11.1 I-adic topology and completion of rings

The I-adic topology on rings uses powers of an ideal to create a basis for open neighborhoods. This topology allows us to define convergence and continuity, setting the stage for ring completions.

Ring completions are built from Cauchy sequences in the I-adic topology. They have useful properties like completeness and continuity of operations, with examples including p-adic integers and formal power series rings.

I-adic Topology and Ring Completion

I-adic topology on rings

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Top images from around the web for I-adic topology on rings

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  Derivations and Integrations on Rings View original

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  Joint Rock Coefficient Estimation Based on Hausdorff Dimension View original

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  New Stone-Weierstrass Theorem View original

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  Derivations and Integrations on Rings View original

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• I-adic topology induces topology on ring R using powers of ideal I
• Basis of open neighborhoods of 0 consists of $\{I^n : n \in \mathbb{N}\}$
• Hausdorff property holds when $\cap_{n=1}^{\infty} I^n = \{0\}$
• First countable topology allows sequential characterization of continuity
• Translation invariance ensures consistency across ring elements
• Convergence occurs when sequence $(a_n)$ approaches $a$ with $a_n - a \in I^k$ for large $n$ and each $k$
• Addition and multiplication maintain continuity in I-adic topology

Completion of rings

• Construction uses Cauchy sequences in R relative to I-adic topology
• Equivalence relation defined by $\lim_{n \to \infty} (a_n - b_n) = 0$
• Completion $\hat{R}$ formed from equivalence classes of Cauchy sequences
• Ring operations on $\hat{R}$ defined componentwise (addition, multiplication)
• Canonical homomorphism $\phi: R \to \hat{R}$ maps $a$ to constant sequence $[(a, a, a, \ldots)]$
• Universal property ensures unique continuous homomorphism $\hat{f}: \hat{R} \to S$ for complete I-adically separated ring S and continuous homomorphism $f: R \to S$

Properties of ring completions

• Completeness of $\hat{R}$ guarantees convergence of all Cauchy sequences
• Diagonal argument proves completeness using representative sequences
• Hausdorff property separates distinct points with disjoint neighborhoods
• Continuity of ring operations extends to $\hat{R}$ in induced topology
• Topological ring structure emerges from continuous operations on $\hat{R}$

Examples of ring completions

• p-adic integers $\mathbb{Z}_p$ complete $\mathbb{Z}$ in $(p)$-adic topology
• Elements represented as $\sum_{i=0}^{\infty} a_i p^i$ with $0 \leq a_i < p$
• Digit-by-digit arithmetic with carry for addition and multiplication
• Power series ring $k[[x]]$ completes polynomial ring $k[x]$ in $(x)$-adic topology
• Formal power series $\sum_{i=0}^{\infty} a_i x^i$ with $a_i \in k$ and term-by-term operations
• Multivariate formal power series complete $k[x_1, \ldots, x_n]$ in $(x_1, \ldots, x_n)$-adic topology
• Algebraic geometry applies completion to local rings of points on varieties