sections etc.
Let's count approximate characters. We'll aim ~1400 characters.
We need to ensure not exceed 2000. We'll keep each paragraph short.
Let's create:
- Title: "Acorn Domains: A Concise Overview"
- Definition
- Basic properties
- Examples
- Applications
- References
Let's produce each section small.
Let's craft:
Acorn Domains
... etc.
Let's count characters roughly.
We'll write.
Better to quickly approximate:
The content:
Acorn Domains
In commutative algebra a one‑dimensional local ring whose only non‑zero prime ideals are maximal is called an acorn domain. Its spectrum consists of a minimal point (0) and a single closed point determined by the maximal ideal.
Basic Properties
- Every acorn domain is integrally closed; its localization at the maximal ideal is a discrete valuation ring (DVR).
- If Noetherian it is a Dedekind domain with a single non‑zero prime.
- Global homological dimension is one; thus all finitely generated torsion‑free modules are projective and every invertible ideal is principal.
Examples
- For a field $K$, $K[[t]]_{(t)}$ is an acorn domain: the maximal ideal $(t)$ is the only non‑zero prime.
- Localizing the integer ring at a single prime, $ \mathbb{Z}_{(p)}$, is a DVR and hence an acorn domain.
Applications
- Local rings of smooth points on algebraic curves are DVRs, providing the standard local model for intersection theory.
- In algebraic number theory, localizations of rings of integers at a prime give acorn domains, simplifying local class‑field calculations.
Acorn Domains
: 23In commutative algebra a one‑dimensional local ring whose only non‑zero prime ideals are maximal is called an acorn domain. Its spectrum consists of a minimal point (0) and a single closed point determined by the maximal ideal.
: This paragraph ~200 chars. Let's approximate: "In commutative algebra a one‑dimensional local ring whose only non‑zero prime ideals are maximal is called an acorn domain. Its spectrum consists of a minimal point (0) and a single closed point determined by the maximal ideal." ~170? We'll count. Better to approximate with 170 characters. ThenBasic Properties
~23- ... items: each
- ~80? Let's estimate 3 li items each ~70? 210.
Examples
~23- two li items ~70 each => 140.
- Every acorn domain is integrally closed; localizing at its unique maximal ideal gives a discrete valuation ring (DVR).
Applications
~23- two li items ~70 each => 140.
References
~23- three li items ~60 each => 180.
Total approximate: 12+23+170+23+210+23+140+23+140+23+180 = about 1154 characters. Good.
Add small overhead for tags: we counted some. Should be under 2000.
Let's produce final HTML. Use self-contained formatting. Ensure no stray characters.
Let's output the content.
Acorn Domains
In commutative algebra a one‑dimensional local ring whose only non‑zero prime ideals are maximal is called an acorn domain. Its spectrum consists of a minimal point (0) and a single closed point determined by the maximal ideal.
Basic Properties
Examples
- For a field \(K\), \(K[[t]]_{(t)}\) is an acorn domain: the maximal ideal \((t)\) is the only non‑zero prime.
- Localizing the integer ring at a prime, \(\mathbb{Z}_{(p)}\), is a DVR and hence an acorn domain.
Applications
- Local rings of smooth points on algebraic curves are DVRs, providing the standard local model for intersection theory.
- Localizations of rings of integers at a single prime give acorn domains, simplifying local class‑field calculations.
References
- Kaplansky, I. Commutative Rings, 1974.
- Atiyah & Macdonald, Introduction to Commutative Algebra, 1969.
- Bruns & Herzog, Cohen–Macaulay Rings, 1998.
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