Subsection 4.2. (0N05) · ScalingStacks

4.2. Motivic extension algebras

We define now the motivic and $\ell$ -adic extension algebras.

Definition 4.3. The motivic extension algebra $E$ is the $\mathbb{Z}$ -graded locally unital algebra

E=\operatorname{End}^{\bullet}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mathcal{T}^{Spr})

which has a grading induced by the autoequivalence $\langle 1\rangle=(1)[2],$ see (2.3). More explicitly, for $n\in\mathbb{Z}$ the $n$ -th graded part of $E$ is

E^{n}=\bigoplus_{i,j}\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n]).

For every prime $\ell\neq 0$ we define the $\ell$ -adic extension algebra $E^{\acute{e}t}_{\ell}$ in the same way, replacing $\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})$ by $\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).$

The realisation functor induces a morphism of algebras $\operatorname{Real}_{\ell}:E\to E^{\acute{e}t}_{\ell}$ . The motivic extension algebra can be understood in terms of $G$ -equivariant Chow groups of the Steinberg varieties $Z_{i,j}=\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}$ equipped with a convolution product.

0MYS

Proposition 4.4. Let $d_{j}=\dim\widetilde{\mathcal{N}}_{j}.$ There is a natural isomorphism

\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])=\operatorname{CH}_{d_{j}-n}^{G}(Z_{i,j})_{\mathbb{Q}}.

0MYT

Proof. Let $Z=Z_{i,j}$ with projections $\pi_{i,j}:Z\to\widetilde{\mathcal{N}}_{i,j}.$ For a variety $X$ denote by $\operatorname{fin}_{X}:X\to\operatorname{pt}$ the structure map. Then by using various adjunctions, base change and $\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{*}=\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{!}(-d_{j})[-2d_{j}]$ since $\widetilde{\mathcal{N}}_{j}$ is smooth, we get

	$\displaystyle\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}($	$\displaystyle\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(\widetilde{\mathcal{N}}_{i},\mathbb{Q})}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}},\mu_{i}^{!}\mu_{j,*}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(\widetilde{\mathcal{N}}_{i},\mathbb{Q})}(\operatorname{fin}_{\widetilde{\mathcal{N}}_{i}}^{}\mathbb{Q},\pi_{i,}\pi_{j}^{!}\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{*}\mathbb{Q}(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{fin}_{\widetilde{\mathcal{N}}_{i},}\pi_{i,}\pi_{j}^{!}\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{!}\mathbb{Q}(n-d_{j})[2(n-d_{j})])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{fin}_{Z,*}\operatorname{fin}_{Z}^{!}\mathbb{Q}(n-d_{j})[2(n-d_{j})])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{M}^{c}(Z)(n-d_{j})[2(n-d_{j})]).$

Now the last term is isomorphic to $\operatorname{CH}^{G}_{d_{j}-n}(Z)_{\mathbb{Q}}$ by [Kel17, Theorem 5.3.14]. ∎

0MYU

Remark 4.5. Since $Z_{i,j}$ is not necessarily equidimensional we need to work with Chow groups indexed by the dimension of cycles here.

The convolution of two cycles $\alpha\in\operatorname{CH}^{G}_{d_{j}-n}(Z_{i,j})_{\mathbb{Q}}$ and $\beta\in\operatorname{CH}^{G}_{d_{k}-m}(Z_{j,k})_{\mathbb{Q}}$ is given by the formula

(4.1)

\displaystyle\alpha\star\beta=p_{*}\delta^{!}(\alpha\times\beta)\in\operatorname{CH}^{G}_{d_{k}-n-m}(Z_{i,k})_{\mathbb{Q}}

where $\alpha\times\beta$ is the exterior product and

	$\displaystyle\delta$	$\displaystyle:\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}\to\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}=Z_{i,j}\times Z_{j,k}\text{ and}$
	$\displaystyle p$	$\displaystyle:\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}\to\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}=Z_{i,k}$

are the diagonal and projection maps.

Convolution of cycles and composition of morphisms are compatible in the obvious way. In the non-equivariant case this is proven and discussed in detail in [Fan16]. The proof can be adapted to the equivariant case by using that both equivariant motivic sheaves and equivariant Chow groups are defined in terms of their non-equivariant versions of approximations of the Borel construction. We will not present the details here.

0MYV

Corollary 4.6. There is an isomorphism of graded algebras

(4.2)

\displaystyle E^{\bullet}\cong\bigoplus_{i,j}\operatorname{CH}_{d_{j}-\bullet}^{G}(Z_{i,j})_{\mathbb{Q}}

Similarly, the $\ell$ -adic extension algebra can be described in terms of $\ell$ -adic Borel–Moore homology, see [CG10, Section 8.6].

0MYW

Proposition 4.7. There is an isomorphism of graded algebras

(4.3)

\displaystyle(E^{\acute{e}t}_{\ell})^{\bullet}\cong\bigoplus_{i,j}H^{BM,G}_{2(d_{j}-\bullet)}(Z_{i,j},\mathbb{Q}_{\ell}(d_{j}-\bullet)).

0MYX

Remark 4.8. The above discussion should be a shadow of the following conjectural general theory. There should be a Chow weight structure on the category $\operatorname{DM}_{G}^{gm}(\mathcal{N})$ similarly to the non-equivariant case, see Example 2.2(3). The heart of this Chow weight structure should be equivalent to a category of equivariant relative Chow motives $\operatorname{Chow}_{G}(\mathcal{N},\mathbb{Q})$ in which the composition of morphisms is defined via convolution as in (4.1). Since by assumption $\widetilde{\mathcal{N}}_{i}$ is smooth and $\mu_{i}$ is projective the motive $\mu_{!}(\mathbb{Q}_{M_{i}})$ should be in the heart and correspond to the relative Borel–Moore motive $M^{BM}(\widetilde{\mathcal{N}}_{i}/\mathcal{N})$ in the category $\operatorname{Chow}_{G}(N,\mathbb{Q}).$

In the non-equivariant case this is shown to be true by Fangzhou [Fan16]. To define a weight structure in the equivariant case, one would need appropriate $G$ -equivariant resolution of singularities or alterations, see [SVW18, Remark II.4.15].

In this article we get around this problem by defining a weight structure on the subcategory $\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})$ by brute force using the conditions (PT) and (FO).

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2