# Regular sequence
A **regular sequence** is an ordered sequence of elements \( f_1, \ldots, f_r \) of a ring \( R \) such that each \( f_i \) is a nonzerodivisor (an element that never sends a nonzero module element to zero) on the module left by the previous ones, and such that the final quotient is not zero. More precisely, for an \( R \)-module \( M \), the sequence is \( M \)-regular when \( f_i \) is a nonzerodivisor on \( M/(f_1, \ldots, f_{i-1})M \) for each \( i = 1, \ldots, r \), and \( M/(f_1, \ldots, f_r)M \neq 0 \). When \( M = R \), the sequence is simply called regular.[1](https://stacks.math.columbia.edu/tag/0AUH) Regular sequences measure how far a module is from being killed by an ideal, and they organize the theory of depth, the [Koszul complex](https://www.edgepedia.org/koszul-complex), Cohen–Macaulay rings, and complete intersections.[2](https://websites.umich.edu/~mmustata/CAnotes.pdf)
| Key fact | Detail |
|---|---|
| Definition | \( f_1, \ldots, f_r \) is \( M \)-regular if each \( f_i \) is a nonzerodivisor on \( M/(f_1, \ldots, f_{i-1})M \) and \( M/(f_1, \ldots, f_r)M \neq 0 \)[1](https://stacks.math.columbia.edu/tag/0AUH) |
| Order matters | In \( k[x,y,z] \), the sequence \( x,\ y(1-x),\ z(1-x) \) is regular, but \( y(1-x),\ z(1-x),\ x \) is not[1](https://stacks.math.columbia.edu/tag/0AUH) |
| Local case | Over a local Noetherian ring, any permutation of an \( M \)-regular sequence of a finitely generated module is again regular[2](https://websites.umich.edu/~mmustata/CAnotes.pdf) |
| Depth | The \( I \)-depth of \( M \) is the maximal length of an \( M \)-regular sequence contained in an ideal \( I \); for a local ring with maximal ideal \( \mathfrak m \), \( \mathrm{depth}(\mathfrak m, M) \) is the depth of \( M \)[3](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/greuel.pdf) |
| Koszul complex | If \( a \) is \( M \)-regular, \( H_k(a, M) = 0 \) for \( k > 0 \); if \( a \) is \( A \)-regular, the Koszul complex \( K(a) \) is a free resolution of \( A/a \)[4](https://www.imath.kiev.ua/~drozd/ComAlg.pdf) |
| Complete intersections | A Noetherian complete local ring \( A \) is a complete intersection when \( A \cong R/\ker \) for a surjection from a regular local ring \( R \) whose kernel is generated by a regular sequence[5](https://stacks.math.columbia.edu/tag/09PY) |
| Regular local rings | A regular local ring is a Noetherian local ring whose maximal ideal is generated by a regular sequence, equivalently \( \dim(\mathfrak m/\mathfrak m^2) = \dim(R) \)[6](https://www.sciencedirect.com/topics/mathematics/regular-sequence) |
## Definition and first properties
The definition is sequential: each element must act as a nonzerodivisor only after the earlier elements have been divided out. The condition \( M/(f_1, \ldots, f_r)M \neq 0 \) prevents degenerate cases in which the sequence already kills the module. Some papers and books drop this nonzero requirement, which has the advantage that being a regular sequence is then preserved under localization; the Stacks project keeps it and uses the definition mainly to define depth over local rings, where the \( f_i \) are required to lie in the maximal ideal.[1](https://stacks.math.columbia.edu/tag/0AUH)
Order is part of the data. The definition depends on the order of the elements. In the polynomial ring \( k[x, y, z] \) over a field, the sequence \( x,\ y(1-x),\ z(1-x) \) is regular, but the permuted sequence \( y(1-x),\ z(1-x),\ x \) is not.[1](https://stacks.math.columbia.edu/tag/0AUH) Mustață's notes give a similar example in \( R = k[x,y,z]/((x-1)z) \): the sequence \( x,\ (x-1)y \) is regular, while \( (x-1)y,\ x \) is not, because \( (x-1)y \) is a zerodivisor there, with \( (x-1)yz = 0 \) and \( z \neq 0 \) in \( R \).[2](https://websites.umich.edu/~mmustata/CAnotes.pdf)
**The local case is better behaved.** If \( M \) is a finitely generated module over a local [Noetherian ring](https://www.edgepedia.org/noetherian-ring) \( (R, \mathfrak m) \) and \( x_1, \ldots, x_n \) is an \( M \)-regular sequence, then any permutation of it is again a regular sequence.[2](https://websites.umich.edu/~mmustata/CAnotes.pdf) The Stacks project records the same conclusion for finite modules over a local Noetherian ring.[1](https://stacks.math.columbia.edu/tag/0AUH) The proof reduces every permutation to adjacent transpositions and uses Krull's Intersection Theorem to show that the second element of a two-term regular sequence is a nonzerodivisor on \( M \).[2](https://websites.umich.edu/~mmustata/CAnotes.pdf) Regular sequences also behave well under localization when the localized quotient stays nonzero: a nonzerodivisor on a module remains a nonzerodivisor on any localization of that module.[2](https://websites.umich.edu/~mmustata/CAnotes.pdf)
Another useful fact is that the successive submodules form a strictly increasing chain \( (x_1)M \subset (x_1, x_2)M \subset \cdots \subset (x_1, \ldots, x_n)M \), and over a Noetherian module any \( M \)-regular sequence of elements of an ideal \( I \) can be extended to a maximal one.[2](https://websites.umich.edu/~mmustata/CAnotes.pdf)
## Depth
**Depth counts regular sequences.** For a Noetherian ring \( A \), an ideal \( I \), and a module \( M \) with \( IM \neq M \), the \( I \)-depth of \( M \), written \( \mathrm{depth}_A(I, M) \), is the maximal length of an \( M \)-regular sequence contained in \( I \); if \( IM = M \) the depth is set to \( \infty \). When \( A \) is local with maximal ideal \( \mathfrak m \), the number \( \mathrm{depth}(\mathfrak m, M) \) is called the depth of \( M \).[3](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/greuel.pdf) After dimension, depth is described as the most important numerical invariant of a module.[2](https://websites.umich.edu/~mmustata/CAnotes.pdf)
It is not clear from the definition alone that every maximal, that is, non-extendable, \( M \)-regular sequence in \( I \) has the same length, though such a length must be finite because \( A \) is Noetherian. This is in fact true, and follows from the depth-sensitivity of the Koszul complex.[3](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/greuel.pdf) In particular, depth \( 0 \) is equivalent to the existence of an associated prime \( \mathfrak p \in \mathrm{Ass}(M) \) with \( I \subset \mathfrak p \), that is, to every element of \( I \) being a zerodivisor on \( M \).[3](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/greuel.pdf) • [12](https://mreid.warwick.ac.uk/MA4J8/Groutides6.pdf)
## The Koszul complex
The Koszul complex \( K_*(a) \) of a sequence \( a = (a_1, \ldots, a_n) \) is the free \( A \)-module complex with basis symbols \( e_{i_1} \cdots e_{i_k} \) for \( 1 \leq i_1 < \cdots < i_k \leq n \), with differential
\[ d_k(e_{i_1} \cdots e_{i_k}) = \sum_{j=1}^{k} (-1)^{j-1} a_{i_j}\, e_{i_1} \cdots \widehat{e_{i_j}} \cdots e_{i_k}, \]
where the hat marks an omitted symbol; for a module \( M \) one sets \( K_*(a, M) = K_*(a) \otimes_A M \).[4](https://www.imath.kiev.ua/~drozd/ComAlg.pdf)
The complex detects regularity. If \( a \) is \( M \)-regular, then \( H_k(a, M) = 0 \) for \( k > 0 \), and in particular, if \( a \) is \( A \)-regular, \( K(a) \) is a free resolution of \( A/a \).[4](https://www.imath.kiev.ua/~drozd/ComAlg.pdf) Conversely, when all \( a_i \) lie in the radical of \( A \) and \( M \) is finitely generated, the three conditions \( a \) is \( M \)-regular, \( H_k(a, M) = 0 \) for \( k > 0 \), and \( H_1(a, M) = 0 \) are equivalent, so vanishing of a single homology group suffices.[4](https://www.imath.kiev.ua/~drozd/ComAlg.pdf) The Encyclopedia of Mathematics states the same vanishing for an \( M \)-regular sequence, with \( H_0 \) computing \( M/\sum_{i=1}^{r} x_i M \), and notes that this makes Koszul complexes an important tool in dimension theory, multiplicity theory, and intersection theory.[7](https://encyclopediaofmath.org/wiki/Koszul_complex)
[Jean-Louis Koszul](https://www.edgepedia.org/jean-louis-koszul) introduced the complex in his 1950 thesis work *Homologie et cohomologie des algèbres de Lie*, published in the Bulletin de la Société mathématique de France, where it served a cohomology theory for Lie algebras; the Institut Fourier's account records that the complex introduced at the 1950 Brussels topology colloquium went on to a career with numerous, sometimes unexpected, applications.[8](https://archive.numdam.org/item/BSMF_1950__78__65_0/) • [9](https://www-fourier.univ-grenoble-alpes.fr/index.php/fr/content/jean-louis-koszul-0)
## Complete intersections and regular local rings
Regular sequences define the class of local rings called complete intersections. For a Noetherian complete local ring \( A \), the [Cohen structure theorem](https://www.edgepedia.org/cohen-structure-theorem) writes \( A \) as a quotient of a regular Noetherian complete local ring \( R \); \( A \) is a complete intersection when there exists a surjection \( R \to A \) with \( R \) regular local whose kernel is generated by a regular sequence. A lemma of the Stacks project shows this notion is independent of the chosen surjection: if it holds for one such surjection, it holds for every one.[5](https://stacks.math.columbia.edu/tag/09PY) Equivalently, a commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence.[10](https://arxiv.org/abs/1109.4921)
The same idea characterizes regular local rings themselves: a regular local ring is a commutative Noetherian local ring \( R \) whose maximal ideal \( \mathfrak m \) is generated by a regular sequence, or equivalently such that \( \dim(\mathfrak m/\mathfrak m^2) = \dim(R) \).[6](https://www.sciencedirect.com/topics/mathematics/regular-sequence) Homological methods built on these notions led Auslander and Buchsbaum to prove in 1959 that every regular local ring is a unique factorization domain.[6](https://www.sciencedirect.com/topics/mathematics/regular-sequence)
Geometrically, a regular sequence corresponds to a hypersurface: cutting a variety by one equation at a time, each new equation meeting the remaining locus properly, is exactly the regular-sequence condition, which is why complete intersections are the algebraic model of varieties defined by the expected number of equations.[11](https://scms.fudan.edu.cn/dfiles/summerschool/lecture_note_day_1-3.pdf)
## References
1. [The Stacks Project, Algebra, Section 10.68: Regular sequences](https://stacks.math.columbia.edu/tag/0AUH)
2. [M. Mustață, Introduction to Commutative Algebra, Chapter 11: Regular sequences and depth](https://websites.umich.edu/~mmustata/CAnotes.pdf)
3. [G.-M. Greuel et al., Commutative Algebra, Appendix B.7: Regular Sequences and Depth](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/greuel.pdf)
4. [Yu. Drozd, Commutative Algebra, Section 21: Regular sequences and depth, the Koszul complex](https://www.imath.kiev.ua/~drozd/ComAlg.pdf)
5. [The Stacks Project, Section 23.8: Local complete intersection rings](https://stacks.math.columbia.edu/tag/09PY)
6. [Regular Sequence, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/regular-sequence)
7. [Koszul complex, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Koszul_complex)
8. [J.-L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. S.M.F. 78 (1950)](https://archive.numdam.org/item/BSMF_1950__78__65_0/)
9. [Jean-Louis Koszul, Institut Fourier](https://www-fourier.univ-grenoble-alpes.fr/index.php/fr/content/jean-louis-koszul-0)
10. [Are complete intersections complete intersections?, arXiv:1109.4921](https://arxiv.org/abs/1109.4921)
11. [Fudan University Summer School, Commutative Algebra Notes: Koszul complexes and regular sequences](https://scms.fudan.edu.cn/dfiles/summerschool/lecture_note_day_1-3.pdf)
12. [mreid.warwick.ac.uk](https://mreid.warwick.ac.uk/MA4J8/Groutides6.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Depth and regular sequences*
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