2.1.1

We provide background on rings of motives over schemes, algebraic spaces, and algebraic stacks. The case of stacks was first studied by Joyce [Reference JoyceJoy07b], who constructed these rings of motives, along with several variations, and called them rings of stack functions.

To avoid repetition, we only state the majority of the definitions and results for stacks, with the understanding that schemes and algebraic spaces are special cases of algebraic stacks.

2.1.2 Stacks with affine stabilizers.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ . We say that $\mathscr{X}\,$ has affine stabilizers if, for any field $k$ and any point $x \in \mathscr{X}\, (k)$ , the stabilizer group of $\mathscr{X}\,$ at $x$ is an affine algebraic group over $k$ .

2.1.3 Rings of motives.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ with affine stabilizers. Define the Grothendieck ring of varieties over $\mathscr{X}\,$ to be the abelian group

(2.1.3.1) \begin{align} K_{{\rm var}} (\mathscr{X}\,) = \mathop {\hat {\bigoplus }}_{Z \to \mathscr{X}} {} \mathbb{Z} \cdot [Z] \, \Big / {\sim } , \end{align}

where we run through all morphisms $Z \to \mathscr{X}$ with $Z$ a $\mathbb{K}$ -variety, and $\hat {\oplus }$ means we take the set of locally finite sums, that is, possibly infinite sums $\sum _{Z \to \mathscr{X}} n_Z \cdot [Z]$ , such that for each quasi-compact open substack $\mathscr{U}\, \subset \mathscr{X}$ , there are only finitely many $Z$ such that $n_Z \neq 0$ and $Z \times _{\mathscr{X}} \mathscr{U}\, \neq \varnothing$ . The relation $\sim$ is generated by $[Z] \sim [Z^{\prime}] + [Z \setminus Z^{\prime}]$ for closed subschemes $Z^{\prime} \subset Z$ .

One can define multiplication on $K_{{\rm var}} (\mathscr{X}\,)$ by taking the fibre product over $\mathscr{X}\,$ , making it into a commutative ring, which may be non-unital when $\mathscr{X}\,$ is not an algebraic space. It is also a (possibly non-unital) commutative $K_{{\rm var}} (\mathbb{K})$ -algebra, where $K_{{\rm var}} (\mathbb{K}) = K_{{\rm var}} (\operatorname {Spec} \mathbb{K})$ , with the action given by the product.

Let $\mathbb{L} = [{\mathbb{A}}^1] \in K_{{\rm var}} (\mathbb{K})$ , and define rings of motives over $\mathscr{X}\,$ ,

(2.1.3.2) \begin{align} {\rm M} (\mathscr{X}\,) & = K_{{\rm var}} (\mathscr{X}\,) \underset {K_{{\rm var}} (\mathbb{K})}{\mathbin {\hat {\otimes }}} K_{{\rm var}} (\mathbb{K}) \, [\mathbb{L}^{-1}] \big / (\mathbb{L} - 1) \textrm{-torsion} , \end{align}

(2.1.3.3) \begin{align} \hat {{\rm M}} (\mathscr{X}\,) & = K_{{\rm var}} (\mathscr{X}\,) \underset {K_{{\rm var}} (\mathbb{K})}{\mathbin {\hat {\otimes }}} K_{{\rm var}} (\mathbb{K}) \, [\mathbb{L}^{-1}, (\mathbb{L}^k - 1)^{-1}],\end{align}

where $\hat {\otimes }$ means we take the set of locally finite sums, that is, possibly infinite sums $\sum _{Z \to \mathscr{X}} [Z] \otimes a_Z$ , such that the family of $Z$ with $a_Z \neq 0$ is locally finite on $\mathscr{X}\,$ , and, in (2.1.3.3), we invert $\mathbb{L}^k - 1$ for all $k \geqslant 1$ .

2.1.4 Motives of algebraic spaces and stacks.

Let $\mathscr{X}\,$ be as above. For an algebraic space $Z$ and a morphism $Z \to \mathscr{X}$ of finite type, one can also assign a class $[Z] \in K_{{\rm var}} (\mathscr{X}\,)$ , extending the usual definition for varieties, since $Z$ can be stratified by varieties.

Furthermore, as in Joyce [Reference JoyceJoy07b] or Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, § 5.3], for any finite type morphism of algebraic stacks $\mathscr{Z} \to \mathscr{X}$ , where $\mathscr{Z}$ has affine stabilizers, one can assign a class $[\mathscr{Z}\,\,] \in \hat {{\rm M}} (\mathscr{X}\,)$ , which agrees with the usual one when $\mathscr{Z}$ is a variety, and satisfies the relation $[\mathscr{Z}\,\,] = [\mathscr{Z}^{\kern2pt\prime}] + [\mathscr{Z} \setminus \mathscr{Z}^{\kern2pt\prime}]$ for closed substacks $\mathscr{Z}^{\kern2pt\prime} \subset \mathscr{Z}$ . In particular, the class $[\mathscr{X}\,] \in \hat {{\rm M}} (\mathscr{X}\,)$ is the multiplicative unit of the ring $\hat {{\rm M}} (\mathscr{X}\,)$ .

For an algebraic stack $\mathscr{X}\,$ over $\mathbb{K}$ of finite type, with affine stabilizers, one thus has a class $[\mathscr{X}\,] \in \hat {{\rm M}} (\mathbb{K})$ , called the motive of $\mathscr{X}\,$ .

2.1.5 Pullbacks and pushforwards.

Let $\mathscr{X}, \mathscr{Y}\,$ be algebraic stacks over $\mathbb{K}$ with affine stabilizers, and let $f \colon \mathscr{X} \to \mathscr{Y}\,$ be a morphism.

There is a pullback map

\begin{align*} f^* \colon \hat {{\rm M}} (\mathscr{Y}{\kern3pt}) \longrightarrow \hat {{\rm M}} (\mathscr{X}\,) \ \end{align*}

which is an $\hat {{\rm M}} (\mathbb{K})$ -algebra homomorphism, given on generators by $f^* [Z] = [Z \times _{\mathscr{Y}} \mathscr{X}\,]$ , where the right-hand side is defined as in § 2.1.4. Pulling back respects composition of morphisms. If, moreover, $f$ is representable, then there are pullback maps

\begin{align*} f^* \colon K_{{\rm var}} (\mathscr{Y}{\kern3pt}) & \longrightarrow K_{{\rm var}} (\mathscr{X}\,) , \\ f^* \colon {\rm M} (\mathscr{Y}{\kern3pt}) & \longrightarrow {\rm M} (\mathscr{X}\,) , \end{align*}

which are $K_{{\rm var}} (\mathbb{K})$ - and ${\rm M} (\mathbb{K})$ -algebra homomorphisms, respectively, and are defined similarly.

On the other hand, if $f$ is of finite type, then there are pushforward maps

\begin{align*} f_! \colon K_{{\rm var}} (\mathscr{X}\,) & \longrightarrow K_{{\rm var}} (\mathscr{Y}{\kern3pt}) , \\ f_! \colon {\rm M} (\mathscr{X}\,) & \longrightarrow {\rm M} (\mathscr{Y}{\kern3pt}) , \\ f_! \colon {\hat {{\rm M}} (\mathscr{X}\,)} & \longrightarrow {\hat {{\rm M}} (\mathscr{Y}{\kern3pt})} , \end{align*}

which are $K_{{\rm var}} (\mathbb{K})$ -, ${\rm M} (\mathbb{K})$ -, and $\hat {{\rm M}} (\mathbb{K})$ -module homomorphisms, respectively, given on generators by $f_! [Z] = [Z]$ . Pushing forward respects composition of morphisms.

In particular, when $\mathscr{X}\,$ is of finite type, pushing forward along the structure morphism $\mathscr{X} \to \operatorname {Spec} \mathbb{K}$ is sometimes called motivic integration, and denoted by

\begin{align*} \int _{\mathscr{X}} {} (-) \colon \hat {{\rm M}} (\mathscr{X}\,) \longrightarrow \hat {{\rm M}} (\mathbb{K}) \ . \end{align*}

2.1.6 Base change and projection formulae.

Suppose we have a pullback diagram

where $\mathscr{X}, \mathscr{Y}, \mathscr{X}^{\kern1.6pt\prime}, \mathscr{Y}^{\,\prime}$ are algebraic stacks over $\mathbb{K}$ with affine stabilizers, and $f$ is of finite type. Then we have the base change formula

(2.1.6.1) \begin{align} g^* \circ f_! = f^{\prime}_! \circ g^{\prime *} \end{align}

on $\hat {{\rm M}} (-)$ , as in [Reference JoyceJoy07b, Theorem 3.5]. Moreover, if $g$ is representable, then this also holds for $K_{{\rm var}} (-)$ and ${\rm M} (-)$ .

Let $f \colon \mathscr{X} \to \mathscr{Y}\,$ be as above. We have the projection formula

(2.1.6.2) \begin{align} f_! (a \cdot f^* (b)) = f_! (a) \cdot b \end{align}

for all $a \in \hat {{\rm M}} (\mathscr{X}\,)$ and $b \in \hat {{\rm M}} (\mathscr{Y}{\kern3pt})$ , which can be verified directly on generators. Moreover, if $f$ is representable, then this also holds for $K_{{\rm var}} (-)$ and ${\rm M} (-)$ .

2.1.7 Motives of principal bundles.

Following Serre [Reference SerreSer58, § 4], an algebraic group $G$ over $\mathbb{K}$ is special if all principal $G$ -bundles over a $\mathbb{K}$ -scheme are Zariski locally trivial. For example, the groups ${\rm GL} (n)$ and $\mathbb{G}_{{\rm a}}$ are special; semidirect products of special groups are special; and disconnected groups are not special. See also Joyce [Reference JoyceJoy07b, Definition 2.1] for related discussions.

For a special group $G$ and a principal $G$ -bundle $\pi \colon \mathscr{Y} \to \mathscr{X}$ , where $\mathscr{X}\,$ is an algebraic stack over $\mathbb{K}$ with affine stabilizers, we have the relation

(2.1.7.1) \begin{align} \pi _! \circ \pi ^* = [G] \cdot {{\rm id}} \end{align}

in $K_{{\rm var}} (\mathscr{X}\,)$ , ${\rm M} (\mathscr{X}\,)$ , and $\hat {{\rm M}} (\mathscr{X}\,)$ , which can be deduced from the definition of a special group. In particular, we have $[\mathscr{Y}] = [G] \cdot [\mathscr{X}\,]$ in $\hat {{\rm M}} (\mathscr{X}\,)$ .

Note, however, that such a principal $G$ -bundle $\pi$ itself is not necessarily Zariski locally trivial, but it becomes so after a base change to any scheme.

The relation (2.1.7.1) is not necessarily true if $G$ is not special. For example, consider the principal $\mathbb{Z}_2$ -bundle $\mathbb{G}_{{\rm m}} \to \mathbb{G}_{{\rm m}}$ given by $t \mapsto t^2$ . Then the equality cannot hold, since $[\mathbb{G}_{{\rm m}}] \neq 2 \cdot [\mathbb{G}_{{\rm m}}]$ .

2.1.8 The Euler characteristic.

As in Joyce [Reference JoyceJoy07b, Example 6.3], there is a ring map

\begin{align*} \chi \colon {\rm M} (\mathbb{K}) \longrightarrow \mathbb{Z} , \end{align*}

sending each generator $[Z]$ to its Euler characteristic, and sending $\mathbb{L}$ to $1$ . This extends naturally to a map $\chi \colon \hat {{\rm M}} (\mathbb{K}) \to \mathbb{Q} \cup \{ \infty \}$ , sending $1 / (1 + \mathbb{L} + \cdots + \mathbb{L}^{k - 1})$ to $1/k$ for each $k \geqslant 1$ , and sending elements not in ${\rm M} (\mathbb{K}) [(1 + \mathbb{L} + \cdots + \mathbb{L}^{k - 1})^{-1} : k \geqslant 1]$ to $\infty$ .

For an algebraic stack $\mathscr{X}\,$ over $\mathbb{K}$ of finite type, with affine stabilizers, we have a map

\begin{align*} \int _{\mathscr{X}} {} (-) \, d \chi \colon \hat {{\rm M}} (\mathscr{X}\,) \longrightarrow \mathbb{Q} \cup \{ \infty \} \ \end{align*}

defined by pushing forward along $\mathscr{X} \to \operatorname {Spec} \mathbb{K}$ and then taking the Euler characteristic. We have $\int _{\mathscr{X}} {} a \, d \chi \in \mathbb{Z}$ for all $a \in {\rm M} (\mathscr{X}\,)$ .

2.1.9 Constructible functions.

For an algebraic stack $\mathscr{X}\,$ over $\mathbb{K}$ , a constructible function on $\mathscr{X}\,$ is a map of sets

\begin{align*} a \colon | \mathscr{X} | \longrightarrow \mathbb{Z} , \end{align*}

where $| \mathscr{X} |$ is the underlying topological space of $\mathscr{X}\,$ , such that for any $c \in \mathbb{Z}$ , the preimage $a^{-1} (c)$ is a locally constructible subset of $| \mathscr{X} |$ . The abelian group of constructible functions on $\mathscr{X}\,$ is denoted by ${\rm CF} (\mathscr{X}\,)$ .

There is an Euler characteristic map

\begin{align*} \chi \colon \tilde {{\rm M}} (\mathscr{X}\,) \longrightarrow {\rm CF} (\mathscr{X}\,) , \end{align*}

where $\tilde {{\rm M}} (\mathscr{X}\,) \subset \hat {{\rm M}} (\mathscr{X}\,)$ is the ${\rm M} (\mathbb{K})$ -subalgebra generated by classes $[\mathscr{Z}\,\,]$ for representable morphisms $\mathscr{Z} \to \mathscr{X}$ , and $\chi$ is given by taking the fibrewise Euler characteristic, as in Joyce [Reference JoyceJoy07b, Definition 3.2], where this map was denoted by $\pi _{\mathscr{X}}^{{{\rm stk}}}$ .

One can also define pullback and pushforward maps on ${\rm CF} (-)$ for representable morphisms, where pushing forward also requires the morphism to be of finite type. These maps are compatible with the map $\chi$ , and satisfy the base change and projection formulae as in § 2.1.6 for representable morphisms.

2.1.10 Rings of monodromic motives.

Let ${{\hat {\mu }}} = \lim \mu _n$ be the projective limit of the groups $\mu _n$ of roots of unity. For a $\mathbb{K}$ -scheme $Z$ , a good ${\hat {\mu }}$ -action on $Z$ is one that factors through $\mu _n$ for some $n$ such that each orbit is contained in an affine open subscheme.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ with affine stabilizers. Define the monodromic Grothendieck ring of varieties over $\mathscr{X}\,$ to be the abelian group

(2.1.10.1) \begin{align} K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,) = \mathop {\hat {\bigoplus }}_{Z \to \mathscr{X}} {} \mathbb{Z} \cdot [Z]^{{\hat {\mu }}} \, \Big / {\sim } , \end{align}

where $\hat {\oplus }$ indicates taking locally finite sums as in § 2.1.3, and we sum over all morphisms $Z \to \mathscr{X}$ with $Z$ a $\mathbb{K}$ -variety with a good ${\hat {\mu }}$ -action that is compatible with the trivial ${\hat {\mu }}$ -action on $\mathscr{X}\,$ . The relation $\sim$ is generated by $[Z]^{{\hat {\mu }}} \sim [Z^{\prime}]^{{\hat {\mu }}} + [Z \setminus Z^{\prime}]^{{\hat {\mu }}}$ for ${\hat {\mu }}$ -invariant closed subschemes $Z^{\prime} \subset Z$ , and $[Z \times V]^{{\hat {\mu }}} \sim [Z \times {\mathbb{A}}^n]^{{\hat {\mu }}}$ for a ${\hat {\mu }}$ -representation $V$ of dimension $n$ , where the projections to $\mathscr{X}\,$ factor through $Z$ , and ${\hat {\mu }}$ acts on ${\mathbb{A}}^n$ trivially. See Looijenga [Reference LooijengaLoo02, § 5] and Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, § 5].

Using the $K_{{\rm var}} (\mathbb{K})$ -module structure on $K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,)$ , we define rings of monodromic motives over $\mathscr{X}\,$ ,

(2.1.10.2) \begin{align} {\rm M}^{{\hat {\mu }}} (\mathscr{X}\,) & = K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,) \underset {K_{{\rm var}} (\mathbb{K})}{\mathbin {\hat {\otimes }}} K_{{\rm var}} (\mathbb{K}) \, [\mathbb{L}^{-1}] \big / (\mathbb{L} - 1) \mbox{-}{\rm torsion} , \end{align}

(2.1.10.3) \begin{align} \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}\,) & = K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,) \underset {K_{{\rm var}} (\mathbb{K})}{\mathbin {\hat {\otimes }}} K_{{\rm var}} (\mathbb{K}) \, [\mathbb{L}^{-1}, (\mathbb{L}^k - 1)^{-1}] \Big / {\approx } , \end{align}

where $\hat {\otimes }$ indicates taking locally finite sums as in § 2.1.3. The ring structures and the relation $\approx$ are defined below.

We consider the multiplication on $K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,)$ denoted by ‘ $\odot$ ’ in [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Definition 5.3]; see there for the definition. We will denote this by ‘ ${} \cdot {}$ ’. This makes $K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,)$ into a ring, possibly non-unital when $\mathscr{X}\,$ is not an algebraic space. Note that this is not given by the fibre product, although the latter does define a different ring structure. The relation $\approx$ is defined as in [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Definitions 5.5 and 5.13], denoted there by ‘ $I^{{\rm st}, {{\hat {\mu }}}}_{{\mathscr{X}}}$ ’, and is imposed so that the map $\Upsilon$ in § 2.1.11 below respects the tensor product.

There is an element

\begin{align*} \mathbb{L}^{1/2} = 1 - [\mu _2]^{{\hat {\mu }}} \in K_{{\rm var}}^{{\hat {\mu }}} (\mathbb{K}) , \end{align*}

where ${\hat {\mu }}$ acts on $\mu _2$ non-trivially. It satisfies $(\mathbb{L}^{1/2})^2 = \mathbb{L}$ . We also write $\mathbb{L}^{-1/2} = \mathbb{L}^{-1} \cdot \mathbb{L}^{1/2} \in {\rm M}^{{\hat {\mu }}} (\mathbb{K})$ .

There are natural maps

\begin{align*} \iota ^{{\hat {\mu }}} \colon K_{{\rm var}} (\mathscr{X}\,) \longrightarrow K_{{\rm var}}^{{\hat {\mu }}} (\mathscr{X}\,) , \quad \iota ^{{\hat {\mu }}} \colon {\rm M} (\mathscr{X}\,) \longrightarrow {\rm M}^{{\hat {\mu }}} (\mathscr{X}\,) , \quad \iota ^{{\hat {\mu }}} \colon \hat {{\rm M}} (\mathscr{X}\,) \longrightarrow \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}\,) , \end{align*}

given on generators by $[Z] \mapsto [Z]$ , with the trivial ${\hat {\mu }}$ -action on $Z$ . These are $K_{{\rm var}} (\mathbb{K})$ -, ${\rm M} (\mathbb{K})$ -, and $\hat {{\rm M}} (\mathbb{K})$ -algebra homomorphisms, respectively.

One can define pullback and pushforward maps on $K_{{\rm var}}^{{\hat {\mu }}} (-)$ , ${\rm M}^{{\hat {\mu }}} (-)$ , and $\hat {{\rm M}}{}^{{\hat {\mu }}} (-)$ , similar to the cases of $K_{{\rm var}} (-)$ , ${\rm M} (-)$ , and $\hat {{\rm M}} (-)$ . These maps satisfy the base change and projection formulae in § 2.1.6, and the principal bundle relation in § 2.1.7.

There is also the Euler characteristic map $\chi \colon {\rm M}^{{\hat {\mu }}} (\mathbb{K}) \to \mathbb{Z}$ , which is a ring homomorphism, defined by taking the Euler characteristic of the underlying non-monodromic motive. From this, one can define analogues of the operations in §§ 2.1.8 and 2.1.9 for ${\rm M}^{{\hat {\mu }}} (-)$ and $\hat {{\rm M}}{}^{{\hat {\mu }}} (-)$ . In particular, we have $\chi (\mathbb{L}^{1/2}) = -1$ .

2.1.11 Motives of double covers.

For a principal $\mu _2$ -bundle $\mathscr{P} \to \mathscr{X}$ , there is a class

\begin{align*} \Upsilon (\mathscr{P}) = \mathbb{L}^{-1/2} \cdot ([\mathscr{X}\,] - [\mathscr{P}\,]^{{\hat {\mu }}}) \in \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}\,) , \end{align*}

where ${\hat {\mu }}$ acts on $\mathscr{P}$ via the $\mu _2$ -action and the class $[\mathscr{P}\,]^{{\hat {\mu }}}$ is defined in a similar way to § 2.1.4 when $\mathscr{P}$ is not a variety. See [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Definitions 5.5 and 5.13] for more details.

Note that $\Upsilon$ commutes with pullbacks, by definition. Also, we have the relation

(2.1.11.1) \begin{align} \Upsilon (\mathscr{P}_1 \otimes \mathscr{P}_2) = \Upsilon (\mathscr{P}_1) \cdot \Upsilon (\mathscr{P}_2) \end{align}

for principal $\mu _2$ -bundles $\mathscr{P}_1, \mathscr{P}_2 \to \mathscr{X}$ , where $\mathscr{P}_1 \otimes \mathscr{P}_2$ is also a principal $\mu _2$ -bundle. See [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Definitions 5.5 and 5.13].

2.2.1

We now discuss the descent properties of the rings of motives defined above. While constructible functions ${\rm CF} (-)$ descend under any reasonable topology, descent for rings of motives such as $\hat {{\rm M}} (-)$ is more restrictive. For example, pulling back along the double cover $(-)^2 \colon \mathbb{G}_{{\rm m}} \to \mathbb{G}_{{\rm m}}$ is not injective on motives, since the classes of the trivial double cover $\mathbb{G}_{{\rm m}} \times \mu _2 \to \mathbb{G}_{{\rm m}}$ and the non-trivial double cover $\mathbb{G}_{{\rm m}} \to \mathbb{G}_{{\rm m}}$ get identified after pulling back. Therefore, rings of motives do not satisfy étale descent.

However, we show in Theorem 2.2.3 below that these rings of motives do satisfy descent under the Nisnevich topology.

2.2.2 The Nisnevich topology.

Recall that for an algebraic space $X$ , a Nisnevich cover of $X$ is a family of étale morphisms $(f_i \colon X_i \to X)_{i \in I}$ , such that for each point $x \in X$ , there exists $i \in I$ and a point $x^{\prime} \in X_i$ such that $f_i (x^{\prime}) = x$ , and $f_i$ induces an isomorphism on residue fields at $x^{\prime}$ and $x$ .

Let $\mathscr{X}\,$ be an algebraic stack. Define a Nisnevich cover of $\mathscr{X}\,$ to be a representable étale cover $(f_i \colon \mathscr{X}_i \to \mathscr{X}\,)_{i \in I}$ such that its base change to any algebraic space is a Nisnevich cover of algebraic spaces. See also Choudhury, Deshmukh and Hogadi [Reference Choudhury, Deshmukh and HogadiCDH23, Definition 1.2 ff.].

For example, for an integer $n \gt 1$ , the morphism $* \to [* / \mu _n]$ is not a Nisnevich cover, since its base change $\mathbb{G}_{{\rm m}} \to \mathbb{G}_{{\rm m}}$ , $t \mapsto t^n$ is not a Nisnevich cover.

Algebraic spaces over $\mathbb{K}$ (assumed locally of finite type) admit Nisnevich covers by affine $\mathbb{K}$ -varieties, as can be deduced from Knutson [Reference KnutsonKnu71, II, Theorem 6.4].

2.2.3 Theorem.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ with affine stabilizers, and let $(f_i \colon \mathscr{X}_i \to \mathscr{X}\,)_{i \in I}$ be a Nisnevich cover. Then the map

\begin{align*} (f_i^*)_{i \in I} \colon \hat {{\rm M}} (\mathscr{X}\,) \longrightarrow \operatorname {eq} \biggl ( \prod _{i \in I} \hat {{\rm M}} (\mathscr{X}_i) \;\substack{\displaystyle\longrightarrow\\[-1pt] \displaystyle\longrightarrow}\; \prod _{i, j \in I} \hat {{\rm M}} (\mathscr{X}_i \times _{\mathscr{X}} \mathscr{X}_j) \biggr ) \end{align*}

is an isomorphism, where the right-hand side is the equalizer of the two maps induced by pulling back along projections from each $\mathscr{X}_i \times _{\mathscr{X}} \mathscr{X}_j$ to $\mathscr{X}_i$ and $\mathscr{X}_j$ , respectively.

The same also holds for $\hat {{\rm M}}{}^{{\hat {\mu }}} (-)$ in place of $\hat {{\rm M}} (-)$ . Moreover, if $\mathscr{X}\,$ is an algebraic space, then the same holds for $K_{{\rm var}} (-),$ ${\rm M} (-),$ $K_{{\rm var}}^{{\hat {\mu }}} (-),$ and ${\rm M}^{{\hat {\mu }}} (-)$ .

2.2.4 Quotient stacks and fundamental stacks.

We now discuss classes of algebraic stacks that can be covered by quotient stacks, which will be used in the sequel. See § 2.2.5 below for examples of stacks satisfying these properties.

A quotient stack over $\mathbb{K}$ is an algebraic stack of the form $[U / G]$ , where $U$ is an algebraic space over $\mathbb{K}$ acted on by an algebraic group $G \simeq {\rm GL} (n)$ for some $n$ . Equivalently, one can allow $G$ to be any linear algebraic group, since if one chooses an embedding $G \hookrightarrow {\rm GL} (n)$ then $[U / G] \simeq [(U \times ^G {\rm GL} (n)) / {\rm GL} (n)]$ .

A fundamental stack over $\mathbb{K}$ is a quotient stack of the form $[U / G]$ , where $U$ is an affine $\mathbb{K}$ -variety acted on by an algebraic group $G \simeq {\rm GL} (n)$ for some $n$ . Equivalently, one can allow $G$ to be any reductive group, by a similar argument to that above; see the proof of [Reference Alper, Hall and RydhAHR20, Corollary 4.14]. This terminology is from Alper, Hall and Rydh [Reference Alper, Hall and RydhAHR25].

An algebraic stack over $\mathbb{K}$ is Nisnevich locally a quotient stack if it admits a Nisnevich cover by quotient stacks. This class of stacks is also discussed by Choudhury, Deshmukh and Hogadi [Reference Choudhury, Deshmukh and HogadiCDH23], who call them cd-quotient stacks.

An algebraic stack over $\mathbb{K}$ is Nisnevich locally fundamental if it admits a Nisnevich cover by fundamental stacks. This implies being Nisnevich locally a quotient stack.

Similarly, we define stacks that are étale locally a quotient stack or étale locally fundamental by requiring representable étale covers with the corresponding properties.

These properties are satisfied by a large class of stacks, which we discuss below.

2.2.5 Local structure theorems.

A series of local structure theorems for algebraic stacks by Alper, Hall and Rydh [Reference Alper, Hall and RydhAHR20, Reference Alper, Hall and RydhAHR25] can be used to produce covers of algebraic stacks by quotient and fundamental stacks.

The first result follows from [Reference Alper, Hall and RydhAHR20, Theorem 1.1], and is stated in Alper, Halpern-Leistner and Heinloth [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Remarks 2.6 and 2.7]. The second result is [Reference Alper, Hall and RydhAHR25, Theorem 6.1].

2.3.1 Idea.

The motivic nearby and vanishing cycle maps considered below are based on the idea of Milnor fibres. It is perhaps more straightforward to explain this in the analytic setting.

For this purpose, let $X$ be a complex manifold, with a smooth metric $d \colon X \times X \to \mathbb{R}_{\geqslant 0}$ , and let $f \colon X \to \mathbb{C}$ be a holomorphic function. Let $x \in X$ be a point such that $f (x)=0$ . Let $0 \lt \delta \ll \varepsilon \ll 1$ be small positive numbers, and consider the map

\begin{align*} X^\times _{\delta , \varepsilon } (x) = B_{\varepsilon } (x) \cap f^{-1} (D^\times _\delta ) \overset {f}{\longrightarrow } D^\times _\delta , \end{align*}

where $D^\times _\delta = \{ z \in \mathbb{C} \mid 0 \lt |z| \lt \delta \}$ . This map is a topological fibration, and its fibre ${\rm MF}_f (x)$ is called the Milnor fibre of $f$ at $x$ . The cohomology of ${\rm MF}_f (x)$ carries the action of the monodromy operator induced by this fibration.

We will consider below the motivic analogue of the above construction, with $X$ a smooth variety over a field $\mathbb{K}$ . The Milnor fibre is then replaced by the motivic Milnor fibre of Denef and Loeser [Reference Denef and LoeserDL98, Reference Denef, Loeser, Miró-Roig, Verdera and Xambó-DescampsDL01, Reference Denef and LoeserDL02], which is a monodromic motive on $X_0$ . This construction is closely related to Donaldson–Thomas theory; see Behrend [Reference BehrendBeh09], Joyce and Song [Reference Joyce and SongJS12, § 4], and Kontsevich and Soibelman [Reference Kontsevich and SoibelmanKS08, § 4] for more details.

2.3.2 The motivic Milnor fibre.

Let $X$ be a smooth, irreducible $\mathbb{K}$ -variety, and let $f \,:\, X \to {{\mathbb{A}}^1}$ be a morphism. Write $X_0 = {f^{-1}} (0)$ . Following Denef and Loeser [Reference Denef and LoeserDL98, Reference Denef, Loeser, Miró-Roig, Verdera and Xambó-DescampsDL01, Reference Denef and LoeserDL02], we define the motivic Milnor fibre of $f$ , which is an element

\begin{align*} {\rm MF}_f \in {\rm M}^{{\hat {\mu }}} (X_0) , \end{align*}

as follows.

If $f$ is constant, define ${{\rm MF}_f} = 0$ . Otherwise, choose a resolution $\pi \colon \tilde {X} \to X$ of $f$ , in the sense that $\tilde {X}$ is a smooth, irreducible $\mathbb{K}$ -variety, $\pi$ is a proper morphism that restricts to an isomorphism on $\pi ^{-1} (X \setminus X_0)$ , and $\pi ^{-1} (X_0)$ is a simple normal crossings divisor in $\tilde {X}$ . See, for example, Kollár [Reference KollárKol07] for the existence and properties of such resolutions.

Let $(E_i)_{i \in J}$ be the irreducible components of $\pi ^{-1} (X_0)$ , and write $N_i$ for the multiplicity of $E_i$ in the divisor of $f \circ \pi$ on $\tilde {X}$ . For a non-empty subset $I \subset J$ , write $E^\circ _I = \bigcap _{i \in I} E_i \setminus \bigcup _{i \notin I} E_i$ . Let $m_I = \gcd _{i \in I} N_i$ , and define an $m_I$ -fold cover $\tilde {E}^\circ _I \to E^\circ _I$ as follows. For each open set $U \subset \tilde {X}$ such that $f \circ \pi = u v^{m_I}$ on $U$ for $u \colon U \to {\mathbb{A}}^1 \setminus \{ 0 \}$ and $v \colon U \to {\mathbb{A}}^1$ , define the restriction of $\tilde {E}^\circ _I$ on $E^\circ _I \cap U$ as

(2.3.2.1) \begin{align} \tilde {E}^\circ _I |_{E^\circ _I \cap U} = \bigl \{ (z, y) \in {\mathbb{A}}^1 \times (E^\circ _I \cap U) \bigm | z^{m_I} = u^{-1} \bigr \} \ . \end{align}

Since $E^\circ _I$ can be covered by such open sets $U$ , (2.3.2.1) can be glued together to obtain a cover $\tilde {E}^\circ _I \to E^\circ _I$ , with a natural $\mu _{{m_I}}$ -action given by scaling the $z$ -coordinate, which induces a ${\hat {\mu }}$ -action on $\tilde {E}^\circ _I$ . The motivic Milnor fibre ${\rm MF}_f$ is then given by

(2.3.2.2) \begin{align} {\rm MF}_f = \sum _{\varnothing \neq I \subset J} {} (1 - \mathbb{L})^{|I| - 1} \, \big[\tilde {E}^\circ _I\big]^{{\hat {\mu }}} \ . \end{align}

It can be shown [Reference Denef, Loeser, Miró-Roig, Verdera and Xambó-DescampsDL01, Definition 3.8] that this is independent of the choice of the resolution $\pi$ .

2.3.3 Nearby and vanishing cycles.

Let $X$ be a $\mathbb{K}$ -variety, and let $f \colon X \to {{\mathbb{A}}^1}$ be a morphism. Write $X_0 = {f^{-1}} (0)$ .

Define the nearby cycle map of $f$ , denoted by

\begin{align*} \Psi _f \colon {\rm M} (X) \longrightarrow {\rm M}^{{\hat {\mu }}} (X_0) , \end{align*}

to be the unique ${\rm M} (\mathbb{K})$ -linear map such that for any smooth, irreducible $\mathbb{K}$ -variety $Z$ and any proper morphism $\rho \colon Z \to X$ , we have

\begin{align*} \Psi _f ([Z]) = (\rho _0)_! \, ({\rm MF}_{f \circ \rho }) \in {\rm M}^{{\hat {\mu }}} (X_0) , \end{align*}

where $\rho _0 \colon Z_0 \to X_0$ is the restriction of $\rho$ to $Z_0 = (f \circ \rho )^{-1} (0)$ , and ${\rm MF}_{f \circ \rho } \in {\rm M}^{{\hat {\mu }}} (Z_0)$ is the motivic Milnor fibre of $f \circ \rho$ . It follows from Bittner [Reference BittnerBit05, Claim 8.2] that the map $\Psi _f$ is well-defined.

Define the vanishing cycle map of $f$ to be the map

\begin{align*} \Phi _f = \Psi _f - \iota ^{{\hat {\mu }}} \circ i^* \colon {\rm M} (X) \longrightarrow {\rm M}^{{\hat {\mu }}} (X_0) , \end{align*}

where $i \colon X_0 \hookrightarrow X$ is the inclusion, and $\iota ^{{\hat {\mu }}} \colon {\rm M} (X_0) \hookrightarrow {\rm M}^{{\hat {\mu }}} (X_0)$ is as in § 2.1.10.

2.3.4 For algebraic spaces.

We now generalize the above construction from varieties to algebraic spaces.

As in Bittner [Reference BittnerBit05, Theorem 8.4], the nearby and vanishing cycle maps are compatible with pulling back along smooth morphisms. In particular, they define morphisms $\Psi , \Phi \colon {\rm M} (-) \to {\rm M}^{{\hat {\mu }}} ((-)_0)$ of sheaves on the category of $\mathbb{K}$ -varieties over ${\mathbb{A}}^1$ , with the Nisnevich topology. Since algebraic spaces admit Nisnevich covers by affine $\mathbb{K}$ -varieties, as mentioned in § 2.2.2, these morphisms of sheaves induce maps on their evaluations on algebraic spaces over $\mathbb{K}$ .

In other words, for an algebraic space $X$ over $\mathbb{K}$ and a morphism $f \colon X \to {\mathbb{A}}^1$ , we have defined nearby and vanishing cycle maps

\begin{align*} \Psi _f, \Phi _f \colon {\rm M} (X) \longrightarrow {\rm M}^{{\hat {\mu }}} (X_0) \ . \end{align*}

We state some of their properties below.

2.3.5 Theorem.

Let $X, Y$ be algebraic spaces over $\mathbb{K}$ .

1. (i) Let $g \colon Y \to X$ be a proper morphism, and $f \colon X \to {\mathbb{A}}^1$ a morphism. Then we have the following commutative diagram.

   (2.3.5.1)

   

2. (ii) Let $g \colon Y \to X$ be a smooth morphism, and $f \colon X \to {\mathbb{A}}^1$ a morphism. Then we have the following commutative diagram.

   (2.3.5.2)

   

2.4.1 Assumptions on the stack.

From now on, we assume that $\mathscr{X}\,$ is an algebraic stack over $\mathbb{K}$ that is Nisnevich locally a quotient stack in the sense of § 2.2.4.

Note that this assumption is weaker than that in Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, § 5], where the stack was assumed to be Zariski locally a quotient stack.

2.4.2 Theorem.

Let $\mathscr{X}\,$ be as in § 2.4.1 , and let $f \colon \mathscr{X} \to {\mathbb{A}}^1$ be a morphism. Write $\mathscr{X}_0 = {f^{-1}} (0)$ . Then there is a unique $\hat {{\rm M}} (\mathbb{K})$ -linear map

\begin{align*} \Psi _f \colon \hat {{\rm M}} (\mathscr{X}\,) \longrightarrow \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}_0) , \end{align*}

called the nearby cycle map of $f$ , such that for any $\mathbb{K}$ -variety $Y$ and any smooth morphism $g \colon Y \to \mathscr{X}$ , we have a commutative diagram

(2.4.2.1)

where the right-hand map is defined in § 2.3.3.

We then define the vanishing cycle map of $f$ to be the map

\begin{align*} \Phi _f = \Psi _f - \iota ^{{\hat {\mu }}} \circ i^* \colon \hat {{\rm M}} (\mathscr{X}\,) \longrightarrow \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}_0) , \end{align*}

where $i \colon \mathscr{X}_0 \hookrightarrow \mathscr{X}$ and $\iota ^{{\hat {\mu }}} \colon \hat {{\rm M}} (\mathscr{X}_0) \hookrightarrow \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}_0)$ are the inclusions.

Proof. Let $(j_i \colon \mathscr{X}_i \to \mathscr{X}\,)_{i \in I}$ be a Nisnevich cover, where each $\mathscr{X}_i \simeq [U_i / G_i]$ , with $U_i$ an algebraic space over $\mathbb{K}$ , acted on by a group $G_i \simeq {\rm GL} (n_i)$ for some $n_i$ . Let $\pi _i \colon U_i \to \mathscr{X}_i$ be the projection.

First, note that the condition on $\Psi _f$ implies that the same condition holds when $Y$ is an algebraic space, with the right-hand map in (2.4.2.1) defined in § 2.3.4. This can be seen by passing to a Nisnevich cover of $Y$ by $\mathbb{K}$ -varieties, and applying Theorem 2.2.3 to this cover.

To define the map $\Psi _f$ , by Theorem 2.2.3 it is enough to define it on each $\mathscr{X}_i$ , and then verify that they agree on overlaps. Let $a \in \hat {{\rm M}} (\mathscr{X}\,)$ be an element. We define the element ${\Psi _f (a)} \in \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}_0)$ by giving its pullbacks ${\Psi _f (a)_i} = j_i^* \circ {\Psi _f (a)} \in \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}_{i, 0})$ for each $i$ , where $\mathscr{X}_{i, 0} = \mathscr{X}_i \times _{\mathscr{X}} \mathscr{X}_0$ . The condition on $\Psi _f$ forces

(2.4.2.2) \begin{align} \Psi _f (a)_i = j_i^* \circ \Psi _f (a) & = [G_i]^{-1} \cdot (\pi _i)_! \circ \pi _i^* \circ j_i^* \circ \Psi _f (a) \nonumber \\ & = [G_i]^{-1} \cdot (\pi _i)_! \circ \Psi _{f \circ j_i \circ \pi _i} \circ \pi _i^* \circ j_i^* (a) , \end{align}

where $[G_i] \in \hat {{\rm M}} (\mathbb{K})$ is the class of $G_i$ and is invertible in $\hat {{\rm M}} (\mathbb{K})$ , and we applied (2.1.7.1) to $\pi _i$ , using the fact that $G_i$ is special. This shows that if the map $\Psi _f$ exists, then it is unique.

To check that the elements $\Psi _f (a)_i$ agree on overlaps, let $1, 2 \in I$ be two indices, and form the pullback squares

(2.4.2.3)

where $U^{\prime}_1$ , $U^{\prime}_2$ , $U^{\prime\prime}$ are algebraic spaces. We need to show that $(j^{\prime}_2)^* ({\Psi _f (a)_1}) = (j^{\prime}_1)^* ({\Psi _f (a)_2})$ . We have

(2.4.2.4) \begin{align} & \quad (j^{\prime}_2)^* (\Psi _f (a)_1) \nonumber \\& \quad = [G_1]^{-1} \cdot (j^{\prime}_2)^* \circ (\pi _1)_! \circ \Psi _{f \circ j_1 \circ \pi _1} \circ (j_1 \circ \pi _1)^* (a) \nonumber \\ & \quad = [G_1]^{-1} \cdot (\pi ^{\prime}_1)_! \circ (j^{\prime\prime}_2)^* \circ \Psi _{f \circ j_1 \circ \pi _1} \circ (j_1 \circ \pi _1)^* (a) \nonumber \\& \quad = [G_1]^{-1} \cdot (\pi ^{\prime}_1)_! \circ \Psi _{f \circ j_1 \circ \pi _1 \circ j^{\prime\prime}_2} \circ (j_1 \circ \pi _1 \circ j^{\prime\prime}_2)^* (a) \nonumber \\& \quad = [G_1]^{-1} \cdot [G_2]^{-1} \cdot (\pi ^{\prime}_1)_! \circ (\pi ^{\prime\prime}_2)_! \circ (\pi ^{\prime\prime}_2)^* \circ \Psi _{f \circ j_1 \circ \pi _1 \circ j^{\prime\prime}_2} \circ (j_1 \circ \pi _1 \circ j^{\prime\prime}_2)^* (a) \nonumber \\& \quad = [G_1]^{-1} \cdot [G_2]^{-1} \cdot (\pi ^{\prime}_1 \circ \pi ^{\prime\prime}_2)_! \circ \Psi _{f \circ j_1 \circ \pi _1 \circ j^{\prime\prime}_2 \circ \pi ^{\prime\prime}_2} \circ (j_1 \circ \pi _1 \circ j^{\prime\prime}_2 \circ \pi ^{\prime\prime}_2)^* (a), \end{align}

where we applied (2.1.6.1) in the second step, Theorem 2.3.5 (ii) in the third and fifth steps, and § 2.1.7 in the fourth step. This expression is now symmetric in the indices $1$ and $2$ , so the element ${\Psi _f} (a)$ is well-defined.

It now remains to show that the map $\Psi _f$ satisfies the required condition. Let $Y$ be a $\mathbb{K}$ -variety and $\pi \colon Y \to \mathscr{X}$ a smooth morphism. For each $i \in I$ , write $Y_i = Y \times _{\mathscr{X}} \mathscr{X}_i$ . Then $(k_i \colon Y_i \to Y)_{i \in I}$ is a Nisnevich cover by algebraic spaces, and it suffices to show that

(2.4.2.5) \begin{align} k_i^* \circ g^* \circ \Psi _f = k_i^* \circ \Psi _{f \circ g} \circ g^* \end{align}

for each $i$ . Consider the diagram

(2.4.2.6)

where all squares are pullback squares. In particular, $\rho _i$ is a principal $G_i$ -bundle. For any $a \in \hat {{\rm M}} (\mathscr{X}\,)$ , we have

(2.4.2.7) \begin{align} &\quad\, k_i^* \circ g^* \circ \Psi _f (a) \nonumber \\ & = g_i^* (\Psi _f (a)_i) \nonumber \\ & = [G_i]^{-1} \cdot g_i^* \circ (\pi _i)_! \circ \Psi _{f \circ j_i \circ \pi _i} \circ (j_i \circ \pi _i)^* (a) \nonumber \\ & = [G_i]^{-1} \cdot (\rho _i)_! \circ (g^{\prime}_i)^* \circ \Psi _{f \circ j_i \circ \pi _i} \circ (j_i \circ \pi _i)^* (a) \nonumber \\ & = [G_i]^{-1} \cdot (\rho _i)_! \circ \Psi _{f \circ j_i \circ \pi _i \circ g^{\prime}_i} \circ (j_i \circ \pi _i \circ g^{\prime}_i)^* (a) \nonumber \\ & = [G_i]^{-1} \cdot (\rho _i)_! \circ \Psi _{f \circ g \circ k_i \circ \rho _i} \circ (g \circ k_i \circ \rho _i)^* (a) \nonumber \\ & = [G_i]^{-1} \cdot (\rho _i)_! \circ \rho _i^* \circ \Psi _{f \circ g \circ k_i} \circ (g \circ k_i)^* (a) \nonumber \\ & = \Psi _{f \circ g \circ k_i} \circ (g \circ k_i)^* (a) \nonumber \\ & = k_i^* \circ \Psi _{f \circ g} \circ g^* (a) , \end{align}

where we applied (2.1.6.1) in the third step, Theorem 2.3.5 (ii) in the fourth, sixth, and eighth steps, and § 2.1.7 in the seventh step. This proves the desired identity (2.4.2.5).

2.4.3 The motivic Milnor fibre.

Let $\mathscr{X}\,$ be as in § 2.4.1, and let $f \colon \mathscr{X} \to {{\mathbb{A}}^1}$ be a morphism. Write $\mathscr{X}_0 = {f^{-1}} (0)$ . The motivic Milnor fibre of $f$ is the element

\begin{align*} {\rm MF}_f = \Psi _f ([\mathscr{X}\,]) \in \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}_0) \ . \end{align*}

When $\mathscr{X}\,$ is smooth, this is closely related to the construction of Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, § 5.4], which we will further discuss and generalize in § 2.5.4 below. The main difference is that the latter construction starts from the critical locus of $f$ instead of $\mathscr{X}\,$ , and can be generalized to stacks glued from such critical loci; it uses $\Phi _{{f}}$ instead of $\Psi _{{f}}$ , and introduces a twist by a power of $\mathbb{L}^{1/2}$ to make gluing possible.

We relate this to the description of the motivic Milnor fibre for schemes in § 2.3.2. Suppose that we are given a resolution of $f$ , which is a representable proper morphism $\pi \colon \tilde {\mathscr{X}} \to \mathscr{X}$ such that it restricts to an isomorphism on $\pi ^{-1} (\mathscr{X} \setminus \mathscr{X}_0)$ and $\pi ^{-1} (\mathscr{X}_0)$ is a simple normal crossings divisor in $\tilde {\mathscr{X}}$ , in the sense that it is so after pulling back along smooth morphisms from schemes. Let $(\mathscr{E}_i)_{{i \in J}}$ be the family of irreducible components of $\pi ^{-1} (\mathscr{X}_0)$ , and define $\mathscr{E}^{\,\circ} _I$ and $\,\tilde {\!\!\mathscr{E}}^{\,\circ} _I$ for non-empty $I \subset J$ similarly to § 2.3.2, where $\,\,\tilde {\!\!\mathscr{E}}^{\,\circ} _I$ carries a natural ${\hat {\mu }}$ -action. We then claim that

(2.4.3.1) \begin{align} {\rm MF}_f = \sum _{\varnothing \neq I \subset J} {} (1 - \mathbb{L})^{|I| - 1} \, \big[\,\,\tilde {\!\!\mathscr{E}}^{\,\circ} _I\big]^{{\hat {\mu }}} \ . \end{align}

Indeed, this can be shown by a similar argument to that used in the proof of Theorem 2.4.2, by first passing to a Nisnevich cover by quotient stacks, then using the relation (2.1.7.1) to further reduce to the case of algebraic spaces, and finally passing to a Nisnevich cover again to reduce to the case of affine varieties.

2.4.4 Theorem.

Let $\mathscr{X}, \mathscr{Y}\,$ be algebraic stacks as in § 2.4.1.

1. (i) Let $g \colon \mathscr{Y} \to \mathscr{X}$ be a proper morphism, and $f \colon \mathscr{X} \to {\mathbb{A}}^1$ a morphism. Then we have the following commutative diagram.

   (2.4.4.1)

   

2. (ii) Let $g \colon \mathscr{Y} \to \mathscr{X}$ be a smooth morphism, and $f \colon \mathscr{X} \to {\mathbb{A}}^1$ a morphism. Then we have the following commutative diagram.

   (2.4.4.2)

   

   In particular, we have ${\rm MF}_{f \circ g} = g^* ({\rm MF}_f)$ .

Proof. For (i), we first restrict to the case when $g$ is representable. By Theorem 2.4.2, the map $\Psi _f$ is determined by pullbacks along smooth morphisms from $\mathbb{K}$ -varieties to $\mathscr{X}\,$ , so we may assume that $\mathscr{X}\,$ is a $\mathbb{K}$ -variety and $\mathscr{Y}\,$ is an algebraic space that is proper over $\mathscr{X}\,$ . This case is covered by Theorem 2.3.5 (i).

For the general case, similarly, we may assume that $\mathscr{X} = X$ is a $\mathbb{K}$ -variety. It suffices to show that $g_! \circ \Psi _{f \circ g} ([Z]) = \Psi _f \circ g_! ([Z])$ for smooth $\mathbb{K}$ -varieties $Z$ mapping to $\mathscr{Y}\,$ , as these classes span $\hat {{\rm M}} (\mathscr{Y}{\kern3pt})$ over $\hat {{\rm M}} (\mathbb{K})$ . Since $\mathscr{Y}\,$ is proper over $X$ and has affine stabilizers, it has finite inertia, and admits a coarse space $\pi _{\mathscr{Y}} \colon \mathscr{Y} \to {{\overline {{Y}\vphantom {X}}}}$ by the Keel–Mori theorem [Reference Keel and MoriKM97, Reference ConradCon]. The morphism $\pi _{\mathscr{Y}}$ is a proper universal homeomorphism.

By Rydh’s compactification theorem for representable morphisms of Deligne–Mumford stacks [Reference RydhRyd, Theorem B], we may choose a relative compactification $\mathscr{Z}$ of $Z$ over $\mathscr{Y}\,$ such that there is a dense open immersion $i \colon Z \hookrightarrow \mathscr{Z}$ and a proper representable morphism $h \colon \mathscr{Z} \to \mathscr{Y}\,$ . In particular, $\mathscr{Z}$ also has finite inertia, and admits a coarse space $\pi _{\mathscr{Z}} \colon \mathscr{Z} \to {{\overline {{Z}\vphantom {X}}}}$ , which can be seen as a relative compactification of $Z$ over ${\overline {{Y}\vphantom {X}}}$ . We have a commutative diagram

(2.4.4.3)

where $\overline {g}$ and ${\overline {{h}\vphantom {X}}}$ are the induced morphisms, and all morphisms except $i$ are proper. It is then enough to show that

(2.4.4.4) \begin{align} (\pi _{\mathscr{Z}})_! \circ \Psi _{f \circ g \circ h} ([Z]) = \Psi _{f \circ \overline {g} \circ {{\overline {{h}\vphantom {X}}}}} \circ (\pi _{\mathscr{Z}})_! ([Z]) , \end{align}

since the compatibility with pushing forward along $h$ and $\overline {g} \circ {{\overline {{h}\vphantom {X}}}}$ is covered by the previous case.

We now apply Bergh–Rydh’s divisorialification theorem [Reference Bergh and RydhBR19, Theorem A] to a desingularization of the pair $(\mathscr{Z}, \mathscr{Z} \setminus Z)$ (see, for example, [Reference Encinas and VillamayorEV98]), which gives a representable proper morphism ${{\overline {{\mathscr{Z}}\vphantom {X}}}} \to \mathscr{Z}$ that is an isomorphism on the preimage of $Z$ , such that ${{\overline {{\mathscr{Z}}\vphantom {X}}}} \setminus Z = \mathscr{D}$ is a simple normal crossings divisor on ${\overline {{\mathscr{Z}}\vphantom {X}}}$ , with smooth irreducible components $\mathscr{D}_i \subset {{\overline {{\mathscr{Z}}\vphantom {X}}}}$ , and for each $x \in {{\overline {{\mathscr{Z}}\vphantom {X}}}}$ , writing $I_x = \{ i \in I \mid x \in \mathscr{D}_i \}$ , étale locally around $x$ , one has ${{\overline {{\mathscr{Z}}\vphantom {X}}}} \sim \prod _{{i \in I_x}} {} [{\mathbb{A}}^1 / {\mu _{n_i}}] \times {\mathbb{A}}^{d - |I_x|}$ , where $d = \dim {{\overline {{\mathscr{Z}}\vphantom {X}}}}$ , each $\mu _{n_i}$ acts on ${\mathbb{A}}^1$ by scaling, and $\mathscr{D}_i$ corresponds to the locus where the $i$ th factor is zero; the number $n_i$ is the order of the generic stabilizer of $\mathscr{D}_i$ .

From now on, we assume that $\mathscr{Z} = {{\overline {{\mathscr{Z}}\vphantom {X}}}}$ , since, again, pushing forward along the representable morphism ${{\overline {{\mathscr{Z}}\vphantom {X}}}} \to \mathscr{Z}$ and the corresponding morphism of coarse spaces is already dealt with.

Now, choose a resolution $\pi \colon \,\tilde {\!\!\mathscr{Z}} \to \mathscr{Z}$ for the morphism $\mathscr{Z} \to {\mathbb{A}}^1$ , which is a composition of blow-ups along smooth centres. Then $\,\tilde {\!\!\mathscr{Z}}$ still has the same local description as before. The local description implies that the coarse space of $\,\tilde {\!\!\mathscr{Z}}$ , denoted by $\tilde {Z}$ , is a smooth algebraic space, and can be seen as a resolution for the morphism ${{\overline {{Z}\vphantom {X}}}} \to {\mathbb{A}}^1$ .

For each $i \in I$ , let $\,\tilde {\!\!\mathscr{D}}_i \subset \,\tilde {\!\!\mathscr{Z}}$ be the strict transform of $\mathscr{D}_i$ , which is a smooth divisor, and let $({\mathscr{E}_j} \subset \,\tilde {\!\!\mathscr{Z}})_{{j \in J}}$ be the family of irreducible components of $\,\tilde {\!\!\mathscr{Z}}_0$ . Then, by construction, all the divisors $\,\tilde {\!\!\mathscr{D}}_i, {\mathscr{E}_j} \subset \,\tilde {\!\!\mathscr{Z}}$ have simple normal crossings, and $\,\tilde {\!\!\mathscr{Z}} \setminus \bigcup _{i \in I} \,\tilde {\!\!\mathscr{D}}_i$ is an algebraic space. Let $\tilde {D}_i, {E_j} \subset \tilde {Z}$ be the corresponding divisors in the coarse spaces. For $I^{\prime} \subset I$ , write $\mathscr{D}_{I^{\prime}} = \bigcap _{i \in I^{\prime}} \mathscr{D}_i$ and $\,\tilde {\!\!\mathscr{D}}_{I^{\prime}} = \bigcap _{i \in I^{\prime}} \,\tilde {\!\!\mathscr{D}}_i$ , with the convention that $\mathscr{D}_{\varnothing } = \mathscr{Z}$ and $\,\tilde {\!\!\mathscr{D}}_{\varnothing } = \,\tilde {\!\!\mathscr{Z}}$ . Then each $\,\tilde {\!\!\mathscr{D}}_{I^{\prime}}$ can be seen as a resolution for the morphism $\mathscr{D}_{I^{\prime}} \to {\mathbb{A}}^1$ . By § 2.4.3, we have

\begin{align*} (\pi _{\mathscr{Z}})_! \circ \Psi _{f \circ g \circ h} ([Z]) & = \sum _{I^{\prime} \subset I} {} (-1)^{|I^{\prime}|} \cdot (\pi _{\mathscr{Z}})_! \circ \Psi _{f \circ g \circ h} ([\mathscr{D}_{I^{\prime}}]) \\ & = \sum _{I^{\prime} \subset I} {} (-1)^{|I^{\prime}|} \cdot \sum _{\varnothing \neq J^{\prime} \subset J} {} (1 - \mathbb{L})^{|J^{\prime}| - 1} \, \big[\,\,\tilde {\!\!\mathscr{E}}^{\,\circ} _{{J^{\prime}}} \cap \,\tilde {\!\!\mathscr{D}}_{I^{\prime}}\big]^{{\hat {\mu }}} \\ & = \sum _{\varnothing \neq J^{\prime} \subset J} {} (1 - \mathbb{L})^{|J^{\prime}| - 1} \, \biggl [ \,\,\tilde {\!\!\mathscr{E}}^{\,\circ} _{{J^{\prime}}} \Bigm \backslash \bigcup _{i \in I} \,\tilde {\!\!\mathscr{D}}_i \biggr ]^{{\hat {\mu }}} \\ & = \sum _{\varnothing \neq J^{\prime} \subset J} {} (1 - \mathbb{L})^{|J^{\prime}| - 1} \, \biggl [ \tilde {E}^\circ _{{J^{\prime}}} \Bigm \backslash \bigcup _{i \in I} \tilde {D}_i \biggl ]^{{\hat {\mu }}} \\ & = \Psi _{f \circ \overline {g} \circ {{\overline {{h}\vphantom {X}}}}} \circ (\pi _{\mathscr{Z}})_! ([Z]) , \end{align*}

where the second last step used the fact that each $\,\,\tilde {\!\!\mathscr{E}}^{\,\circ} _{{J^{\prime}}} \setminus \bigcup _{i \in I} \,\tilde {\!\!\mathscr{D}}_i$ is an algebraic space.

For (ii), similarly, the case when $g$ is representable follows from Theorem 2.3.5 (ii). For the general case, we may assume that $\mathscr{X}\,$ is a $\mathbb{K}$ -variety. Since $\Psi _{f \circ g}$ is determined by pullbacks along smooth morphisms from schemes to $\mathscr{Y}\,$ , we can also assume that $\mathscr{Y}\,$ is a $\mathbb{K}$ -variety, and the result follows from Theorem 2.3.5 (ii).

The final statement follows from applying (ii) to the element $[\mathscr{X}\,] \in \hat {{\rm M}} (\mathscr{X}\,)$ .

2.4.5 Remark.

The length of the proof of Theorem 2.4.4 is primarily due to the case of pushing forward along proper morphisms that are not necessarily representable. We will indeed need this general case in the proof of one of our main results, Theorem 4.1.1, where $g$ will be taken to be a weighted blow-up in the sense of § 4.1.2.

2.5.1 Shifted symplectic and d-critical stacks.

Let $\mathfrak{X}$ be a $(-1)$ -shifted symplectic stack over $\mathbb{K}$ (see § 1.6.1), and let $\mathscr{X} = \mathfrak{X}_{{\rm cl}}$ be its classical truncation. Assume that $\mathscr{X}\,$ is an algebraic stack over $\mathbb{K}$ .

Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, § 3.3] define a d-critical structure on $\mathscr{X}\,$ induced from the shifted symplectic structure on $\mathfrak{X}$ , so that $\mathscr{X}\,$ is a d-critical stack. See Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15 and Joyce [Reference JoyceJoy15] for the precise definitions. For our purposes, it suffices to know the following properties:

1. (i) For a smooth $\mathbb{K}$ -variety $U$ and a morphism $f \colon U \to {\mathbb{A}}^1$ , the critical locus ${\rm Crit} (f) \subset U$ carries a canonical d-critical structure.

2. (ii) d-critical structures can be pulled back along smooth morphisms of algebraic stacks over $\mathbb{K}$ .

3. (iii) If a $\mathbb{K}$ -scheme $X$ carries a d-critical structure, then it can be covered by open subschemes called critical charts, each of which, with the induced d-critical structure, has the form ${\rm Crit} (f)$ as in (i). We denote such a critical chart by $i \colon {\rm Crit} (f) \hookrightarrow X$ .

2.5.2 Orientations.

Let $\mathfrak{X}$ be an $s$ -shifted symplectic stack over $\mathbb{K}$ , where $s$ is odd. An orientation of $\mathfrak{X}$ is a line bundle $K_{\mathfrak{X}}^{{1/2}} \to \mathfrak{X}$ , together with an isomorphism $(K_{\mathfrak{X}}^{{1/2}})^{{\otimes 2}} \simeq K_{\mathfrak{X}}$ , where $K_{\mathfrak{X}}$ is the canonical bundle of $\mathfrak{X}$ , defined as the determinant line bundle of the cotangent complex of $\mathfrak{X}$ .

Note the unfortunate clash of terminology with the unrelated notion of orientations in the sense of Pantev, Toën, Vaquié and Vezzosi [Reference Pantev, Toën, Vaquié and VezzosiPTVV13, Definition 2.4]. The latter notion will not be used in this paper except in the proof of Theorem 3.1.6.

Now let $s=-1$ , and let $\mathscr{X}\,$ be the associated d-critical stack of $\mathfrak{X}$ , as in § 2.5.1. Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Theorem 3.18] show that the restriction $K_{\mathfrak{X}} |_{\mathscr{X}^{\kern2.1pt{\rm{red}}}}$ is determined by the d-critical structure on $\mathscr{X}\,$ , where $\mathscr{X}^{\kern2.1pt{\rm{red}}}$ is the reduction of $\mathscr{X}\,$ . We denote this restriction simply by $K_{\mathscr{X}}$ , and call it the canonical bundle of the d-critical stack $\mathscr{X}\,$ . Similarly, as in [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Definition 3.6], an orientation of a d-critical stack $\mathscr{X}\,$ is a line bundle $K_{\mathscr{X}}^{{1/2}} \to \mathscr{X}^{\kern2.1pt{\rm{red}}}$ , together with an isomorphism $(K_{\mathscr{X}}^{{1/2}})^{{\otimes 2}} \simeq K_{\mathscr{X}}$ .

The d-critical scheme ${\rm Crit} (f)$ in § 2.5.1 (i) has a canonical orientation given by $K_{{\rm Crit} (f)}^{{1/2}} = K_U |_{{\rm Crit} (f)^{{{\rm red}}}}$ .

By [Reference JoyceJoy15, Lemma 2.58], for a smooth morphism $g \colon \mathscr{Y} \to \mathscr{X}$ of d-critical stacks, compatible with the d-critical structures, an orientation $K_{\mathscr{X}}^{{1/2}}$ of $\mathscr{X}\,$ induces an orientation of $\mathscr{Y}\,$ given by $K_{\mathscr{Y}}^{{1/2}} = g^* (K_{\mathscr{X}}^{{1/2}}) \otimes \det \mathbb{L}_{\mathscr{Y} / \mathscr{X}} |_{\mathscr{Y}^{{{\,\rm red}}}}$ .

2.5.3 Definition for schemes.

Let $X$ be an oriented d-critical $\mathbb{K}$ -scheme. Its motivic Behrend function $\nu ^{{{\rm mot}}}_X \in {\rm M}^{{\hat {\mu }}} (X)$ is defined by the following property:

1. − For any critical chart $i \colon {\rm Crit} (f) \hookrightarrow X$ , where $f \colon U \to {\mathbb{A}}^1$ and $U$ is a smooth $\mathbb{K}$ -variety, we have

   (2.5.3.1) \begin{align} i^* (\nu ^{{{\rm mot}}}_X) = -\mathbb{L}^{-{\dim U / 2}} \cdot \Phi _f ([U]) \cdot \Upsilon \big( i^* \bigr(K_X^{{1/2}}\bigr) \otimes K_U^{-1} |_{{\rm Crit} (f)^{{{\rm red}}}} \big) , \end{align}
   in ${\rm M}^{{\hat {\mu }}} ({\rm Crit} (f))$ , where $\Phi _{{f}}$ is the vanishing cycle map defined in § 2.3.3, and $\Phi _{{f}} ([U])$ is supported on ${\rm Crit} (f)$ . The map $\Upsilon$ is as in § 2.1.11, and the part inside $\Upsilon ({\cdots })$ is a line bundle on ${\rm Crit} (f)^{{{\rm red}}}$ whose square is trivial, so it can be seen as a $\mu _2$ -bundle.

This is well-defined, from the result of Bussi, Joyce and Meinhardt [Reference Bussi, Joyce and MeinhardtBJM19, Theorem 5.10].

For $X$ as above, and a smooth morphism $g \colon Y \to X$ of relative dimension $d$ , where $Y$ is equipped with the induced oriented d-critical structure, we have the relation

(2.5.3.2) \begin{align} g^* (\nu ^{{{\rm mot}}}_X) = \mathbb{L}^{d / 2} \cdot \nu ^{{{\rm mot}}}_Y , \end{align}

which follows from [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Theorem 5.14].

2.5.4 Definition for stacks.

Let $\mathscr{X}\,$ be an oriented d-critical stack over $\mathbb{K}$ , and assume that $\mathscr{X}\,$ is Nisnevich locally a quotient stack in the sense of § 2.2.4.

We define the motivic Behrend function of $\mathscr{X}\,$ , slightly generalizing the construction of Ben-Bassat, Brav, Bussi and Joyce [Reference Ben-Bassat, Brav, Bussi and JoyceBBBB⁺15, Theorem 5.14], who only considered stacks that are Zariski locally quotient stacks.

Theorem. Let $\mathscr{X}\,$ be as above. Then there exists a unique element

\begin{align*} \nu ^{{{\rm mot}}}_{\mathscr{X}} \in \hat {{\rm M}}{}^{{\hat {\mu }}} (\mathscr{X}\,) , \end{align*}

called the motivic Behrend function of $\mathscr{X}\,$ , such that for any $\mathbb{K}$ -variety $Y$ and any smooth morphism $f \colon Y \to \mathscr{X}$ of relative dimension $d$ , we have

(2.5.4.1) \begin{align} f^* (\nu ^{{{\rm mot}}}_{\mathscr{X}}) = \mathbb{L}^{d / 2} \cdot \nu ^{{{\rm mot}}}_Y \end{align}

in $\hat {{\rm M}}{}^{{\hat {\mu }}} (Y)$ , where $Y$ is equipped with the induced oriented d-critical structure.

In particular, we write $\nu ^{{{\rm mot}}}_{\mathfrak{X}} = \nu ^{{{\rm mot}}}_{\mathscr{X}}$ if the d-critical structure on $\mathscr{X}\,$ comes from a $(-1)$ -shifted symplectic stack $\mathfrak{X}$ with $\mathscr{X} \simeq \mathfrak{X}_{{\rm cl}}$ .

2.5.5 Compatibility with smooth pullbacks.

We now show that the smooth pullback relation (2.5.4.1) holds for all smooth morphisms of d-critical stacks.

Theorem. Let $\mathscr{X}, \mathscr{Y}\,$ be oriented d-critical stacks over $\mathbb{K}$ that are Nisnevich locally quotient stacks, and let $f \colon \mathscr{Y} \to \mathscr{X}$ be a smooth morphism of relative dimension $d$ which is compatible with the oriented d-critical structures. Then we have the relation

(2.5.5.1) \begin{align} f^* (\nu ^{{{\rm mot}}}_{\mathscr{X}}) = \mathbb{L}^{d / 2} \cdot \nu ^{{{\rm mot}}}_{\mathscr{Y}} \ . \end{align}

2.5.6 The numerical Behrend function.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ that is Nisnevich locally a quotient stack, equipped with an oriented d-critical structure. The Behrend function of $\mathscr{X}\,$ is the constructible function

\begin{align*} \nu _{\mathscr{X}} = \chi (\nu ^{{{\rm mot}}}_{\mathscr{X}}) \in {\rm CF} (\mathscr{X}\,) , \end{align*}

where $\chi$ denotes taking the pointwise Euler characteristic, as in § 2.1.9.

In fact, one can still define $\nu _{\mathscr{X}}$ even if $\mathscr{X}\,$ is only étale locally a quotient stack, and without the orientability assumption. Indeed, the relation (2.5.5.1) implies that the numerical Behrend function is compatible with smooth morphisms preserving the d-critical structure (not necessarily orientations), up to a sign $(-1)^d$ , where $d$ is the relative dimension. This is because changing the orientation only affects the term $\Upsilon ({\cdots })$ in (2.5.3.1), which always has Euler characteristic $1$ , and the sign is due to the fact that $\chi (\mathbb{L}^{1/2}) = -1$ . Now, to define $\nu _{\mathscr{X}}$ , one can pass to a smooth cover of $\mathscr{X}\,$ by $\mathbb{K}$ -varieties, and apply smooth descent of constructible functions.

When $\mathbb{K} = \mathbb{C}$ , the Behrend function $\nu _{\mathscr{X}}$ agrees with the original definitions of Behrend [Reference BehrendBeh09] and Joyce and Song [Reference Joyce and SongJS12, § 4.1]. This follows from the compatibility of both versions with smooth pullbacks, namely Theorem 2.5.5 and [Reference Joyce and SongJS12, Theorem 4.3], and the case of critical loci on smooth varieties, which follows from [Reference Denef and LoeserDL02, Theorem 3.10] and [Reference Joyce and SongJS12, Theorem 4.7].

3.1.1 Definition.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ . Following Halpern-Leistner [Reference Halpern-LeistnerHLa22], we define the stack of graded points and the stack of filtered points of $\mathfrak{X}$ , respectively, as the mapping stacks over $\mathbb{K}$ ,

\begin{align*} {{\rm Grad}} (\mathscr{X}\,) & = {\rm Map} ([* / \mathbb{G}_{{\rm m}}], \mathscr{X}\,) , \\ {\rm Filt} (\mathscr{X}\,) & = {\rm Map} ([{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}], \mathscr{X}\,) , \end{align*}

where $\mathbb{G}_{{\rm m}}$ acts on ${\mathbb{A}}^1$ by scaling.

Consider the morphisms

where $\operatorname {{{\rm pr}}}$ is induced by the projection ${\mathbb{A}}^1 \to *$ . These morphisms induce morphisms of stacks

where the notation ‘ ${{\rm op}}$ ’, ‘ ${\rm gr}$ ’, ‘ ${\rm sf}$ ’, and ‘ ${\rm tot}$ ’ is used for the opposite graded point, the associated graded point, the split filtration, and the total point, respectively.

The stacks ${{\rm Grad}} (\mathscr{X}\,)$ and ${\rm Filt} (\mathscr{X}\,)$ are algebraic stacks over $\mathbb{K}$ . If, moreover, $\mathscr{X}\,$ has affine stabilizers, then so do ${{\rm Grad}} (\mathscr{X}\,)$ and ${\rm Filt} (\mathscr{X}\,)$ , and, by [Reference Halpern-LeistnerHLa22, Lemma 1.3.8], the morphism ${\rm gr}$ is quasi-compact, and induces a bijection $\pi _0 ({\rm Filt} (\mathscr{X}\,)) \mathrel {\overset {{{\scriptstyle \sim }}\mspace {4mu}}{\to }} \pi _0 ({{\rm Grad}} (\mathscr{X}\,))$ .

3.1.2 The derived version.

Now, consider a derived algebraic stack $\mathfrak{X}$ over $\mathbb{K}$ . Similarly, we can consider the derived mapping stacks

\begin{align*} {\rm dGrad} (\mathfrak{X}) & = {\rm dMap} ([* / \mathbb{G}_{{\rm m}}], \mathfrak{X}) , \\ {\rm dFilt} (\mathfrak{X}) & = {\rm dMap} ([{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}], \mathfrak{X}) , \end{align*}

where ${\rm dMap} (-, -)$ denotes the mapping stack in the $\infty$ -category of derived stacks. These are again derived algebraic stacks over $\mathbb{K}$ , by Halpern-Leistner and Preygel [Reference Halpern-Leistner and PreygelHLP23, Theorem 5.1.1] or Halpern-Leistner [Reference Halpern-LeistnerHLb21, Theorem 1.2.1]. Moreover, they are locally finitely presented whenever $\mathfrak{X}$ is, by an argument similar to that in [Reference Halpern-Leistner and PreygelHLP23, § 5.1.5].

We use the same notation ( ${\rm gr}$ , ${\rm ev}_1$ , etc.,) as in § 3.1.1 for the induced morphisms between these derived stacks.

We often write ${{\rm Grad}} (\mathfrak{X}), {\rm Filt} (\mathfrak{X})$ for ${\rm dGrad} (\mathfrak{X}), {\rm dFilt} (\mathfrak{X})$ when there is no ambiguity. We adopt the convention that ${{\rm Grad}} (\mathscr{X}\,)$ and ${\rm Filt} (\mathscr{X}\,)$ refer to the classical versions when $\mathscr{X}\,$ is assumed to be classical, and the derived versions otherwise.

Note that for a classical algebraic stack $\mathscr{X}\,$ , the stacks ${\rm dGrad} (\mathscr{X}\,)$ and ${\rm dFilt} (\mathscr{X}\,)$ can have non-trivial derived structure. See [Reference Halpern-LeistnerHLb21, Example 1.6.4] for an example of this phenomenon. However, we always have ${\rm dGrad} (\mathscr{X}\,)_{{\rm cl}} \simeq {{\rm Grad}} (\mathscr{X}\,)$ and ${\rm dFilt} (\mathscr{X}\,)_{{\rm cl}} \simeq {\rm Filt} (\mathscr{X}\,)$ , where $(-)_{{\rm cl}}$ denotes the classical truncation. More generally, by [Reference Halpern-LeistnerHLa22, Lemma 1.2.1], for a derived algebraic stack $\mathfrak{X}$ , we have ${\rm dGrad} (\mathfrak{X})_{{\rm cl}} \simeq {{\rm Grad}} (\mathfrak{X}_{{\rm cl}})$ and ${\rm dFilt} (\mathfrak{X})_{{\rm cl}} \simeq {\rm Filt} (\mathfrak{X}_{{\rm cl}})$ .

3.1.3 For quotient stacks.

Let $\mathscr{X} = [U / G]$ be a quotient stack, where $U$ is an algebraic space over $\mathbb{K}$ , acted on by a reductive group $G$ . The stacks of graded and filtered points of $\mathscr{X}\,$ can be described very explicitly, following Halpern-Leistner [Reference Halpern-LeistnerHLa22, § 1.4].

Let $\lambda \colon \mathbb{G}_{{\rm m}} \to G$ be a cocharacter, that is, a morphism of algebraic groups. Define the Levi subgroup and the parabolic subgroup of $G$ associated to $\lambda$ by

\begin{align*} L_\lambda & = \{ g \in G \mid g = \lambda (t) \, g \, \lambda (t)^{-1} \,\mathrm{for\ all}\ t \}, \\ P_\lambda & = \{ g \in G \mid \lim _{t \to 0} \lambda (t) \, g \, \lambda (t)^{-1} \,\mathrm{exists}\}, \end{align*}

respectively. Define the fixed locus and the attractor associated to $\lambda$ by

\begin{align*} U^{\lambda , 0} & = {\rm Map}^{\mathbb{G}_{{\rm m}}} (*, U) , \\ U^{\lambda , +} & = {\rm Map}^{\mathbb{G}_{{\rm m}}} ({\mathbb{A}}^1, U) , \end{align*}

where ${\rm Map}^{\mathbb{G}_{{\rm m}}} (-, -)$ denotes the $\mathbb{G}_{{\rm m}}$ -equivariant mapping space; $\mathbb{G}_{{\rm m}}$ acts on $U$ via $\lambda$ , and on ${\mathbb{A}}^1$ by scaling. These are algebraic spaces by Drinfeld and Gaitsgory [Reference Drinfeld and GaitsgoryDG14, Proposition 1.3.4 and Theorem 1.5.2] or Halpern-Leistner [Reference Halpern-LeistnerHLa22, Proposition 1.4.1]. There is a closed immersion $U^{\lambda , 0} \hookrightarrow U$ , an unramified morphism $U^{\lambda , +} \to U$ given by evaluation at $1$ , and an affine morphism $U^{\lambda , +} \to U^{\lambda , 0}$ given by evaluation at $0$ .

The $G$ -action on $U$ induces a $P_\lambda$ -action on $U^{\lambda , +}$ and an $L_\lambda$ -action on $U^{\lambda , 0}$ . Moreover, by [Reference Halpern-LeistnerHLa22, Theorem 1.4.8], we have

\begin{align*} {{\rm Grad}} (\mathscr{X}\,) & \simeq \coprod _{\lambda \colon \mathbb{G}_{{\rm m}} \to G} [U^{\lambda , 0} / L_\lambda ] , \\ {\rm Filt} (\mathscr{X}\,) & \simeq \coprod _{\lambda \colon \mathbb{G}_{{\rm m}} \to G} [U^{\lambda , +} / P_\lambda ] , \end{align*}

where the disjoint union is over all conjugacy classes of cocharacters $\lambda$ .

3.1.4 Deformation theory.

For a derived algebraic stack $\mathfrak{X}$ locally of finite presentation over $\mathbb{K}$ , one can express the tangent complexes of ${{\rm Grad}} (\mathfrak{X})$ and ${\rm Filt} (\mathfrak{X})$ in terms of that of $\mathfrak{X}$ . Concretely, by Halpern-Leistner and Preygel [Reference Halpern-Leistner and PreygelHLP23, Proposition 5.1.10] or Halpern-Leistner [Reference Halpern-LeistnerHLa22, Lemma 1.2.2], we have

(3.1.4.1) \begin{equation} \mathbb{T}_{{{\rm Grad}} (\mathfrak{X})} \simeq {\rm tot}^* (\mathbb{T}_{\mathfrak{X}})_0 ,\end{equation}

(3.1.4.2) \begin{equation} \mathbb{T}_{{\rm Filt} (\mathfrak{X})} \simeq q_* \circ {\rm ev}^* (\mathbb{T}_{\mathfrak{X}}) , \end{equation}

where $(-)_0$ denotes the weight $0$ part with respect to the natural $\mathbb{G}_{{\rm m}}$ -action, ${\rm ev} \colon [{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}] \times {\rm Filt} (\mathfrak{X}) \to \mathfrak{X}$ is the evaluation morphism, and $q \colon [{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}] \times {\rm Filt} (\mathfrak{X}) \to {\rm Filt} (\mathfrak{X})$ is the projection.

3.1.5 Oriented Lagrangian correspondences.

Let $\mathfrak{X}$ and $\mathfrak{Y}$ be oriented $s$ -shifted symplectic stacks over $\mathbb{K}$ , as in § 2.5.2, where $s$ is odd, and let

(3.1.5.1) \begin{equation} \mathfrak{X} \overset {f}{\longleftarrow } \mathfrak{L} \overset {g}{\longrightarrow } \mathfrak{Y} \end{equation}

be an $s$ -shifted Lagrangian correspondence, in the sense of [Reference Calaque, Haugseng and ScheimbauerCHS22, § 2.4]. We thus have an exact triangle

(3.1.5.2) \begin{equation} \mathbb{T}_{\mathfrak{L}} \longrightarrow f^* (\mathbb{T}_{\mathfrak{X}}) \oplus g^* (\mathbb{T}_{\mathfrak{Y}}) \longrightarrow \mathbb{L}_{\mathfrak{L}} [s] \longrightarrow \mathbb{T}_{\mathfrak{L}} [1] \end{equation}

of perfect complexes on $\mathfrak{L}$ . An orientation of the shifted Lagrangian correspondence (3.1.5.1) is an isomorphism $K_{\mathfrak{L}} \simeq f^* (K_{\mathfrak{X}}^{{1/2}}) \otimes g^* (K_{\mathfrak{Y}}^{{1/2}})$ such that it squares to the canonical isomorphism $K_{\mathfrak{L}}^{{\otimes 2}} \simeq f^* (K_{\mathfrak{X}}) \otimes g^* (K_{\mathfrak{Y}})$ induced by the exact triangle (3.1.5.2).

3.1.6 Theorem.

Let $\mathfrak{X}$ be an $s$ -shifted symplectic stack over $\mathbb{K}$ , with symplectic form $\omega$ . Then we have an induced $s$ -shifted symplectic structure ${\rm tot}^* (\omega )$ on ${{\rm Grad}} (\mathfrak{X})$ , and an $s$ -shifted Lagrangian correspondence

(3.1.6.1) \begin{align} {{\rm Grad}} (\mathfrak{X}) \overset {{\rm gr}}{\longleftarrow } {\rm Filt} (\mathfrak{X}) \overset {{\rm ev}_1}{\longrightarrow } \mathfrak{X} \ . \end{align}

Moreover, if $s$ is odd and $\mathfrak{X}$ has an orientation $K_{\mathfrak{X}}^{{1/2}}$ , then ${{\rm Grad}} (\mathfrak{X})$ has an induced orientation $K_{{{\rm Grad}} (\mathfrak{X})}^{{1/2}}$ , and the Lagrangian correspondence is oriented.

Proof. The stacks ${{\rm Grad}} (\mathfrak{X})$ and ${\rm Filt} (\mathfrak{X})$ are derived algebraic stacks locally of finite presentation over $\mathbb{K}$ , as mentioned in § 3.1.1.

To prove that (3.1.6.1) is an $s$ -shifted Lagrangian correspondence, by Calaque [Reference CalaqueCal15, Theorem 4.8] it is enough to show that the cospan

(3.1.6.2) \begin{align} [* / \mathbb{G}_{{\rm m}}] \overset {0}{\longrightarrow } [{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}] \overset {1}{\longleftarrow } * \end{align}

is a $0$ -oriented cospan, in the sense of [Reference CalaqueCal15, § 4.2] and [Reference Calaque, Haugseng and ScheimbauerCHS22, § 2.5]. Indeed, $*$ carries a natural $0$ -orientation, and the $0$ -orientation on $[* / \mathbb{G}_{{\rm m}}]$ is given by the isomorphism $\mathbb{R} \Gamma (\mathscr{O}_{[* / \mathbb{G}_{{\rm m}}]}) \mathrel {\overset {{{\scriptstyle \sim }}\mspace {4mu}}{\to }} \mathbb{K}$ . To see that this is indeed a $0$ -orientation, we check the condition in [Reference Pantev, Toën, Vaquié and VezzosiPTVV13, Definition 2.4]. For $A \in {\textsf {CdgA}}_{\mathbb{K}}^{{\leqslant 0}}$ and a perfect complex $\mathscr{E} \in {\textsf {Perf}} (\operatorname {Spec} A \times [* / \mathbb{G}_{{\rm m}}])$ , one has $p_* (\mathscr{E}^{\,\vee} )^\vee \simeq p_* (\mathscr{E}\,)$ on $\operatorname {Spec} A$ , where $p \colon \operatorname {Spec} A \times [* / \mathbb{G}_{{\rm m}}] \to \operatorname {Spec} A$ is the projection, since both sides are the weight $0$ part of the induced $\mathbb{G}_{{\rm m}}$ -action on $\pi ^* (\mathscr{E}\,)$ , where $\pi \colon \operatorname {Spec} A \to \operatorname {Spec} A \times [* / \mathbb{G}_{{\rm m}}]$ is the projection.

To see that (3.1.6.2) is a $0$ -oriented cospan, we check the condition in [Reference Calaque, Haugseng and ScheimbauerCHS22, Lemma 2.5.5]. For any $A \in {\textsf {CdgA}}_{\mathbb{K}}^{{\leqslant 0}}$ and $\mathscr{E} \in {\textsf {Perf}} (\operatorname {Spec} A \times [{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}])$ , we need to show that the induced commutative diagram

(3.1.6.3)

in ${\textsf {Perf}} (A)$ is cartesian, where $p$ and $q$ are the projections from $\operatorname {Spec} A \times [* / \mathbb{G}_{{\rm m}}]$ and $\operatorname {Spec} A \times [{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}]$ to $\operatorname {Spec} A$ , respectively. Indeed, as in Halpern-Leistner [Reference Halpern-LeistnerHLb21, Proposition 1.1.2 ff.], such an object $\mathscr{E}$ can be seen as a filtered object in ${\textsf {Perf}} (A)$ , that is, a sequence of maps

\begin{align*} {\cdots } \longrightarrow E_{\geqslant 1} \longrightarrow E_{\geqslant 0} \longrightarrow E_{\geqslant -1} \longrightarrow {\cdots } \end{align*}

in ${\textsf {Perf}} (A)$ , where all but finitely many arrows are isomorphisms, such that $E_{\geqslant n} = 0$ for $n \gg 0$ . Write $E_n = \operatorname {cofib} (E_{\geqslant n+1} \to E_{\geqslant n})$ , and write $E = {\rm colim}_{n \to -\infty } E_{\geqslant n}$ . Then $0^* (\mathscr{E}\,) \simeq \bigoplus _n E_n$ , with the natural $\mathbb{G}_{{\rm m}}$ -action having weight $n$ on $E_n$ . One can deduce from [Reference Halpern-LeistnerHLb21, Proposition 1.1.2 ff.] that we have natural identifications

\begin{align*} q_* (\mathscr{E}\,) & \simeq E_{\geqslant 0} , \\ p_* \circ 0^* (\mathscr{E}\,) & \simeq E_0 , \\ 1^* (\mathscr{E}\,) & \simeq E , \\ q_* (\mathscr{E}^{\,\vee} )^\vee & \simeq ((E^\vee )_{\geqslant 0})^\vee \simeq E_{\leqslant 0} , \end{align*}

where $E_{\leqslant 0} = \operatorname {cofib} (E_{\geqslant 1} \to E)$ , and the arrows in the diagram (3.1.6.3) are the natural ones. This implies that (3.1.6.3) is cartesian.

For the final statement, observe that

\begin{align*} {\rm tot}^* (K_{\mathfrak{X}}) & \simeq \det ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^0 ) \otimes \det ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^+ ) \otimes \det ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^- ) \\ & \simeq K_{{{\rm Grad}} (\mathfrak{X})} \otimes \det ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^+ ) \otimes \det ( ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^+ )^\vee [-s] ) \\ & \simeq K_{{{\rm Grad}} (\mathfrak{X})} \otimes \det ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^+ )^{{2}} , \end{align*}

where $(-)^0$ , $(-)^+$ , $(-)^-$ denote the parts with zero, positive, and negative weights, respectively, with respect to the natural $\mathbb{G}_{{\rm m}}$ -action. Therefore, we may define

(3.1.6.4) \begin{align} K_{{{\rm Grad}} (\mathfrak{X})}^{{1/2}} = {\rm tot}^* \big(K_{\mathfrak{X}}^{{1/2}}\big) \otimes \det ( {\rm tot}^* (\mathbb{L}_{\mathfrak{X}})^+ )^{-1} , \end{align}

and this gives an orientation on ${{\rm Grad}} (\mathfrak{X})$ . To see that the $s$ -shifted Lagrangian correspondence is oriented, consider the cartesian diagram

(3.1.6.5)

in ${\textsf {Perf}} ({\rm Filt} (\mathfrak{X}))$ , witnessing the $s$ -shifted Lagrangian correspondence structure. Write $\mathscr{E} = {\rm ev}^* (\mathbb{T}_{\mathfrak{X}})$ , where ${\rm ev} \colon [{\mathbb{A}}^1 / \mathbb{G}_{{\rm m}}] \times {\rm Filt} (\mathfrak{X}) \to \mathfrak{X}$ is the evaluation morphism. As in the argument above, $\mathscr{E}$ can be seen as a filtered object in ${\textsf {Perf}} ({\rm Filt} (\mathfrak{X}))$ , and the terms in (3.1.6.5) can be identified with $E_{\geqslant 0}$ , $E_0$ , $E$ , and $E_{\leqslant 0}$ , respectively. In particular, one has $K_{{\rm Filt} (\mathfrak{X})} \simeq {\rm gr}^* (K_{{{\rm Grad}} (\mathfrak{X})}^{{1/2}}) \otimes {\rm ev}_1^* (K_{\mathfrak{X}}^{{1/2}})$ , as both sides can be identified with $\det (E_{\geqslant 0})^{-1}$ .

See also Kinjo, Park and Safronov [Reference Kinjo, Park and SafronovKPS24, Corollary 3.18] for a more detailed proof of the first part of this theorem.

3.1.7 Lemma

Let $\mathfrak{X}$ be an $s$ -shifted symplectic stack over $\mathbb{K}$ . Then we have an isomorphism

\begin{align*} {\rm sf}^* (\mathbb{T}_{{\rm Filt} (\mathfrak{X})}) \simeq {{{\rm op}}}^* \circ {\rm sf}^* (\mathbb{L}_{{\rm Filt} (\mathfrak{X})} [s]) \end{align*}

of perfect complexes on ${{\rm Grad}} (\mathfrak{X})$ .

3.2.1

We now study the stacks of graded and filtered points of algebraic stacks admitting étale covers by quotient stacks in the sense of §§ 2.2.4 and 2.2.5, and give local descriptions of these stacks using such étale covers.

3.2.2 Theorem.

Let $\mathscr{X}\,$ be an algebraic stack over $\mathbb{K}$ , and let $(\mathscr{X}_i \to \mathscr{X}\,)_{i \in I}$ be a representable étale cover, where each $\mathscr{X}_i \simeq [S_i / G_i]$ , with $S_i$ an algebraic space over $\mathbb{K}$ , and $G_i$ a reductive group. Then there are commutative diagrams

(3.2.2.1)

where all vertical arrows are representable and étale, $\lambda \colon \mathbb{G}_{{\rm m}} \to G_i$ is a cocharacter, and the left-hand square is a pullback square. Moreover, the families

\begin{align*} \bigl ( [S_i^{\lambda , 0} / L_{i, \lambda }] & \longrightarrow {{\rm Grad}} (\mathscr{X}\,) \bigr ) _{i \in I, \ \lambda \colon \mathbb{G}_{{\rm m}} \to G_i} , \\ \bigl ( [S_i^{\lambda , +} / P_{i, \lambda }] & \longrightarrow {\rm Filt} (\mathscr{X}\,) \bigr ) _{i \in I, \ \lambda \colon \mathbb{G}_{{\rm m}} \to G_i} \end{align*}

are representable étale covers of ${{\rm Grad}} (\mathscr{X}\,)$ and ${\rm Filt} (\mathscr{X}\,)$ , respectively.

3.2.3 ${\mathbb{A}}^1$ -action retracts.

We define a notion of ${\mathbb{A}}^1$ -action retracts for algebraic stacks, which are ${\mathbb{A}}^1$ -deformation retracts that also give rise to ${\mathbb{A}}^1$ -actions. This will help us to prove Lemma 3.2.4 below.

More precisely, for a morphism $\pi \colon \mathscr{Y} \to \mathscr{X}$ of algebraic stacks, the structure of an ${\mathbb{A}}^1$ -action retract consists of a monoid action $r \colon {\mathbb{A}}^1 \times \mathscr{Y} \to \mathscr{Y}\,$ , a section $s \colon \mathscr{X} \to \mathscr{Y}\,$ of $\pi$ , and an equivalence $r (0, -) \simeq s \circ \pi$ . Here, the monoid structure on ${\mathbb{A}}^1$ is given by multiplication, and we note that a monoid action requires extra coherence data. We say that $\pi$ is an ${\mathbb{A}}^1$ -action retract if such a structure exists.

3.2.4 Lemma.

Let $\mathscr{X}, \mathscr{Y}\,$ be algebraic stacks over $\mathbb{K}$ with affine stabilizers, and let $f \colon \mathscr{Y} \to \mathscr{X}$ be an étale morphism. Then there is a pullback diagram as follows.

(3.2.4.1)

3.2.5 Lemma.

Let $\mathscr{X}, \mathscr{Y}_1, \mathscr{Y}_2$ be algebraic stacks over $\mathbb{K}$ , and let

be a commutative diagram such that $p_1$ and $p_2$ are ${\mathbb{A}}^1$ -action retracts and $f$ is $\mathbb{A}^1$ -equivariant and étale. Then $f$ is an isomorphism.

Proof. It is enough to show that $f$ is representable and bijective. Choose a point $y \in \mathscr{Y}_2 (\mathbb{K})$ , and form the pullback diagram

where $g$ is given by $g (t) = t \cdot y$ . Then $\mathscr{U}\,$ admits an ${\mathbb{A}}^1$ -action retract to a point, and $f^{\prime}$ is ${\mathbb{A}}^1$ -equivariant and étale. It is enough to show that $f^{\prime}$ is an isomorphism.

The morphism $f^{\prime}$ is surjective since its image is a $\mathbb{G}_{{\rm m}}$ -equivariant open subset of ${\mathbb{A}}^1$ containing $0$ , and hence all of ${\mathbb{A}}^1$ . Let $u \in \mathscr{U}\, (\mathbb{K})$ be a point such that $f^{\prime} (u)=1$ . Then the map $h \colon {\mathbb{A}}^1 \to \mathscr{U}\,$ given by $h (t) = t \cdot u$ is a section of $f^{\prime}$ , and hence is also étale. Moreover, such a point $u$ is unique, since another point $u^{\prime}$ with $f^{\prime} (u^{\prime}) = 1$ would give a section $h^{\prime} \colon {\mathbb{A}}^1 \to \mathscr{U}\,$ with $h^{\prime} (t) = t \cdot u^{\prime}$ , but $h$ and $h^{\prime}$ agree on an open neighbourhood of $0$ , and hence everywhere. It follows that $f^{\prime}$ is bijective on $\mathbb{K}$ -points, and it is enough to show that $\mathscr{U}\,$ is an algebraic space.

Consider the fibre product $V = \mathbb{A}^1 \times _{h, \mathscr{U}\,, h} \mathbb{A}^1$ , which is an algebraic space. Then $V$ also admits an ${\mathbb{A}}^1$ -action retract to a point, and the projection $f^{\prime\prime} \colon V \to \mathbb{A}^1$ to either factor is étale and ${\mathbb{A}}^1$ -equivariant. Repeating the above argument with $V$ in place of $\mathscr{U}\,$ , we see that $f^{\prime\prime}$ is bijective on $\mathbb{K}$ -points, and hence an isomorphism. It follows that $\mathbb{K}$ -points of $\mathscr{U}\,$ cannot have non-trivial automorphisms.

This lemma is based on a joint piece of work [Reference Bu, Halpern-Leistner, Núñez and KinjoBHLINK25] with D. Halpern-Leistner, A. Ibáñez Núñez, and T. Kinjo, and is an improvement of a similar result in a previous version of this paper.

4.1.1 Theorem.

Suppose that we are given the following data:

1. − A finite-dimensional $\mathbb{G}_{{\rm m}}$ -representation $V$ over $\mathbb{K}$ . Let

   \begin{align*} V = \bigoplus _{k \in \mathbb{Z}} V_k \end{align*}
   be the decomposition into weight spaces. Write $V_+ = \bigoplus _{k \gt 0} V_k$ .

2. − A $\mathbb{K}$ -variety $U$ acted on by $\mathbb{G}_{{\rm m}}$ , and a $\mathbb{G}_{{\rm m}}$ -equivariant étale morphism $\iota \colon U \to V$ . Write $U^0 = U^{\mathbb{G}_{{\rm m}}}$ for the fixed locus, and $U^+ = {\rm Map}^{\mathbb{G}_{{\rm m}}} ({\mathbb{A}}^1, U)$ for the attractor as in § 3.1.3 . For a point $u_0 \in U^0 (\mathbb{K})$ , write

   \begin{align*} U^+ (u_0) = \big \{ u \in U \bigm | \lim _{t \to 0} t \cdot u=u_0 \big \} \end{align*}
   for the fibre of the limit map $U^+ \to U^0$ at $u_0$ . We have a canonical isomorphism $U^+ (u_0) \simeq V_+$ by Lemma 3.2.5.

3. − A $\mathbb{G}_{{\rm m}}$ -invariant function $f \colon U \to {\mathbb{A}}^1$ , with $f (u_0)=0$ .

Then we have the identities

(4.1.1.1) \begin{align} \int \limits _{u \in U^+ (u_0)} \Psi _f ([U]) (u) & = \mathbb{L}^{\dim V_+} \cdot \Psi _{f} ([U^0]) (u_0) , \end{align}

(4.1.1.2) \begin{align} \int \limits _{u \in U^+ (u_0)} \Phi _f ([U]) (u) & = \mathbb{L}^{\dim V_+} \cdot \Phi _{f} ([U^0]) (u_0) \ . \end{align}

Moreover, these hold as identities in ${\rm M}^{{\hat {\mu }}} (U^0)$ , where we vary $u_0 \in U^0$ .

This theorem can be seen as a generalization of the integral identity conjectured by Kontsevich and Soibelman [Reference Kontsevich and SoibelmanKS08, Conjecture 4], and proved by Lê [Reference LêLê15], who restricted to the case when the $\mathbb{G}_{{\rm m}}$ -action on $V$ only has weights $-1$ , $0$ , and $1$ . Compare also Joyce and Song [Reference Joyce and SongJS12, Theorem 5.11], where a similar identity involving Euler characteristics is proved, with the same restriction on the weights.

The rest of this subsection is devoted to the proof of Theorem 4.1.1. In the following, we first provide preliminaries on weighted projective spaces and weighted blow-ups, and prove some preparatory results. Then, in Lemma 4.1.8, we establish a weaker version of the theorem, using the theory of motivic nearby cycles for stacks developed in § 2.4. Finally, in § 4.1.9, we show that the weaker version implies the stronger version.

4.1.2 Weighted projective spaces.

Let $V$ be a finite-dimensional $\mathbb{G}_{{\rm m}}$ -representation over $\mathbb{K}$ , with only positive weights. The weighted projective space of $V$ is the quotient stack

\begin{align*} {{^{{\rm w}}}\mathbb{P}} (V) = [(V \setminus \{ 0 \}) / \mathbb{G}_{{\rm m}}] \ . \end{align*}

This is a proper Deligne–Mumford stack over $\mathbb{K}$ , since we have the identification

\begin{align*} {{^{{\rm w}}}\mathbb{P}} (V) \simeq \Biggl [ \mathbb{P} (V) \bigg / \prod _{k=1}^{\dim V} \mu _{n_k} \Biggr ] , \end{align*}

where $\mathbb{P} (V)$ is the usual projective space, and, using a basis of eigenvectors of $V$ , each $n_k$ is the weight of the $k$ th coordinate and $\mu _{n_k}$ acts by scaling the $k$ th coordinate.

By § 2.1.7, the motive of ${{^{{\rm w}}}\mathbb{P}} (V)$ is given by

(4.1.2.1) \begin{align} [{{^{{\rm w}}}\mathbb{P}} (V)] = \frac {\mathbb{L}^{\dim V} - 1}{\mathbb{L} - 1} , \end{align}

and is independent of the choice of weights on $V$ .

We also consider the coarse space ${{^{{\rm cw}}}\mathbb{P}} (V)$ of ${{^{{\rm w}}}\mathbb{P}} (V)$ , which is given by

\begin{align*} {{^{{\rm cw}}}\mathbb{P}} (V) = \operatorname {Proj} \mathbb{K} [V] , \end{align*}

where $\mathbb{K} [V]$ is the free polynomial algebra on $V$ , with $\mathbb{Z}$ -grading given by the weights of $V$ . It is an integral, normal, projective $\mathbb{K}$ -variety.

4.1.3 Weighted blow-ups.

Let $V$ be a finite-dimensional $\mathbb{G}_{{\rm m}}$ -representation over $\mathbb{K}$ , with only positive weights. Let $U$ be a smooth $\mathbb{K}$ -scheme and $U_0 \subset U$ a reduced closed subscheme, and let $p \colon U \to V$ be a smooth morphism such that $U_0 = p^{-1} (0)$ .

Define the weighted blow-up of $U$ along $U_0$ , with weights given by those of $V$ , as the quotient stack

\begin{align*} {{^{{\rm w}}}{\rm Bl}}_{U_0} (U) = [ \{ (t, v, u) \in {\mathbb{A}}^1 \times (V \setminus \{ 0 \}) \times U | p (u) = t \cdot v \} / \mathbb{G}_{{\rm m}} ] , \end{align*}

where $t \cdot (-)$ denotes the $\mathbb{G}_{{\rm m}}$ -action naturally extended to $t \in {\mathbb{A}}^1$ , and $\mathbb{G}_{{\rm m}}$ acts with weight $-1$ on ${\mathbb{A}}^1$ , with the given weights on $V$ , and trivially on $U$ . Note that we have an isomorphism ${{^{{\rm w}}}{\rm Bl}}_{U_0} (U) \simeq U \times _V {{^{{\rm w}}}{\rm Bl}}_{\{ 0 \}} (V)$ .

The natural projection ${{^{{\rm w}}}{\rm Bl}}_{U_0} (U) \to U$ is proper. It restricts to an isomorphism over $U \setminus U_0$ , and has fibres ${{^{{\rm w}}}\mathbb{P}} (V)$ over points in $U_0$ . In particular, we have the relation

(4.1.3.1) \begin{align} [{{^{{\rm w}}}{\rm Bl}}_{U_0} (U)] = \frac {\mathbb{L}^{\dim V} - 1}{\mathbb{L} - 1} \cdot [U_0] + [U \setminus U_0] \end{align}

of motives on $U$ .

4.1.4 Lemma.

Let $U$ be a separated algebraic space of finite type over $\mathbb{K}$ , acted on by a torus $T \simeq \mathbb{G}_{{\rm m}}^n$ for some $n$ , such that points in $U$ have finite stabilizers. Let $\mathscr{X} = [U / T]$ be the quotient stack.

Then $\mathscr{X}\,$ admits a coarse space $\pi \colon \mathscr{X} \to X$ which is a proper universal homeomorphism, and we have an isomorphism

(4.1.4.1) \begin{align} \pi _! = (\pi ^*)^{-1} \colon \hat {{\rm M}} (\mathscr{X}\,) \mathrel {\overset {{{\scriptstyle \sim }}\mspace {3mu}}{\longrightarrow }} \hat {{\rm M}} (X) \ . \end{align}

A similar statement holds for ${\rm M}^{{\hat {\mu }}} (\mathscr{X}\,)$ .

4.1.5 Lemma.

In the situation of Theorem 4.1.1, the locus in $U$ where the morphism $\iota$ preserves $\mathbb{G}_{{\rm m}}$ -stabilizers is open.