Keel–Mori theorem
In algebraic geometry, the Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. The theorem was proved by {{harvs|txt|last1=Keel|first1=Sean|last2=Mori|first2=Shigefumi|author2-link=Shigefumi Mori|year=1997}}.
A consequence of the Keel–Mori theorem is the existence of a coarse moduli space of a separated algebraic stack, which is roughly a "best possible" approximation to the stack by a separated algebraic space.
Statement
All algebraic spaces are assumed of finite type over a locally Noetherian base. Suppose that j:R→X×X is a flat groupoid whose stabilizer j−1Δ is finite over X (where Δ is the diagonal of X×X). The Keel–Mori theorem states that there is an algebraic space that is a geometric and uniform categorical quotient of X by j, which is separated if j is finite.
A corollary is that for any flat group scheme G acting properly on an algebraic space X with finite stabilizers there is a uniform geometric and uniform categorical quotient X/G which is a separated algebraic space. {{harvs|txt|last=Kollár|first= János|authorlink=János Kollár |year=1997}} proved a slightly weaker version of this and described several applications.
References
- {{citation|last=Conrad|first=Brian|authorlink=Brian Conrad|title=The Keel–Mori theorem via stacks|year=2005|url=http://math.stanford.edu/~conrad/papers/coarsespace.pdf}}
- {{citation|mr=1432041|last=Keel|first=Seán|last2= Mori|first2= Shigefumi|author2-link=Shigefumi Mori|title=Quotients by groupoids|journal=Annals of Mathematics|series= 2|volume= 145 |year=1997|issue= 1|pages= 193–213|doi=10.2307/2951828}}
- {{citation|mr=1432036|last=Kollár|first= János|authorlink=János Kollár |title=Quotient spaces modulo algebraic groups|journal=Annals of Mathematics|series= 2|volume=145 |year=1997|issue= 1|pages= 33–79|doi=10.2307/2951823|arxiv=alg-geom/9503007}}
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