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Arithmetic Compactifications of PEL-Type Shimura Varieties$
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Kai-Wen Lan

Print publication date: 2013

Print ISBN-13: 9780691156545

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691156545.001.0001

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Representability of Moduli Problems

Representability of Moduli Problems

Chapter:
(p.91) Chapter Two Representability of Moduli Problems
Source:
Arithmetic Compactifications of PEL-Type Shimura Varieties
Author(s):

Kai-Wen Lan

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691156545.003.0002

This chapter elaborates on the representability of the moduli problems defined in the previous chapter. The treatment here is biased towards the prorepresentability of local moduli and Artin's criterion of algebraic stacks. The geometric invariant theory or the theory of Barsotti–Tate groups has been set aside: the argument is very elementary and might be considered outdated by the experts in this area. The chapter, however, discusses the Kodaira–Spencer morphisms of abelian schemes with PEL structures, which are best understood via the study of deformation theory. It also considers the proof of the formal smoothness of local moduli functors, illustrating how the linear algebraic assumptions are used.

Keywords:   representability, moduli problems, prorepresentability, algebraic stacks, Kodaira–Spencer morphisms, deformation theory, local moduli functors, linear algebraic assumptions

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