M-regular sequence

Definition

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.

5 Must Know Facts For Your Next Test

1. An m-regular sequence can be thought of as a way to build up to a maximal ideal, highlighting the relationships between elements in modules and their quotients.
2. If you have an m-regular sequence of length n in a module, it implies that the depth of that module is at least n.
3. Every regular sequence is an m-regular sequence if you are working in a local ring where m is the maximal ideal.
4. If you have a sequence that is not m-regular, it means at least one element acts as a zero divisor on some quotient related to previous elements.
5. The property of being an m-regular sequence can also impact the resolution of modules, particularly when considering projective or injective resolutions.

Review Questions

• How does an m-regular sequence influence the depth of a module?
  • An m-regular sequence directly impacts the depth of a module since the length of this sequence determines how many non-zero divisors exist within the module. Specifically, if you have an m-regular sequence of length n, it indicates that the depth of that module is at least n. This relationship helps to understand how 'deep' or complex the module's structure is concerning its elements and their interactions.
• In what ways does the concept of non-zero divisors relate to identifying an m-regular sequence within a module?
  • The identification of an m-regular sequence relies heavily on understanding non-zero divisors. For a sequence to be considered m-regular, each element must act as a non-zero divisor on the quotient formed by preceding elements. If any element fails this criterion, it disqualifies the sequence from being classified as regular. Thus, examining non-zero divisors provides crucial insight into whether a given sequence fulfills the requirements of being an m-regular sequence.
• Evaluate how understanding m-regular sequences can enhance your overall grasp of module theory and its applications.
  • Understanding m-regular sequences significantly enriches your comprehension of module theory by revealing deeper relationships between elements and ideals within modules. This concept aids in exploring properties like depth and dimensionality in algebraic structures, which can be crucial when applying these principles to more complex problems in algebraic geometry or commutative algebra. Moreover, analyzing these sequences allows for better insights into resolutions and homological dimensions, making them foundational tools for advanced studies and applications within mathematics.

Related terms

Depth: The depth of a module or ring is defined as the length of the longest regular sequence contained in it, providing insight into the structure and properties of the module.

Non-zero divisor: An element in a ring that, when multiplied by any non-zero element of the ring, does not yield zero; it is significant for determining whether sequences are regular.

Maximal ideal: An ideal m in a ring R such that there are no other ideals contained between m and R, playing an important role in localizing rings and studying their properties.