2022/02/10 by Quoc P. Ho, Penghui Li, Ho, Quoc P. +1 · 3 citations
Mathematics · #14A30. Secondary 14M15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 14F08 #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2202.04833
openalex publication_date 2022/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type Y over \mathbbFq a symmetric monoidal DG-category Shvgr, c(Y) of constructible graded sheaves on Y along with the six-functor formalism, a perverse t-structure, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse t-structures, and Frobenius weights on the category of (mixed) ℓ-adic sheaves. Classically, mixed versions were only constructed in very special cases due to the non-semisimplicity of Frobenius. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, the category Shvgr, c(Y) agrees with those previously constructed when they are available. For example, for any reductive group G with a fixed pair T⊂ B of a maximal torus and a Borel subgroup, we have an equivalence of monoidal DG weight categories Shvgr, c(B\backslash G/B) ≃ Chb(SBimW), where Chb(SBimW) is the monoidal DG-category of bounded chain complexes of Soergel bimodules and W is the Weyl group of G.