Local ring

A local ring is a commutative ring with exactly one maximal ideal.

A homomorphism $\phi :(A,\mathfrak{m})\to (B,\mathfrak{n})$ of local rings is local if $\phi (\mathfrak{m})\subseteq \mathfrak{n}$, equivalently ${\phi }^{-1}(\mathfrak{n})=\mathfrak{m}$. The equivalence uses the fact that all elements outside $\mathfrak{m}$ are units. It induces an embedding of residue fields $A/\mathfrak{m}\to B/\mathfrak{n}$.

A local ring that is also an integral domain. Multiplication by any nonzero element is injective. This permits cancellation in Artin-Rees lemma length-growth arguments.

For a local ring $(R,\mathfrak{m})$ with residue field $\kappa =R/\mathfrak{m}$, its cotangent space is the $\kappa$-vector space $\mathfrak{m}/{\mathfrak{m}}^2$. In a Noetherian local ring, its dimension is the minimal number of generators of $\mathfrak{m}$, by the Nakayama lemma. Equality of this dimension with the Krull dimension defines a regular local ring.

The embedding dimension of a Noetherian local ring $(R,\mathfrak{m})$ is ${\dim }_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)$. It counts the smallest number of local generators needed for the maximal ideal. It is at least the Krull dimension; equality characterizes a regular local ring.

A Noetherian local ring $(R,\mathfrak{m})$ is regular when the minimal number of generators of its maximal ideal equals its Krull dimension. Every regular local ring is an integral domain and a unique factorization domain.

For a regular local ring of embedding dimension $e$, a minimal maximal-ideal generating set induces the displayed polynomial-algebra isomorphism. It is surjective in every degree. A nonzero homogeneous kernel relation would bound the cumulative Hilbert function by $O(t^{e-1})$, contradicting the prime-chain lower bound for local length at $\dim R=e$. The graded ring is a domain; the Krull intersection theorem then shows the original ring is a domain by multiplying nonzero initial forms.

A valuation ring is an integral domain $R$ such that, for every nonzero element $x$ of its fraction field $K$, either $x\in R$ or $x^{-1}\in R$. Equivalently, its ideals are totally ordered by inclusion.

Every local subring $A$ of a field $K$ is dominated by a valuation ring with fraction field $K$. Order local overrings by inclusion and contraction of their maximal ideals; chain unions are local, so Zorn's lemma gives a maximal one. For each $z\in K^{\times }$, at least one of the extensions of its maximal ideal to $V[z]$ and $V[z^{-1}]$ is proper: otherwise two least-degree relations for $1$ with maximal-ideal coefficients reduce one another to a smaller-degree relation. Localizing the proper extension gives a dominating overring, so maximality forces either $z$ or $z^{-1}$ into $V$.

If an element $x$ of the fraction field is absent from a valuation ring $V$, its inverse $y$ belongs to the maximal ideal. A monic equation for $x$, multiplied by a power of $y$, would put $1$ in that maximal ideal. Thus $V$ is an integrally closed domain, without a discrete or Noetherian hypothesis.

A discrete valuation ring is a principal ideal domain with exactly one nonzero maximal ideal. Every nonzero element of its fraction field is a unit times a unique integer power of a uniformizer.

A complete discrete valuation ring is a discrete valuation ring complete for its maximal-ideal-adic topology. Equivalently, the canonical map to the inverse limit of the residue rings is an isomorphism: compatible residue classes determine a Cauchy sequence, and completeness gives its unique limit. Standard examples are ${\mathbb{Z}}_p$ and $k[[t]]$. This completeness is the hypothesis that makes Newton iteration over a valued field converge in the ring in the usual Hensel lemma.

A discrete valuation on a field $K$ is a surjective group homomorphism $v:K^{\times }\to \mathbb{Z}$ satisfying $v(x+y)\ge \min (v(x),v(y))$ whenever $x+y\ne 0$. Its valuation ring is $\{0\}\cup \{x:v(x)\ge 0\}$.

A discretely valued field is a field equipped with a discrete valuation. It is complete when every Cauchy sequence for the induced metric converges.

The value group of a valued field is the ordered abelian group formed by the values of its nonzero elements. A valuation is discrete when its value group is infinite cyclic with the order inherited from $\mathbb{Z}$.

A uniformizer of a discrete valuation ring $R$ is a generator $\pi$ of its unique nonzero maximal ideal. Every nonzero $x$ in the fraction field has a unique expression $x=u{\pi }^n$ with $u$ a unit and $n\in \mathbb{Z}$.