## 16.1 Introduction

The main result of this chapter is the following:

This theorem is due to Popescu, see [popescu-letter]. A readable exposition of Popescu's proof was given by Richard Swan, see [swan] who used notes by André and a paper of Ogoma, see [Ogoma].

Our exposition follows Swan's, but we first prove an intermediate result which lets us work in a slightly simpler situation. Here is an overview. We first solve the following “lifting problem”: A flat infinitesimal deformation of a filtered colimit of smooth algebras is a filtered colimit of smooth algebras. This result essentially says that it suffices to prove the main theorem for maps between reduced Noetherian rings. Next we prove two very clever lemmas called the “lifting lemma” and the “desingularization lemma”. We show that these lemmas combined reduce the main theorem to proving a Noetherian, geometrically regular algebra $\Lambda $ over a field $k$ is a filtered colimit of smooth $k$-algebras. Next, we discuss the necessary local tricks that go into the Popescu-Ogoma-Swan-André proof. Finally, in the last three sections we give the proof.

We end this introduction with some pointers to references. Let $A$ be a henselian Noetherian local ring. We say $A$ has the *approximation property* if for any $f_1, \ldots , f_ m \in A[x_1, \ldots , x_ n]$ the system of equations $f_1 = 0, \ldots , f_ m = 0$ has a solution in the completion of $A$ if and only if it has a solution in $A$. This definition is due to Artin. Artin first proved the approximation property for analytic systems of equations, see [Artin-Analytic-Approximation]. In [Artin-Algebraic-Approximation] Artin proved the approximation property for local rings essentially of finite type over an excellent discrete valuation ring. Artin conjectured (page 26 of [Artin-Algebraic-Approximation]) that every excellent henselian local ring should have the approximation property.

At some point in time it became a conjecture that every regular homomorphism of Noetherian rings is a filtered colimit of smooth algebras (see for example [Raynaud-Rennes], [popescu-global], [Artin-power-series], [Artin-Denef]). We're not sure who this conjecture^{1} is due to. The relationship with the approximation property is that if $A \to A^\wedge $ is a colimit of smooth algebras with $A$ as above, then the approximation property holds (insert future reference here). Moreover, the main theorem applies to the map $A \to A^\wedge $ if $A$ is an excellent local ring, as one of the conditions of an excellent local ring is that the formal fibres are geometrically regular. Note that excellent local rings were defined by Grothendieck and their definition appeared in print in 1965.

In [Artin-power-series] it was shown that $R \to R^\wedge $ is a filtered colimit of smooth algebras for any local ring $R$ essentially of finite type over a field. In [Rotthaus-Artin] it was shown that $R \to R^\wedge $ is a filtered colimit of smooth algebras for any local ring $R$ essentially of finite type over an excellent discrete valuation ring. Finally, the main theorem was shown in [popescu-GND], [popescu-GNDA], [popescu-letter], [Ogoma], and [swan] as discussed above.

Conversely, using some of the results above, in [Rotthaus-excellent] it was shown that any Noetherian local ring with the approximation property is excellent.

The paper [Spivakovsky] provides an alternative approach to the main theorem, but it seems hard to read (for example [Lemma 5.2, Spivakovsky] appears to be an incorrectly reformulated version of [Lemma 3, Elkik]). There is also a Bourbaki lecture about this material, see [Teissier].

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