Parity sheaves and Smith theory
| Autor: | Spencer Leslie, Gus Lonergan |
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| Rok vydání: | 2021 |
| Předmět: |
Derived category
Pure mathematics Functor Applied Mathematics General Mathematics Categorification Prime number Algebraic variety Reductive group Mathematics::Category Theory FOS: Mathematics Equivariant map Representation Theory (math.RT) Mathematics - Representation Theory Affine Grassmannian 20G05 55P43 Mathematics |
| Zdroj: | Journal für die reine und angewandte Mathematik (Crelles Journal). 2021:49-87 |
| ISSN: | 1435-5345 0075-4102 |
| Popis: | Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. Under certain assumptions, we use this to define a functor from the category of parity sheaves on $X$ to the category of parity sheaves on the fixed-point locus $X^{\mathbb{Z}/p\mathbb{Z}}$. This may be thought of as a categorification of Smith theory. When $X$ is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence. Comment: 34 pages;v3: substantial edits throughout, improved exposition and introduction, main application to the Frobenius-contraction functor strengthened; v4: accepted version, to appear in Journal f\"{u}r die Reine und Angewandte Mathematik |
| Databáze: | OpenAIRE |
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