LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Formal power-series substitution converges in a complete local algebra

Statement

Let $(A, m)$ be a complete local ring, let $k \to A$ be a ring map, and let $x_{1}, \dots, x_{n} \in m$ . For a formal series $F = \sum_{α} a_{α} X^{α} \in k ⟦ X_{1}, \dots, X_{n} ⟧,$ the partial sums ordered by total degree, $S_{N} = \sum_{∣ α ∣ < N} a_{α} x^{α},$ form an $m$ -adically Cauchy sequence in $A$ and hence converge.

Facts & Assumptions

Given: A complete local ring $(A, m)$ , a ring map $k \to A$ , and elements $x_{1}, \dots, x_{n} \in m$ .

[L1]

Completeness means that every $m$ -adically Cauchy sequence in $A$ converges (Separated and complete filtered modules).

Proof

technique · degree-$N$ tails land in $\mathfrak m^N$

1.1

If $M > N$ , then $S_{M} - S_{N} = \sum_{N \leq ∣ α ∣ < M} a_{α} x^{α} .$ Every monomial $x^{α}$ appearing here is a product of $∣ α ∣ \geq N$ elements of $m$ , so by the definition of the ideal power $m^{N}$ one has $x^{α} \in m^{N}$ . Hence $S_{M} - S_{N} \in m^{N}$ .

givenalgebra

2.1

Step 1.1 is exactly the $m$ -adic Cauchy condition for $(S_{N})$ . By [L1], the partial sums therefore converge in $A$ .

L1step 1.1

3.1

Thus substitution of maximal-ideal elements into a multivariable formal power series converges in a complete local ring.

step 2.1∎

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Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Theorem 22.32 (standard reference, not scraped)
Melvin Hochster, The structure theory of complete local rings (standard reference, not scraped)