Singular points and local rings

For an affine algebraic set X with coordinate ring A and point p, let m_p be the ideal of functions vanishing at p. The local ring is A localized away from m_p. Its elements are fractions f/g with g(p) nonzero, interpreted as germs of regular functions near p. Two fractions represent the same germ if they agree after an allowed localization relation. When A is a domain this is ordinary fraction equality, but with zero divisors localization equality must use a multiplier from the allowed denominator set.

The localized ring has one maximal ideal, consisting of germs vanishing at p, and its residue field is k in our algebraically closed setting. A germ with nonzero value is a unit because its numerator is then an allowed denominator. This gives a practical way to simplify a neighbourhood. For example on xy=0 near (1,0), x is invertible, so the relation forces y=0. The other component disappears locally because it does not pass through that point, even though it remains part of the global algebraic set.

The quotient m/m² discards products of functions that both vanish at the point. It is a finite-dimensional k-vector space, the cotangent space. Its dual is the Zariski tangent space. Passing to m/m² therefore extracts exactly the first-order information retained by the Jacobian, now without reference to an embedding. Nakayama's lemma says that the vector-space dimension is the minimum number of generators of m as an ideal in the Noetherian local ring. This number is called embedding dimension.

Every Noetherian local ring has Krull dimension no greater than its embedding dimension. It is regular when equality holds. For finite-type reduced geometry over our algebraically closed field, regular local rings characterize nonsingular points. A local equation with only quadratic and higher terms leaves all ambient linear classes independent in the cotangent quotient, often creating an embedding dimension larger than the actual dimension. The criterion does not require choosing a preferred parameterization, and is preserved by any isomorphism of the local rings.

At the origin of the cusp y²=x³, the relation has no linear term. The germs x and y remain independent modulo m², although the curve has dimension one. This excess cannot be removed by an invertible coordinate change. Away from the origin in characteristic zero, one Jacobian derivative is nonzero and the local ring becomes regular. A singular curve may therefore have a dense nonsingular open subset; calling the curve singular does not mean every point has the same defect.

Nonreduced structure can produce a similar dimension excess for a different reason. The doubled line defined by y² has ordinary support equal to a smooth line, but its local ring retains a nonzero nilpotent transverse class. Its embedding dimension at an ordinary point is two while its Krull dimension is one. By contrast the reduced line y=0 has embedding dimension one. For curves, normal Noetherian local domains are regular, but normality and regularity diverge in higher dimensions; no tangent calculation should be replaced by a blanket claim that integral closure resolves every singularity.