A DESCENT THEOREM FOR FORMAL SMOOTHNESS

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

Representability of Moduli Problems

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or the theory of Barsotti–Tate groups has been set aside: the argument is very elementary and might be considered outdated by the experts in this area.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.

Separable Functors and Formal Smoothness

The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of ad-(co)invariant integrals for a Hopf algebra H is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double D(H) over H. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π : E → H is a bialgebra surjection with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then this section can be chosen to be H-bicolinear. We also deal with the dual case.