## 38.1 Introduction

In this chapter, we discuss some advanced results on flat modules and flat morphisms of schemes and applications. Most of the results on flatness can be found in the paper [GruRay] by Raynaud and Gruson.

Before reading this chapter we advise the reader to take a look at the following results (this list also serves as a pointer to previous results):

General discussion on flat modules in Algebra, Section 10.38.

The relationship between $\text{Tor}$-groups and flatness, see Algebra, Section 10.74.

Criteria for flatness, see Algebra, Section 10.98 (Noetherian case), Algebra, Section 10.100 (Artinian case), Algebra, Section 10.127 (non-Noetherian case), and finally More on Morphisms, Section 37.16.

Generic flatness, see Algebra, Section 10.117 and Morphisms, Section 29.26.

Openness of the flat locus, see Algebra, Section 10.128 and More on Morphisms, Section 37.15.

Flattening, see More on Algebra, Sections 15.16, 15.17, 15.18, 15.19, and 15.20.

Additional results in More on Algebra, Sections 15.21, 15.22, 15.25, and 15.26.

As applications of the material on flatness we discuss the following topics: a non-Noetherian version of Grothendieck's existence theorem, blowing up and flatness, Nagata's theorem on compactifications, the h topology, blow up squares and descent, weak normalization, descent of vector bundles in positive characteristic, and the local structure of perfect complexes in the h topology.

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