# The Riemann-Hilbert Correspondence for Unit \(F\)-crystals

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*Matthew Emerton; Mark Kisin*

A publication of the Société Mathématique de France

Let \(\mathbb {F}_q\) denote the finite field of order \(q\) (a
power of a prime \(p\)), let \(X\) be a smooth scheme over a field
\(k\) containing \(\mathbb {F}_q\), and let \(\Lambda \)
be a finite \(\mathbb {F}_q\)-algebra. We study the relationship between
constructible \(\Lambda \)-sheaves on the étale site of \(X\),
and a certain class of quasi-coherent \(\mathcal {O_X}\otimes _{\mathbb
F_q}{\Lambda }\)-modules equipped with a "unit" Frobenius structure. The
authors show that the two corresponding derived categories are anti-equivalent
as triangulated categories, and that this anti-equivalence is compatible with
direct and inverse images, tensor products, and certain other operations.

They also obtain analogous results relating complexes of constructible
\(\mathbb {Z}/p^n\mathbb {Z}\)-sheaves on smooth
\(W_n(k)\)-schemes, and complexes of Berthelot's arithmetic
\(\mathcal {D}\)-modules, equipped with a unit Frobenius.

The volume is suitable for graduate students and researchers interested in algebra and algebraic geometry.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.