$$\underline{\operatorname{Hom}}$$-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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References:

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