Push-out of Schemes By Kezheng Li∗

Abstract. We study the existence of a push-out for two mor-phisms Z → X and Z → Y in the category of schemes. Push-out is a generalization of quotient of groupoid. We give a necessary and suf-ficient 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 par-ticular, 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. 0