Hom-stacks and restriction of scalars

Abstract

Abstract. 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 Hom S (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 Hom S (X , Y) is an Artin stack with quasi-compact and separated diagonal. Statements of resultsFix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over S with finite diagonals. Define Hom S (X , Y) to be the fibered category over the category of S-schemes, which to any T → S associates the groupoid of functors X T → Y T over T , whereTheorem 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 Hom S (X , Y) is an Artin stack locally of finite presentation over S with separated and quasi-compact diagonal. If Y is a DeligneMumford stack (resp. algebraic space) then Hom S (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 thatétale locally on S there exists a finite flat cover Z → X as in (1

Weil and Grothendieck approaches to adelic points

“…This is proved by anétale descent argument to upgrade from schemes to algebraic spaces, and is explained in (the proof of) [Ols,Prop. 2.2] apart from the property of having fibers non-empty and geometrically connected of pure dimension d. So now we address this latter fibral property.…”

Flattening stratification and the stack of partial stabilizations of prestable curves

“…These examples are derived from an example worked out using the formalism of Hom-stacks of Olsson [17]. Anétale cover of the moduli stack of prestable curves of genus 0 is exhibited, whose connected components are global quotient stacks, while as predicted by Edidin and Fulghesu [6] no global quotient presentation exists for sufficiently large finite-type open substacks of the moduli stack itself.…”

A boundedness theorem for Hom-stacks

“…Let Y → W be a quasi-finite proper surjection over S to a separated algebraic space W over S of finite presentation. By [Ol,1.1] we then have Deligne-Mumford stacks Hom S (X , Y) and Hom S (X , W ). is of finite type.…”

“…Let X and Y be separated Deligne-Mumford stacks of finite presentation over an algebraic space S and define Hom S (X , Y) as in [Ol,1.1]. Assume that X is flat and proper over S, and that locally in the fppf topology on S, there exists a finite flat surjection Z → X from an algebraic space Z.…”

“…Note that in the case when S is the spectrum of a field, then X has a coarse moduli space π : X → X and the formation of this moduli space commutes with arbitrary base change (see [Ol,2.11] for a discussion of this). In this case the right side of 1.1.1 is canonically isomorphic to Hom S (X, W ) by the universal property of coarse moduli spaces.…”