Glossary

12.2 Depth of modules and rings

2 min readโ€ขjuly 25, 2024

Depth in commutative algebra measures how "nice" a module is over a local ring. It's calculated using regular sequences or free resolutions, and is closely tied to the module's structure and properties.

Depth relates to a module's dimension, with the setting an upper bound. Cohen-Macaulay modules are special cases where depth equals dimension, showcasing ideal behavior in algebraic geometry and ring theory.

Depth Fundamentals

Definition of module depth

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  • M over local ring (R, m) measures number of elements in maximal within m
  • M-regular sequence consists of elements x1,...,xnx_1, ..., x_n in m where each xix_i acts as on M/(x1,...,xiโˆ’1)MM/(x_1, ..., x_{i-1})M
  • Denoted [depthR(M)](https://www.fiveableKeyTerm:depthr(m))[depth_R(M)](https://www.fiveableKeyTerm:depth_r(m)) or depth(M)depth(M) when R is clear from context (polynomial rings, power series rings)

Depth computation via free resolution

  • of M forms exact sequence of free R-modules: ...โ†’F2โ†’F1โ†’F0โ†’Mโ†’0... \to F_2 \to F_1 \to F_0 \to M \to 0
  • Depth formula expresses depth(M)=min{iโˆฃExtRi(R/m,M)โ‰ 0}depth(M) = min\{i | Ext^i_R(R/m, M) \neq 0\}
  • Computation involves:
    1. Constructing minimal free resolution of M
    2. Calculating ExtRi(R/m,M)Ext^i_R(R/m, M) for increasing i
    3. Identifying smallest i where ExtRi(R/m,M)Ext^i_R(R/m, M) is non-zero

Depth vs maximal regular sequence

  • represents longest sequence of elements in m forming M-regular sequence
  • establishes depth of M equals length of any maximal M-regular sequence
  • Maximal regular sequences exhibit:
    • Uniform length regardless of chosen sequence
    • Length independence from specific sequence elements

Depth and Dimension Relationships

Depth-dimension relationship in modules

  • (R, m) defined as measuring length of longest chain of prime ideals
  • Depth-dimension inequality states depth(M)โ‰คdim(R)depth(M) \leq dim(R) for finitely generated R-modules M
  • Cohen-Macaulay modules satisfy depth(M)=dim(R)depth(M) = dim(R) (complete intersections, )
  • have associated primes of same dimension (integral domains, reduced rings)
  • Depth exhibits sensitivity to quotients potentially decreasing while dimension remains unchanged under quotients by regular elements

Key Terms to Review (15)

Cohen-Macaulay Module: A Cohen-Macaulay module is a type of module over a ring that satisfies certain depth conditions, specifically that the depth of the module equals the Krull dimension of the ring. This property connects the structure of the module to the underlying ring, showing that modules with this property can behave well in terms of their homological properties and dimensions, which is crucial for understanding both depth and the characteristics of Cohen-Macaulay rings.
Cohen-Macaulay Rings: Cohen-Macaulay rings are a special class of rings where the depth of every ideal equals the height of that ideal. This property is significant because it implies that the ring has a well-behaved structure, particularly in terms of its associated primes and the relationships between various prime ideals. Cohen-Macaulay rings often arise in algebraic geometry and commutative algebra, providing a bridge between algebraic properties and geometric intuition.
Depth of module: The depth of a module is a fundamental concept in commutative algebra that measures the 'size' or 'richness' of a module in terms of the length of its longest chain of prime ideals. It provides important information about the structure and properties of both the module and the ring over which it is defined, revealing insights into aspects like regularity, dimension, and homological characteristics.
Depth_r(m): The term depth_r(m) refers to the depth of a module m over a ring r, which measures the length of the longest regular sequence contained in the ideal associated with the module. This concept is crucial in understanding the structure of modules and rings, particularly in determining their homological properties and dimensions. Depth gives insight into how far we can go into the module using elements from the ring without hitting zero, connecting it closely with the concepts of dimension and regular sequences.
Depth-dimension inequality: The depth-dimension inequality is a fundamental result in commutative algebra that relates the depth of a module to its dimension. Specifically, for a Noetherian local ring, the depth of a module is always less than or equal to its dimension, establishing a crucial connection between these two important properties of modules over rings.
Depth-Regular Sequence Theorem: The Depth-Regular Sequence Theorem states that if a sequence of elements in a Noetherian ring forms a regular sequence, then the depth of the quotient of the ring by the ideal generated by these elements equals the depth of the original ring minus the length of the sequence. This theorem connects the concept of regular sequences with the depth of modules, highlighting how certain sequences impact the structure of a module or ring and its properties.
Dimension of Local Ring: The dimension of a local ring is a fundamental concept in commutative algebra that represents the length of the longest chain of prime ideals contained within that ring. This dimension gives insight into the 'size' or 'complexity' of the ring's structure, as it relates to properties like depth, height, and the behavior of modules over that ring.
Ext Functor: The Ext functor is a tool in homological algebra that measures the extent to which a module fails to be projective, essentially quantifying extensions of modules. It can be thought of as a way to study the relationships between modules by capturing how they can be extended by other modules, providing insights into their structure. Specifically, when examining the depth of modules and rings, the Ext functor reveals crucial information about the syzygies and resolutions of modules, shedding light on their homological properties.
Gorenstein Rings: Gorenstein rings are a special class of commutative rings that have nice duality properties, particularly in the context of homological algebra. These rings exhibit both Cohen-Macaulay properties and a finite injective dimension, making them significant in understanding depth and dimensions of modules. Gorenstein rings can be seen as a generalization of regular local rings, and their study plays an essential role in the characterization of Cohen-Macaulay rings and their depth.
Krull dimension: Krull dimension is a fundamental concept in commutative algebra that measures the 'size' of a ring by considering the maximum length of chains of prime ideals. This dimension helps to understand the structure of rings and their prime ideals, which connects to various important properties and theorems in algebraic geometry and ring theory.
M-regular sequence: An m-regular sequence is a sequence of elements in a module over a ring that maintains a certain homological property with respect to the maximal ideal m. Specifically, a sequence is considered m-regular if each element of the sequence is a non-zero divisor on the quotient of the module by the ideal generated by the previous elements in the sequence. This concept is crucial for understanding depth, which reflects the number of non-zero divisors that can be found in the module.
Maximal regular sequence: A maximal regular sequence is a sequence of elements in a ring or module that is both regular (meaning that each element in the sequence is not a zero-divisor on the quotient by the ideal generated by the previous elements) and maximal with respect to inclusion. This concept connects deeply with the notions of depth and homological dimensions, reflecting how many times we can extract a non-zero-divisor from a given module before reaching a state where further extraction fails.
Minimal Free Resolution: A minimal free resolution is a specific type of projective resolution of a module that is obtained by applying the concept of minimality to the sequence of projective modules. It represents the 'simplest' way to express a given module as a combination of free modules, where no unnecessary elements are present, making it particularly useful for understanding the structure and properties of the module. This concept connects deeply with the notions of depth and homological algebra, highlighting how a module can be broken down into simpler components while retaining essential characteristics like depth.
Non-zero-divisor: A non-zero-divisor is an element in a ring that, when multiplied by a non-zero element of the ring, does not result in zero. This concept is essential in understanding the structure of rings and modules, particularly in determining properties such as depth and projectivity. Non-zero-divisors help identify elements that maintain the integrity of multiplication within a ring, which connects deeply to the study of modules over rings and their dimensions.
Unmixed Modules: Unmixed modules are modules over a ring whose associated primes all have the same height, typically indicating that they are related to the depth of the module. This concept is crucial as it helps in understanding the structure and properties of modules, particularly in relation to Cohen-Macaulay rings and their depth. Unmixed modules often arise in situations where the depth reflects a certain regularity or uniformity among the prime ideals associated with the module.
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