$ for the image of $a_{ij}$ in $\mathcal O_X \otimes B$ . Letting $K_A(f)$ be the exterior algebra on $\{e_1, \dots , e_n\}$ and $K_B(T-t)$ the exterior algebra on $\left \{e_1^{\prime }, \dots , e_n^{\prime }\right \}$ , the map $\zeta $ is specified by $\zeta (e_i) = \sum _j \bar a_{ij} e_j^{\prime }$ . The isomorphism

$$\begin{align*}\operatorname{Hom}_A(B, A) \simeq h^n \operatorname{Hom}_{X}(K_B(T-t)^{\bullet}, \mathcal O_X) \simeq h^n \operatorname{Hom}_{X}(K_A(f)^{\bullet}, \mathcal O_X) \simeq B \end{align*}$$

is thus given by

$$ \begin{gather*} \operatorname{Hom}_A(B, A) \xrightarrow{\xi} \operatorname{Hom}_{X}(K_B(T-t)^{n}, \mathcal O_X) \xrightarrow{\det\left(a_{ij}\right)^*} \operatorname{Hom}_{X}(K_A(f)^{n}, \mathcal O_X) \\ \simeq \operatorname{Hom}_{X}(\mathcal O_X, \mathcal O_X) \simeq \mathcal O_X \to B. \end{gather*} $$

Write the image of $\det (a_{ij})$ in $B \otimes B$ as $\sum _k b_k \otimes b_k^{\prime }$ . Tracing through the definitions, we find that this composite sends $\alpha \in \operatorname {Hom}_A(B, A)$ to $\sum _k \alpha (b_k)b_k^{\prime }$ . By Remark 2.16, this is precisely $\tilde \Delta $ .

This concludes the proof.□

Definition 2.20. If $X=U \subset \mathbb A^n_S$ , $(T_i)$ are the standard coordinates, and $F=(f_1, \dots , f_n)$ , we denote the symmetric bilinear form already constructed by

$$\begin{align*}\langle - \vert - \rangle^{\mathrm{SS}} = \langle - \vert - \rangle^{\mathrm{SS}}(U, F, S). \end{align*}$$

This form was first constructed, without explicitly using coherent duality, by Scheja and Storch [Reference Ananyevskiy2, Reference Scheja and Storch67].

2.4 The Poincaré–Hopf Euler number with respect to a section

In this subsection, we recall the Euler class defined in [Reference Kass and Wickelgren49, Section 4] and prove Theorem 1.1. To distinguish this Euler class from the others under consideration, here we call it the Poincaré–Hopf Euler number, because it is a sum of local indices as in the Poincaré–Hopf theorem for the Euler characteristic of a manifold. It is defined using local coordinates.

Let k be a field, and let X be an n-dimensional smooth k-scheme. Let z be a closed point of X.

As before, let V be a relatively oriented, rank n vector bundle on X. Let $\sigma $ be a section with only isolated zeros, and let $Z \hookrightarrow X$ denote the closed subscheme given by the zero locus of $\sigma $ . Let z be a point of Z. The Poincaré–Hopf local index or degree

$$ \begin{align*}\mathrm{ind}_z^{\mathrm{PH}} \sigma \in \mathrm{GW}(k)\end{align*} $$

was defined in [Reference Kass and Wickelgren49, Definition 30] as follows: Choose a system of Nisnevich coordinates $\varphi : U \to \mathbb A^n_k$ around z. After possibly shrinking U, the restriction of V to U is trivial and we may choose an isomorphism $\psi : V \rvert _{U} \to \mathcal {O}_{U}^n$ of V. The local trivialization $\psi $ induces a distinguished section of $\det V(U)$ . The system of local coordinates $\varphi $ induces a distinguished section of $\det T_X (U)$ , and we therefore have a distinguished section of $(\det V \otimes \omega _X)(U)$ . As in [Reference Kass and Wickelgren49, Definition 19], the local coordinates $\varpi $ and local trivialization $\psi $ are said to be compatible with the relative orientation if the distinguished element is the tensor square of a section of $\mathcal L(U)$ . By multiplying $\psi $ by a section in $\mathcal {O}(U)$ , we may assume this compatibility.

Under $\psi $ , the section $\sigma $ can be identified with an n-tuple $(f_1, \ldots , f_n)$ of regular functions, $\psi (\sigma ) = (f_1, \ldots , f_n) \in \oplus _{i=1}^n \mathcal {O}_{U}$ . Let m denote the maximal ideal of $\mathcal {O}_{U}$ corresponding to z. Since z is an isolated zero, there is an integer n such that $m^n=0$ in $\mathcal {O}_{Z,z}$ . For any N, it is possible to choose $(g_1, \ldots , g_n)$ in $\oplus _{i=1}^n m^{N}$ such that $(f_i + g_i)\rvert _{U}$ is in the image of $\varphi ^*: \mathcal {O}_{\mathbb A^r_S} \to \varphi _* \mathcal {O}_U$ after possibly shrinking U [Reference Kass and Wickelgren49, Lemma 22]. For $N=2n$ , choose such $(g_1, \ldots , g_n)$ and let $F_i$ in $\mathcal {O}_{\mathbb A^d_k}\left (\mathbb A^d_k\right )$ be the functions such that $\varphi ^*(F_i) = f_i + g_i$ . Then $\varphi $ induces an isomorphism $\mathcal {O}_{Z,z} \cong k[t_1, \dots , t_n]_{m_{\varphi (z)}}/(F_1, \dots , F_n)$ [Reference Kass and Wickelgren49, Lemma 25], and $\mathrm {ind}_z^{\mathrm {PH}} \sigma $ is defined to be the associated Scheja–Storch form $\langle - \vert - \rangle ^{\mathrm {SS}}(\varphi (U), F, k)$ (see §2.3 and Definition 2.20 for the definition of $\langle - \vert - \rangle ^{\mathrm {SS}}(\varphi (U), F, k)$ ). The local index $\mathrm {ind}_z^{\mathrm {PH}} \sigma $ is well defined by [Reference Kass and Wickelgren49, Lemma 26]. Then the Poincaré–Hopf Euler number is defined to be the sum of the local indices:

One can extend the comparison of local degrees $\mathrm {ind}_z^{\mathrm {PH}} \sigma = \mathrm {ind}_z \sigma $ to work over a more general base scheme S. This was done for $S = \mathbb A^1_k$ with k a field in [Reference Kass and Wickelgren49, Lemma 33], but in more generality, it is useful to pick the local coordinates using knowledge of both $\sigma $ and X, as follows:

Suppose that X has dimension n over S, so that the rank of V is also n. Let $Z \subset Z(\sigma )$ be a clopen component and write $\varpi : Z \to S$ for the structure map. The local index generalizes straightforwardly from Definition 2.14:

Definition 2.29. We call

$$\begin{align*}\mathrm{ind}_{Z}(\sigma) = \mathrm{ind}_{Z}(V, \sigma, \rho) = \varpi_*\left(i^* \mathcal L \otimes i^* \mathcal L \to i^* \mathcal L^{\otimes 2}\right) \in \mathrm{BL}^{\mathrm{naive}}(S) \end{align*}$$

the local index at Z.

A system of coordinates for $(V,X,\sigma ,\rho ,Z)$ determines a presentation $Z=Z(\sigma \rvert _U)=Z(\sigma ^{\prime }) \subset \mathbb A^n_S$ , where $\sigma ^{\prime }: \mathbb A^n_S \supset \phi (U) \to \mathbb A^n$ . Hence Definition 2.20 supplies us with a symmetric bilinear form $\langle - \vert - \rangle ^{\mathrm {SS}}(\varphi (U), \sigma , S) \in \mathrm {BL}^{\mathrm {naive}}(S)$ .

In contrast, our proof that the Euler number (sum of indices) is independent of the choice of section (i.e., Proposition 2.3) does not generalize immediately; in fact, this will not hold in $\mathrm {BL}^{\mathrm {naive}}(S)$ but rather in some quotient (like $\mathrm {GW}(k)$ in the case of fields). As indicated in Remark 2.7, one situation in which it is easy to see this independence is if the quotient group satisfies homotopy invariance. This suggests studying Euler numbers valued in more general homotopy-invariant cohomology theories for algebraic varieties, which is what the remainder of this work is concerned with.

3.2 Features of cohomology theories

Many cohomology theories that occur in practice satisfy additional properties beyond the basic ones of our definition, and many come with more data. We list here some of those relevant to this paper.

• Morphisms of theories: Cohomology theories form a category in an evident way, with morphisms given by natural transformations.

• Trivial bundles: We usually abbreviate $E^{\mathcal O^n}(X)$ to $E^n(X)$ .

• Additive and multiplicative structure: Often, E takes values in abelian groups. Moreover, often $E^0(X)$ is a ring and $E^\xi (X)$ is a module over $E^0(X)$ . Typically all of this structure is preserved by the pullback maps.

• Disjoint unions: Usually E converts finite disjoint unions into products–that is, $E(\emptyset ) = *$ and $E(X \coprod Y) = E(X) \times E(Y)$ . If E takes values in abelian groups, this is usually written as $E(X \coprod Y) = E(X) \oplus E(Y)$ .

• Orientations: In many cases, the cohomology theory E factors through a quotient of the category $\mathcal C^L$ , built using a quotient $q: K(X) \to K^{\prime }(X)$ of the K-theory groupoid. In other words, one has canonical isomorphisms $E^\xi (X) \simeq E^{\xi ^{\prime }}(X)$ for certain K-theory points $\xi , \xi ^{\prime }$ . More specifically:

  • $\mathrm {GL}$ -orientations: If $K^{\prime }(X) = \mathbb {Z}$ and q is the rank map, then we speak of a $\mathrm {GL}$ -orientation. In other words, in this situation we canonically have $E^\xi (X) \simeq E^{rk(\xi )}(X)$ . In particular, Warning 3.3 does not apply: all automorphisms of vector bundles act trivially on E. This happens, for example, if $E = \mathrm {CH}$ (see Example 3.2(1)).

  • $\mathrm {SL}$ -orientations: If instead $K^{\prime }(X) = \mathrm {Pic}(D(X))$ via the determinant, then we speak of an $\mathrm {SL}$ -orientation. In other words, in this situation $E^\xi (X)$ depends only on the rank and (ungraded) determinant of $\xi $ . We write $E^{rk(\xi )}(X, \det (\xi ))$ for this common group. This happens, for example, if $E=\mathrm {GW}$ (see Example 3.2(1)).

  • $\mathrm {SL}^c$ -orientations: This is a further strengthening of the concept of an $\mathrm {SL}$ -orientation, where in $K^{\prime }(X) = \mathrm {Pic}(D(X))$ we mod out (in the sense of groupoids) by the squares of line bundles. In other words, if $\mathcal L_1, \mathcal L_2, \mathcal L_3$ are line bundles on X, then any isomorphism $\mathcal L_1 \simeq \mathcal L_2 \otimes \mathcal L_3^{\otimes 2}$ induces

    $$\begin{align*}E^n(X, \mathcal L_1) \simeq E^n(X, \mathcal L_2). \end{align*}$$
    Note that then, in particular, $E^n(X, \mathcal L) \simeq E^n(X, \mathcal L^*)$ . This also happens for $E = \mathrm {GW}$ , essentially by construction.

• Supports: Often, for $Z \subset X$ closed there is a cohomology with support, denoted $E^\xi _Z(X)$ . It enjoys further functorialities which we do not list in detail here.

• Transfers: In many theories, for appropriate morphisms $p: X \to Y$ and $\xi \in K(Y)$ , there exists $tw(p, \xi ) \in K(X)$ and a transfer map

  $$\begin{align*}p_*: E^{tw\left(p, \xi\right)}(X) \to E^\xi(Y), \end{align*}$$
  compatible with composition. Typically p is required to be lci, and
  $$\begin{align*}tw(p, \xi) = p^*\xi + L_p, \end{align*}$$
  where $L_p$ is the cotangent complex [Reference Illusie46]. Furthermore, typically p is required to be proper, or else we need to fix $Z \subset X$ closed and proper over Y and obtain $p_*: E^{tw\left (p, \xi \right )}_Z(X) \to E^\xi (Y)$ . Finally, usually E takes values in abelian groups and satisfies the disjoint union property, and transfer from a disjoint union is just the sum of the transfers.

3.3 Some cohomology theories

We now introduce a number of cohomology theories that can be used in this context.

• Hermitian K-theory $\mathrm {GW}$ : This is the theory from Example 3.2(2). It is $\mathrm {SL}^c$ -oriented. We believe that it has transfers for (at least) smooth, proper morphisms and regular immersions, but we are not aware of a reference for this in adequate generality. If X is regular and $1/2 \in X$ , one can use the comparison with $\mathrm {KO}$ -theory (see later).

• Naive derived bilinear forms $\mathrm {BL}_{\mathrm {naive}}$ : See §2.2.

• Cohomology theories represented by motivic spectra: Let $\mathcal {SH}(S)$ denote the motivic stable $\infty $ -category. Then any $E \in \mathcal {SH}(S)$ defines a cohomology theory on $\mathrm {S}\mathrm {ch}{}_S$ , automatically satisfying many good properties; for example, they always have transfers along smooth and proper morphisms, as well as regular immersions. For a lucid introduction, see [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31]. We recall some of the main points in §4.

• Orthogonal K-theory spectrum $\mathrm {KO}$ : This spectrum is defined and stable under arbitrary base change if $1/2 \in S$ [ Reference Spitzweck71, Reference Panin and Walter65]. Over regular bases, it represents Hermitian K-theory $\mathrm {GW}$ ; in general it represents a homotopy-invariant version.

• Generalized motivic cohomology $\mathrm {H}\tilde {\mathbb {Z}}$ : This can be defined as (see, e.g., [Reference Bachmann5]). Over fields (of characteristic not $2$ ) it represents generalized motivic cohomology in the sense of Calmès and Fasel [Reference Calmès and Fasel20, Reference Bachmann and Hopkins9, Reference Calmès and Fasel20]; it is unclear whether this theory is useful in this form over more general bases.

3.4 The yoga of Euler numbers

Let E be a cohomology theory.

Definition 3.5. We will say that E has Euler classes if, for each scheme X over S and each vector bundle V on X, we are supplied with a class

$$\begin{align*}e(V, E) \in E^{V^*}(X). \end{align*}$$

Typically, the Euler classes will satisfy further properties, such as stability under base change; we do not formalize this here.

Now suppose that $\pi : X \to S$ is smooth and proper, V is relatively oriented, and E has transfers for smooth proper maps and is $\mathrm {SL}^c$ -oriented. In this case we have a transfer map

$$\begin{align*}E^{V^*}(X) \simeq E^n(X, \det V) \stackrel{\rho}{\simeq} E^n(X, \omega_{X/S}) \simeq E^{L_\pi}(X) \xrightarrow{\pi_*} E^0(S). \end{align*}$$

Definition 3.7. In the foregoing situation, we call

$$\begin{align*}n(V, \rho, E) = \pi_* e(V, E) \end{align*}$$

the Euler number of V in E with respect to the relative orientation $\rho $ .

Example 3.8. Let $E = \mathrm {GW}$ . We can define a family of Euler classes by

$$\begin{align*}e(V) = [K(V, 0)] \in \mathrm{GW}\left(X, \widetilde\det V\right) \simeq \mathrm{GW}^{V^*}(X); \end{align*}$$

here we use the Koszul complex from §2.1. This depends initially on a choice of section, but we shall show that the Grothendieck–Witt class often does not. In any case, here we chose the zero section for definiteness. Assuming that $\mathrm {GW}$ has transfers (of the expected form) in this context, we find that

$$\begin{align*}n(V, \rho, \mathrm{GW}) = n^{\mathrm{GS}}(V, 0, \rho). \end{align*}$$

Now let $\sigma $ be a nondegenerate section of V (in the sense of Definition 2.25) and write $i: Z=Z(\sigma ) \hookrightarrow X$ for the inclusion of the zero scheme. Thus i is a regular immersion. In this case one has (see Example 2.27)

$$\begin{align*}[L_i] \simeq -\left[N_{Z/X}^*\right] \simeq -[V^*\rvert_Z], \end{align*}$$

and consequently, if E has push-forwards along regular immersions, there is a transfer map

$$\begin{align*}i_*: E^0(Z) \simeq E^{[V^*]\rvert_Z - \left[V^*\rvert_Z\right]}(Z) \simeq E^{i^*V^* + L_i}(Z) \to E^{V^*}(X). \end{align*}$$

The following result is true in all cases that we know of; but of course it cannot be proved from the weak axioms that we have listed:

Meta-Theorem 3.9. Let $\sigma $ be a nondegenerate section of a vector bundle V over a scheme X. Let E be a cohomology theory with Euler classes and push-forwards along regular immersions, such that $E^0(S)$ has a distinguished element $1$ (e.g., is a ring). Then

$$\begin{align*}e(V, E) = i_*(1), \end{align*}$$

where we use the identification from before for the push-forward.

Going back to the situation where X is smooth and proper over S, V is relatively oriented, and E is $\mathrm {SL}^c$ -oriented and has transfers along proper lci morphisms, we also have the push-forward

$$\begin{align*}\varpi_*: E^0(Z) \simeq E^0\left(Z, \mathcal L^{\otimes 2}\middle\rvert_Z\right) \stackrel{\rho}{\simeq} E^0\left(Z, \det V^* \otimes \omega_{X/S}\rvert_Z\right) \simeq E^{L_\varpi}(Z) \to E^0(S). \end{align*}$$

More generally, if $Z^{\prime } \subset Z$ is a clopen component, then we have a similar transfer originating from $E^0(Z^{\prime })$ .

Definition 3.10. For $V, \sigma , X, E$ as before, for any clopen component $Z^{\prime } \subset Z$ we denote by

$$\begin{align*}\mathrm{ind}_{Z^{\prime}}(\sigma, \rho, E) = \varpi^{\prime}_*(1) \in E^0(S) \end{align*}$$

the local index of $\sigma $ around $Z^{\prime }$ in E. Here $\varpi ^{\prime }: Z^{\prime } \to S$ is the restriction of $\varpi $ to $Z^{\prime }$ .

Meta-Corollary 3.11. Let $\sigma $ be a nondegenerate section of a relatively oriented vector bundle V over $\pi : X \to S$ . Let E be an $\mathrm {SL}^c$ -oriented cohomology theory with Euler classes and push-forwards along proper lci morphisms, such that Meta-Theorem 3.9 applies. Then

$$\begin{align*}n(V, \rho, E) = \sum_{Z^{\prime}} \mathrm{ind}_{Z^{\prime}}(\sigma, \rho, E). \end{align*}$$

Recall the notion of coordinates from Definition 2.28. The following result states that indices may be computed in local coordinates:

Proposition 3.13. Let E be an $\mathrm {SL}^c$ -oriented cohomology theory with Euler classes and push-forwards along proper lci morphisms. Let $(\psi ,\varphi ,\sigma _2)$ be a system of coordinates for $(V,X,\sigma _1,\rho _1,Z)$ . Then

$$\begin{align*}\mathrm{ind}_Z(\sigma_1, \rho_1, E) = \mathrm{ind}_Z(\sigma_2, \rho_2, E), \end{align*}$$

where $\rho _2$ is the canonical relative orientation of $\mathcal O_{\mathbb A^n}^n/\mathbb A^n$ .

Proof. Let $\varpi : Z \to S$ denote the canonical map. Then both sides are obtained as $\varpi _*(1)$ , but conceivably the orientations used to define the transfer could be different; we shall show that they are not. In other words, we are given two isomorphisms

$$\begin{align*}\widetilde \det L_\varpi \stackrel{\alpha_i}{\simeq} \mathcal L_i^{\otimes 2} \end{align*}$$

and we need to exhibit $\mathcal L_1 \stackrel {\beta }{\simeq } \mathcal L_2$ such that $\alpha _1^{-1} \alpha _2 = \beta ^{\otimes 2}$ . The isomorphisms $\alpha _i$ arise as

$$\begin{align*}\det L_\varpi \simeq \det N_{Z/X} \otimes \omega_{X/S}\rvert_Z \stackrel{\sigma_i}{\simeq} \det V \otimes \omega_{X/S}\rvert_Z \stackrel{\rho_i}{\simeq} \mathcal L_i^{\otimes 2}\rvert_Z, \end{align*}$$

where $\mathcal L_2 = \mathcal O$ and for $i=2$ we implicitly use $\varphi $ and $\psi $ as well. We first check that the two isomorphisms $\det N_{Z/X} \simeq \det V\rvert _Z$ are the same. Indeed, $V\rvert _U \simeq \mathcal O^n$ via $\psi $ , and up to this trivialization the isomorphism is given by the trivialization of $C_{Z/X}$ by $\sigma _i$ ; these are the same by Definition 2.28. Now we deal with the second half of the isomorphism. By construction, we have an isomorphism $\mathcal O \simeq \mathcal O^{\otimes 2} \stackrel {\alpha _1^{-1}\alpha _2}{\simeq } \mathcal L_1^{\otimes 2}$ ; what we need to check is that the corresponding global section of $\mathcal L_1^{\otimes 2}$ is a tensor square. Unwinding the definitions, this follows from Definition 2.28.

4.1 Aspects of the six-functors formalism

We recall some aspects of the six-functors formalism for the motivic stable categories $\mathcal {SH}(\mathord -)$ , following the exposition in [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31].

4.1.1 Adjunctions

For every scheme X, we have a symmetric monoidal, stable category $\mathcal {SH}(X)$ . For every morphism $f: X \to Y$ of schemes we have an adjunction

$$\begin{align*}f^*: \mathcal{SH}(Y) \leftrightarrows \mathcal{SH}(X): f_*. \end{align*}$$

If no confusion can arise, we sometimes write $E_Y := f^* E$ . If f is smooth, there is a further adjunction

$$\begin{align*}f_\#: \mathcal{SH}(X) \leftrightarrows \mathcal{SH}(Y): f^*. \end{align*}$$

If f is locally of finite type, then there is the exceptional adjunction

$$\begin{align*}f_!: \mathcal{SH}(X) \leftrightarrows \mathcal{SH}(Y): f^!. \end{align*}$$

There is a natural transformation $\alpha : f_! \to f_*$ . If f is proper, then $\alpha $ is an equivalence.

The assignments $f \mapsto f^*, f_*, f^!, f_!, f_\#$ are functorial. In particular, given composable morphisms $f, g$ of the appropriate type, we have equivalences $(fg)_* \simeq f_*g_*$ , and so on.

4.1.2 Exchange transformations

Suppose a commutative square of categories

which is a natural isomorphism $\gamma : FG^{\prime } \simeq GF^{\prime }$ . If the functors $G^{\prime }, G$ have right adjoints $H^{\prime }, H$ , then we have the natural transformation

$$\begin{align*}F^{\prime} H^{\prime} \xrightarrow{\mathrm{unit}} HG F^{\prime}H^{\prime} \stackrel{\gamma}{\simeq} HFG^{\prime}H^{\prime} \xrightarrow{\mathrm{counit}} HF, \end{align*}$$

called the associated exchange transformation. Similarly, if $G^{\prime }, G$ have left adjoints $K^{\prime }, K$ , then we have the exchange transformation

$$\begin{align*}KF \xrightarrow{\mathrm{unit}} KF G^{\prime}K^{\prime} \stackrel{\gamma}{\simeq} KGFK^{\prime} \xrightarrow{\mathrm{counit}} FK^{\prime}. \end{align*}$$

Suppose we have a commutative square of schemes

(4)

Then we have an induced commutative square of categories

Passing to the right adjoints, we obtain the exchange transformation

$$\begin{align*}\mathrm{Ex}^{*}_{*}: f^{*}g_{*} \to g^{\prime}_* f^{\prime*}. \end{align*}$$

Similarly, there is $\mathrm {Ex}_\#^*: g^{\prime }_\# f^{\prime *} \to f^*g_\#$ (for g smooth; this is in fact an equivalence if diagram (4) is cartesian), and so on.

4.1.3 Exceptional exchange transformation

Given a cartesian square of schemes as in diagram (4), with g (and hence $g'$ ) locally of finite type, there is a canonical equivalence

$$\begin{align*}\mathrm{Ex}^*_!: f^*g_! \simeq g^{\prime}_! f^{\prime*}. \end{align*}$$

Passing to right adjoints, we obtain

$$\begin{align*}\mathrm{Ex}^{*!}: f^{\prime*}g^! \to g^{\prime!}f^*. \end{align*}$$

4.1.4 Thom transformation

Given a perfect complex $\mathcal E$ of vector bundles on X, the motivic J-homomorphism $K(X) \to \mathrm {Pic}(\mathcal {SH}(X))$ [Reference Bachmann and Hoyois10, §16.2] provides us with an invertible spectrum . We denote by the associated invertible endofunctor. If $\mathcal E$ is a vector bundle (concentrated in degree $0$ ), then is the suspension spectrum on the Thom space $\mathcal E^*/\mathcal E^* \setminus 0$ .Footnote ⁶

Lemma 4.1. The functor $f^!$ commutes with Thom transforms.

Proof. This follows from the projection formula [Reference Cisinski and Déglise23, A.5.1(6)] and the invertibility of

:

4.1.5 Purity transformation

Let $f: X \to Y$ be a smoothable [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31, §2.1.21] lci morphism. Then the cotangent complex $\mathrm {L}_f$ is perfect, and there exists a canonical purity transformation

$$\begin{align*}\mathfrak{p}_f: \Sigma^{\mathrm{L}_f} f^* \to f^!. \end{align*}$$

4.2 Cohomology groups and Gysin maps

4.2.1

Let S be a scheme and set $E \in \mathcal {SH}(S)$ . Given $(\pi : X \to S) \in \mathrm {S}\mathrm {ch}{}_S$ , $i: Z \hookrightarrow X$ closed, and $\xi \in K(Z)$ , we define the $\xi $ -twisted E-cohomology of X with support in Z as

This assignment forms a cohomology theory in the sense of §3.1. It takes values in abelian groups, has supports, and satisfies the disjoint union property. We shall see that it has transfers for proper lci maps. It need not be orientable in general.

If $Z = X$ , we may omit it from the notation and just write $E^\xi (X)$ . As before, if $\xi $ is a trivial virtual vector bundle of rank $n \in \mathbb {Z}$ , then we also write $E^n_Z(X)$ instead of $E^\xi _Z(X)$ .

Example 4.2. Suppose that $\xi = i^*V$ , where V is a vector bundle on X. We have

where we have used the fact that $i_* \simeq i_!$ and Lemma 4.1. Using the localization sequence $j_\#j^* \to \operatorname {id} \to i_*i^*$ [Reference Hoyois44, Theorem 6.18(4)] to identify

, we find that

$$\begin{align*}E_Z^\xi(X) \simeq \left[ X/(X \setminus Z), \Sigma^\xi \pi^* E\right]_{\mathcal{SH}(X)}. \end{align*}$$

Remark 4.3. The final expression in Example 4.2 depends only on $Z \subset X$ as a subset, not a subscheme. This also follows directly from the definition, since $\mathcal {SH}(Z) \simeq \mathcal {SH}(Z_{\mathrm {red}})$ ; this is another consequence of localization.

4.2.3 Contravariant functoriality in X

Define $f: X^{\prime } \to X \in \mathrm {S}\mathrm {ch}{}_S$ . Then there is a pullback map

$$\begin{align*}f^*: E^\xi_Z(X) \to E^{f^* \xi}_{f^{-1}(Z)}(X^{\prime}), \end{align*}$$

coming from the morphism

$$ \begin{gather*} \pi_* i_! \Sigma^\xi i^! \pi^* E \xrightarrow{\mathrm{unit}} \pi_* f_*f^* i_! \Sigma^\xi i^! \pi^* E \simeq \pi^{\prime}_* f^* i_! \Sigma^\xi i^! \pi^* E \stackrel{\mathrm{Ex}_!^*}{\simeq} \pi^{\prime}_* i^{\prime}_! f^{\prime*} \Sigma^\xi i^! \pi^* E \\ \simeq \pi^{\prime}_* i^{\prime}_! \Sigma^{f^*\xi} f^{\prime*} i^! \pi^* E \xrightarrow{\mathrm{Ex}^{*!}} \pi^{\prime}_* i^{\prime}_! \Sigma^{f^*\xi} i^{\prime*} f^* \pi^* E \simeq \pi^{\prime}_* i^{\prime}_! \Sigma^{f^*\xi} i^{\prime*} \pi^{\prime*} E. \end{gather*} $$

Here the $\mathrm {Ex}_!^*$ and $\mathrm {Ex}^{*!}$ come from the cartesian square

Lemma 4.4. Set $f: X^{\prime } \to X \in \mathrm {S}\mathrm {ch}{}_X$ , $Z \subset X$ , $\xi \in K(Z)$ , and $\alpha : E \to F \in \mathcal {SH}(S)$ . The following square commutes:

where $\xi ^{\prime } = f^*(\xi )$ and $Z^{\prime } = f^{-1}(Z)$ .

Proof. This is just an expression of the fact that the exchange transformations used to build $f^*$ are indeed natural transformations.

4.2.4 Covariant functoriality in X

Suppose a commutative square in $\mathrm {S}\mathrm {ch}{}_S$ ,

(5)

where f is smoothable lci, $i, k$ are closed immersions, and g is proper. For every $\xi \in K(Z_2)$ , there is a Gysin map

$$\begin{align*}f_*:E_{Z_1}^{g^* \xi + i^* \mathrm{L}_f}(X) \to E_{Z_2}^\xi(Y) \end{align*}$$

coming from the morphism

$$\begin{align*}f_* i_! \Sigma^{g^*\xi + i^* \mathrm{L}_f} i^! f^* E_Y &\xrightarrow{\mathfrak{p}_f} f_* i_! \Sigma^{g^* \xi} i^! f^! E_Y \simeq k_! g_! \Sigma^{g^* \xi} g^!k^! E_Y \simeq k_! g_! g^!\Sigma^{\xi} k^! E_Y\\ &\xrightarrow{\mathrm{counit}} k_! \Sigma^\xi k^! E_Y, \end{align*}$$

where we used Lemma 4.1 to move $\Sigma ^{L_f}$ through $i^!$ and $\Sigma ^\xi $ through $g^!$ , and also used $i_* \simeq i_!, k_* \simeq k_!, g_* \simeq g_!$ .

Remark 4.5. In [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31], the Gysin map is denoted by $f_!$ , to emphasize that it involves the purity transform. We find the notation $f_*$ more convenient.

Lemma 4.6. The following hold:

1. (1) Suppose a square as in diagram (5) and $\alpha : E \to F \in \mathcal {SH}(S)$ . Then the following square commutes:

   

2. (2) Suppose a square as in diagram (5) and $s: Y^{\prime } \to Y$ such that $s, f$ are tor-independent. Set $X^{\prime } = X \times _Y Y^{\prime }$ , $Z_i^{\prime } = Z_i \times _Y Y^{\prime }$ , $i^{\prime }, g^{\prime }, k^{\prime }, f^{\prime }$ the induced maps, and so on. Then the following square commutes:

   

3. (3) Suppose a commutative diagram in $\mathrm {S}\mathrm {ch}{}_S$ as follows

   

   Then given $\xi \in K(Z_3)$ , we have

   $$\begin{align*}f^{\prime}_* f_* = (f^{\prime}f)_*: E^{\left(g^{\prime}g\right)^* \xi + i^* L_{f^{\prime}f}}_{Z_1}(X) \to E_{Z_3}^\xi(W). \end{align*}$$
   Here we use the equivalence $L_{f^{\prime }f} \simeq f^{\prime *} L_f + L_{f^{\prime }} \in K(X)$ , coming from the cofiber sequence $f^{\prime *} L_f \to L_{f^{\prime }f} \to L_{f^{\prime }}$ .

Proof. (1) Since $f_*: E_{Z_1}^{g^* \xi + i^* \mathrm {L}_f}(X) \to E_{Z_2}^\xi (Y)$ is obtained as $[1, \zeta _E]$ , where $\zeta $ is a natural transformation of endofunctors of $\mathcal {SH}(S)$ , this is clear.

(2) Consider the following diagram:

Here $s_X: X^{\prime } \to X$ is the canonical map and $\xi ^{\prime } = \left (Z_2^{\prime } \to Z_2\right )^* \xi $ . All the unlabeled equivalences arise from moving Thom transforms through (various) pullbacks, the compatibility of pullbacks and push-forwards with composition, and equivalences of the form $p_* \simeq p_!$ for p proper. All the maps labeled ‘ $\mathrm {Ex}$ ’ are exchange transformations expressing the compatibility of $p_*, p_!, p^!$ with base change. Denote the diagram by $\mathcal D$ . Then

yields a diagram of abelian groups. The outer square of that diagram identifies with the square which we are trying to show commutes. It thus suffices to show that $\mathcal D$ commutes. All cells commute for trivial reasons, except for (a), which commutes by [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31, Proposition 2.2.2(ii)] and (b), which commute by the stability of the counit transformations under base change.

(3) Consider the following diagram:

All the unlabeled equivalences arise from moving Thom transforms through (various) pullbacks, the compatibility of pullbacks and push-forwards with composition, and equivalences of the form $p_* \simeq p_!$ for p proper. Denote the diagram by $\mathcal D$ . Then

yields a diagram of abelian groups. Going from the top to the bottom middle via the leftmost path, we obtain $f^{\prime }_* f_*$ ; going instead via the rightmost path we obtain $(f^{\prime }f)_*$ . It hence suffices to show that $\mathcal D$ commutes. All cells commute for trivial reasons, except for (a), which commutes by [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31, Proposition 2.2.2(i)], and (b), which commutes by the compatibility of the counit transformations with composition.

Example 4.7. Consider a commutative square as in diagram (5), with $X=Y$ and $f = \operatorname {id}$ , so that $g: Z_1 \hookrightarrow Z_2$ is a closed immersion. Then $f_*: E^{g^* \xi }_{Z_1}(X) \to E^\xi _{Z_2}(X)$ is the ‘extension of support’ map. In particular taking $Z_2 = X$ as well, we obtain the map $E^{i^* \xi }_{Z_1}(X) \to E^\xi (X)$ ‘forgetting the support’. Lemma 4.6(3) now in particular tells us that given a proper map $f: X \to Y$ , a closed immersion $i: Z \hookrightarrow X$ , and $\xi \in K(Y)$ , the following diagram commutes:

where the upper horizontal map forgets the support.

4.3 Orientations

4.3.1 Product structures

By a ring spectrum (over S) we an object $E \in \mathcal {SH}(S)$ together with homotopy classes of maps and $m: E \wedge E \to E$ satisfying the evident identities. If E is a ring spectrum, then there are multiplication maps

$$\begin{align*}E^\xi_{Z_1}(X) \times E^{\xi^{\prime}}_{Z_2}(X) \to E^{\xi + \xi^{\prime}}_{Z_1 \cap Z_2}(X) \end{align*}$$

induced by

$$\begin{align*}\left(\Sigma^\xi E\right) \wedge \left(\Sigma^{\xi^{\prime}} E\right) \simeq \Sigma^{\xi + \xi^{\prime}} E \wedge E \xrightarrow{m} \Sigma^{\xi + \xi^{\prime}} E \end{align*}$$

and the diagonalFootnote ⁷

$$\begin{align*}X/X \setminus (Z_1 \cap Z_2) \to X/X \setminus Z_1 \wedge X/X \setminus Z_2. \end{align*}$$

Lemma 4.8. The multiplicative structure on E-cohomology is compatible with pullback: given $Z_1, Z_2 \subset X$ , $\xi , \xi ^{\prime } \in K(X)$ , and $f: X^{\prime } \to X$ , the following diagram commutes:

Proof. This is immediate from the definitions.

4.3.2 Thom spectra

Let $G = (G_n)_n$ be a family of finitely presented S-group schemes, equipped with a morphism of associative algebras $G \to \left (\mathrm {GL}_{nk,S}\right )_n$ (for the Day convolution symmetric monoidal structure on $\mathrm {Fun}(\mathbb {N}, \mathrm {Grp}(\mathrm {S}\mathrm {ch}{}_S))$ ). Then there is a notion of a (stable) vector bundle with structure group G, the associated K-theory space $K^G(X)$ , and the associated Thom spectrum $\mathrm {M} G$ , which is a ring spectrum [Reference Bachmann and Hoyois10, Example 16.22].

Example 4.9. If $G_n = \mathrm {GL}_n$ , then $K^G(X) = K(X)$ and $\mathrm {M} \mathrm {GL}$ is the algebraic cobordism spectrum [Reference Bachmann and Hoyois10, Theorem 16.13]. If $G_n = \mathrm {SL}_n$ (resp., $\mathrm {Sp}_n$ ), then $K^G(X)$ is the K-theory of oriented (resp., symplectic) vector bundles in the usual sense, and $\mathrm {M} \mathrm {SL}$ (resp., $\mathrm {M} \mathrm {Sp}$ ) is the Thom spectrum as defined in [Reference Panin and Walter63].

In order to work effectively with $\mathrm {M} G$ , one needs to know that it is stable under base change. This is easily seen to be true for $\mathrm {M}\mathrm {GL}$ , $\mathrm {M}\mathrm {SL}$ , and $\mathrm {M}\mathrm {Sp}$ [Reference Bachmann and Hoyois10, Example 16.23]. We record the following more general result for future reference:

Proposition 4.10. The Thom spectrum $\mathrm {M} G$ is stable under base change, provided that each $G_n$ is flat and quasi-affine.

Proof. We have a presheaf $K^G \in \mathcal {P}(\mathrm {S}\mathrm {ch}{}_S)$ and a map $K^G \to K$ . For $f: X \to S \in \mathrm {S}\mathrm {ch}{}_S$ , denote by $K^G_X \in \mathcal {P}({\mathrm {S}\mathrm {m}}_X)$ and $j_X: K^G_X \to K\rvert _{{\mathrm {S}\mathrm {m}}_X}$ the restrictions. Then by definition, $\mathrm {M} G_X = M_X(j_X)$ , where $M_X: \mathcal {P}({\mathrm {S}\mathrm {m}}_X)_{/K} \to \mathcal {SH}(X)$ is the motivic Thom-spectrum functor [Reference Bachmann and Hoyois10, §16.1]. Let $L K^G_S \in \mathcal {P}(\mathrm {S}\mathrm {ch}{}_S)$ denote the left Kan extension of $K^G_S$ . We claim that $L K^G_S \to K^G$ is a Nisnevich equivalence. Assuming this, we deduce that $f^* K^G_S \simeq \left .\left (L K^G_S\right )\right \rvert _{{\mathrm {S}\mathrm {m}}_X} \to K^G_X$ is a Nisnevich equivalence. Since $M_X$ inverts Nisnevich equivalences [Reference Bachmann and Hoyois10, Proposition 16.9], this implies that $f^* \mathrm {M} G_S \simeq \mathrm {M} G_X$ , which is the desired result.

To prove the claim, we first note that by [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson32, Lemma 3.3.9], we may assume S is affine, and it suffices to prove that the restriction of $K^G$ to ${\mathrm {A}\mathrm {ff}}_S$ is left Kan extended from smooth affine S-schemes. By definition, $K^G = \left (\mathrm {Vect}^G\right )^{gp}$ , where $\mathrm {Vect}^G = \coprod _{n \ge 0} BG_n$ (here the coproduct is as stacks i.e., fppf sheaves). The desired result now follows from [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson32, Proposition A.0.4 and Lemma A.0.5] (noting that the coproduct of stacks is the same as the coproduct of $\Sigma $ -presheaves, and Kan extension preserves $\Sigma $ -presheaves).

Now set $\xi \in K^G(X)$ . Then there is a canonical equivalence [Reference Bachmann and Hoyois10, Example 16.29]

$$\begin{align*}\Sigma^\xi \mathrm{M} G_X \simeq \Sigma^{\lvert\xi\rvert} \mathrm{M} G_X. \end{align*}$$

We denote by $t_\xi \in \mathrm {M} G^{\xi - \lvert \xi \rvert }(X)$ the class of the map

4.3.3 Oriented ring spectra

Definition 4.11. Let $E \in \mathcal {SH}(S)$ be a ring spectrum and $G = (G_n)_n$ a family of group schemes as in §4.3.2. By a strong G -orientation of E we mean a ring map $\mathrm {M} G \to E$ .

Example 4.12. The spectrum $\mathrm {KO}$ is strongly $\mathrm {SL}$ -oriented (see Corollary A.3).

Note that if $E \in \mathcal {SH}(S)$ is strongly G-oriented, then there is no reason a priori why $E_X$ should be strongly $G_X$ -oriented. This is true if $\mathrm {M} G$ is stable under base change, so for most reasonable G by Proposition 4.10. We will not talk about strong G-orientations unless $\mathrm {M} G$ is stable under base change, so assume this throughout.

Given $\xi \in K^G(X)$ , the map $\mathrm {M} G \to E$ provides us with $t_\xi = t_\xi (E) \in E^{\xi - \lvert \xi \rvert }(X)$ .

Proposition 4.13. Let E be strongly G-oriented and set $\xi \in K^G(X)$ .

1. (1) The classes $t_\xi (E)$ are stable under base change: for $f: X^{\prime } \to X$ we have $t_{f^*\xi }(E) = f^*t_\xi (E)$ .

2. (2) Multiplication by $t_\xi (E)$ induces an equivalence $\Sigma ^{\lvert \xi \rvert } E \simeq \Sigma ^\xi E$ and an isomorphism $t: E^{\lvert \xi \rvert }_Z(X) \simeq E^\xi _Z(X)$ , called the Thom isomorphism.

3. (3) The Thom isomorphism is compatible with base change: $f^*(t(x)) = t(f^*(x))$ .

In particular, E is G-oriented.

Proof. (1) follows from the same statement for $\mathrm {M} G$ , where it holds by construction. For the first half of (2), it suffices to show that $t_\xi (E)$ is a unit in the Picard-graded homotopy ring of E. This follows from the same statement for $\mathrm {M} G$ . The second half of (2) follows. (3) immediately follows from (1).

Example 4.14. Let E be strongly $\mathrm {GL}$ -oriented. Then for any $\xi \in K(X)$ we obtain $E^\xi (X) \simeq E^{rk(\xi )}(X)$ , so E is oriented in the sense of §3.2.

Definition 4.15. For $X \in \mathrm {S}\mathrm {ch}{}_S$ and $\mathcal L$ a line bundle on X, put

$$\begin{align*}E^n_Z(X, \mathcal L) = E^{n-1+\mathcal L}_Z(X). \end{align*}$$

Example 4.16. Let E be strongly $\mathrm {SL}$ -oriented and set $\xi \in K(X)$ . Then $\xi ^{\prime } := \xi - (\lvert \xi \rvert - 1 + \det \xi ) \in K(X)$ lifts canonically to $K^{\mathrm {SL}}(X)$ , whence by Proposition 4.13 we get a canonical (Thom) isomorphism $E^\xi (X) \simeq E^{\lvert \xi \rvert }(X, \det \xi )$ . In particular, E is $\mathrm {SL}$ -oriented in the sense of §3.2.

Remark 4.17. If E is strongly $\mathrm {SL}$ -oriented, then since $\det (\mathcal L_1 \oplus \mathcal L_2) \simeq \mathcal L_1 \otimes \mathcal L_2$ , by Example 4.16 the product structure on E-cohomology twisted by line bundles takes the form $E^n(X, \mathcal L_1) \times E^m(X, \mathcal L_2) \to E^{n+m}(X, \mathcal L_1 \otimes \mathcal L_2)$ .

Remark 4.18. Strong G-orientations have better permanence properties than ordinary ones (provided that $\mathrm {M} G$ is stable under base change): they are stable under base change and taking (very) effective covers, for example.

4.4 $\mathrm {SL}^c$ -orientations

A. Ananyevskiy has done important work on $\mathrm {SL}$ and $\mathrm {SL}^c$ orientations. We shall make use of the following result (see, e.g., [Reference Ananyevskiy1, Theorem 1.1]):

Proof. Note that (2) follows from (1). Let $\mathcal L = \mathcal L_3$ . It suffices to exhibit a canonical equivalence $\Sigma ^{\mathcal L^{\otimes 2}} E \simeq \Sigma ^{\mathcal O} E$ . We have canonical equivalences

$$\begin{align*}\Sigma^{\mathcal L + \mathcal L} E \simeq \Sigma^{\mathcal O + \mathcal L^{\otimes 2}} E, \qquad \Sigma^{\mathcal L + \mathcal L^*} E \simeq \Sigma^{\mathcal O^2}E, \qquad \Sigma^{\mathcal L + \mathcal L} \simeq \Sigma^{\mathcal L + \mathcal L^*}, \end{align*}$$

by [Reference Ananyevskiy1, Corollary 3.9, Lemma 4.1]. Consequently, $\Sigma ^{\mathcal O + \mathcal L^{\otimes 2}} E \simeq \Sigma ^{\mathcal O^2}E$ , whence the claim.

5.1 Tautological Euler class

Let $E \in \mathcal {SH}(S)$ be a ring spectrum, set $X \in \mathrm {S}\mathrm {ch}{}_S$ , and let V be a vector bundle on X.

Definition 5.1. We denote by $e(V) = e(V, E) \in E^{V^*}(X)$ the tautological Euler class of V, defined as the composite

Lemma 5.2. Define $f: X^{\prime } \to X \in \mathrm {S}\mathrm {ch}{}_S$ . Then

$$\begin{align*}f^* e(V, E) = e(f^*V, E) \in E^{f^* V}(X^{\prime}). \end{align*}$$

If E is strongly $\mathrm {SL}$ -oriented in the sense of §4.3.3, and hence $\mathrm {SL}^c$ -oriented in the sense of §3.2, then for any relatively oriented vector bundle V over a smooth and proper scheme $X/S$ we obtain an Euler number $n(V, \rho , E) \in E^0(S)$ (see §3.4).

5.2 Integrally defined Euler numbers

Corollary 5.3 Euler numbers are stable under base change

Let E be a strongly $\mathrm {SL}$ -oriented cohomology theory and let V be vector bundle V over a smooth and proper scheme $X/S$ , relatively oriented by $\rho $ . Let $f: S^{\prime } \to S$ be a morphism of schemes. Then

$$\begin{align*}f^* n(V, \rho, E) = n(f^*V, f^*o, f^*E) \in E^0(S^{\prime}). \end{align*}$$

Proposition 5.4. Let d be even or $d=1$ , $X/\mathbb {Z}[1/d]$ smooth and proper, and $V/X$ a relatively oriented vector bundle. Then for any field k with $2d \in k^\times $ , we have

$$\begin{align*}n\left(V_k, \rho, \mathrm{H}\tilde{\mathbb{Z}}\right) \in \mathbb{Z}[\langle -1 \rangle, \langle 2 \rangle, \dots, \langle d \rangle] \subset \mathrm{GW}(k). \end{align*}$$

In fact, there is a formula

$$\begin{align*}n\left(V_k, \rho, \mathrm{H}\tilde{\mathbb{Z}}\right) = \sum_{a \in \mathbb{Z}[1/d!]^\times} n_a \langle a \rangle, \end{align*}$$

which holds over any such field, with the coefficients $n_a \in \mathbb {Z}$ independent of k (and zero for all but finitely many a).

Note that here the assumption that the rank of V equals the dimension of X is included in the hypothesis that $V/X$ is a relatively oriented vector bundle (see Definition 1.5).

Proof. Recall the very effective cover functor $\tilde f_0$ and the truncation in the effective homotopy t-structure $\pi _0^{\mathrm {eff}}$ , for example from [Reference Bachmann5, §§3,4]. We have a diagram of spectra

$$\begin{align*}\mathrm{KO}_k \leftarrow \tilde f_0 \mathrm{KO}_k \to \pi_0^{\mathrm{eff}} \mathrm{KO}_k \leftarrow \pi_0^{\mathrm{eff}} \mathrm{MSL}_k \simeq H\tilde{\mathbb{Z}}; \end{align*}$$

see [Reference Bachmann and Hoyois10, Example 16.34] for the last equivalence. The functors $\tilde f_0$ and $\pi _0^{\mathrm {eff}}$ are lax monoidal in an appropriate sense, so this is a diagram of ring spectra. Moreover, all of the ring spectra are strongly $\mathrm {SL}$ -oriented (via the ring map $\mathrm {MSL}_k \to \pi _0^{\mathrm {eff}} \mathrm {MSL}_k$ ; see also Remark 4.18). Finally, all the maps induce isomorphisms on , essentially by construction. It follows that $n(V_k, \rho , \mathrm {KO}) = n\left (V_k, \rho , H\tilde {\mathbb {Z}}\right ) \in \mathrm {GW}(k)$ . We may thus just as well prove the result for $n(V_k, \rho , \mathrm {KO})$ instead.

If d is even, then we have $\mathrm {KO} \in \mathcal {SH}(\mathbb {Z}[1/d])$ , and by Corollary 5.3 we see that $n(V_k, \rho , \mathrm {KO}) \in im(\mathrm {GW}(\mathbb {Z}[1/d]) \to \mathrm {GW}(k))$ . The result thus follows from Lemma 5.6. If $d=1$ , we use the $\mathrm {SL}$ -oriented ring spectrum $\mathrm {KO}^{\prime } \in \mathcal {SH}(\mathbb {Z})$ from [Reference Bachmann and Hopkins9, §3.8.3]. We find that $n(V_k, \rho , \mathrm {KO})$ is the image of $n(V, \rho , \mathrm {KO}^{\prime }) \in \mathrm {KO}^{\prime {^0}}(\mathbb {Z})$ . This latter group is isomorphic to $\mathrm {GW}(\mathbb {Z})$ by [Reference Bachmann and Hopkins9, Lemma 3.38(2)], whence the result.

Proof. Let $A = \mathbb {Z}[1/d]$ and $U=\operatorname {Spec} A$ . If $d=1$ , this is well known [Reference Milnor and Husemoller59, Theorem II.4.3]. Hence from now on, $1/2 \in A$ . Using Lemma B.2, it suffices to prove the analogous statement for $W(A)$ . Let $I \subset W$ be the fundamental ideal and denote by $I^*$ its powers; we view these as presheaves on U. Consider the commutative graded ring $k_* = H^*_{\mathrm {Nis}}\left (U, I^*/I^{*+1}\right )$ . Of course $k_0 = \mathbb {Z}/2$ . It follows from [Reference Bachmann and Østvær11, Corollary 4.9], [Reference Bachmann7, Theorem 2.1, Lemma 2.7], and [Reference Spitzweck71, Theorem 3.9] that $k_1 \simeq \mathcal O^\times (U)/2$ and $k_n \simeq \mathbb {Z}/2\{(-1)^n\}$ for $n \ge 2$ . We have the classes $\langle a \rangle -1 \in I(U)$ (for $a \in \mathcal O^\times (U)$ ) showing that $H^0_{\mathrm {Nis}}(U, I^n) \to k_n$ is surjective. The short exact sequences

$$\begin{align*}0 \to H^0_{\mathrm{Nis}}\left(U, I^{n+1}\right) \to H^0_{\mathrm{Nis}}(U, I^{n}) \to k_n \end{align*}$$

thus show that $H^0_{\mathrm {Nis}}(U, W)$ is generated by the $\langle a \rangle $ together with $H^0_{\mathrm {Nis}}(U, I^n)$ , for any n. For n sufficiently large, $H^0_{\mathrm {Nis}}(U, I^n) \simeq 2^nH^0_{ret}(U,\mathbb {Z}) \simeq 2^n\mathbb {Z}$ [Reference Bachmann7, Proposition 2.3] is generated by $(\langle -1 \rangle -1)^n$ ; thus $H^0_{\mathrm {Nis}}(U, W)$ is generated by the $\langle a \rangle $ . It remains to observe that $H^0_{\mathrm {Nis}}(U, W) = W(U)$ . This follows from the descent spectral sequence for computing $W(U) = [U, \mathrm {KW}]$ , using the fact that the motivic spectrum $\mathrm {KW}$ has $\underline \pi _* \mathrm {KW} = a_{\mathrm {Nis}} W$ for $* \equiv 0\ \,\pmod {4}$ and $\underline \pi _* \mathrm {KW} = 0$ .

Notation 5.7. Set $A \hookrightarrow \mathbb {R}$ and let V be a relatively oriented, rank n vector bundle on a smooth, proper n-dimensional scheme X over A. We have $\mathrm {GW}(\mathbb {R}) \simeq \mathbb {Z} \oplus \mathbb {Z}\langle -1 \rangle $ . There are thus unique integers $n_{\mathbb {R}}, n_{\mathbb {C}} \in \mathbb {Z}$ such that

$$\begin{align*}n\left(V_{\mathbb{R}}, \rho, H\tilde{\mathbb{Z}}\right) = \frac{ n_{\mathbb{C}} + n_{\mathbb{R}} }{2} + \frac{n_{\mathbb{C}} - n_{\mathbb{R}}}{2}\langle -1 \rangle. \end{align*}$$

The integers $n_{\mathbb {R}}$ and $n_{\mathbb {C}}$ are the Euler numbers of the corresponding real and complex topological vector bundles, respectively, at least when X is projective, justifying the notation. To show this, consider the cycle class map $\mathrm {CH}^\ast (X) \to H^\ast (X(\mathbb {C}),\mathbb {Z})$ from the Chow ring of a smooth $\mathbb {C}$ -scheme X to the singular cohomology of the complex manifold $X(\mathbb {C})$ [Reference Fulton37, Chapter 19]. Furthermore, there are real cycle class maps from oriented Chow rings of $\mathbb {R}$ -smooth schemes X to the singular cohomology of the real manifold $X(\mathbb {R})$ , discussed in [Reference Hornbostel, Wendt, Xie and Zibrowius43] (as well as more refined real cycle class maps defined in [Reference Benoist and Wittenberg13]): for a smooth $\mathbb {R}$ -scheme X and a line bundle $\mathcal {L} \to X$ , consider the real cycle class map

$$ \begin{align*}\widetilde{\mathrm{CH}}^\ast (X, \mathcal{L})\cong H^n\left(X, \mathrm{K}^{\mathrm{MW}}_n\left(\omega_{X/k}\right)\right) \cong H\tilde{\mathbb{Z}}^{L_{\pi}}(X) \to H^\ast(X(\mathbb{C}),\mathbb{Z}(\mathcal{L}))\end{align*} $$

from the oriented Chow groups twisted by $\mathcal {L}$ to the singular homology of the associated local system on $X(\mathbb {R})$ . We use results on the real cycle class map due to Hornbostel, Wendt, Xie, and Zibrowius [Reference Hornbostel, Wendt, Xie and Zibrowius43], including compatibility with push-forwards. Since they only had need of this compatibility in the case of the push-forward by a closed immersion, we first extend it slightly:

Lemma 5.8. Let $\pi : X \to \operatorname {Spec} k$ be the structure map of a smooth, projective scheme X of dimension n over the real numbers $\mathbb {R}$ . Then the following square commutes:

where the vertical maps are the canonical push-forwards and the horizontal maps are the real cycle class maps.

Proof. By [Reference Hornbostel, Wendt, Xie and Zibrowius43, Proposition 6.1], the $\mathbb A^1$ -Euler class $e\left (V, H\tilde {\mathbb {Z}}\right )$ of $V_{\mathbb {R}}$ in the oriented Chow group $H\tilde {\mathbb {Z}}^{V^*}(X_{\mathbb {R}}) $ maps to the topological Euler class of $V(\mathbb {R})$ under the real cycle class map. By Lemma 5.8, it follows that the image of the Euler number $n\left (V,H\tilde {\mathbb {Z}}\right )$ under the real cycle class map is the topological Euler number of $V(\mathbb {R})$ . Under the canonical isomorphism $H^0(\ast , \mathbb {Z}) \cong \mathbb {Z}$ , the real cycle class map $\mathrm {GW}(\mathbb {R}) \to H^0(\ast , \mathbb {Z}) \cong \mathbb {Z}$ is the map taking a bilinear form over $\mathbb {R}$ to its signature. It follows that $n_{\mathbb {R}}$ is the topological Euler number of $V(\mathbb {R})$ .

Let $\gamma : \widetilde {\mathrm {CH}}^\ast (X, \det V^*) \cong H\tilde {\mathbb {Z}}^{V^*}(X_{\mathbb {R}}) \to \mathrm {CH}^{\ast }(X) \to \mathrm {CH}^{\ast }(X_{\mathbb {C}}) \to H^{\ast }(X(\mathbb {C}), \mathbb {Z})$ denote the composition of the canonical map to Chow followed by the (usual) cycle class map, and we similarly have

$$ \begin{align*}\gamma: \mathrm{GW}(\mathbb{R}) \cong \widetilde{\mathrm{CH}}^{0}(\operatorname{Spec} \mathbb{R}) \to H^{0}(\operatorname{Spec} \mathbb{R} (\mathbb{C}), \mathbb{Z}) \cong H^0(\ast, \mathbb{Z}) \cong \mathbb{Z},\end{align*} $$

which sends the class of a bilinear form in $\mathrm {GW}(\mathbb {R})$ to its rank. The cycle class map is compatible with Chern classes ([Reference Fulton37, Proposition 19.1.2]), whence the image of $e\left (V, H\tilde {\mathbb {Z}}\right )$ under $\gamma $ is the topological Euler class of $V(\mathbb {C})$ . The cycle class map commutes with the relevant push-forwards and pullbacks [Reference Fulton37, Corollary 19.2, Example 19.2.1], so we have that $\gamma \left (n\left (V,H\tilde {\mathbb {Z}}\right )\right ) = n_{\mathbb {C}}$ is the topological Euler number of $V(\mathbb {C})$ .

Theorem 5.11. Suppose X is smooth and proper over $\mathbb {Z}[1/2]$ . Let V be a relatively oriented vector bundle on X and let $V_k$ denote the base change of V to k for any field k. Then one of the following is true:

(6) $$ \begin{align} n^{\mathrm{GS}}(V_k, \rho) = \frac{ n_{\mathbb{C}} +n_{\mathbb{R}} }{2} + \frac{n_{\mathbb{C}} - n_{\mathbb{R}}}{2}\langle -1 \rangle\end{align} $$

or

(7) $$ \begin{align} n^{\mathrm{GS}}(V_k, \rho) = \frac{ n_{\mathbb{C}} +n_{\mathbb{R}} }{2} + \frac{n_{\mathbb{C}} - n_{\mathbb{R}}}{2}\langle -1 \rangle + \langle 2 \rangle -1,\end{align} $$

where the same formula holds for all fields k of characteristic $\ne 2$ .

If instead X is smooth and proper over $\mathbb {Z}$ , then equation (6) holds for any field k (including fields k of characteristic $2$ ).

Recall that for the last claim regarding X smooth and proper over $\mathbb {Z}$ , we rely on [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18] (see Remark 5.5).

Proof. First assume that $char(k) \ne 2$ . By Corollary 1.4, we have $n^{\mathrm {GS}}(V_k, \rho ) = n\left (V_k, \rho , H\tilde {\mathbb {Z}}\right )$ . If the base is $\mathbb {Z}$ , then we learn from Proposition 5.4 that there exist $a, b \in \mathbb {Z}$ (independent of k!) such that

$$\begin{align*}n\left(V_k, \rho, \mathrm{H}\tilde{\mathbb{Z}}\right) = a + b\langle -1 \rangle. \end{align*}$$

In $\mathrm {GW}(\mathbb {Z}[1/2]) \hookrightarrow \mathrm {GW}(\mathbb {Q})$ we have the relations

$$\begin{align*}\langle -2 \rangle = 1 + \langle -1 \rangle - \langle 2 \rangle \quad \text{and} \quad 2\langle 2 \rangle = 2 \end{align*}$$

(the former because $\langle a \rangle + \langle -a \rangle = \langle 1 \rangle + \langle -1 \rangle $ for any a, and see, e.g., [Reference Bachmann6, Lemma 42] for the latter). Hence if the base is $\mathbb {Z}[1/2]$ , there exist $a, b \in \mathbb {Z}, c \in \{0,1\}$ (independent of the choice of field k!) such that

$$\begin{align*}n\left(V_k, \rho, \mathrm{H}\tilde{\mathbb{Z}}\right) = a + b\langle -1 \rangle + c \langle 2 \rangle. \end{align*}$$

If the base is $\mathbb {Z}$ , let us put $c=0$ . By construction we have

$$ \begin{align*}n_{\mathbb{R}} = \operatorname{sign} n\left(V_{\mathbb{R}}, \rho, \mathrm{H}\tilde{\mathbb{Z}}\right)=(a+c)-b\end{align*} $$

and

$$ \begin{align*}n_{\mathbb{C}} = \operatorname{rank} n\left(V_{\mathbb{C}}, \rho, \mathrm{H}\tilde{\mathbb{Z}}\right)=(a+c)+b,\end{align*} $$

which determines $a+c$ and b, so that there remain only at most two possible values for $n\left (V_k,\rho ,\mathrm {H}\tilde {\mathbb {Z}}\right )$ .

Now suppose that $char(k) = 2$ (so that in particular the base is $\mathbb {Z}$ ). Since $\langle 1 \rangle = \langle -1 \rangle $ over fields of characteristic $2$ , we need to show that $n^{\mathrm {GS}}(V_k, \rho ) = n_{\mathbb {C}}$ . We may as well assume that $k=\mathbb {F}_2$ . The rank induces an isomorphism $\mathrm {GW}(\mathbb {F}_2) \cong \mathbb {Z}$ (see, e.g., Corollary B.4). Considering the canonical maps

$$\begin{align*}\mathrm{GW}(\mathbb{F}_2) \to \mathrm{KGL}^0(\mathbb{F}_2) \leftarrow \mathrm{KGL}^0(\mathbb{Z}) \to \mathrm{KGL}^0(\mathbb{Z}[1/2]) \leftarrow \mathrm{GW}(\mathbb{Z}[1/2]), \end{align*}$$

in which all but the right-most one are isomorphisms, we get the string of equalities

$$\begin{align*}n\left(V_{\mathbb{F}_2}\right) = n\left(V_{\mathbb{F}_2}, \mathrm{KGL}\right) = n(V_{\mathbb{Z}}, \mathrm{KGL}) = n\left(V_{\mathbb{Z}\left[1/2\right]}, \mathrm{KGL}\right) = rk\left(n\left(V_{\mathbb{Z}\left[1/2\right]}, \mathrm{KO}\right)\right). \end{align*}$$

The result follows.

5.3 Refined Euler classes and numbers

Definition 5.13 refined Euler class

Let $E \in \mathcal {SH}(S)$ be a homotopy ring spectrum, set $X \in \mathrm {S}\mathrm {ch}{}_S$ , and let $V \to X$ be a vector bundle and $\sigma : X \to V$ a section with zero scheme $Z = Z(\sigma )$ . We denote by $e(V, \sigma ) = e(V, \sigma , E) \in E^{V^*}_Z(X)$ the class corresponding to the composite

(see Example 4.2).

Remark 5.15. In [Reference Levine and Raksit56, Definition 3.9] the authors define for an $\mathrm {SL}$ -oriented ring spectrum E and a vector bundle $p: V \to X$ the canonical Thom class $th(V) \in E^{p^*V}_0(V)$ , which one can check coincides with $e(p^*V, \sigma _0)$ , where $\sigma _0$ is the tautological section of $p^*V$ . Since $e(V, \sigma ) = \sigma ^* e(p^*V, \sigma _0)$ , we deduce that

$$\begin{align*}e(V, \sigma) = \sigma^* th(V). \end{align*}$$

Proof. This follows from the commutative diagram

Indeed the composite from the bottom left to the top right along the top left represents $e(V)$ , by Definition 5.1 (note that $\sigma $ is a homotopy inverse to the projection $V \to X$ ), whereas the composite along the bottom right represents $e(V, \sigma )$ with support forgotten, by Definition 5.13 and Example 4.2.

Definition 5.17 refined Euler number

Suppose that $\pi : X \to S$ is smoothable lci, $Z \to S$ is finite, and V is relatively oriented (so in particular, $rk(V) = rk(\mathrm {L}_\pi )$ ). Suppose further that E is $\mathrm {SL}$ -oriented. Then we put

$$\begin{align*}n(V, \sigma, \rho) = n(V, \sigma, \rho, E) = \pi_* e(V, \sigma) \in E^0(S). \end{align*}$$

5.4 Refined Euler classes and the six-functors formalism

We now relate our Euler classes to the six-functors formalism. The following result shows that our refined Euler class coincides with the one defined by Déglise, Jin, and Khan [Reference Déglise, Jin and Khan26, Remark 3.2.10]:

Proposition 5.19. Let $E \in \mathcal {SH}(S)$ be a homotopy ring spectrum, set $X \in {\mathrm {S}\mathrm {m}}_S$ , and let V be a vector bundle over X and $\sigma $ a section of V. Then

$$\begin{align*}e(V, \sigma, E) = \sigma^*z_*(1), \end{align*}$$

where $z: X \to V$ is the zero section and we use the canonical isomorphism $N_z \simeq V$ to form the push-forward $z_*$ .

Proof. Let $p: V \to X$ be the projection and $s_0: V \to p^* V$ the canonical section. Then $\sigma ^*(p^*V, \sigma _0) = (V, s)$ , and hence, using Remark 5.14, it suffices to show that $z_*(1) = e(p^*V, \sigma _0, E)$ . By [Reference Elmanto, Hoyois, Khan, Sosnilo and Yakerson31, last sentence of §2.1.1], we know that

$$\begin{align*}z_*: E^0(X) \simeq E^{z^* p^*V^* + L_z} \to E^{p^*V^*}_X(V) \end{align*}$$

is the purity equivalence. In the case of the zero section of a vector bundle, it just takes the tautological form

and hence indeed sends $1$ to $e(p^*V, \sigma _0)$ .

It follows from the foregoing that Euler classes of vector bundles are determined by Euler classes of vector bundles with nondegenerate sections.