We give an elementary theory of Henselian local rings and construct the Henselization of a local ring. All our theorems have an algorithmic content.

Let $(K,v)$ be a valued field and let $(K^h,v^h)$ the henselization determined by choice of an extension $v$ to algebraic closure $K$. Consider embedding $v(K^*)\hookrightarrow\Lambda$ value group into divisible ordered abelian group. $T(K,\Lambda)$, $T(K^h,\Lambda)$ trees formed all $\Lambda$-valued extensions $v$, $v^h$ polynomial rings $K[x]$, $K^h[x]$, respectively. We show that natural res...

We define and note some properties of k H-pairs (k Henselian pairs), k N-pairs, and k N’-pairs. It is shown that the 2-Henselization and the 3Henselization of a pair exist. Characterizations of quasi-local 2H-pairs are given, and an equivalence to the chain conjecture is proved.

An introduction to the theme of local rings and Henselian local rings is given through numerous examples. We outline an elementary and effective construction of the Henselization of a local ring and an effective proof of about a classical result in Henselian local rings. This paper anounces a joint work of the three authors.

Given a smooth projective variety X over a field k of characteristic zero, we consider the composition of the de Rham cohomology cycle class map over k from the Chow group CH q (X × k K), where K is the field of fractions of henselization A h of the local ring of a smooth closed point of a variety over the field k with an appropriate projection: