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Published online by Cambridge University Press:  16 July 2021

Tom Bachmann
Mathematisches Institut, LMU Munich, Theresienstr. 39, D-80333 München, Germany (
Kirsten Wickelgren
Department of Mathematics, Duke University, 120 Science Drive, 117 Physics Building, Campus Box 90320, Durham, NC 27708-032, USA (
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We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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1 Introduction

For algebraic vector bundles with an appropriate orientation, there are Euler classes and numbers enriched in bilinear forms. We will start over a field k and then discuss more general base schemes, obtaining integrality results. Let $\mathrm {GW}(k)$ denote the Grothendieck–Witt group of k, defined to be the group completion of the semi-ring of nondegenerate, symmetric, k-valued, bilinear forms (see, e.g., [Reference Lam53]). Let $\langle a \rangle $ in $\mathrm {GW}(k)$ denote the class of the rank $1$ bilinear form $(x,y) \mapsto axy$ for a in $k^*$ .

For a smooth, proper k-scheme $f: X \to \operatorname {Spec} k $ of dimension n, coherent duality defines a trace map $\eta _f: \mathrm {H}^n(X, \omega _{X/k}) \to k$ , which can be used to construct the following Euler number in $\mathrm {GW}(k)$ . Let $V \to X$ be a rank n vector bundle equipped with a relative orientation, meaning a line bundle $\mathcal L$ on X and an isomorphism

$$ \begin{align*}\rho: \det V \otimes \omega_{X/k} \to \mathcal L^{\otimes 2}.\end{align*} $$

For $0 \leq i, j \leq n$ , let $\beta _{i,j}$ denote the perfect pairing

(1) $$ \begin{align} \beta_{i,j}: \mathrm{H}^i\left(X, \wedge^j V^* \otimes \mathcal L\right) \otimes \mathrm{H}^{n-i}\left(X, \wedge^{n-j} V^* \otimes \mathcal L\right)\to k\end{align} $$

given by the composition

$$ \begin{align*}\mathrm{H}^i\left(X, \wedge^j V^* \otimes \mathcal L\right) \otimes \mathrm{H}^{n-i}\left(X, \wedge^{n-j} V^* \otimes \mathcal L\right) \stackrel{\cup}{\to} \mathrm{H}^n\left(X, \wedge^n V^* \otimes \mathcal L^{\otimes 2}\right) \stackrel{\rho}{\to} \mathrm{H}^n(X, \omega_{X/k}) \stackrel{\eta_f}{\to} k.\end{align*} $$

For $i = n-i$ and $j=n-j$ , note that $\beta _{i,j}$ is a bilinear form on $ \mathrm {H}^i\left (X, \wedge ^j V^* \otimes \mathcal L\right )$ . Otherwise, $\beta _{i,j} \oplus \beta _{n-i,n-j}$ determines the bilinear form on $\mathrm {H}^i\left (X, \wedge ^j V^* \otimes \mathcal L\right ) \oplus \mathrm {H}^{n-i}\left (X, \wedge ^{n-j} V^* \otimes \mathcal L\right )$ . The alternating sum

$$ \begin{align*}n^{\mathrm{GS}}(V) : = \sum_{0 \leq i, j \leq n} (-1)^{i+j}\beta_{i,j}\end{align*} $$

thus determines an element of $\mathrm {GW}(k)$ , which we will call the Grothendieck–Serre duality or coherent duality Euler number. Note that $\beta _{i,j} \oplus \beta _{n-i,n-j}$ in $\mathrm {GW}(k)$ is an integer multiple of h, where h denotes the hyperbolic form $h = \langle 1 \rangle + \langle -1 \rangle $ , with Gram matrix

$$ \begin{align*}h= \left[ {\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}}\right].\end{align*} $$

This notion of the Euler number was suggested by M. J. Hopkins, J.-P. Serre, and A. Raksit, and developed by M. Levine and Raksit for the tangent bundle in [Reference Levine and Raksit56].

For a relatively oriented vector bundle V equipped with a section $\sigma $ with only isolated zeros, an Euler number $n^{\mathrm {PH}}(V, \sigma )$ was defined in [Reference Kass and Wickelgren49, Section 4] as a sum of local indices:

$$ \begin{align*}n^{\mathrm{PH}}(V, \sigma) = \sum_{x : \sigma(x) = 0} \mathrm{ind}^{\mathrm{PH}}_x \sigma .\end{align*} $$

The index $\mathrm {ind}^{\mathrm {PH}}_x \sigma $ can be computed explicitly with a formula of Scheja and Storch [Reference Scheja and Storch67] or of Eisenbud and Levine/Khimshiashvili [Reference Eisenbud and Levine29] (see §§2.4 and 2.3) and is also a local degree [Reference Kass and Wickelgren48] (this is discussed further in §7). For example, when x is a simple zero of $\sigma $ with $k(x)=k$ , the index is given by a well-defined Jacobian $\mathrm {Jac}\sigma $ of $\sigma $ ,

$$ \begin{align*}\mathrm{ind}^{\mathrm{PH}}_x \sigma= \langle \mathrm{Jac}\sigma(x) \rangle,\end{align*} $$

illustrating the relation with the Poincaré–Hopf formula for topological vector bundles. (For the definition of the Jacobian, see the beginning of §6.2.) In [Reference Kass and Wickelgren49, Section 4, Corollary 36], it was shown that $n^{\mathrm {PH}}(V, \sigma ) = n^{\mathrm {PH}}(V, \sigma ')$ when $\sigma $ and $\sigma '$ are in a family over $\mathbb A^1_L$ of sections with only isolated zeros, where L is a field extension with $[L:k]$ odd. We strengthen this result by equating $n^{\mathrm {PH}}(V, \sigma )$ and $n^{\mathrm {GS}}(V)$ ; this is the main result of §2.

Theorem 1.1 see §2.4

Let k be a field and $V \to X$ be a relatively oriented, rank n vector bundle on a smooth, proper k-scheme of dimension n. Suppose V has a section $\sigma $ with only isolated zeros. Then

$$ \begin{align*}n^{\mathrm{PH}}(V, \sigma) = n^{\mathrm{GS}}(V).\end{align*} $$

In particular, $n^{\mathrm {PH}}(V, \sigma )$ is independent of the choice of $\sigma $ .

Remark 1.2 Theorem 1.1 strengthens [Reference Bethea, Kass and Wickelgren15], removing its hypothesis (2) entirely. It also simplifies the proofs of [Reference Kass and Wickelgren49, Theorem 1] and [Reference Srinivasan and Wickelgren72, Theorems 1 and 2]: it is no longer necessary to show that the sections of certain vector bundles with nonisolated isolated zeros are of codimension $2$ , as in [Reference Kass and Wickelgren49, Lemmas 54, 56, and 57] and in [Reference Srinivasan and Wickelgren72, Lemma 1], because $n^{\mathrm {PH}}(V, \sigma )$ is independent of $\sigma $ .

1.1 Sketch proof and generalizations

The proof of Theorem 1.1 proceeds in three steps:

  1. (0) For a section $\sigma $ of V, we define an Euler number relative to the section using coherent duality and denote it by $n^{\mathrm {GS}}(V, \sigma , \rho )$ . If $\sigma = 0$ , we recover the absolute Euler number $n^{\mathrm {GS}}(V, \rho )$ , essentially by construction.

  2. (1) For two sections $\sigma _1, \sigma _2$ , we show that $n^{\mathrm {GS}}(V, \sigma _1, \rho ) = n^{\mathrm {GS}}(V, \sigma _2, \rho )$ . To prove this, one can use homotopy invariance of Hermitian K-theory or show that $n^{\mathrm {GS}}(V, \sigma _1, \rho ) = n^{\mathrm {GS}}(V, \rho )$ by showing an instance of the principle that alternating sums, like Euler characteristics, are unchanged by passing to the homology of a complex.

  3. (2) If a section $\sigma $ has isolated zeros, then $n^{\mathrm {GS}}(V, \sigma , \rho )$ can be expressed as a sum of local indices $\mathrm {ind}_{Z/S}(\sigma )$ , where Z is (a clopen component of) the zero scheme of $\sigma $ .

  4. (3) For Z a local complete intersection in affine space–that is, in the presence of coordinates – we compute the local degree explicitly and identify it with the Scheja–Storch form [Reference Ananyevskiy2, Reference Scheja and Storch67].

Taken together, these steps show that $n^{\mathrm {GS}}(V, \rho )$ is a sum of local contributions given by Scheja–Storch forms, which is essentially the definition of $n^{\mathrm {PH}}(V, \rho )$ .

These arguments can be generalized considerably, replacing the Grothendieck–Witt group $\mathrm {GW}$ by a more general cohomology theory E. We need E to admit transfers along proper lci morphisms of schemes, and an $\mathrm {SL}^c$ -orientation (see §3 for more details). Then for step (0) one can define an Euler class $e(V, \sigma , \rho )$ as $z^*\sigma _*(1)$ , where z is the zero section. Step (2) is essentially formal; the main content is in steps (1) and (3). Step (1) becomes formal if we assume that E is $\mathbb A^1$ -invariant. In particular, steps (0)–(2) can be performed for $\mathrm {SL}$ -oriented cohomology theories represented by motivic spectra; this is explained in §§3, 4, and 5.

It remains to find a replacement for step (3). We offer two possibilities: in §7 we show that, again in the presence of coordinates, the local indices can be identified with appropriate $\mathbb A^1$ -degrees. On the other hand, in §8 we show that for $E = \mathrm {KO}$ the motivic spectrum corresponding to Hermitian K-theory, the local indices are again given by Scheja–Storch forms. This implies the following:

Corollary 1.3 see Corollary 8.2 and Definition 3.10

Let $S=Spec(k)$ , where k is a field of characteristic $\ne 2$ .Footnote 1 Let $\pi : X \to k$ be smooth and $V/X$ a relatively oriented vector bundle with a nondegenerate section $\sigma $ . Write $\varpi : Z=Z(\sigma ) \to k$ for the vanishing scheme (which need not be smooth). Then

$$\begin{align*}n^{\mathrm{PH}}(V, \sigma) = \varpi_*(1) \in \mathrm{KO}^0(k) = \mathrm{GW}(k). \end{align*}$$

Here we have used the lci push-forward

$$\begin{align*}\varpi_*: \mathrm{KO}^0(Z) \stackrel{\rho}{\simeq} \mathrm{KO}^{L_\varpi}(Z) \to \mathrm{KO}^0(k) \end{align*}$$

of Déglise, Jin, and Khan [Reference Déglise, Jin and Khan26]. If, moreover, X is proper, then $\varpi _*(1)$ also coincides with $\pi _*z^*z_*(1)$ , where $z: X \to V$ is the zero section (see Corollaries 5.18 and 5.21 and Proposition 5.19). This provides an alternative proof that $n^{\mathrm {PH}}(V,\sigma )$ is independent of the choice of $\sigma $ (under our assumption on k).

Another important example is when E is taken to be the motivic cohomology theory representing Chow–Witt groups. This recovers the Barge–Morel Euler class $e^{\mathrm {BM}}(V)$ in $\widetilde {\mathrm {CH}}^r(X, \det V^*)$ [Reference Barge and Morel12], which is defined for a base field of characteristic not $2$ . Suppose that $\rho $ is a relative orientation of V and $\pi : X \to \operatorname {Spec} k$ is the structure map.

Corollary 1.4. Let k be a field of characteristic $\ne 2$ . Then $\pi _* e^{\mathrm {BM}}(V, \rho ) = n^{\mathrm {GS}}(V, \rho )$ in $\mathrm {GW}(k)$ .

Proof. We have $e^{\mathrm {BM}}(V, \rho ) = e\left (V, \rho , H\tilde {\mathbb {Z}}\right )$ ; indeed, by Proposition 5.19, $e\left (V, \rho , H\tilde {\mathbb {Z}}\right )$ can be computed in terms of push-forward along the zero section of V, and exactly the same is true for $e^{\mathrm {BM}}$ by definition [Reference Barge and Morel12, §2.1]. We also have $n^{\mathrm {GS}}(V,\rho ) = n(V, \rho , \mathrm {KO})$ ; indeed, the right-hand side is represented by the natural symmetric bilinear form on the cohomology of the Koszul complex by Example 8.1, and this is essentially the definition of $n^{\mathrm {GS}}(V,\rho )$ .

It thus suffices to prove that $n\left (V, \rho , H\tilde {\mathbb {Z}}\right ) = n(V, \rho , \mathrm {KO}) \in \mathrm {GW}(k)$ . Consider the span of ring spectra $H\tilde {\mathbb {Z}} \leftarrow \tilde f_0 \mathrm {KO} \to \mathrm {KO}$ as in the proof of Proposition 5.4. It induces an isomorphism on $\pi _0(\mathord -)(k)$ , namely with $\mathrm {GW}(k)$ in all cases. The desired equality follows from the naturality of the Euler numbers.

(An alternative argument proceeds as follows: It suffices to prove that $\pi _* e^{\mathrm {BM}}(V,\rho )$ and $n^{\mathrm {GS}}(V, \rho )$ have the same image in $\mathrm {W}(k)$ and $\mathbb {Z}$ . The image of $n^{\mathrm {GS}}(V, \rho )$ in $\mathrm {W}(k$ ) is given by $n(V, \rho , \mathrm {KW})$ ; for this we need only show that $e(V,\rho ,\mathrm {KW})$ is represented by the Koszul complex, which is Example 5.20. It will thus be enough to show that $n(V, \rho , H\mathbb {Z}) = n(V, \rho , \mathrm {KGL})$ and $n\left (V, \rho , \underline {W}\left [\eta ^{\pm }\right ]\right ) = n(V, \rho , \mathrm {KW})$ ; this follows as before by considering the spans $H\mathbb {Z} \leftarrow \mathrm {kgl} \to \mathrm {KGL}$ and $\underline {W}\left [\eta ^{\pm }\right ] \leftarrow \mathrm {KW}_{\ge 0} \to \mathrm {KW}$ .)Footnote 2

The left-hand side is the Euler class studied by M. Levine in [Reference Levine55]. We do not compare these Euler classes with the obstruction-theoretic Euler class of [Reference Morel61, Chapter 8]. Asok and Fasel show that the latter agrees with $\pi _* e^{\mathrm {BM}}(V, \rho )$ up to a unit in $\mathrm {GW}(k)$ [Reference Asok and Fasel3].

1.2 Applications

It is straightforward to see that Euler numbers for cohomology theories are stable under base change (see Corollary 5.3). This implies that in considering vector bundles on varieties which are already defined over, for example, $\operatorname {Spec}(\mathbb {Z}[1/2])$ , the possible Euler numbers are constrained to live in $\mathrm {GW}(\mathbb {Z}[1/2]) = \mathbb {Z}[\langle -1 \rangle , \langle 2 \rangle ] \subset \mathrm {GW}(\mathbb {Q})$ . Using novel results on Hermitian K-theory [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18] allows one to use the base scheme $\operatorname {Spec} \mathbb {Z}$ as well. Proposition 5.4 contains both of these cases, and the $\mathbb {Z}[1/2]$ case is independent of [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18]. It follows that for relatively oriented bundles over $\mathbb {Z}$ the Euler numbers can be read off from topological computations (Proposition 5.9). Over $\mathbb {Z}[1/2]$ the topological Euler numbers of the associated real and complex vector bundles, together with one further algebraic computation over some field in which $2$ is not a square, determine the Euler number (and this is again independent of [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18]); see Theorem 5.11.

We use this to compute a weighted count of d-dimensional hyperplanes in a general complete intersection

$$ \begin{align*}\left\{f_1 = \cdots = f_j \right\} \hookrightarrow \mathbb P^n_k\end{align*} $$

over a field k. This count depends only on the degrees of the polynomials $f_i$ and not on the $f_i$ themselves: it is determined by associated real and complex counts, for any d and degrees such that the expected variety of d-planes is $0$ -dimensional and the associated real count is defined. This is Corollary 6.9. For example, combining with results of Finashin and Kharlamov over $\mathbb {R}$ [Reference Finashin and Kharlamov34], we have that $160,839 \langle 1 \rangle + 160,650 \langle -1 \rangle $ and

$$ \begin{align*} &32,063,862,647,475,902,965,720,976,420,325 \langle 1 \rangle\\ &\quad {}+ 32,063,862,647,475,902,965,683,320,692,800 \langle -1 \rangle\end{align*} $$

are arithmetic counts of the $3$ -planes in a $7$ -dimensional cubic hypersurface and in a $16$ -dimensional degree $5$ hypersurface, respectively (see Example 6.13). This builds on results of Finashin and Kharlamov [Reference Finashin and Kharlamov34], J. L. Kass and the second author of the present paper [Reference Kass and Wickelgren49], M. Levine [Reference Levine54], S. McKean [Reference McKean58], Okonek and Teleman [Reference Okonek and Teleman62], S. Pauli [Reference Pauli66], J. Solomon [Reference Solomon70], P. Srinivasan and the second author [Reference Srinivasan and Wickelgren72], and M. Wendt [Reference Wendt74].

1.3 Notation and conventions

1.3.1 Grothendieck duality

We believe that if $f: X \to Y$ is a morphism of schemes which is locally of finite presentation, then there is a well-behaved adjunction

$$\begin{align*}f_!: D_{\mathrm{qcoh}}(X) \leftrightarrows D_{\mathrm{qcoh}}(Y): f^! \end{align*}$$

between the associated derived ( $\infty $ -)categories of unbounded complexes of $\mathcal O_X$ -modules with quasi-coherent homology sheaves. Unfortunately, we are not aware of any references in this generality. Instead, whenever mentioning a functor $f^!$ , we implicitly assume that X and Y are separated and of finite type over some Noetherian scheme S. In this situation, the functor $f^!$ is constructed for homologically bounded-above complexes in [73, Tag 0A9Y] (see also [ Reference Conrad24, Reference Hartshorne40]), and this is all we will use.

1.3.2 Vector bundles

We identify locally free sheaves and vector bundles covariantly, via the assignment

$$\begin{align*}\mathcal E \leftrightarrow \operatorname{Spec}(\operatorname{Sym}(\mathcal E^*)). \end{align*}$$

While it can be convenient to (not) pass to duals here (as in, e.g., [Reference Déglise, Jin and Khan26]), we do not do this, because it confuses the first author terribly.

1.3.3 Regular sequences and immersions

Following, for example, [Reference Berthelot, Grothendieck et, Illusie, Jouanolou, Jussila, Kleiman, Raynaud et and Serre14], by a regular immersion of schemes we mean what is called a Koszul-regular immersion in [73, Tag 0638]–that is, a morphism which is locally a closed immersion cut out by a Koszul-regular sequence. Moreover, by a regular sequence we will always mean a Koszu-regular sequence [73, Tag 062D], and we reserve the term strongly regular sequence for the usual notion. A strongly regular sequence is regular [73, Tag 062F], whence a strongly regular immersion is regular. In locally Noetherian situations, regular immersions are strongly regular [73, Tags 063L].

1.3.4 Cotangent complexes

For a morphism $f: X \to Y$ , we write $L_f$ for the cotangent complex. Recall that if f is smooth, then $L_f \simeq \Omega _f$ , whereas if f is a regular immersion, then $L_f \simeq C_f[1]$ , where $C_f$ denotes the conormal bundle.

1.3.5 Graded determinants

We write $\widetilde \det : K(X) \to \mathrm {Pic}(D(X))$ for the determinant morphism from Thomason–Trobaugh K-theory to the groupoid of graded line bundles. If C is a perfect complex, then we write $\widetilde \det C$ for the determinant of the associated K-theory point. We write $\det C \in \mathrm {Pic}(X)$ for the ungraded determinant.

Given an lci morphism f, we set $\omega _f = \det L_f$ and $\widetilde \omega _f = \widetilde \det L_f$ .

We systematically use graded determinants throughout the article; for example, we have the following compact definition of a relative orientation:

Definition 1.5. Let $\pi : X \to S$ be an lci morphism and V a vector bundle on X. By a relative orientation of $V/X/S$ we mean a choice of line bundle $\mathcal L$ on X and an isomorphism

$$\begin{align*}\rho: \underline{\operatorname{Hom}}\left(\widetilde\det V^*, \widetilde \omega_{X/S}\right) \xrightarrow{\simeq} \mathcal L^{\otimes 2}. \end{align*}$$

Note that if $\pi $ is smooth, this just means that the locally constant functions $x \mapsto \operatorname {rank}(V_x)$ and $x \mapsto \dim \pi ^{-1}(\pi (x))$ on X agree, and that we are given an isomorphism $\mathcal L^{\otimes 2} \simeq \omega _{X/S} \otimes \det V$ . Hence we recover the definition from [Reference Kass and Wickelgren49, Definition 17].

2 Equality of coherent duality and Poincaré–Hopf Euler numbers

We prove Theorem 1.1 in this section.

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 $\bigoplus _{i,j} H^i\left (X, \Lambda ^j V^* \otimes \mathcal L\right )$ .

Proposition 2.3. For any section $\sigma $ , we have $n^{\mathrm {GS}}(V, \sigma , \rho ) = n^{\mathrm {GS}}(V, \rho ) \in \mathrm {GW}(k)$ .

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

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


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

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.

Remark 2.7. Admitting a version of Hermitian K-theory which is $\mathbb A^1$ -invariant on regular schemes and has proper push-forwards, one can give an alternative proof of Proposition 2.3 by considering the Koszul complex with respect to the section $t\sigma $ on $\mathbb A^1 \times X$ . While we believe such a theory exists, at the time of writing there is no reference for this in characteristic $2$ , so we chose to present our argument instead.

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.

Remark 2.10. A symmetric bilinear form $\phi $ on the derived category $D(S)$ is usually not a very sensible notion. We offer three ways around this:

  1. (1) If $1/2 \in S$ , we could look at the image of $\phi $ in the Balmer–Witt group of S.

  2. (2) If $\phi $ happens to be concentrated in degree $0$ , it corresponds to a symmetric bilinear form on a vector bundle on S, which is a sensible invariant.

  3. (3) If $S = \operatorname {Spec}(k)$ is the spectrum of a field, then $D(S)$ is equivalent to the category of graded vector spaces, and we can split $\phi $ into components by degree and consider

    $$\begin{align*}cl(\phi) := \left[H^0(\phi)\right] + \sum_{i> 0} (-1)^i \left[H^i(\phi) \oplus H^{-i}(\phi)\right] \in \mathrm{GW}(k). \end{align*}$$

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 $(f g)_* \simeq f_* g_*$ .

Proof. The main point is that $\eta _{f, \mathcal L} \circ f_*\left (\eta _{g, \mathcal M [n]}\right ) = \eta _{fg, \mathcal L}$ . The categorical details are worked out in the reference.

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

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

Proof. By Lemma 2.12 we have $\varpi _* = f_* i_*$ . Proposition 2.11 and the projection formula imply that $i_*\left (i^* \mathcal L \otimes i^* \mathcal L \to i^* \mathcal L^{\otimes 2}\right ) = \beta _{V,\sigma ,\rho }$ . We conclude by Example 2.9.

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 $Z = Z(f_1, \dots , f_n)$ .

Remark 2.15. Since $Z \to X$ is quasi-finite, Lemma B.5 shows that $f_1, \dots , f_n$ is a regular sequence and $Z \to X$ is flat, so finite locally free (being finite and finitely presented [73, Tag 02KB]).

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 $(f_i)$ .

Remark 2.19. The theorem asserts in particular that the isomorphism $\tilde \Delta $ , and hence the section $\Delta $ , is independent of the choice of the $a_{ij}$ .

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].

Example 2.21. Suppose that $Z \to S$ is an isomorphism (where $Z=Z(F)$ as before, so that the diagonal $\delta : Z \to Z \times _S Z$ is also an isomorphism. Then $\langle - \vert - \rangle ^{\mathrm {SS}}$ is just the rank $1$ bilinear form corresponding to multiplication by $\delta ^*(\Delta ) \in \mathcal O_Z \simeq \mathcal O_S$ . In other words, using Remark 2.17, $\langle - \vert - \rangle ^{\mathrm {SS}}$ identifies with $(x,y) \mapsto (\mathrm {Jac} F) xy$ .

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.

Definition 2.22 compare [Reference Kass and Wickelgren49, Definition 17]

By a system of Nisnevich coordinates around z we mean a Zariski open neighborhood U of z in X, and an étale map $\varphi : U \to \mathbb A^n_k$ such that the extension of residue fields $k(\varphi (z)) \subseteq k(z)$ is an isomorphism.

Proposition 2.23. When $n>0$ , there exists a system of Nisnevich coordinates around every closed point z of X.

Proof. When k is infinite, this follows from [Reference Knus52, Chapter 8. Proposition 3.2.1]. When $k(z)/k$ is separable, for instance when k is finite, this is [Reference Kass and Wickelgren49, Lemma 18].

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:

Definition 2.24. The Poincaré–Hopf Euler number $n^{\mathrm {PH}}(V, \sigma )$ of V with respect to $\sigma $ is $n^{\mathrm {PH}}(V, \sigma ) = \sum _{z : \sigma (z) = 0} \mathrm {ind}_z^{\mathrm {PH}} \sigma $ .

Proof of Theorem 1.1. By Proposition 2.3 we have $n^{\mathrm {GS}}(V) = n^{\mathrm {GS}}(V, \sigma )$ , where the orientation has been suppressed from the notation but is indeed present. Using equation (3), it is thus enough to show that $\mathrm {ind}_z(\sigma ) = \mathrm {ind}_z^{\mathrm {PH}}(\sigma )$ . This follows from Theorem 2.18. One needs to be careful about the trivializations used in defining the various push-forward maps; this is ensured precisely by the condition that the tautological section is a square. The details of this argument are spelled out more carefully in the proof of Proposition 3.13.□

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:

Definition 2.25. Let X be a scheme, V a vector bundle on X, and $\sigma $ a section of V.

  1. (1) We call $\sigma $ nondegenerate if it locally corresponds to a regular sequence.

  2. (2) Given another scheme S and a morphism $\pi : X \to S$ , we call $\sigma $ very nondegenerate (with respect to $\pi $ ) if it is nondegenerate and the zero locus $Z(\sigma )$ is finite and locally free over S.

Remark 2.26. Suppose that X is smooth over S and $rk(V) = \dim X/S$ .

  1. (1) If $S=\operatorname {Spec}(k)$ is the spectrum of a field, then $Z(\sigma ) \to \operatorname {Spec}(k)$ is quasi-finite if and only if it is finite locally free, if and only if $\sigma $ is locally given by a regular sequence. In other words, $\sigma $ is nondegenerate if and only if it is very nondegenerate, if and only if $Z(\sigma ) \to Z$ is quasi-finite.

  2. (2) In general, $\sigma $ is nondegenerate as soon as $Z(\sigma ) \to S$ is quasi-finite, and very nondegenerate if and only if $Z(\sigma ) \to S$ is finite (see Lemma B.5).

Example 2.27. If $\sigma $ is a nondegenerate section, then precomposition with $\sigma $ induces an isomorphism $\operatorname {Hom}(V, \mathcal O) \simeq C_{Z/X}$ . In particular, $N_{Z/X} \simeq V$ and $L_{Z/X} \simeq V^*[1]$ .

Definition 2.28. Let X be a smooth S-scheme and let $V \to X$ be a vector bundle, relatively oriented by $\rho $ . Let $\sigma $ be a very nondegenerate section of V and let Z be a closed and open subscheme of the zero locus $Z(\sigma )$ of $\sigma $ . By a system of coordinates for $(V,X,\sigma ,\rho ,Z)$ we mean an open neighborhood U of Z in X, an étale map $\varphi : U \to \mathbb A^n_S$ , a trivialization $\psi : V\rvert _U \simeq \mathcal O_U^n$ , and a section $\sigma ^{\prime } \in \mathcal O_{\mathbb A_S^n}^n(\varphi (U))$ , such that the following conditions hold:

  1. (1) $Z = Z(\sigma \rvert _U) \cong Z(\sigma ^{\prime })$ ,

  2. (2) $\det (\sigma \rvert _Z) = \det (\varphi ^* \sigma ^{\prime }\rvert _Z) \in \det N_{Z/X}$ , and

  3. (3) the canonical section of $\omega _{X/S} \otimes \det V\rvert _Z \cong \mathcal L^{\otimes 2}\rvert _Z$ determined by $\psi $ and $\varphi $ corresponds to the square of a section of $\mathcal L\rvert _Z$ .

Here for conditions (2) and (3) we used Example 2.27.

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.

Remark 2.30. Since $\varpi $ is finite locally free, $\varpi _*$ preserves vector bundles. In particular, $\mathrm {ind}_{Z}(\sigma ) \in \mathrm {BL}^{\mathrm {naive}}(S)$ is a symmetric bilinear form on a vector bundle, as opposed to just on a complex up to homotopy. (See also Remark 2.10.)

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 $\langle - \vert - \rangle ^{\mathrm {SS}}(\varphi (U), \sigma , S) \in \mathrm {BL}^{\mathrm {naive}}(S)$ .

Proposition 2.31. The form $\langle - \vert - \rangle ^{\mathrm {SS}}(\varphi (U), \sigma ^{\prime }, S)$ coincides (up to isomorphism) with $\mathrm {ind}_Z(\sigma )$ . In particular, its isomorphism class is independent of the choice of coordinates.

Proof. The argument is the same as in the proofs of Theorem 1.1 and Proposition 3.13.

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 Cohomology theories for schemes

3.1 Introduction

In order to generalize the results from the previous sections, we find it useful to introduce the concept of a cohomology theory twisted by K-theory. We do not seek here to axiomatize all the relevant data, just to introduce a common language for similar phenomena.

Definition 3.1. Let S be a scheme and $\mathcal C \subset \mathrm {S}\mathrm {ch}{}_S$ a category of schemes. Denote by $\mathcal C^L$ the category of pairs $(X, \xi )$ , where $X \in \mathcal C$ and $\xi \in K(X)$ (i.e., a point in the K-theory space of X), and morphisms those maps of schemes compatible with the K-theory points.Footnote 5 By a cohomology theory E over S (for schemes in $\mathcal C$ ) we mean a presheaf of sets on $\mathcal C^L$ –that is, a functor

$$\begin{align*}E: \left(\mathcal C^L\right)^{\mathrm{op}} \to \mathrm{S}\mathrm{et}, (X, \xi) \mapsto E^\xi(X). \end{align*}$$

To illustrate the flavor of cohomology theory we have in mind, we begin with two examples.

Example 3.2. We can set either of the following:

  1. (1) $E^\xi (X) = \mathrm {CH}^{rk(\xi )}(X)$ , the Chow group of algebraic cycles up to rational equivalence of the appropriate codimension, or

  2. (2) $E^\xi (X) = \mathrm {GW}\left (X, \widetilde \det \xi \right )$ , the Grothendieck–Witt group of symmetric bilinear perfect complexes for the duality $\underline {\operatorname {Hom}}\left (\mathord -, \widetilde \det \xi \right )$ (see, e.g., [Reference Schlichting68]).

Warning 3.3. For cohomology theories with values in a $1$ -category (like sets), in this definition we can safely replace $K(X)$ by its truncation $K(X)_{\le 1}$ –that is, the ordinary $1$ -groupoid of virtual vector bundles. However, we can in general not replace it by just the set $K_0(X)$ . In other words, if (say) V is a vector bundle on X and $\phi $ an automorphism of V, then there is an induced automorphism

$$\begin{align*}E(\phi): E^{V}(X) \to E^{V}(X) \end{align*}$$

which may or may not be trivial. For example, in the case $E = \mathrm {GW}$ as before, if $V = \mathcal O$ is trivial and $\phi $ corresponds to $a \in \mathcal O^\times (X)$ , then $E(\phi )$ is given by multiplication by $\langle a \rangle \in \mathrm {GW}(X)$ .

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.

Remark 3.4.

  1. (1) We have defined a morphism of cohomology theories as a natural transformation of functors valued in sets. Whether or not such a transformation respects additional structure (abelian group structures, orientations, transfers, etc.) must be investigated in each case.

  2. (2) In many cases (in particular in the presence of homotopy invariance), $\mathrm {SL}$ -oriented theories are also canonically $\mathrm {SL}^c$ -oriented (see Proposition 4.19).

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 Fasel20Reference 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

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

Remark 3.6. The twist by $V^*$ (instead of V) in this definition may seem peculiar. It ultimately comes from our choice of covariant (instead of contravariant) equivalence between locally free sheaves and vector bundles, whereas a contravariant equivalence is used in the motivic Thom spectrum functor and hence in the definition of twists.

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

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

Proof. By assumption, transfers are compatible with composition and additive along disjoint unions. The result follows.

Example 3.12. If $S = \operatorname {Spec}(k)$ is the spectrum of a field, then Z is $0$ -dimensional and hence decomposes into a finite disjoint union of ‘fat points’. In particular, the Euler number is expressed as a sum of local indices, in bijection with the zeros of our nondegenerate section.

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 Cohomology theories represented by motivic spectra

We recall some background material about motivic extraordinary cohomology theories–that is, theories represented by motivic spectra. We make essentially no claim to originality.

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


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


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.

4.2.2 Functoriality in E

If $\alpha : E \to F \in \mathcal {SH}(S)$ is any morphism, then there is an induced morphism $\alpha _*: E^\xi _Z(X) \to F^\xi _Z(X)$ . This just follows from the fact that $\pi ^*$ and so on are functors.

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 $\mathrm {S}\mathrm {ch}{}_S$ ,


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]):

Proposition 4.19 Ananyevskiy

Let $E \in \mathcal {SH}(S)$ be $\mathrm {SL}$ -oriented and $\mathcal L_1, \mathcal L_2, \mathcal L_3$ be line bundles on X. Suppose an isomorphism $\mathcal L_1 \simeq \mathcal L_2 \otimes \mathcal L_3^{\otimes 2}$ .

  1. (1) There is a canonical equivalence $\Sigma ^{\mathcal L_1} E \simeq \Sigma ^{\mathcal L_2} E$ , compatible with base change.

  2. (2) There is a canonical isomorphism $E^n_Z(X, \mathcal L_1) \simeq E^n_Z(X, \mathcal L_2)$ , compatible with base change.

In particular, the cohomology theory represented by E is $\mathrm {SL}^c$ -oriented.

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 Euler classes for representable theories

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

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

Proof. This is immediate.

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

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

Proof. This holds because all our constructions are stable under base change–see in particular Lemma 4.6(2) (for the compatibility of Gysin maps with pullback, which applies since $X \to S$ is smooth), Propositions 4.19 and 4.13(3) (ensuring that the identification $E^{V^*}(X) \simeq E^{L_\pi }(X)$ is compatible with base change), and Lemma 5.2 (for the compatibility of Euler classes with base change).

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).

Remark 5.5. If $d=1$ , Proposition 5.4 relies on the novel results about Hermitian K-theory of the integers from [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18]. In the following proof, this is manifested in the dependence of [Reference Bachmann and Hopkins9, Lemma 3.38(2)] on these results. We will later use Proposition 5.4 for the $d=1$ case of Theorem 5.11, whence this result is also using [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18] in an essential way. For $d \geq 2$ , the proof is independent of [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle18].

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.

Lemma 5.6. Let d be even or