Depth_r(m)

Definition

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.

5 Must Know Facts For Your Next Test

1. depth_r(m) is always less than or equal to the dimension of the support of m, providing a limit on how deep we can go within the structure of the module.
2. If m is finitely generated over a Noetherian ring r, then depth_r(m) can also be interpreted as an invariant that reflects the local behavior of m at prime ideals.
3. For a module m, if depth_r(m) = 0, this indicates that there are no non-zero elements of r that form a regular sequence in m.
4. The concept of depth_r(m) plays a significant role in various depth conditions like Cohen-Macaulayness, where modules have depth equal to their Krull dimension.
5. Computing depth_r(m) often involves examining minimal prime ideals and using techniques from local cohomology to understand how they interact with the module.

Review Questions

• How does depth_r(m) relate to regular sequences and what implications does this have for understanding the structure of modules?
  • depth_r(m) is intrinsically linked to regular sequences since it measures how many elements from a ring can be used in succession without causing multiplication by zero in the module. A longer regular sequence indicates a deeper structure and more robust interactions between the module and its associated ring. Understanding this relationship helps us analyze whether modules have desirable properties like being Cohen-Macaulay or simply connected.
• In what ways can computing depth_r(m) provide insights into the homological properties of modules over Noetherian rings?
  • Computing depth_r(m) reveals critical information about a module's homological properties, especially in Noetherian rings where finite generation plays a crucial role. The depth can indicate whether certain resolutions exist, guide towards finding projective or injective dimensions, and help understand relationships between modules through Ext and Tor functors. It also aids in determining when two modules are equivalent or when they exhibit similar structural behaviors.
• Evaluate how changes in depth_r(m) affect other invariants associated with modules, especially in light of homological algebra principles.
  • Changes in depth_r(m) directly influence other invariants such as dimension, projective dimension, and homological dimension. For instance, an increase in depth might suggest that the module has more intricate structure or relationships with prime ideals, potentially leading to higher projective dimensions. Conversely, decreasing depth could indicate simpler interactions within the module or potential collapses in its resolution structures. Analyzing these shifts provides deeper insights into how modules behave under various algebraic operations and transformations.

Related terms

Regular Sequence: A sequence of elements in a ring such that each element is not a zero-divisor on the quotient by the ideal generated by the preceding elements.

Depth: The minimum number of generators of an ideal that defines a prime ideal in a given ring, reflecting how far we can go before reaching a certain point.

Homological Dimension: A measure of the complexity of modules over a ring, often related to projective, injective, or flat resolutions.