##
**Split embedding problems over complete domains.**
*(English)*
Zbl 1247.12005

The paper under review is concerned with the absolute Galois group \(G_K\) of a field \(K\). This group is profinite, hence is determined in some sense by which finite embedding problems are solvable. (Here a finite embedding problem for \(K\) consists of a finite Galois extension \(L/K\) and an epimorphism of finite groups \(f: B\twoheadrightarrow \mathrm{Gal}(L/K)\). A solution is a continuous epimorphism \(g: G_K\twoheadrightarrow B\) such that \(f\circ g : G_K\to \mathrm{Gal}(L/K)\) is the restriction of automorphisms map.)

Two technical notions that will be needed are of Frattini and of split embedding problems, the former being that \(\ker f\) is contained in the Frattini subgroup of \(B\) and the latter being that \(f\) has a group theoretical section. A general principle in profinite groups says that the study of solvability of finite embedding problems can be reduced to the study of Frattini and of split embedding problems. The former is govern by Galois cohomology. For split embedding problems, say for \(K=\mathbb{Q}\), there is no obvious obstruction for solvability. In fact, P. Dèbes and B. Dechamps [Lond. Math. Soc. Lect. Note Ser. 243, 119–138 (1997; Zbl 0905.12004)] conjecture that every finite split embedding problem over \(\mathbb{Q}\) is solvable. This conjecture in particular implies an affirmative answer to the inverse Galois problem since \(B\to 1\) splits for any finite group \(B\).

Dèbes-Dechamps conjecture, which we call here DD1 in short, is more general in the sense that \(\mathbb{Q}\) is replaced by an arbitrary Hilbertian field \(K\). By definition \(K\) is Hilbertian if for every irreducible \(f(X,Y) \in K[X,Y]\) that is separable in \(Y\) there exists \(a\in K\) such that \(f(a,Y)\) is irreducible. (The Hilbertianity of \(\mathbb{Q}\) is the content of Hilbert’s irreducibility theorem.)

Over an arbitrary field DD1 clearly fails, e.g. for \(K=\mathbb{C}\). Thus Dèbes and Dechamps has a regular version of the conjecture suitable for any field, which we call DD2: For every finite Galois extension \(L/K\) and every finite split embedding problem \(B\twoheadrightarrow \mathrm{Gal}(L(x)/K(x))\) for \(K(x)\), there is a regular solution \(\phi : G_{K(x)}\twoheadrightarrow B\). Here \(\phi\) is regular in the sense that the fixed field \(F\) of the kernel of \(\phi\) in an algebraic closure of \(K(x)\) is regular over \(L\). A standard specialization argument gives that DD2 implies DD1 (if \(K\) is Hilbertian).

In the paper under review the author proves these conjectures for quotient fields of certain complete rings:

Theorem. Let \(K\) be the quotient field of a Noetherian integrally closed domain \(A\). Assume that there is a minimal nonzero prime ideal \(\mathfrak{p}\) of \(A\) such that \(A\) is complete in the \(\mathfrak{p}\)-adic topology and the \(\mathfrak{p}\)-adic valuation on \(A\) extends to a discrete valuation on \(K\). Then:

1. DD2 holds true.

2. If in addition the Krull dimension of \(A\) is at least \(2\), then DD1 holds true.

To prove this theorem the author significantly extends the method of algebraic patching that was introduced by D. Haran and H. Völklein [Isr. J. Math. 93, 9–27 (1996; Zbl 0869.12006)] and further developed by Haran-Jarden in a series of papers.

Let us conclude the review with several remarks:

1. Both parts of the theorem applies to the rings \(A=k[[x_1, \dots, x_n]]\), where \(k\) is any field and \(n\geq 2\), and to \(A=Z[[x_1, \dots, x_n]]\), where \(Z\) is a Noetherian integrally closed domain which is not a field (e.g. \(Z=\mathbb{Z}\)) and \(n\geq 1\).

2. The special case \(A=k[[x_1, x_2]]\) is due to D. Harbater and K. F. Stevenson [Adv. Math. 198, No. 2, 623–653 (2005; Zbl 1104.12003)] using formal patching. Their method fails to extend to higher dimensions.

3. The second part of the theorem does not hold in general in dimension \(1\), as an example take \(K=\mathbb{C}((t))\) and the embedding problem \((\mathbb{Z}/2 \mathbb{Z})^2\to 1\).

4. By a theorem of R. Weissauer [J. Reine Angew. Math. 334, 203–220 (1982; Zbl 0477.12029)], \(K\) as in the theorem is Hilbertian if the Krull dimension of \(A\) is at least \(2\). Hence the second part of the theorem follows from the first.

5. It was unknown whether the fields for which the theorem applies to are ample or not. F. Pop [Ann. Math. (2) 172, No. 3, 2183–2195 (2010; Zbl 1220.12001)] settled that point by showing they are ample, hence he gives a new proof the above theorem.

Two technical notions that will be needed are of Frattini and of split embedding problems, the former being that \(\ker f\) is contained in the Frattini subgroup of \(B\) and the latter being that \(f\) has a group theoretical section. A general principle in profinite groups says that the study of solvability of finite embedding problems can be reduced to the study of Frattini and of split embedding problems. The former is govern by Galois cohomology. For split embedding problems, say for \(K=\mathbb{Q}\), there is no obvious obstruction for solvability. In fact, P. Dèbes and B. Dechamps [Lond. Math. Soc. Lect. Note Ser. 243, 119–138 (1997; Zbl 0905.12004)] conjecture that every finite split embedding problem over \(\mathbb{Q}\) is solvable. This conjecture in particular implies an affirmative answer to the inverse Galois problem since \(B\to 1\) splits for any finite group \(B\).

Dèbes-Dechamps conjecture, which we call here DD1 in short, is more general in the sense that \(\mathbb{Q}\) is replaced by an arbitrary Hilbertian field \(K\). By definition \(K\) is Hilbertian if for every irreducible \(f(X,Y) \in K[X,Y]\) that is separable in \(Y\) there exists \(a\in K\) such that \(f(a,Y)\) is irreducible. (The Hilbertianity of \(\mathbb{Q}\) is the content of Hilbert’s irreducibility theorem.)

Over an arbitrary field DD1 clearly fails, e.g. for \(K=\mathbb{C}\). Thus Dèbes and Dechamps has a regular version of the conjecture suitable for any field, which we call DD2: For every finite Galois extension \(L/K\) and every finite split embedding problem \(B\twoheadrightarrow \mathrm{Gal}(L(x)/K(x))\) for \(K(x)\), there is a regular solution \(\phi : G_{K(x)}\twoheadrightarrow B\). Here \(\phi\) is regular in the sense that the fixed field \(F\) of the kernel of \(\phi\) in an algebraic closure of \(K(x)\) is regular over \(L\). A standard specialization argument gives that DD2 implies DD1 (if \(K\) is Hilbertian).

In the paper under review the author proves these conjectures for quotient fields of certain complete rings:

Theorem. Let \(K\) be the quotient field of a Noetherian integrally closed domain \(A\). Assume that there is a minimal nonzero prime ideal \(\mathfrak{p}\) of \(A\) such that \(A\) is complete in the \(\mathfrak{p}\)-adic topology and the \(\mathfrak{p}\)-adic valuation on \(A\) extends to a discrete valuation on \(K\). Then:

1. DD2 holds true.

2. If in addition the Krull dimension of \(A\) is at least \(2\), then DD1 holds true.

To prove this theorem the author significantly extends the method of algebraic patching that was introduced by D. Haran and H. Völklein [Isr. J. Math. 93, 9–27 (1996; Zbl 0869.12006)] and further developed by Haran-Jarden in a series of papers.

Let us conclude the review with several remarks:

1. Both parts of the theorem applies to the rings \(A=k[[x_1, \dots, x_n]]\), where \(k\) is any field and \(n\geq 2\), and to \(A=Z[[x_1, \dots, x_n]]\), where \(Z\) is a Noetherian integrally closed domain which is not a field (e.g. \(Z=\mathbb{Z}\)) and \(n\geq 1\).

2. The special case \(A=k[[x_1, x_2]]\) is due to D. Harbater and K. F. Stevenson [Adv. Math. 198, No. 2, 623–653 (2005; Zbl 1104.12003)] using formal patching. Their method fails to extend to higher dimensions.

3. The second part of the theorem does not hold in general in dimension \(1\), as an example take \(K=\mathbb{C}((t))\) and the embedding problem \((\mathbb{Z}/2 \mathbb{Z})^2\to 1\).

4. By a theorem of R. Weissauer [J. Reine Angew. Math. 334, 203–220 (1982; Zbl 0477.12029)], \(K\) as in the theorem is Hilbertian if the Krull dimension of \(A\) is at least \(2\). Hence the second part of the theorem follows from the first.

5. It was unknown whether the fields for which the theorem applies to are ample or not. F. Pop [Ann. Math. (2) 172, No. 3, 2183–2195 (2010; Zbl 1220.12001)] settled that point by showing they are ample, hence he gives a new proof the above theorem.

Reviewer: Lior Bary-Soroker (Tel Aviv)

### Keywords:

Galois theory; embedding problems; patching; complete domains; field arithmetic; Hilbertian fields### References:

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