A regular sequence is a sequence of elements in a ring that is both a sequence of non-zero divisors and such that the ideal generated by the first $k$ elements in the sequence has depth $k$ for all $k$. This concept plays a crucial role in understanding the structure of modules over rings, especially in relation to prime ideals and Cohen-Macaulay rings.
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Regular sequences are important for establishing properties of modules, such as Cohen-Macaulayness and projectivity.
In a regular local ring, every system of parameters is a regular sequence, which ties back to the depth and height of prime ideals.
The ideal generated by a regular sequence forms a perfect complex, which means it behaves well in terms of homological algebra.
If a sequence is regular, it can also be used to compute the depth of the module generated by those elements.
Regular sequences have applications in algebraic geometry, particularly in defining smooth varieties and understanding their cohomological properties.
Review Questions
How does a regular sequence relate to the concepts of depth and height in prime ideals?
A regular sequence helps define the depth of an ideal. Specifically, if you have a sequence that generates an ideal, its length gives the depth, which reflects how many non-zero divisors can be included before hitting zero. The relationship between depth and height is crucial because it connects the algebraic properties of prime ideals with geometric interpretations, indicating how many dimensions are 'missing' from the space defined by these ideals.
Discuss the role of Koszul complexes when analyzing regular sequences in commutative algebra.
Koszul complexes arise naturally from regular sequences and are used to study their homological properties. By constructing a Koszul complex from a regular sequence, one can derive important information about the module's structure, such as computing Ext and Tor functors. This connection emphasizes how regular sequences not only influence algebraic properties but also provide tools for deeper investigations into module theory.
Evaluate how identifying regular sequences contributes to determining whether a ring is Cohen-Macaulay.
Identifying regular sequences is essential for determining if a ring is Cohen-Macaulay because such rings have depth equal to their dimension. When a regular sequence can be found that reaches this maximum length corresponding to the ring's dimension, it confirms that the ring satisfies Cohen-Macaulay conditions. This relationship between regular sequences and Cohen-Macaulayness highlights how understanding basic sequences can lead to broader conclusions about the geometric properties of algebraic varieties.
Related terms
Depth: The length of the longest regular sequence contained in an ideal, which provides insight into the dimensions of the ring and its associated modules.
A type of ring that has the same depth as its Krull dimension, indicating a balanced relationship between the ideal structure and the ring's geometric properties.
A complex constructed from a regular sequence, which provides tools for studying homological properties and can be used to compute Ext and Tor functors.