LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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The Cohen map is surjective by completeness

Statement

Let $(A, m)$ be a complete equicharacteristic Noetherian local ring, let $k \subseteq A$ be a coefficient field, let $x_{1}, \dots, x_{e} \in m$ lift a $k$ -basis of $m / m^{2}$ , and let $ϕ : k ⟦ X_{1}, \dots, X_{e} ⟧ \to A$ be the continuous $k$ -algebra homomorphism with $ϕ (X_{i}) = x_{i}$ . Then $ϕ$ is surjective.

Facts & Assumptions

Given: A complete equicharacteristic Noetherian local ring $(A, m)$ , a coefficient field $k \subseteq A$ , lifts $x_{1}, \dots, x_{e} \in m$ of a $k$ -basis of $m / m^{2}$ , and the resulting continuous $k$ -algebra map $ϕ$ .

[L1]

Degreewise formal-series substitution converges in a complete local ring (Formal power-series substitution converges in a complete local algebra).

[L2]

For every $n \geq 1$ , each class in $m^{n} / m^{n + 1}$ has a homogeneous degree- $n$ polynomial preimage under the Cohen map (The Cohen map is surjective modulo every power of the maximal ideal).

Proof

technique · build a compatible inverse system of polynomial approximations

1.1

Let $a \in A$ . Because $k \to A / m$ is an isomorphism, choose $F_{0} \in k$ with $ϕ (F_{0}) \equiv a (m o d m)$ . Inductively, if $F_{n - 1}$ is chosen with $ϕ (F_{n - 1}) \equiv a (m o d m^{n})$ , then [L2] applied to the error class in $m^{n} / m^{n + 1}$ gives a homogeneous polynomial correction $H_{n}$ of degree $n$ such that $ϕ (F_{n}) \equiv a (m o d m^{n + 1}), F_{n} : = F_{n - 1} + H_{n} .$

L2givenchooseinduction

2.1

The formal sum $F = \sum_{n \geq 0} H_{n}$ defines an element of $k ⟦ X_{1}, \dots, X_{e} ⟧$ . By [L1], the series $ϕ (F) = \sum_{n \geq 0} ϕ (H_{n})$ converges in $A$ , and step 1.1 says its partial sums are congruent to $a$ modulo arbitrarily high powers of $m$ . Since $A$ is separated, the limit must equal $a$ .

L1step 1.1algebra

3.1

Therefore every $a \in A$ lies in the image of $ϕ$ , so $ϕ$ is surjective.

step 2.1∎

Depends on

Used by

A complete equicharacteristic Noetherian local ring is a power-series quotient Corollary

Dependency tree · two levels

7 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

Melvin Hochster, The structure theory of complete local rings (standard reference, not scraped)
The Stacks Project, Section 10.160: The Cohen structure theorem (standard reference, not scraped)