Finite quotients in algebraic geometry

Abstract

In the 1960s Alexander Grothendieck and Michel Demazure organized the seminar, S eminaire de G eom etrie Alg ebrique du Bois Marie (SGA), in which they developed the theory of group schemes. My thesis primarily focuses on the construction of quotients of schemes in various contexts. First, we analyze [1] to understand how the construction of quotients of schemes is related to the construction of quotients of more general locally ringed spaces. Speci cally, we'll see how to pass to the quotient scheme by a at groupoid. Also, we'll nd the quotient induced by a nite group action on an a ne scheme. Then we discuss how to make sense of X=G when there is no categorical quotient. Working on the relationship between quotients taken in the category of locally ringed spaces and the category of algebraic spaces, we investigate the comparison map between those quotients. Finally, we examine Hironaka's Example of a smooth proper complex variety with an involution which has no quotient in the category of schemes to provide a special case for the comparison map context.