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Codimension

In mathematics, codimension quantifies the dimensional deficiency of a geometric object embedded within a larger space, typically defined as the difference between the dimension of the ambient space and that of the object itself. This concept arises in various contexts, such as linear algebra, topology, differential geometry, and algebraic geometry, where it measures how "codimensionally" a subspace, submanifold, or subset sits inside its ambient structure. For instance, a line in three-dimensional Euclidean space has codimension 2, as the ambient dimension is 3 and the subspace dimension is 1.^[1] In linear algebra, for a subspace M of a vector space X, the codimension of M in X is the dimension of any complementary subspace N such that X = M \oplus N, which equals \dim X - \dim M. This value is well-defined and independent of the choice of complement, reflecting the minimal number of coordinates needed to specify directions outside M. Subspaces of codimension 1 are known as hyperplanes, which are kernels of nonzero linear functionals.^[1]^[2] In differential geometry, the codimension of a smooth submanifold N of dimension k in a manifold M of dimension n (with k \leq n) is n - k, indicating the number of independent constraints defining N locally as a level set. Hypersurfaces, which have codimension 1, are particularly significant, as they locally resemble the zero set of a single smooth function. Boundaries of manifolds also exemplify codimension-1 submanifolds. Examples include the unit sphere S^{n-1} in \mathbb{R}^n, which has codimension 1.^[3] In topology and algebraic geometry, codimension generalizes to arbitrary topological spaces or schemes. For an irreducible closed subset Y in a topological space X, the codimension \operatorname{codim}_X(Y) is the supremum of the lengths of chains of irreducible closed subsets Y = Y_0 \subset Y_1 \subset \cdots \subset Y_e = X, where each inclusion is strict; this takes values in \{0, 1, 2, \dots\} \cup \{\infty\}. In the context of affine schemes, for an irreducible closed subscheme Z \subset Z' corresponding to prime ideals, \operatorname{codim}_{Z'}(Z) = \dim Z' - \dim Z, where dimensions are measured by maximal chains of prime ideals. This notion underpins key results like Krull's principal ideal theorem, which bounds the codimension of zero sets of single elements.^[4]^[5]^[6] Codimension plays a crucial role in intersection theory, singularity analysis, and embedding theorems, where low-codimension objects often exhibit generic behaviors, such as transversality in intersections. For example, in algebraic geometry, varieties of codimension greater than 1 may require more equations than their codimension suggests, highlighting non-trivial global constraints.^[5]

Fundamentals

Definition

In linear algebra, the codimension of a subspace A of a finite-dimensional vector space V over a field k is defined as the difference between the dimensions of V and A, that is,

\codim_V(A) = \dim V - \dim A.

^[7] This quantity equals the dimension of the quotient space V/A.^[8] The notation \codim_V(A) explicitly indicates the ambient space V, though \codim(A) is often used when the context is clear.^[2] For example, the trivial subspace \{0\} (which has dimension 0) in \mathbb{R}^n has codimension n.^[7] Similarly, a hyperplane in \mathbb{R}^n, being a subspace of dimension n-1, has codimension 1.^[2] In the more general setting of infinite-dimensional vector spaces, the codimension of a subspace A in V is defined as the dimension of a complementary subspace to A in V, or equivalently, the dimension of the quotient space V/A, which need not be finite.^[1] For instance, in a Hilbert space, the codimension of a closed subspace equals the dimension of its orthogonal complement, which may be countably infinite (as in the case of a finite-dimensional subspace) or even uncountable.^[1]

Basic Properties

In linear algebra, the codimension of a subspace A in a finite-dimensional vector space V over a field satisfies the dimension counting formula \dim(A) + \codim(A) = \dim(V). This relation follows directly from the definition of codimension as the dimension of the quotient space V/A, combined with the rank-nullity theorem applied to the canonical projection map \pi: V \to V/A. To prove it via basis extension, suppose \dim(V) = n and \dim(A) = k < n. Let \{v_1, \dots, v_k\} be a basis for A. By Zorn's lemma or the basis extension theorem, this extends to a basis \{v_1, \dots, v_k, w_1, \dots, w_{n-k}\} for V. The images \{\pi(w_1), \dots, \pi(w_{n-k})\} form a basis for V/A, since any element in V/A is \pi(x) for some x = a + \sum c_i w_i with a \in A, and linear independence holds as the w_i are independent modulo A. Thus, \dim(V/A) = n - k = \codim(A).^[9]^[10] A key property is the identification \codim_V(A) = \dim(V/A), where V/A is the quotient space consisting of cosets v + A. This equivalence underscores codimension's role in measuring the "deficiency" of A relative to V. The isomorphism theorems further imply that for a linear map T: V \to W with kernel K, the induced isomorphism V/K \cong \operatorname{im}(T) yields \dim(V/K) = \dim(\operatorname{im}(T)), linking codimension to image dimensions; similarly, the third isomorphism theorem states that for subspaces A \subset B \subset V, (V/A)/(B/A) \cong V/B, preserving dimensions and thus codimensions in quotient structures.^[9]^[10] Codimension exhibits additivity along chains of subspaces. For A \subset B \subset V with all finite-dimensional, \codim_V(A) = \codim_V(B) + \codim_B(A). This holds because \codim_V(A) = \dim(V) - \dim(A), \codim_V(B) = \dim(V) - \dim(B), and \codim_B(A) = \dim(B) - \dim(A) = \dim(B/A), so the equation simplifies algebraically via the dimension counting formula. More generally, for a flag A_0 = \{0\} \subset A_1 \subset \cdots \subset A_m = V, the successive quotients satisfy \dim(A_{i+1}/A_i) = \codim_{A_{i+1}}(A_i), and the total dimension adds up telescopically.^[10] Under direct sum decompositions, codimension behaves additively in product spaces. If V = X \oplus Y and A \subset X, B \subset Y, then \codim_{X \oplus Y}(A \oplus B) = \codim_X(A) + \codim_Y(B). This follows since \dim(X \oplus Y) = \dim(X) + \dim(Y) and \dim(A \oplus B) = \dim(A) + \dim(B), so \codim_{X \oplus Y}(A \oplus B) = [\dim(X) - \dim(A)] + [\dim(Y) - \dim(B)]. The quotient (X \oplus Y)/(A \oplus B) \cong (X/A) \oplus (Y/B) is an isomorphism of vector spaces, confirming the dimension additivity.^[9]^[10]

Interpretations

Dimension Complement

Codimension serves as a complementary measure to dimension within a given space, quantifying the extent to which a subspace or subvariety "falls short" of filling the ambient space completely. For a subspace W embedded in a finite-dimensional vector space V, the codimension is defined as \codim_V W = \dim V - \dim W, which equals the dimension of the quotient space V/W. This perspective highlights the "room" available outside the subspace: in \mathbb{R}^3, for instance, a line has dimension 1 and thus codimension 2, corresponding to the 2-dimensional family of directions perpendicular to the line, allowing movement away from it in the surrounding space.^[11]^[12] The notion of codimension emerged in early 20th-century geometry, particularly within the developing framework of algebraic geometry, where it provided a tool for classifying subvarieties according to their dimensional deficiency relative to the ambient variety. Pioneering work by André Weil in the 1930s formalized divisors as effective cycles of codimension 1 on varieties, building on earlier efforts by Oscar Zariski to rigorize dimension in higher-dimensional spaces through ideal theory and birational invariants. This historical development addressed the need to measure how subobjects contribute to the overall structure of geometric spaces beyond mere size.^[13]^[14] In coordinate geometry over affine spaces, codimension manifests clearly through linear constraints. The solution set to a system of k linearly independent linear equations in \mathbb{R}^n forms an affine subspace of codimension k, provided the equations span a k-dimensional space of functionals; for example, one equation defines a hyperplane of codimension 1, while two independent equations generally yield a plane of codimension 2 in \mathbb{R}^3. This reflects the direct reduction in freedom imposed by each independent constraint.^[11]^[12] Codimension facilitates the study of transversality in geometric configurations, setting the stage for intersection theory by enabling predictions about the expected dimension of overlaps between subobjects. When two subvarieties intersect transversally, their intersection inherits a codimension equal to the sum of their individual codimensions, ensuring the result remains a well-defined subvariety of the anticipated size.^[15]

Dual Perspective

In linear algebra, the codimension of a subspace admits a dual perspective through the theory of dual spaces and annihilators, providing an algebraic reinterpretation that connects subspaces to their "vanishing sets" in the dual space. For a vector space V over a field F and a subspace A \subseteq V, the annihilator of A in the dual space V^* (the algebraic dual, consisting of all linear functionals V \to F) is defined as

A^0 = \{ \phi \in V^* \mid \phi(a) = 0 \ \forall a \in A \}.

This annihilator is itself a subspace of V^*.^[16] When V is finite-dimensional, the codimension of A equals the dimension of its annihilator: \operatorname{codim}_V A = \dim A^0. More precisely, \dim A + \dim A^0 = \dim V. This follows from the rank-nullity theorem applied to the evaluation map or dual basis arguments, where a basis for A extends to a basis for V, and the dual basis elements corresponding to the complement span A^0.^[16]^[17] This relation arises from a natural duality isomorphism between the dual of the quotient space and the annihilator. Specifically, there is a canonical linear isomorphism (V/A)^* \cong A^0, given explicitly by the composition with the quotient map \pi: V \to V/A: for \psi \in (V/A)^*, the corresponding element in A^0 is \psi \circ \pi \in V^*, which vanishes on A since \pi(A) = \{0\}. This map is injective because if \psi \circ \pi = 0, then \psi vanishes on the image of \pi, which is all of V/A; it is surjective because any \phi \in A^0 factors through the quotient as \phi = \tilde{\phi} \circ \pi for some \tilde{\phi}, by the universal property of quotients. In finite dimensions, this yields \dim(V/A) = \dim A^0 = \operatorname{codim}_V A.^[17]^[18] A concrete example occurs in \mathbb{R}^n with the standard Euclidean inner product, which identifies (\mathbb{R}^n)^* \cong \mathbb{R}^n via \phi_v(w) = v \cdot w. Under this identification, the annihilator A^0 corresponds to the orthogonal complement A^\perp = \{ v \in \mathbb{R}^n \mid v \cdot a = 0 \ \forall a \in A \}, so \operatorname{codim}_{\mathbb{R}^n} A = \dim A^\perp. For instance, if A is a hyperplane (codimension 1), its orthogonal complement is a line spanned by the normal vector.^[19] In infinite-dimensional vector spaces, the isomorphism (V/A)^* \cong A^0 persists algebraically, but the dimension equality \operatorname{codim}_V A = \dim A^0 requires care: it holds when the codimension is finite, as A^0 is then finitely generated by extending a basis of a complement, but fails in the infinite case where \dim A^0 (as a cardinal) typically exceeds \operatorname{codim}_V A. In the context of topological vector spaces like Banach spaces, with continuous duals and closed subspaces, additional structure such as reflexivity (where the natural embedding V \hookrightarrow V^{**} is surjective) is needed to ensure relations like A = (A^0)^0 hold, preserving dimension alignments via the bidual. Without reflexivity, subspaces may not recover under double annihilation, complicating codimension interpretations.^[18]

Applications

In Manifolds

In the context of smooth manifolds, the codimension of a submanifold provides a measure of how it sits within the ambient space. For an embedded submanifold S of dimension k in a smooth manifold M of dimension n \geq k, the codimension is defined as \codim_M(S) = n - k. This value remains constant along S, as the embedding ensures that the dimension of S is uniform everywhere, allowing local charts around points in S to reflect this fixed difference in dimensions.^[20]^[3] The geometric structure underlying this codimension is captured by the normal bundle of S in M, denoted \nu(S) \to S. This vector bundle is constructed as the quotient TM|_S / TS, where TM|_S is the restriction of the tangent bundle of M to S. The rank of \nu(S), which equals the dimension of each fiber, is precisely n - k, matching the codimension; thus, the normal bundle encodes the "transverse directions" perpendicular to S within M. Near points of S, a tubular neighborhood can be diffeomorphic to the total space of this normal bundle, facilitating local analysis of embeddings and transversality.^[21]^[20] A key global embedding property tied to codimension arises from the Whitney embedding theorem, which guarantees that any smooth k-dimensional manifold embeds as a closed submanifold into \mathbb{R}^{2k}. In this embedding, the codimension in the target Euclidean space is at least k, since $2k - k = k; this minimal codimension bound highlights the intrinsic dimensionality constraints and enables the realization of manifolds in high-dimensional ambient spaces without self-intersections. The theorem's proof relies on approximating immersions and resolving double points, ensuring the embedding is smooth and proper for compact manifolds.^[20] Illustrative examples clarify these concepts in low dimensions. Consider a smooth curve (1-dimensional submanifold) embedded in a surface (2-dimensional manifold), such as a closed geodesic on a torus; here, the codimension is 1, making the curve a hypersurface whose complement consists of two connected components locally near regular points. In contrast, the same curve embedded in 3-dimensional Euclidean space has codimension 2, where the normal bundle fibers are 2-dimensional planes transverse to the curve, allowing for more flexibility in perturbations without intersecting the submanifold. These cases demonstrate how codimension influences local topology and the number of independent normal directions available.^[3]^[20]

In Algebraic Geometry

In algebraic geometry, the codimension of an irreducible subvariety Z of a variety X is defined as \operatorname{codim}(Z) = \dim X - \dim Z, where the dimension is the Krull dimension, equivalently the transcendence degree of the function field of the variety over the base field.^[22] This notion extends naturally to schemes: for a scheme X and an irreducible closed subscheme Y \subset X, the codimension is \operatorname{codim}_X(Y) = \dim X - \dim Y, with the dimension of X being the Krull dimension of its structure sheaf.^[6] In the affine case, if X = \operatorname{Spec} A for a domain A and Y = V(\mathfrak{p}) for a prime ideal \mathfrak{p} \subset A, then \operatorname{codim}_X(Y) equals the height of \mathfrak{p} in A, the length of the longest chain of prime ideals contained in \mathfrak{p}.^[6] A fundamental result bounding codimensions is Krull's principal ideal theorem, which states that in a Noetherian ring A, the minimal primes over a principal ideal (f) generated by a non-unit f have height at most 1; geometrically, this implies that any irreducible component of a hypersurface defined by a single equation in a variety has codimension at most 1.^[22] For example, in affine space \mathbb{A}^n_k over a field k, a point has dimension 0 and thus codimension n, while in projective space \mathbb{P}^3_k, a curve (dimension 1) typically has codimension 2.^[23] In projective embeddings, codimension influences key invariants captured by the Hilbert polynomial of a subscheme X \subset \mathbb{P}^n_k: the degree of this polynomial equals \dim X = n - \operatorname{codim}(X), and its leading coefficient, scaled by (\dim X)!, gives the degree of X as an embedded variety, measuring its intersection multiplicity with a general linear subspace of complementary dimension.^[24] Higher codimension thus corresponds to lower-degree polynomials with potentially larger leading coefficients for varieties of fixed geometric degree, reflecting denser embeddings.^[24]

In Topology

In topology, the codimension of a subset A of a topological space X is generally defined as the difference between the topological dimension of X and that of A, where the topological dimension is measured using either the small inductive dimension \operatorname{ind}(X) or the Lebesgue covering dimension \dim(X). These two notions of dimension coincide for separable metric spaces and provide a way to approximate \operatorname{codim}(A, X) = \dim(X) - \dim(A) for closed subsets A, though the exact equality holds under additional regularity conditions on A.^[25] The inductive dimension is defined recursively: \operatorname{ind}(\emptyset) = -1, and for a non-empty space X, \operatorname{ind}(X) = n if every point has arbitrarily small neighborhoods whose boundaries have inductive dimension at most n-1. A key role for codimension arises in embedding theory within geometric topology. The Haefliger-Hauptvermutung results establish that, for embeddings of an n-dimensional manifold into an m-dimensional manifold with codimension m - n \geq 3, the embedding is unique up to homeomorphism in appropriate ranges, resolving aspects of the classical Hauptvermutung conjecture on triangulation equivalence for such high-codimension settings. This contrasts with lower codimensions, where exotic phenomena like non-trivial isotopy classes persist. In knot theory, codimension illustrates the complexity of embeddings. Classical knots are embeddings of a 1-dimensional circle S^1 into 3-dimensional space S^3 or \mathbb{R}^3, yielding codimension 2, where non-trivial knots abound and are classified by invariants like the knot group or Jones polynomial. Higher-codimension knots generalize this to embeddings of S^q into S^n with n - q \geq 3; in the PL category, such knots are often trivial for codimension greater than 2, but smooth and topological versions reveal non-trivial examples in dimensions congruent to 3 modulo 4, as classified by Haefliger.^[26] The metastable range in geometric topology further highlights codimension's importance for existence and classification of embeddings. According to extensions of the Smale-Hirsch theory, in the metastable range, where n > \frac{3}{2}k, an embedding of a k-dimensional manifold into an n-dimensional manifold exists provided certain obstructions in the homotopy groups of the orthogonal group vanish; this range allows reduction to homotopy-theoretic data via the embedding calculus. Outside this range, additional invariants like Haefliger invariants detect knottedness. Codimension also underpins general position arguments in topological spaces. For generic embeddings or maps, if two submanifolds of dimensions d_1 and d_2 are placed in an ambient space of dimension d with d_1 + d_2 - d < 0, their intersections are empty by transversality; more generally, the expected dimension of intersections is d_1 + d_2 - d, and high codimension ensures these are of minimal possible dimension, facilitating inductive proofs and avoiding unexpected overlaps in inductive dimension theory.

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