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Successive Hensel corrections are Cauchy

Statement

Let $(g_{r}, h_{r})_{r \geq 1}$ be a sequence of Hensel corrections such that $g_{r + 1} - g_{r} \in I^{r} [T]$ and $h_{r + 1} - h_{r} \in I^{r} [T]$ for every $r \geq 1$ . Then each coefficient sequence of $(g_{r})$ and of $(h_{r})$ is Cauchy for the $I$ -adic topology on $A$ .

Facts & Assumptions

Given: Successive lifts $(g_{r}, h_{r})$ with differences in $I^{r} [T]$ at stage $r$ .

[L1]

One Hensel correction step changes each factor by a polynomial whose coefficients lie in the current ideal power (One correction step raises factor lifting by one ideal power).

Proof

technique · telescope the stagewise corrections

1.1

Fix a coefficient index $j$ . If $s > r$ , then the coefficient of $T^{j}$ in $g_{s} - g_{r}$ is a sum of coefficients from the increments $g_{k + 1} - g_{k}$ for $k = r, \dots, s - 1$ . By [L1], each summand lies in $I^{k} \subseteq I^{r}$ , so the whole difference lies in $I^{r}$ . Thus the $j$ th coefficients of the $g_{r}$ form an $I$ -adic Cauchy sequence.

L1givenalgebra

1.2

The same argument applied to the increments $h_{k + 1} - h_{k} \in I^{k} [T]$ shows that each coefficient sequence of the $h_{r}$ is also $I$ -adically Cauchy.

L1givenalgebra

2.1

Hence the iterative Hensel corrections are coefficientwise Cauchy.

step 1.1step 1.2∎

Depends on

One correction step raises factor lifting by one ideal power

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Chapter 22 (standard reference, not scraped)