In this chapter we prove some important results on smooth homomorphisms. Starting from basic definitions (Section 2.2), we interpret the first cohomology module in terms of infinitesimal extensions (Section 2.1) to characterize formal smoothness in terms of homology.

The first key result is the Jacobian criterion of formal smoothness (2.3.5), a homological characterization of this property. This result and some other results on the homology of field extensions in Section 2.4 (in particular a homological characterization of separability) will allow us to prove the main theorems in this section:

• Grothendieck's Theorems 2.5.8 and 2.5.9, which assert that formal smoothness over a field is equivalent to geometric regularity [EGA, 0IV, 22.5.8]. Another ingenious proof by Faltings of this result can be seen in Matsumura's book [Mt, Theorem 28.7].

• Corollary 2.6.5, which reduces formal smoothness over a noetherian local ring to formal smoothness (or geometric regularity) over a field. This result was proved also by Grothendieck [EGA, 0IV 19.7.1] and it is stated without proof in [Mt, Theorem 28.9]. The more difficult part is to prove that a formally smooth homomorphism is flat. Note that this is much easier if the homomorphism is of finite type (see, e.g., [Bo, Chap. X, §7.10, lemma 5], but the general case is needed.

Our proofs follow those by André [An1, 7.27, 16.18], but we need a few changes in order to avoid the use of simplicial methods or homology modules in higher dimensions.