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The monodromy groups of lisse sheaves and overconvergent F-isocrystals

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D’Addezio,  Marco
Max Planck Institute for Mathematics, Max Planck Society;

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D’Addezio, M. (2020). The monodromy groups of lisse sheaves and overconvergent F-isocrystals. Selecta Mathematica, 26(3): 45. doi:10.1007/s00029-020-00569-3.


Cite as: http://hdl.handle.net/21.11116/0000-0006-BC41-D
Abstract
It has been proven by Serre, Larsen-Pink and Chin, that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have "the same" $\pi_0$ and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent $F$-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent $F$-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent $F$-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects.