A construction of a quotient tensor category

Abstract

For a rigid tensor abelian category $T$ over a field $k$ we introduce a notion of a normal quotient $q:T\to Q$. In case $T$ is a Tannaka category, our notion is equivalent to Milne's notion of a normal quotient. More precisely, if $T$ is the category of finite dimensional representations of a groupoid scheme $G$ over $k$, then $Q$ is equivalent to the representation category of a normal subgroupoid scheme of $G$. We describe such a quotient in terms of the subcategory $S$ of $T$ consisting of objects which become trivial in $Q$. We show that, under some condition on $S$, $Q$ is uniquely determined by $S$. If $S$ is an 'etale finite tensor category, we show that the quotient of $T$ by $S$ exists. In particular we show the existence of the base change of $T$ with respect to finite separable field extensions. As an application, we obtain a condition for the exactness of sequences of groupoid schemes in terms of the representation categories.