Hom–stacks and Restriction of Scalars

Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack HomS(X ,Y) classifying morphisms between X and Y. Assume that X is proper and flat over S, and fppf–locally on S there exists a finite finitely presented flat cover Z → X with Z an algebraic space. Then we show that HomS(X ,Y) is an Artin stack with quasi–compact and separated diagonal. 1. Statements of results Fix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over S with finite diagonals. Define HomS(X ,Y) to be the fibered category over the category of S–schemes, which to any T → S associates the groupoid of functors XT → YT over T , where XT (resp. YT ) denotes X ×S T (resp. Y ×S T ). Theorem 1.1. Let X and Y be finitely presented separated Artin stacks over S with finite diagonals. Assume in addition that X is flat and proper over S, and that locally in the fppf topology on S there exists a finite and finitely presented flat surjection Z → X from an algebraic space Z. Then the fibered category HomS(X ,Y) is an Artin stack locally of finite presentation over S with separated and quasi–compact diagonal. If Y is a Deligne– Mumford stack (resp. algebraic space) then HomS(X ,Y) is also a Deligne–Mumford stack (resp. algebraic space). Remark 1.2. If S is the spectrum of a field and X is a Deligne–Mumford stack which is a global quotient stack and has quasi–projective coarse moduli space, then by ([15], 2.1) there always exists a finite flat cover Z → X . Remark 1.3. When S is arbitrary and X is a twisted curve in the sense of ([1]), it is also true that étale locally on S there exists a finite flat cover Z → X as in (1.1). This is shown in ([19]). Remark 1.4. It is possible to prove that HomS(X ,Y) is an Artin stack under weaker assumptions on the diagonals of X and Y (though for the diagonal of HomS(X ,Y) to have reasonable properties the assumptions of (1.1) seem necessary). This has recently been shown by Aoki. Theorem 1.1 will be deduced from another result about pushforwards of stacks. Let S/S be a separated Artin stack locally of finite presentation and with finite diagonal. For any morphism of algebraic spaces f : S → T , define f∗S to be the fibered category over T which to any T ′/T associates the groupoid S(T ′ ×T S), with the natural notion of pullback. We call f∗S the restriction of scalars of S from S to T . Theorem 1.5. Let f : S → T be a proper, finitely presented, and flat morphism of algebraic spaces. Then the fibered category f∗S is an Artin stack locally of finite presentation over T 1