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Regular sequence

In commutative algebra, a regular sequence is a sequence of elements f_1, \dots, f_r in a commutative ring R that act as independently as possible on an R-module M, specifically where each f_i is a non-zero-divisor on the quotient module M / (f_1, \dots, f_{i-1})M for i = 1, \dots, r, and the final quotient M / (f_1, \dots, f_r)M is nonzero.^[1] When M = R, the sequence is simply called a regular sequence in R.^[1] This notion captures algebraic independence in a homological sense, generalizing the idea of algebraically independent elements over fields to arbitrary rings.^[2] The concept of a regular sequence, originally termed an "R/I-sequence," was introduced by Maurice Auslander and David A. Buchsbaum in their 1956 paper on homological dimensions in Noetherian rings, where it was defined using colon ideals to ensure no element creates zero-divisors in successive quotients.^[3] Around the same time, Jean-Pierre Serre independently developed related ideas in his work on algebraic geometry and homological algebra, contributing to the characterization of regular local rings.^[4] The modern formulation in terms of non-zero-divisors became standard in subsequent texts, emphasizing its role in exactness of Koszul complexes.^[1] Regular sequences are fundamental for defining key invariants in commutative algebra, such as the grade of an ideal (the length of a maximal regular sequence generating it) and the depth of a module (the length of a maximal regular sequence on it).^[1] They underpin the theory of Cohen-Macaulay rings and modules, where the depth equals the dimension, providing a bridge between algebraic properties and geometric singularities.^[5] Applications extend to homological algebra, where regular sequences ensure vanishing of certain Tor groups, and to algebraic geometry, where they relate to complete intersections and resolution of singularities.^[6]

Fundamental Definitions

Non-Zero-Divisors

In commutative algebra, an element r \in R of a commutative ring R is called a non-zero-divisor, or regular element, on an R-module M if the multiplication map M \to M given by m \mapsto r \cdot m is injective. Equivalently, r \cdot m = 0 implies m = 0 for every m \in M.^[7] This condition ensures that r does not annihilate any nonzero element of M, preserving the structure without introducing torsion in this specific sense. An equivalent formulation is that the annihilator of r in M, defined as \mathrm{Ann}_M(r) = \{ m \in M \mid r \cdot m = 0 \}, is the zero submodule. Thus, r lies outside the annihilator ideal \mathrm{Ann}_R(M) if M itself is considered, but more generally, the kernel of the endomorphism induced by multiplication by r vanishes.^[8] This perspective highlights the role of regular elements in maintaining exactness in sequences and avoiding zero kernels in module homomorphisms. A basic property arises in the context of integral domains: if R is an integral domain, then every nonzero element of R acts as a non-zero-divisor on R itself as a module over R, since there are no nontrivial zero-divisors in R.^[9] This underscores the absence of torsion elements annihilated by nonzero ring elements in such settings. The term "regular element" originates from developments in early 20th-century algebra, particularly in the study of ideals and modules, and was formalized in influential texts on commutative algebra, such as those by Bourbaki.

M-Regular Sequences

In commutative algebra, an M-regular sequence for a module M over a commutative ring R is a finite sequence of elements r_1, \dots, r_d \in R such that r_1 is a non-zero-divisor on M (i.e., multiplication by r_1 is injective on M), and for each i = 2, \dots, d, r_i is a non-zero-divisor on the quotient module M / (r_1, \dots, r_{i-1}) M.^[10] Some definitions further require that the final quotient module M / (r_1, \dots, r_d) M \neq 0.^[11] When M = R, the sequence is called an R-regular sequence, or simply a regular sequence in R.^[10] Such sequences are often denoted by \mathbf{r} = (r_1, \dots, r_d), with the ideal they generate written as (\mathbf{r}) = (r_1, \dots, r_d).^[11] The concept, originally termed an "R/I-sequence" using colon ideals, was introduced by Auslander and Buchsbaum in 1956.^[3] Some later texts refer to related notions as "A-sequences."^[10] A basic property is that if \mathbf{r} is an M-regular sequence, then for each i = 1, \dots, d, the colon ideal (r_1, \dots, r_{i-1}) : r_i = (r_1, \dots, r_{i-1}) in R, indicating that r_i introduces no additional relations beyond the previous elements.^[11] The length of a maximal M-regular sequence equals the depth of M.^[10]

Depth of Modules

In commutative algebra, the depth of an R-module M with respect to an ideal I \subseteq R is defined as the supremum of the lengths of M-regular sequences contained in I, taking the value \infty if no such sequence exists because IM = M.^[12] For a local ring (R, \mathfrak{m}), the depth is typically considered with respect to the maximal ideal \mathfrak{m} and denoted \mathrm{depth}_R M.^[12] For finitely generated modules over a Noetherian ring, depth exhibits several key properties: it is nonnegative, so $0 \leq \mathrm{depth}_R M; under finite direct sums, \mathrm{depth}_R (M \oplus N) = \min(\mathrm{depth}_R M, \mathrm{depth}_R N); and \mathrm{depth}_R M \leq \dim M, where \dim M is the Krull dimension of M.^[13] A fundamental homological characterization holds in Noetherian local rings: for a nonzero finitely generated module M, \mathrm{depth}_R M = \inf \{ i \mid \mathrm{Ext}^i_R (R/\mathfrak{m}, M) \neq 0 \}.^[14] This links depth directly to Ext modules and underscores its role as a homological invariant measuring the "regularity" of M relative to the residue field. For the specific case where M = R/I with I an ideal of R, the depth \mathrm{depth}_R (R/I) coincides with the grade of I, defined as the minimal integer i such that \mathrm{Ext}^i_R (R/I, R) \neq 0. This equivalence highlights how regular sequences quantify the homological complexity of quotient modules. The concept of depth was formalized in the 1960s by Auslander and Bass, building on foundational work by Serre on local rings and homological dimensions.

Properties of Regular Sequences

Independence and Order

In commutative algebra, the elements r_1, \dots, r_d of a regular sequence in a ring R exhibit algebraic independence in the quotient module, meaning that the ideal I = (r_1, \dots, r_d) they generate has grade equal to its length d, where the grade is the supremum of the lengths of regular sequences contained in I.^[15] This property ensures that the sequence imposes the maximal possible constraints without introducing dependencies that would shorten the effective length.^[10] The regularity of a sequence depends on its order, as the definition requires each element to be a nonzerodivisor on the successive quotients. However, in a Noetherian local ring, if \mathbf{r} = (r_1, \dots, r_d) is a regular sequence contained in the maximal ideal, then every permutation of \mathbf{r} is also regular.^[15] Counterexamples to permutation invariance exist in non-Noetherian rings; for instance, in the polynomial ring k[x, y, z] over a field k, the sequence x, y(1 - x), z(1 - x) is regular, but the permutation y(1 - x), z(1 - x), x is not.^[1] A key extension property holds in commutative Noetherian rings: if \mathbf{r} = (r_1, \dots, r_d) is a regular sequence and s \in R satisfies (\mathbf{r}) : s = (\mathbf{r}), then \mathbf{r}, s forms a regular sequence.^[15] This condition on the colon ideal ensures that s acts independently on the quotient by (\mathbf{r}).^[10] The Koszul complex provides a homological criterion for regularity: a sequence \mathbf{r} = (r_1, \dots, r_d) in R is regular on a module M if and only if the Koszul complex K(\mathbf{r}; M) is acyclic in positive degrees, yielding a free resolution of M / (\mathbf{r}) M.^[15] Ideals generated by regular sequences are perfect, meaning the projective dimension of the quotient module R / I is finite and equal to the length d of the sequence; moreover, such ideals require exactly d minimal generators, as the grade equals the height and the embedding dimension constraints.^[10] When maximal, regular sequences achieve the depth of the module.^[15]

Length and Dimension

In commutative algebra, the length of a regular sequence on a finitely generated module M over a Noetherian ring R is bounded above by the Krull dimension of M, defined as the supremum of the lengths of chains of prime ideals in the support of M. This bound arises because each element in the sequence reduces the dimension of the quotient module by at most one, and the process terminates when the dimension reaches zero. Equality holds precisely when M is a Cohen-Macaulay module, meaning a nonzero module where the depth equals the dimension; here, the depth is the length of a maximal regular sequence on M.^[12]^[16]^[17] A key result relating regular sequences to ideal heights is a variant of Krull's principal ideal theorem: if \mathbf{r} = (r_1, \dots, r_d) is a regular sequence in R, then the height of the ideal (\mathbf{r}) is exactly d. This follows from iteratively applying the principal ideal theorem, which states that the height of a prime minimal over an ideal generated by k elements is at most k, combined with the fact that each successive quotient by a regular element preserves the necessary minimality conditions for height increase. In Cohen-Macaulay rings, this extends to the general case where the height of any proper ideal I equals the supremum of lengths of regular sequences in I.^[18] The fundamental inequality \mathrm{depth}_R M \leq \dim_R M holds for any finitely generated module M over a Noetherian local ring (R, \mathfrak{m}), with equality characterizing Cohen-Macaulay modules. In the local setting, a system of parameters for R—a sequence of elements generating \mathfrak{m} whose length equals \dim R—forms a regular sequence if and only if R is Cohen-Macaulay. This equivalence underscores the role of regular sequences in measuring how "deep" the ring is relative to its dimension.^[16]^[19] For polynomial rings over fields, such as k[x_1, \dots, x_n] localized at the maximal ideal (x_1, \dots, x_n), the variables x_1, \dots, x_n form a regular sequence generating the maximal ideal, with length exactly equal to the Krull dimension n. This illustrates the bound in a regular local ring, where minimal generators of the maximal ideal always constitute a regular sequence.^[20]

Examples and Counterexamples

Standard Examples

In integral domains, a sequence of non-zero elements forms a regular sequence provided that each element is non-zero in the quotient by the ideal generated by the previous elements, as the quotients remain integral domains and thus every non-zero element is a non-zero-divisor.^[21] For example, in the ring of integers \mathbb{Z}, the sequence $2, 3 is regular because multiplication by 2 is injective on \mathbb{Z} (as \mathbb{Z} is a domain), the quotient \mathbb{Z}/(2) \cong \mathbb{Z}/2\mathbb{Z} is a field (hence a domain), and the image of 3 in this quotient is non-zero, yielding a final non-zero quotient.^[6] In polynomial rings over a field, the variables themselves provide a canonical regular sequence. Specifically, in the ring k[x_1, \dots, x_n] where k is a field, the sequence x_1, \dots, x_n is regular, as each x_i acts as a non-zero-divisor on the quotient by the previous variables (which is isomorphic to a polynomial ring in the remaining variables, hence a domain), and the sequence generates the maximal ideal at the origin with depth equal to n.^[1] The localization construction preserves regular sequences in many cases, yielding standard examples in local rings. In the local ring \mathbb{Z}_{(p)} obtained by localizing \mathbb{Z} at the prime ideal (p), the sequence consisting of the single element p is regular of length 1, as \mathbb{Z}_{(p)} is a discrete valuation ring of dimension 1, p generates its maximal ideal, and the quotient \mathbb{Z}_{(p)}/(p) is the field \mathbb{F}_p.^[21] Power series rings over fields exhibit similar behavior to polynomial rings due to their completeness. In the formal power series ring k[[x_1, \dots, x_n]] where k is a field, the sequence x_1, \dots, x_n forms a regular sequence, as this ring is a complete local ring of dimension n, and the variables generate the maximal ideal while satisfying the non-zero-divisor condition iteratively in the quotients (each of which is again a power series ring in fewer variables).^[21] In quotient rings, regular sequences can arise from elements avoiding the relations in the ideal. For instance, in k[x,y]/(xy), the sequence (x, y) fails to be regular because x is a zero-divisor (as x \cdot y = 0 but y \neq 0), but single elements that are non-zero-divisors, such as in related quotients like k[x,y]/(x^2) where y acts injectively, form regular sequences of length 1.^[6]

Non-Examples

A common failure of regularity occurs when the first element of the sequence is a zero-divisor in the ring. For instance, consider the ring R = k[x, y] / (x^2, xy) over a field k. Here, the image \bar{x} satisfies \bar{x} \cdot \bar{y} = 0 but \bar{y} \neq 0, so \bar{x} is a zero-divisor, and the sequence (\bar{x}) is not regular.^[22] The iterative condition can also fail even if individual elements are non-zero-divisors. In the polynomial ring k[x, y, z] over a field k, the sequence y(1 - x), z(1 - x), x is not regular because, in the quotient k[x, y, z] / (y(1 - x)), the element z(1 - x) annihilates y modulo the ideal but y \not\equiv 0. A similar breakdown occurs in \mathbb{C}[x, y] with the sequence (xy, x^2): while xy is a non-zero-divisor, in the quotient \mathbb{C}[x, y] / (xy), x^2 annihilates y since x^2 y = x (xy) = 0 but y \neq 0.^[1] In non-Noetherian rings, the order of elements can critically affect regularity, unlike in Noetherian local rings where permutations preserve regularity. Consider the ring R = [k](/page/K)[x_1, x_2, \dots ] / (x_1^2, x_2^2, \dots ) over a field k. The infinite sequence (x_1, x_2, \dots ) is regular, as each x_i acts without non-trivial kernel on the successive quotients. However, reordering to (x_2, x_1, x_3, \dots ) fails: in R / (x_2), x_1 satisfies x_1 \cdot x_1 = 0 but x_1 \neq 0, making x_1 a zero-divisor. Such order dependence highlights failures beyond Noetherian settings.^[23] Sequences containing redundant elements, such as the zero element or a unit, trivially fail regularity in any ring, as the zero element annihilates everything non-trivially, and a unit would require the module to be zero for the multiplication map to be injective. Geometrically, non-regular sequences correspond to hypersurfaces whose intersections are not transverse, such as when multiple components share common factors in the defining equations, leading to higher-dimensional intersections than expected.^[24] These non-examples, including those in non-local and non-Noetherian contexts, demonstrate situations where the depth of the module is less than the length of the proposed sequence.

Applications

Koszul Complexes and Resolutions

The Koszul complex associated to a sequence \mathbf{r} = (r_1, \dots, r_d) in a commutative ring R is the chain complex K_\bullet(\mathbf{r}; R) whose k-th term is the exterior power \wedge^k R^d, with differentials defined by d_k(e_{i_1} \wedge \cdots \wedge e_{i_k}) = \sum_{j=1}^k (-1)^{j-1} r_{i_j} e_{i_1} \wedge \cdots \widehat{e_{i_j}} \cdots \wedge e_{i_k}, where \{e_i\} is the standard basis of R^d. This complex is exact if and only if \mathbf{r} is an R-regular sequence.^[25] A fundamental theorem states that a sequence \mathbf{r} is R-regular if and only if K_\bullet(\mathbf{r}; R) is a free resolution of the quotient module R/(\mathbf{r}), meaning the complex is acyclic with homology zero in all positive degrees and H_0(K_\bullet(\mathbf{r}; R)) \cong R/(\mathbf{r}).^[25] This equivalence provides a homological criterion for regularity, linking algebraic properties of sequences to the exactness of the associated complex.^[25] This construction generalizes to modules: for an R-module M, the Koszul complex K_\bullet(\mathbf{r}; M) = K_\bullet(\mathbf{r}; R) \otimes_R M resolves M/(\mathbf{r})M precisely when \mathbf{r} is an M-regular sequence.^[25] The homology groups of the Koszul complex satisfy H_i(K_\bullet(\mathbf{r}; M)) = 0 for all i > 0 if and only if \mathbf{r} is M-regular; in this case, the higher Tor groups \Tor_i^R(M, R/(\mathbf{r})) also vanish.^[25] These vanishing conditions enable explicit computations in homological algebra. Koszul complexes are instrumental in determining projective dimensions: if \mathbf{r} is a regular sequence of length d, then the projective dimension of R/(\mathbf{r}) (or M/(\mathbf{r})M) is exactly d.^[26] In particular, for regular local rings, where a system of parameters forms a regular sequence generating the maximal ideal, the Koszul complex yields a free resolution of the residue field of length equal to the Krull dimension, establishing that the global dimension equals the Krull dimension.^[26] The Koszul complex was originally developed by Jean-Louis Koszul in the 1950s to study the homology and cohomology of Lie algebras.^[27] It was subsequently adapted to commutative algebra, where it plays a central role in the study of regular sequences and resolutions.

Cohen-Macaulay Rings and Regularity

A local Noetherian ring (R, \mathfrak{m}) is Cohen-Macaulay if \mathrm{depth}_R R = \dim R.^[19] This condition is equivalent to every system of parameters of R forming a regular sequence.^[19] The Cohen-Macaulay property captures rings where homological invariants align with geometric dimension, ensuring no "embedded components" in associated primes.^[28] A local Noetherian ring R is regular if its maximal ideal \mathfrak{m} is generated by a regular sequence.^[20] Equivalently, R is locally a complete intersection, meaning its completion is a quotient of a regular local ring by a regular sequence ideal.^[29] Regular rings form the "smoothest" class beyond fields, with minimal generators of \mathfrak{m} matching the Krull dimension.^[20] Regular rings are Cohen-Macaulay, as any minimal generating set for \mathfrak{m} serves as a regular system of parameters.^[30] Moreover, the global dimension of a regular local ring equals its Krull dimension.^[31] This equality reflects optimal homological behavior, where projective resolutions achieve minimal length.^[30] Geometrically, regular sequences define complete intersection subschemes in affine space, cutting down dimensions transversely without excess structure.^[29] The Cohen-Macaulay condition ensures such schemes have pure dimension at singularities, avoiding lower-dimensional components that complicate intersection theory.^[28] For instance, in an affine scheme over a field, quotienting by a regular sequence yields a complete intersection of expected codimension, preserving equidimensionality if the original is Cohen-Macaulay.^[32] Polynomial rings over fields are regular, as their maximal ideals at prime localizations are generated by algebraically independent elements forming regular sequences.^[33] Localizations of polynomial rings at prime ideals remain regular.^[33] If R is Cohen-Macaulay and I is generated by a regular sequence of length equal to its codimension, then the quotient R/I is also Cohen-Macaulay.^[32] Gorenstein rings refine the Cohen-Macaulay class: a local Cohen-Macaulay ring is Gorenstein if its injective dimension as a module over itself equals its depth (hence its dimension).^[34] This duality property strengthens homological symmetry, with canonical modules being the ring itself.^[34] In algebraic geometry, Gorenstein conditions classify mild singularities, such as hypersurface singularities where the defining equation yields a complete intersection.^[35] Applications extend to singularity theory, where maximal Cohen-Macaulay modules over hypersurface rings classify representations and resolve geometric quotients.^[35] The term Cohen-Macaulay honors Irvin Cohen's 1946 work on complete local rings and Francis S. Macaulay's 1916 unmixedness theorem for polynomial ideals.^[36] The modern framework, including regular sequences and depth, was formalized in Hideyuki Matsumura's Commutative Ring Theory (1986).^[15]

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