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.
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Maximal regular sequences help determine the depth of modules, as they correspond to the longest chain of non-zero-divisors within a module.
In a Noetherian ring, every ideal has a maximal regular sequence, which implies that one can always find such sequences for finitely generated modules over these rings.
The concept of maximal regular sequences is essential in understanding Cohen-Macaulay rings, which have certain desirable properties in terms of their depth.
If you have a maximal regular sequence of length $d$, it indicates that the depth of the module or ring is at least $d$.
Maximal regular sequences can also be utilized to study the vanishing of local cohomology modules, providing insights into the structure of the ring or module.
Review Questions
How does a maximal regular sequence relate to the concept of depth in modules?
A maximal regular sequence directly determines the depth of a module since it signifies the longest possible chain of non-zero-divisors that can be formed. If you have a maximal regular sequence of length $d$, then the depth of that module is at least $d$. This relationship helps in assessing how 'deep' or structured a module is within its corresponding ring.
Compare and contrast maximal regular sequences with regular sequences in terms of their properties and implications.
While all elements in a maximal regular sequence are part of a regular sequence, not all regular sequences are maximal. A regular sequence may not reach its potential length due to existing zero-divisors, whereas a maximal regular sequence has no further elements that can extend it without encountering zero-divisors. This distinction is crucial in determining the properties and dimensions associated with modules, especially in rings like Noetherian rings where maximal regular sequences always exist.
Evaluate the significance of maximal regular sequences in understanding Cohen-Macaulay rings and their applications in algebraic geometry.
Maximal regular sequences play a pivotal role in characterizing Cohen-Macaulay rings, which are rings where the depth equals the Krull dimension. The presence of such sequences indicates a well-behaved structure in terms of projective dimension and allows for more manageable computations in algebraic geometry. These sequences help researchers understand singularities and resolutions in varieties, making them essential for deeper explorations into algebraic structures and their geometric interpretations.
Related terms
regular sequence: A regular sequence is a sequence of elements in a ring or module such that each element is not a zero-divisor on the quotient by the ideal generated by the preceding elements.
Depth is a numerical invariant associated with a module that indicates the length of the longest regular sequence of elements that can be formed within it.
zero-divisor: A zero-divisor is an element of a ring that can multiply with another non-zero element to yield zero.