Keel–Mori theorem

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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 Sean Keel and Shigefumi Mori (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.


All algebraic spaces are assumed of finite type over a locally Noetherian base. Suppose that j:RX×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. János Kollár (1997) proved a slightly weaker version of this and described several applications.


  • Conrad, Brian (2005), The Keel–Mori theorem via stacks (PDF)
  • Keel, Seán; Mori, Shigefumi (1997), "Quotients by groupoids", Annals of Mathematics, 2, 145 (1): 193–213, doi:10.2307/2951828, MR 1432041
  • Kollár, János (1997), "Quotient spaces modulo algebraic groups", Annals of Mathematics, 2, 145 (1): 33–79, arXiv:alg-geom/9503007, doi:10.2307/2951823, MR 1432036