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\(\underline{\operatorname{Hom}}\)-stacks and restriction of scalars. (English) Zbl 1114.14002

The author proves that the stack of morphisms between suitable algebraic stacks is again algebraic. Olsson interprets such \(\underline{\operatorname{Hom}}\) stacks as Weil restrictions, and proves more generally that, under certain assumptions, the Weil restriction of an algebraic stack is algebraic as well. The latter is proved by a careful application of M. Artin’s criterion [Invent. Math. 27, 165–189 (1974; Zbl 0317.14001)]. Similar statements hold true for Deligne-Mumford stacks and algebraic spaces.
The precise statement about \(\underline{\operatorname{Hom}}\) stacks is as follows: Let \(S\) be an algebraic space, and \(\mathcal{X}\) and \(\mathcal{Y}\) be two separated, finitely presented algebraic \(S\)-stacks whose diagonal is finite. Suppose that \(\mathcal{X}\) is flat, proper, and admits fppf-locally a finite, flat, finitely presented surjection \(Z\to\mathcal{Y}\) from an algebraic space \(Z\). Then the fibered \(S\)-groupoid \(\underline{\operatorname{Hom}}_S(\mathcal{X},\mathcal{Y})\) is an algebraic stack, which is locally of finite presentation.
The result about Weil restrictions is: Let \(f:S\to T\) be a flat, proper, finitely presented morphism of algebraic spaces, and \(\mathcal{X}\) be a a separated, finitely presented algebraic \(S\)-stack. Then the fibered \(T\)-groupoid \(f_*(\mathcal{X})\) is an algebraic \(T\)-stack. The latter has as fibers over \(T'\) the groupoid \(\mathcal{X}(T'\times_T S)\).
Related results about Hom stacks were obtained by M. Aoki [Manuscr. Math. 119, No. 1, 37–56 (2006; Zbl 1094.14001)]. He made weaker assumptions on \(\mathcal{X}\) and \(\mathcal{Y}\); then the diagonal of the resulting \(\underline{\operatorname{Hom}}\) stack, however, is only locally of finite type. In Olsson’s paper, the diagonal is actually separated and quasicompact, so the \(\underline{\operatorname{Hom}}\) stack is a true algebraic stack in the sense of G. Laumon and L. Moret-Bailly [“Algebraic spaces”, Ergebn. Math. Grenzgeb. 3, 39 (2000; Zbl 0945.14005)].

MSC:

14A20 Generalizations (algebraic spaces, stacks)
14D20 Algebraic moduli problems, moduli of vector bundles
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