tags: non-Archimedean geometry, log structures

Let \(k\) be a non-Archimedean valued field. We want to define a natural metric on the module of Kähler differentials \(\Omega_{Y/X}\) for a morphism of \(k\)-analytic spaces \(X \to Y\). This is presented in the utmost generality in this paper, see also the survey at the NATG’15 conference.

A morphism of normed modules \(f: B \to A\) is called non-expansive if \(||f(b)|| \leq ||b||\).

Given a morphism of normed rings \(A \to B\) one can define a norm on the module of differentials \(\Omega_{B/A}\) so that it is the maxmial norm making the morphism \(d: B \to \Omega_{B/A}\) non-expansive.

In particular, given a \(k\)-ring \(A\), the module of Kähler differentials \(\Omega_{A/k}\) is equipped with the Kähler norm as follows: \[ ||x||_\Omega = \inf_{x = \sum c_i db_i} \max |c_i| |b_i| \] The Kährler semi-norm on \(\Omega_B\) is characterized by the fact that it is the maximal semi-norm making the differential a non-expansive map.

Let \(A\) be an Banach algebra over \(k\), and let \(v \in {\mathcal{M}}(A)\) be a point of its Banach spectrum. Then an element \(f \in A\) is a function on \(A\) with values in residue fields of each point \(v\), \({\operatorname{\mathcal{H}}}(v)\), which is naturally normed, and \(|f|\) is a real-valued function, defined as \(|f|(v) = ||f||_{{\operatorname{\mathcal{H}}}(v)}\).

If \(v \in X\) is a point of a Berkovich analytic space, we want to look at the Kähler semi-norm on the complete residue fields \({\operatorname{\mathcal{H}}}(v)\), so that we can define a real-valued function \(|\omega|\) by putting it to be \(||\omega||_{{\operatorname{\mathcal{H}}}(v)}\) at a point \(v\).

A log structure on a ring \(A\) is a morphism of multiplictive monoids \(\alpha: M \to A\) that induces an isomorphism \(M^\times \to A^\times\).

If \(A\) is a log \(k\)-ring then the module of log differentials is defined as \(\Omega_{A/k} \oplus (A \otimes M^{gp})\) (where \(M^{gp}\) is a groupification of the monoid) module the relations \[ \begin{array}{c} (0, 1 \otimes c)\\ (da, -a \otimes a) = 0 \\ \end{array} \]

Along with the usual derivative \(d\) we have a log derivative \(\delta\), \(\delta a = (0, 1 \otimes a)\), so that elements \(\delta a\) should be thought of as \(d\log a\), with the above relation meaning \[ \begin{array}{c} d\log c = 0, c \in k\\ a d\log a = da, a \in M^{gp}\\ \end{array} \]

If \(K\) is a valued field with a valued ring \(K^\circ\), then \(K^\circ {\setminus}\{0\} \to K^\circ\) is a log structure on \(K^\circ\). We denote \(\Omega^{\log}_{K^\circ/A}\) the module of log differentials, where the log structure on \(K^\circ\) is as above, and the log structure on \(A\) is \(A {\setminus}{\mathrm{Ker}}(A \to K)\).

Adic seminorm: given a \(K^\circ\)-submodule \(M\) is defined on \(V=M\otimes K\) as: \[ ||v|| = \inf {\{ |a| \mid a \in K^\circ, v \in aM \}} \]

One checks that \(\Omega^{\log}_{K^\circ/A^\circ} \otimes_{K^\circ} K = \Omega_{K/A}\), so the image \(\Omega^{\log}_{K^\circ/A^\circ} \to \Omega^{\log}_{K^\circ/A^\circ} \otimes K=\Omega_{K/A}\) is a lattice.

Theorem. The adic semi-norm on \(\Omega^{log}_{K^\circ/A^\circ}\) is the maximal norm making the differential non-expansive.

Let \((V, ||\cdot||)\) be a finite-dimensional normed vector space. Then a basis \(e_1, \ldots, e_n\) is called \(r\)-orthogonal if for any \(v=\sum a_i e_i\) the inequality \(||v|| \geq r\max (|a_i|\cdot||e_i||)\) holds. If \(r=1\) and \(||e_i||=1\) then the basis is called orthogonal.

The valuation \[ v(\sum a_i t^i) = \max |a_i| \] on \(T_1\) is called Gauss valuation (there is a more general expression for more variables, and with radii \(\neq 1\), but let us forfeit it here).

Let \(K/k\) be an extension of valued fields. Let \(t=(t_1, \ldots, t_n)\) be a tuple of elements of \(L\). We call a valuation on \(K\) \(t\)-monomial if its restriction to \(k[t_1, \ldots, t_n]\) is a Gauss valuation.

Lemma. Let \(\xi\) be the Gauss point on \({\mathcal{M}}(T_1)\), the unit disc. Then \(||dt/t||_{\xi}\) is 1.

Proof. From the definition one immediately gets \[ ||dt/t||_{\xi} = ||df/(f' t)||_\xi = \inf_f |1/f' t|\cdot|f| = 1 \] where \(f=\sum a_i t^i\), \(f \neq const\). Indeed, \[ f't=\sum i a_i t^i \] so its norm coincides with that of \(f\).

Corollary. The function \(x \mapsto ||dt||_{x}: {\mathcal{M}}(T_1) \to {\mathbb{R}}\) is the radius function \(r(x)\), i.e. the infimum of a ball containing the point \(x\).

Proof. Indeed, any point in the unit disc is the gauss point of a disc of radius \(r(x)\) and coordinate function of the form \(T-a\), and \(dt=d(t-a)\), so from the previous lemma \(1=||dt/t||_x \leq ||dt||_x \cdot |1/t|\).

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