Some results of algebraic geometry over Henselian rank one valued fields

Abstract

We develop geometry of affine algebraic varieties in K n over Henselian rank one valued fields K of equicharacteristic zero. Several results are provided including: the projection K n × P m (K ) → K n and blowups of the K -rational points of smooth K -varieties are definably closed maps; a descent property for blowups; curve selection for definable sets; a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the K -topology; and extending continuous hereditarily rational functions, established for the real and p-adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field R was developed by Fichou-Huisman-Mangolte-Monnier. The main results here are regulous versions of Nullstellensatz and Cartan's theorems A and B.

Piecewise-regular maps

“…2. Stratified-regular maps and functions are thoroughly investigated, in a more restrictive framework, in [4,29,30,44,45,50,52,54,55,58,[60][61][62]66,74,75,84], where they sometimes appear under different names (cf. Remark 2.5).…”

Maximal ideals of regulous functions are not finitely generated

“…Then the function g • f : R n → R l is also k-regulous (cf. [2,Corollary 4.14] and [5,Corollary 11.7]).…”

“…It can be proven, by means of regular stratification (Theorem 2.6), that the k-regulous topology is noetherian (cf. [2,Theorem 4.3], [5,Proposition 11.10]).…”

Regulous vector bundles

“…It is fair to say that to regulous geometry belong all the results in which regulous functions, stratified‐regular functions or continuous rational functions (these classes of functions are all identical under certain assumptions) play an essential role. For a more complete picture of regulous geometry, the reader may consult .…”

“…all identical under certain assumptions) play an essential role. For a more complete picture of regulous geometry, the reader may consult [3,9,10,[16][17][18][19][20][21][22][23][24][25][26][27][28]32,33,39].…”