Stacks of fiber functors and Tannaka's reconstruction

2020/10/22 by Tonini, Fabio
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.12445

Abstract

Given a quasi-compact category fibered in groupoids X and a monoidal subcategory C of its category of locally free sheaves Vect(X), we are going to introduce the stack of fiber functors Fib_X,C with source C, which comes equipped with a map P_C\colonX\toFib_X,C and a functor G\colonC\toVect(Fib_X,C). If C generates QCoh(X) and X is an fpqc stack with quasi-affine diagonal, we show that P_C\colonX\toFib_X,C is an equivalence, as it happens by Tannaka's reconstruction when X is an affine gerbe over a field. In general, under mild assumption on C, e.g. C=Vect(X), we show that Fib_X,C is a quasi-compact fpqc stack with affine diagonal and that the image G(C) generates QCoh(Fib_X,C).

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