ScalingStacks

0MX4

Theorem (Theorem 4.9). Assuming (PT) and (FO) there is an equivalence of categories between the category of Springer motives and the perfect derived category of graded modules of the motivic extension algebra

DMGS​p​r⁡(𝒩,ℚ){\lx@inpgf@ignorespaces\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})}Dperfℤ⁡(E).{\lx@inpgf@ignorespaces{\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(E)}.}∼\scriptstyle{\lx@inpgf@ignorespaces\sim}

Moreover, for all primes ℓ≠p\ell\neq p the ℓ\ell-adic realisation functor Realℓ\operatorname{Real}_{\ell} gives an isomorphism E⊗ℚℚℓ≅Eℓe´​tE\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\cong E^{\acute{e}t}_{\ell} and acts as a degrading functor with respect to the Tate-twist (1)(1) in the sense of [BGS96]

DMGS​p​r⁡(𝒩,ℚℓ){\lx@inpgf@ignorespaces\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}DGS​p​r⁡(𝒩,ℚℓ).{\lx@inpgf@ignorespaces{\operatorname{D}^{Spr}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).}}Realℓ\scriptstyle{\lx@inpgf@ignorespaces\operatorname{Real}_{\ell}}

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2