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A Riemann-Hilbert correspondence in positive characteristic. (English) Zbl 1458.14030

The main result of this paper is the following Riemann-Hilbert correspondence in positive characteristic:
Theorem 1.0.2. Let \(R\) be a commutative \(\mathbb{F}_p\)-algebra. Then there is a fully faithful embedding of abelian categories: \[\{\substack{p\text{-torsion étale sheaves}\\ \text{on Spec}(R)}\} \xrightarrow{RH} \{\substack{R\text{-modules }M\text{ wit a}\\ \text{Frobenius-semilinear automorpism } \varphi_M}\}.\]
Moreover, the essential image of \(RH\) consists of those \(R\)-modules \(M\) equipped with a Frobenius-semilinear automorphism \(\varphi_M: M\to M\) (i.e., \(M\) is a perfect Frobenius module) which satisfies the following condition: every element \(x\in M\) satisfies an equation of the form \[\varphi_M^nx+a_1\varphi_M^{n-1}x+\dots +a_nx=0\] for some coefficients \(a_1,\dots,a_n\in R\) (these are called algebraic Frobenius modules).
The functor \(RH\) as above is called the Riemann-Hilbert functor. The above theorem can be viewed as a positive characteristic analog of the Riemann-Hilbert correspondence in characteristic zero (that is, the equivalence between the category of perverse sheaves and the category of holonomic \(D\)-modules with regular singularities). In fact, the author also showed that the functor \(RH^c\) obtained by restricting \(RH\) to the subcategory \(Shv_{\text{ét}}^c(\text{Spec}(R), \mathbb{F}_p)\subseteq Shv_{\text{ét}}(\text{Spec}(R), \mathbb{F}_p)\) of constructible étale sheaves on Spec\((R)\), induces an equivalence of categories between \(Shv_{\text{ét}}^c(\text{Spec}(R), \mathbb{F}_p)\) and the category of {holonomic} Frobenius modules (see Definition 4.1.1, basically these are perfect Frobenius modules that satisfy certain finiteness property). The algebraic Frobenius modules are basically colimits of holonomic Frobenius modules, thus this equivalence is naturally enlarged to the equivalence in Theorem 1.0.2.
To prove Theorem 1.0.2, the author first construct the solution functor \(Sol\) that assigns each Frobenius module \(M\) the kernel of the map \(\mathrm{id}-\varphi_M: M\to M\), formed in the categroy of \(Shv_{\text{ét}}(\text{Spec}(R), \mathbb{F}_p)\) (this turns out to be the inverse of the functor \(RH^c\) when restricted to holonomic Frobenius modules). To construct the functor \(RH\), the key step is to construct a pull-back functor \(f^{\diamond}\) on perfect Frobenius modules for any morphism of \(\mathbb{F}_p\)-algebras \(A\to B\), and a functor called the compactly supported direct images of Frobenius modules, \(f_!\), where \(f: A\to B\) is an étale morphism of \(\mathbb{F}_p\)-algebras (under \(RH\), \(f^{\diamond}\) and \(f_!\) correspond to the usual functors \(f^*\) and \(f_!\) on \(Shv_{\text{ét}}(\text{Spec}(R), \mathbb{F}_p))\). The functor \(RH\) applied to an étale sheaf \(F\) is constructed by presenting \(F\) as the cokernel of two étale sheaves of the form \(j_!\mathbb{F}_p\) (after a limiting argument), and then using the observation that the perfection of \(R\) is \(RH(\mathbb{F}_p)\), and the fact that \(RH\) is supposed to be compatible with \(j_!\). These are done in sections 5 and 6. Theorem 1.0.2 is proved in section 7, where it is shown that the functor is fully faithful and identifies its essential image.
In the second half of the paper, the authors consider several refinements of Theorem 1.0.2: in section 8 it is shown that \(RH\) is compatible with tensor product; in section 9, Theorem 1.0.2 is upgraded to an equivalence between étale sheaves with \(\mathbb{Z}/p^n\)-coefficients and Frobenius modules over \(W_n(R)\), the ring of Witt vectors of length \(n\); in section 10, Theorem 1.0.2 is generalized to an arbitrary \(\mathbb{F}_p\)-scheme \(X\) (by glueing the affines).
Finally, in section 11 and 12, the authors compare their Riemann-Hilbert correspondence with previous work of Emerton-Kisin, who estabished an anti-equivalence between the bounded derived category of locally finitely generated unit \(R[F]\)-modules \(D^b_{lfgu}(R[F])\) and the bounded derived category of constructible étale sheaves with \(\mathbb{F}_p\)-coefficients (for \(R\) smooth algebra over a field \(k\) of characteristic \(p\)). The authors show that the functor \(RH\) extends to a functor of derived categories \(D_{\text{ét}}(\text{Spec}(R, \mathbb{F}_p)\to D_{perf}(R[F])\) and identifies its essential image (the solution functor \(Sol\) also extends to the derived categories). The author further construct a duality functor \(\mathbb{D}\): \(D^b_{hol}(R[F])\to D^b_{lfgu}(R[F])^{op}\) that is compatible with the solution functors (the solution functor \(Sol\) constructed here and the solution functor constructed by Emerton-Kisin). In this way, Emerton-Kisin’s Riemann-Hilbert correspondence is recovered and extended to singular \(R\) (after defining locally finitely generated unit \(R[F]\)-modules properly).

MSC:

14G17 Positive characteristic ground fields in algebraic geometry
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14F20 Étale and other Grothendieck topologies and (co)homologies
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