Time：14:00-15:00, Tuesday, February 10 2026
Venue：E14-212
Speaker：Lei Zhang, Sun Yat-Sen University
Title：Local systems in characteristic p
Abstract：Let X be a proper scheme over the field of complex numbers. There is an equivalence between the category finite dimensional complex representations of the topological fundamental group of X^an and the category of stratified bundles on X. Thus stratified bundles are local systems and they are equivalent to O-coherent D-modules if X is moreover smooth. In "Représentations linéaires et compactification profinie des groupes discrets" Grothendieck proved that if a map f: Y->X of connected complex varieties induces an isomorphism of the étale fundamental groups, then fˆ* induces an equivalence of stratified bundles. Now let X be a proper scheme over an algebraically closed field of characteristic p. Grothendieck constructed in Dix Exposé several other candidates for local systems on X, among them stratified bundles, F-divided sheaves, and infinitesimal crystals. Bhatt later showed that they are all equivalent and satisfy h-descent. In this talk, we will prove Grothendieck's theorem also holds true in this setting, namely, if f: Y->X is a map of connected normal proper schemes over k which induces an isomorphism of the étale fundamental groups, then it induces an equivalence of F-divided sheaves. This is a joint work in progress with Andres Fernandez Herrero, Dario Weißmann, Xiaotao Sun and Xucheng Zhang.
