## Push-out of Schemes

J. Math. Sci. Univ. Tokyo
Vol. 14 (2007), No. 4, Page 531--565.

Li, KEZHENG
Push-out of Schemes
We study the existence of a push-out for two morphisms $Z \to X$ and $Z \to Y$ in the category of schemes. Push-out is a generalization of quotient of groupoid. We give a necessary and sufficient condition for the existence of a push-out in the flat projective case. We also give a sufficient condition for the existence of a push-out in the finite normal case, which does not assume any flatness. In particular, this gives a sufficient condition for the existence of a quotient of a finite groupoid on a normal scheme, which does not assume any flatness.