# Depth (ring theory)

In commutative algebra, the **depth** of a module M over a Noetherian local ring (R, m) with residue field k = R/m is the smallest integer i such that \( \operatorname{Ext}^i_R(k, M) \neq 0 \); if no such integer exists the depth is set to \( \infty \)<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>. Equivalently, and more elementarily, the depth of M is the length of a maximal M-regular sequence contained in the [maximal ideal](https://www.edgepedia.org/maximal-ideal) m, where a sequence \( a_1, \ldots, a_n \) is M-regular when each \( a_i \) is a nonzerodivisor on \( M/(a_1, \ldots, a_{i-1})M \)<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup><sup> • </sup><sup>[2](https://stacks.math.columbia.edu/tag/0AUH)</sup>. Depth was first introduced as a homological invariant under the name homological codimension<sup>[3](https://www.sciencedirect.com/science/article/pii/S0021869320301848)</sup>, and it measures in some sense the torsion of the module M<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Homological definition | \( \operatorname{depth} M = \min\{ i : \operatorname{Ext}^i_R(k, M) \neq 0 \} \) for M finitely generated over a Noetherian local ring (R, m), k = R/m<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup> |
| Regular sequence definition | Depth equals the length of every maximal M-regular sequence in m; all maximal M-regular sequences have the same length<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup> |
| Local cohomology form | \( \operatorname{depth} M = \min\{ d : H^d_{\mathfrak m}(M) \neq 0 \} \)<sup>[3](https://www.sciencedirect.com/science/article/pii/S0021869320301848)</sup> |
| Dimension bound | For a nonzero finitely generated module over a local ring, \( \operatorname{depth} M \leq \dim M \)<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup> |
| Auslander–Buchsbaum formula | If M is finitely generated over a local ring R with \( \operatorname{pd}_R(M) < \infty \), then \( \operatorname{pd}_R(M) + \operatorname{depth}(M) = \operatorname{depth}(R) \)<sup>[4](https://ncatlab.org/nlab/show/Auslander-Buchsbaum%2Bformula)</sup> |
| Cohen–Macaulay case | A module is Cohen–Macaulay when depth equals Krull dimension; depth appears in Serre's criteria \( (S_k) \)<sup>[5](https://citeseerx.ist.psu.edu/document?doi=7e958960cd17204b46e2cde477b6d9db873675ba&repid=rep1&type=pdf)</sup><sup> • </sup><sup>[3](https://www.sciencedirect.com/science/article/pii/S0021869320301848)</sup> |
| Convention | The depth of the zero module is set to \( +\infty \)<sup>[6](https://stacks.math.columbia.edu/download/book.pdf)</sup> |

## Regular sequences

The elementary definition rests on the notion of an M-regular sequence. An element x of R is M-regular if multiplication by x on M is injective, that is, xz = 0 for z in M implies z = 0<sup>[7](https://www.cambridge.org/core/books/abs/cohenmacaulay-rings/regular-sequences-and-depth/1B04C495C2B4915A13633CD5E6780EA1)</sup>. A sequence \( x_1, \ldots, x_n \) is an M-regular sequence, or M-sequence, when each \( x_i \) is regular on \( M/(x_1, \ldots, x_{i-1})M \) and the quotient \( M/xM \) is not zero<sup>[7](https://www.cambridge.org/core/books/abs/cohenmacaulay-rings/regular-sequences-and-depth/1B04C495C2B4915A13633CD5E6780EA1)</sup><sup> • </sup><sup>[2](https://stacks.math.columbia.edu/tag/0AUH)</sup>. A weak M-sequence drops the requirement that the final quotient be nonzero<sup>[7](https://www.cambridge.org/core/books/abs/cohenmacaulay-rings/regular-sequences-and-depth/1B04C495C2B4915A13633CD5E6780EA1)</sup>.

**Order matters in general.** The definition depends on the order of the elements. In the ring \( k[x, y, z] \) over a field k, the sequence \( x, y(1-x), z(1-x) \) is regular, but the permuted sequence \( y(1-x), z(1-x), x \) is not<sup>[2](https://stacks.math.columbia.edu/tag/0AUH)</sup>. Over a local [Noetherian ring](https://www.edgepedia.org/noetherian-ring) the situation improves: if \( x_1, \ldots, x_c \) is an M-regular sequence, then any permutation of the \( x_i \) is again a regular sequence<sup>[2](https://stacks.math.columbia.edu/tag/0AUH)</sup>.

**Depth from sequences.** The two definitions of depth agree. If a is an element of m that is a nonzerodivisor on M, then \( \operatorname{depth}(M/aM) = \operatorname{depth} M - 1 \), and by induction \( \operatorname{depth} M/(a_1, \ldots, a_i)M = \operatorname{depth} M - i \) along any M-regular sequence<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>. Depth zero has a direct description: M has depth zero if and only if every element of m is a zerodivisor on M, which holds exactly when m is an associated prime of M<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>. A regular sequence is maximal precisely when every element of m acts as a zerodivisor on the quotient, and the depth is the length of every maximal M-regular sequence<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>.

The Rees theorem connects regular sequences with Ext vanishing more generally: for a Noetherian ring R, an ideal I, and a finitely generated nonzero module M with \( IM \neq M \), the existence of an M-regular sequence of length n with every element in I is equivalent to the vanishing of \( \operatorname{Ext}^i_R(R/I, M) \) for all \( i < n \)<sup>[8](https://firsching.ch/FormalBook/docs/Mathlib/RingTheory/Depth/Rees.html)</sup>.

## Relation to projective dimension

The projective dimension \( \operatorname{pd}_R(M) \), the minimum length of a projective resolution of M, is the other half of the central formula involving depth. Over a local ring, \( \operatorname{pdim}(M) \) can be characterized as \( \sup\{ i : \operatorname{Tor}_i(M, R/\mathfrak m) \neq 0 \} \)<sup>[5](https://citeseerx.ist.psu.edu/document?doi=7e958960cd17204b46e2cde477b6d9db873675ba&repid=rep1&type=pdf)</sup>.

The **Auslander–Buchsbaum formula** states that if M is a finitely generated module over a local ring R and \( \operatorname{pd}_R(M) < \infty \), then

\[ \operatorname{pd}_R(M) + \operatorname{depth}(M) = \operatorname{depth}(R). \]

It follows that the depth of R is an upper bound for the finite values of \( \operatorname{pd}_R(M) \)<sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>. The formula was proved by [Maurice Auslander](https://www.edgepedia.org/maurice-auslander), a mathematician who published his first joint paper with [David Buchsbaum](https://www.edgepedia.org/david-buchsbaum), *Homological dimension in Noetherian rings*, in 1956<sup>[10](https://mathshistory.st-andrews.ac.uk/Biographies/Auslander/)</sup>, and for a complete local ring R and M ∈ P0(R), the formula holds, in the form apdR M = depth R − depthR M, even without the finiteness condition on M<sup>[11](https://citeseerx.ist.psu.edu/document?doi=4e19070232fb005b4317f9cd60f27fd0b20e4c97&repid=rep1&type=pdf)</sup>.

## Depth, dimension, and the Cohen–Macaulay property

Depth is always bounded by dimension: for a nonzero finitely generated module over a local ring, \( \operatorname{depth} M \leq \dim M \)<sup>[1](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>. The gap between the two numbers measures how far the module is from being Cohen–Macaulay. A module is **Cohen–Macaulay** when its depth equals its [Krull dimension](https://www.edgepedia.org/krull-dimension)<sup>[5](https://citeseerx.ist.psu.edu/document?doi=7e958960cd17204b46e2cde477b6d9db873675ba&repid=rep1&type=pdf)</sup>; in this case there is a particularly close connection between the projective dimension and the geometry of the associated primes<sup>[5](https://citeseerx.ist.psu.edu/document?doi=7e958960cd17204b46e2cde477b6d9db873675ba&repid=rep1&type=pdf)</sup>.

Depth also enters Serre's finiteness conditions. A Noetherian ring R has property \( (S_k) \) if for every prime p the local ring \( R_{\mathfrak p} \) has depth at least \( \min\{k, \dim(R_{\mathfrak p})\} \)<sup>[12](https://stacks.math.columbia.edu/tag/031O)</sup>. These conditions characterize standard properties: a Noetherian ring is reduced if and only if it has \( (R_0) \) and \( (S_1) \), and normal if and only if it has \( (R_1) \) and \( (S_2) \), where \( (R_k) \) requires \( R_{\mathfrak p} \) to be regular for every prime of height at most k<sup>[12](https://stacks.math.columbia.edu/tag/031O)</sup>. A module has no embedded associated prime exactly when it satisfies \( (S_1) \)<sup>[12](https://stacks.math.columbia.edu/tag/031O)</sup>.

The Cohen–Macaulay condition yields the **unmixedness theorem**: if R is Cohen–Macaulay and I is an ideal generated by n elements with codimension n, then every associated prime of I is minimal over I<sup>[13](https://www.math.ens.psl.eu/~benoist/refs/Eisenbud.pdf)</sup>. A Noetherian ring is Cohen–Macaulay if and only if the unmixedness theorem holds for it<sup>[14](https://people.math.sc.edu/kustin/teaching/CAII/CAII.pdf)</sup>. The second statement of the theorem was proved by Macaulay for polynomial rings and by Cohen for regular local rings, which is the source of the name<sup>[13](https://www.math.ens.psl.eu/~benoist/refs/Eisenbud.pdf)</sup>.

## Regular local rings

Depth interacts with regularity through homological dimension. In commutative algebra, a regular local ring R is a Noetherian local ring whose maximal ideal m can be generated by a regular sequence—equivalently, it satisfies \( \dim(\mathfrak m/\mathfrak m^2) = \dim(R) \)—and these rings serve as the local coordinate rings of smooth algebraic varieties.<sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup> In 1956, Auslander and Buchsbaum and, independently, Serre used homological methods to characterize regular local rings as those Noetherian local rings with finite global dimension<sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>. Serre proved that if R is local with residue field k and \( \dim_k(\mathfrak m/\mathfrak m^2) = n \), then \( \operatorname{Tor}^R_n(k, k) \neq 0 \), so \( \operatorname{gl.dim}(R) \geq n \) and \( n \geq \dim(R) \geq \operatorname{depth}(R) \)<sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>. Combined with the Auslander–Buchsbaum bound on projective dimensions, this forces \( \operatorname{gl.dim}(R) = \dim_k(\mathfrak m/\mathfrak m^2) = \dim(R) \) whenever the global dimension is finite<sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>.

## Computation and applications

Finding a concrete description of regular sequences is difficult in practice, which motivates computable substitutes. For modules of the form R/I over a polynomial ring R, an **initially regular sequence** on R/I has a tractable concrete description, and when I is a homogeneous ideal the length of such a sequence gives a lower bound for \( \operatorname{depth} R/I \)<sup>[3](https://www.sciencedirect.com/science/article/pii/S0021869320301848)</sup>. Since projective dimension is a computationally amenable quantity, the Auslander–Buchsbaum formula converts depth bounds into projective dimension bounds<sup>[15](https://math.mit.edu/documents/spur/2014brookner.pdf)</sup>.

Depth also has a purely combinatorial analog. For a monomial ideal I in a polynomial ring S, the **Stanley depth** sdepth S/I is defined in terms of certain vector space decompositions of S/I<sup>[16](https://www.math.utep.edu/faculty/duval/papers/MCA17geometrytalk.pdf)</sup>, and a construction of a non-partitionable Cohen–Macaulay simplicial complex disproved the conjecture that the Stanley depth of a monomial ideal is always at least its depth<sup>[17](https://arxiv.org/abs/1504.04279)</sup>.

## References

1. [The CRing Project, Chapter 16, §1: Depth over local rings (David S. Metzler et al.)](https://math.uchicago.edu/~amathew/CRing.pdf)
2. [The Stacks Project, Section 10.68: Regular sequences](https://stacks.math.columbia.edu/tag/0AUH)
3. [Initially regular sequences and depths of ideals, Journal of Algebra (ScienceDirect)](https://www.sciencedirect.com/science/article/pii/S0021869320301848)
4. [Auslander–Buchsbaum formula, nLab](https://ncatlab.org/nlab/show/Auslander-Buchsbaum%2Bformula)
5. [Computational Algebraic Geometry, Chapter 10: Projective Dimension, Cohen–Macaulay Modules (CiteSeerX)](https://citeseerx.ist.psu.edu/document?doi=7e958960cd17204b46e2cde477b6d9db873675ba&repid=rep1&type=pdf)
6. [The Stacks Project, Algebra, Section 10.72: Depth (book PDF)](https://stacks.math.columbia.edu/download/book.pdf)
7. [Bruns and Herzog, Cohen–Macaulay Rings, Chapter 1: Regular sequences and depth (Cambridge Core)](https://www.cambridge.org/core/books/abs/cohenmacaulay-rings/regular-sequences-and-depth/1B04C495C2B4915A13633CD5E6780EA1)
8. [Mathlib.RingTheory.Depth.Rees (Rees theorem, formalized)](https://firsching.ch/FormalBook/docs/Mathlib/RingTheory/Depth/Rees.html)
9. [Regular Sequence, an overview, ScienceDirect Topics (from Weibel, History of Homological Algebra)](https://www.sciencedirect.com/topics/mathematics/regular-sequence)
10. [Maurice Auslander (1926–1994), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Auslander/)
11. [Cohen–Macaulay Local Rings (CiteSeerX)](https://citeseerx.ist.psu.edu/document?doi=4e19070232fb005b4317f9cd60f27fd0b20e4c97&repid=rep1&type=pdf)
12. [The Stacks Project, Section 10.157: Serre's criterion for normality](https://stacks.math.columbia.edu/tag/031O)
13. [Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Chapter 18: Depth, Codimension, and Cohen–Macaulay Rings](https://www.math.ens.psl.eu/~benoist/refs/Eisenbud.pdf)
14. [Kustin, Commutative Algebra II course notes, University of South Carolina](https://people.math.sc.edu/kustin/teaching/CAII/CAII.pdf)
15. [Brookner, Cohen–Macaulayness of Certain Rings, MIT SPUR (2014)](https://math.mit.edu/documents/spur/2014brookner.pdf)
16. [Duval et al., A non-partitionable Cohen–Macaulay simplicial complex (talk slides)](https://www.math.utep.edu/faculty/duval/papers/MCA17geometrytalk.pdf)
17. [Duval, Goeckner, Klivans, Martin (2015). A non-partitionable Cohen–Macaulay simplicial complex. arXiv.](https://arxiv.org/abs/1504.04279)

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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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