Spaces of valuations and spaces of types

Posted on May 17, 2015 by Dima

tags: non-Archimedean geometry, valuations

Let \(k\) be a field. Recall that a valuation \(v: k \to \Gamma\), for \(\Gamma\) an ordered Abelian group, is a group homomorphism \(v: k^\times \to \Gamma\) such that \[ v(x + y) \geq \min\{v(x), v(y) \} \] When \(\Gamma=\mathbb{Z}^n\) then the valuation is called rank \(n\) valuation. In classical terminology rank 1 valuations are just ``valuations’’. If the homomorphism is partial, then \(v\) is called a semi-valuation.

Let \(X\) be an algebraic variety over a field \(k\). For a fixed valuation \(v_0\) on \(k\), the space of valuations on \(k(X)\) that restrict to \(v_0\) is an interesting object to study. If \(v_0\) is trivial (i.e. \(v(f)=0\) for all \(f \in k^\times\)) then the space of all rank 1 valuations is called Riemann-Zariski space. It is topologized as follows: basic opens are sets of valuations \(v\) such that \(v(f_i) \geq 0\) for a some fixed \(f_1, \ldots, f_n\). Incidentally, RZ space is the inverse limit of all schemes over \(k\) with the given function field.

One can also fix the value group \(\Gamma\) and consider spaces of valuations with values in \(\Gamma\), this way one gets Berkovich spaces (\(\Gamma = \mathbb{R}\)). They are topologized in the way similar to Riemann-Zariski spaces: basic opens are sets of valuations that are non-negative on some finite sets of elements of the field. Hrushovski-Loeser spaces (or spaces of stably dominated types) is a way to regard such valuations from a ``semi-algebraic’’ point of view. The setting is much more general here, valuations are with values in \(\Gamma\) arbitrary (which embeds into some very big ordered group, which is fixed).

Stably dominated types

If \(p\) is an arbitrary type and \(f: X \to Y\) is a definable map such that \(p\) is supported on \(X\) then, regarding \(p\) is a finitely additive measure on definable sets that takes values in {0,1}, and map \(f\) being naturally measurable, one can define the pushforward type=measure.

A type \(p\) is called stably dominated by a (stable) type \(q\) via a definable map \(f\) if \(p\) is the unique type such that \(f_*(p)=q\). (caveat: this definition is only correct for types defined over maximally complete fields, full definition glossed over)

One can show that stably definable types are definable (since they are generated by formalus that are constructed from definition of the stable — hence, definable — stable types). Formula definitions of definable stably dominated types allow for a somewhat more explicit description of these types.

Let \(K\) be a valued field with a value ring \({\cal O}\), and let \(V\) be a \(K\)-vector space. An \({\cal O}\)-submodule \(M\) is a semi-lattice if it intersects every one-dimensional \(K\)-subspace of \(V\) in a submodule of the form \(K\), \({\cal O}\), or the trivial module. A semi-lattice \(M\) is a lattice if \(M \otimes K \cong V\).

One easily observes that a definable stably dominated type supported on an affine variety \(X\) can be encoded by a certain set of lattices in the (infinite-dimensional) vector space \(H^0(X, {\cal O}_X)\). The lattice \(\Lambda(p)\) associated to \(p\) is \[ \Lambda(p) := \{ f \in H^0(X, {\cal O}_X) \mid p \models val(f(x)) \geq 0 \} \]

The type definition makes this definition uniform in coefficients of \(f\).

Theorem. The set of lattices \(\Lambda(p)\) that correspond to stably dominated types \(p\) is pro-definable, even definable.

The set of definable stably dominated types that concentrate on \(X\) is denoted \(\hat X\).

If \(p\) is a stably dominated type supported on an affine variety \(X\) then for any function \(f \in H^0(X, {\cal K}_X)\), \(val(f_*(p))\) is well-defined, this defines a valuation on \(K(X)\) which restricts to the standard valuation on \(K\).

Conversely, if \(K\) is maximally complete, then for any \(v: K(X) \to \Gamma\) a valuation there exists the type \[ p_v := \{ val(f) = v(p) \mid f \in H^0(X, {\cal O}_X) \} \] Showing that this type is definable stably dominated is a non-trivial theorem (therem 12.18 in the book of Haskell-Hrushovski-Macpherson).

One consequence of definabity of the space of types is that it makes sense to consider definable maps fram definable sets to sets of definable stably dominated types.

For any definable set \(V\) there is an embedding \(s_V: V \to \hat{V}\) that maps points to types that concentrate on points. The image of \(s_V\) is called the set of simple points.

If \(f: X \to Y\) is a definable map, then \(\widehat{X/Y}\) is the subset of \(\hat X\) of points that project to simple points of \(\hat Y\).

Let \(A\) be definable domain such that the valuation associated to a type \(p\) is \[ v(f) = \inf_{x \in A} val(f(x)) \] terminology: \(p\) is in the Shilov boundary of \(A\), \(p\) is strongly stably dominated.

Alternatively, \(p \in \hat V\) is a strongly stably dominated if there exists a map \(f: V \to \mathbb{A}^n\) such that \(f_* p = p_{\mathcal O}^n\) where \(p_{\cal O}\) is the generic type of the ball \(\{ x \mid v(x) \geq 0\}\).

Let \(U\) be a definable set (typically, a subset of \(\Gamma^n\)) and let \(f: U \hookrightarrow \hat X\) be a definable map. If a definable type \(p\) is a type concentrated on \(U\) one can define the following type concentrated on \(X\) \[ \int_p f := f(a) \textrm{ where } a \models p \] the definition in fact does not depend on the choice of the realization \(a\) and it also only depends on the germ of \(f\) (I personally think that limit would be a better notation).

It turns out that any definable type is of the form \(\int_p f\) where \(p\) is concentrated on \(\Gamma\).