Tame stacks in positive characteristic. (English) Zbl 1222.14004

Ann. Inst. Fourier 58, No. 4, 1057-1091 (2008); corrigendum ibid. 64, No. 3, 945-946 (2014).
A tame stack is an algebraic stack \({\mathcal M}\) with finite inertia and such that the pushforward functor \(\rho_{*} :\mathrm{QCoh}({\mathcal M}) \to \mathrm{QCoh}(M)\) from quasi-coherent sheaves on \({\mathcal M}\) to quasi-coherent sheaves on its coarse moduli space \(M\), is exact. The exactness of the pushforward functor \(\rho_{*}\) is a property that Deligne-Mumford stacks enjoy in characteristic zero, but which fails in positive characteristic.
Another, equivalent, description of a tame stack is given in the main result of the article, Theorem 3.2, stating that a stack with finite inertia is tame if and only if the coarse moduli space of the stack is étale locally the quotient of a scheme by a finite, flat, linearly reductive group. It follows from the main result that tame stacks have several other nice properties as well, such as flatness of its moduli space and that the formation of the coarse moduli commutes with base change.
An important ingredient in the proof of the main result is the classification of finite, flat, linearly reductive groups that the authors give. They show that a finite, flat group \(G\) is linearly reductive if and only if each geometric fiber of \(G\) is an extension of a tame and étale group by a diagonalizable group.
In positive characteristic several interesting moduli problems are not covered by Deligne-Mumford stacks, and the fact that Deligne-Mumford stacks do not always behave well with respect to the above mentioned properties, makes the notion of tame stacks interesting and important.


14A20 Generalizations (algebraic spaces, stacks)
14L15 Group schemes
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