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Homomorphisms of local algebras in positive characteristic. (English) Zbl 1189.13022

Let \(k\) be a field of characteristic \(p>0\).
Let \(A\) resp. \(B\) be local \(k\)-algebras, \(\mathfrak{m}_A\) resp. \(\mathfrak{m}_B\) their maximal ideals. Let \(\varphi:A\to B\) be a homomorphism of local \(k\)-algebras. Assume that \(A\) is an integral domain and \(B\) is regular. Consider the valuation \(v=v_{\mathfrak{m}_B}\circ \varphi\) defined on \(\mathrm{frac}(A/\text{Ker}(\varphi))\) and let \(A_v\) be the valuation ring associated to \(v\) and \(\mathfrak{m}_v\) its maximal ideal. Let \(\text{trdeg}_kv\) be the transcendence degree of the field extension \(k\to A_v/\mathfrak{m}_v\). Here \(v_{\mathfrak{m}_B}\) denotes the \(\mathfrak{m}_B\)-adic order.
If \(\text{Ker}(\varphi)\neq \mathfrak{m}_A\) then \(\text{trdeg}_kv+1\) is called the geometric rank of \(\varphi\) denoted by \(\text{grk}(\varphi)\). If \(\text{Ker}(\varphi)=\mathfrak{m}_A\) then \(\text{grk} (\varphi):=0\).
Under the assumption that \(\widehat{A}\) is an integral domain it is proved that the following conditions are equivalent:
(1)
\(\text{grk}(\varphi)=\dim (A)\)
(2)
There exist \(a, b\in \mathbb{R}\) such that \(a v_{\mathfrak{m}_A}(f)+b\geq v_{\mathfrak{m}_B} (\varphi(f))\) for any \(f\in A\).
Homomorphisms satisfying these equivalent conditions are called regular homomorphisms.
The main tool used is a monomialization theorem for homomorphisms of power series rings in positive characteristic.
Furthermore it is proved that (also in characteristic \(0\)) for regular Henselian \(k\)-algebras \(A,B\) the property \(\text{grk}(\varphi)=\dim(A)\) implies that \(\varphi\) is strongly injective (i.e., \(\widehat{\varphi}^{-1}(B)=A\)).

MSC:

13H99 Local rings and semilocal rings
13B10 Morphisms of commutative rings
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