Green groupoids of 2-Calabi–Yau categories, derived Picard actions, and hyperplane arrangements

نویسندگان

چکیده

We present a construction of (faithful) group actions via derived equivalences in the general categorical setting algebraic 2-Calabi–Yau triangulated categories. To each category C \mathscr {C} satisfying standard mild assumptions, we associate groupoid G Subscript script mathvariant="script">G {G}_{ \mathscr {C} } , named green defined an intrinsic homological way. Its objects are given by set representatives alttext="m r i g m r i g encoding="application/x-tex">mrig\mathscr equivalence classes basic maximal rigid arrows mutation, and relations equating monotone (green) paths silting order. In this generality construct homomorphsim from to Picard collection endomorphism rings Frobenius model ; latter canonically acts triangle between categories rings. prove that constructed representation is faithful if index chamber decompositions split Grothendieck groups come hyperplane arrangements. If alttext="normal Sigma squared approximately-equals d"> mathvariant="normal">Σ 2 ≅ d encoding="application/x-tex">\Sigma ^2 \cong id has finitely many objects, isomorphic Deligne arrangement faithful.

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ژورنال

عنوان ژورنال: Transactions of the American Mathematical Society

سال: 2022

ISSN: ['2330-0000']

DOI: https://doi.org/10.1090/tran/8770