11.3 Homological dimensions
4 min read•august 7, 2024
measure the complexity of modules and rings using resolutions. They connect algebraic properties to homological ones, providing insights into module and ring structures.
Projective, injective, and flat dimensions quantify how close modules are to having these properties. measures a ring's homological complexity, while relates to its algebraic structure.
Projective and Injective Dimensions
Measuring Resolutions with Projective and Injective Dimensions
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- measures the length of the shortest of a module
- Defined as the smallest integer such that there exists a projective resolution of length
- If no finite projective resolution exists, the projective dimension is infinite
- Projective dimension of means the module is projective itself
- measures the length of the shortest of a module
- Defined as the smallest integer such that there exists an injective resolution of length
- If no finite injective resolution exists, the injective dimension is infinite
- Injective dimension of means the module is injective itself
- Projective and injective dimensions provide a way to quantify how far a module is from being projective or injective respectively
- Higher dimensions indicate a module is further from having the respective property
- Dimensions give insight into the structure and complexity of a module
Relating Projective and Injective Dimensions
- Auslander-Buchsbaum formula relates the projective dimension of a finitely generated module over a local ring to the depth of
- Formula states:
- Connects the homological notion of projective dimension to the algebraic notion of depth
- Allows for the computation of one invariant from the other
- Highlights the interplay between homological and algebraic properties of modules and rings
Global and Flat Dimensions
Measuring Rings with Global Dimension
- Global dimension is an invariant of a ring that measures the maximum projective dimension of its modules
- Defined as the supremum of the projective dimensions of all modules over the ring
- If the global dimension is finite, it is the maximum projective dimension attained by any module
- Rings with global dimension are called (all modules are projective)
- Rings with finite global dimension are called regular rings
- Global dimension provides a measure of the homological complexity of a ring
- Lower global dimension indicates a simpler homological structure
- are rings with global dimension at most (every submodule of a projective module is projective)
- Global dimension can be used to classify and study rings based on their homological properties
Flat Dimension and its Properties
- is another invariant that measures the length of the shortest flat resolution of a module
- Defined as the smallest integer such that there exists a flat resolution of length
- If no finite flat resolution exists, the flat dimension is infinite
- Flat dimension of means the module is flat itself
- Flat modules are modules that preserve injectivity of maps when tensored
- Flat resolutions are resolutions by flat modules
- Flat dimension measures how far a module is from being flat
- Flat dimension is related to the projective and injective dimensions
- Projective modules are flat, so projective dimension is always less than or equal to flat dimension
- If a ring has finite global dimension, then the flat dimension of any module is also finite
Dimension Invariants
Bass Numbers and their Applications
- are a sequence of invariants associated to a module over a local ring
- Defined as the ranks of the free modules in a minimal injective resolution of the module
- Provide a measure of the complexity of the injective resolution
- Can be used to study the structure of modules and local rings
- Bass numbers have connections to other invariants and properties
- The zeroth Bass number is the minimal number of generators of the module
- The first Bass number is related to the indecomposability of the injective hull of the module
- Higher Bass numbers are linked to the structure of the syzygies in the minimal injective resolution
- Applications of Bass numbers include:
- Computing the injective (it is the index of the last nonzero Bass number)
- Studying the local cohomology modules of a ring
- Investigating the structure of artinian modules over a complete local ring
Krull Dimension and its Relationship with other Invariants
- Krull dimension is a notion of dimension for commutative rings and their modules
- Defined as the supremum of lengths of chains of prime ideals in the ring
- Measures the "size" or "complexity" of the ring
- For a module, it is defined as the Krull dimension of its support (the set of prime ideals containing its annihilator)
- Krull dimension is related to other dimension invariants
- For a local ring, the Krull dimension is always less than or equal to the global dimension
- If a local ring is Cohen-Macaulay, then its Krull dimension equals its depth
- For finitely generated modules over a local ring, the Auslander-Buchsbaum formula relates Krull dimension, projective dimension, and depth
- Krull dimension is a fundamental invariant in commutative algebra
- Used to classify and study commutative rings and their modules
- Plays a role in many important results and theorems (e.g., Krull's Principal Ideal Theorem, dimension theory)
Key Terms to Review (20)
Auslander-Buchsbaum Theorem: The Auslander-Buchsbaum Theorem states that for a finitely generated module over a Noetherian ring, the projective dimension of the module plus its depth equals the Krull dimension of the ring. This theorem provides a deep relationship between the homological properties of modules and their geometric aspects, linking algebraic and topological concepts.
Bass numbers: Bass numbers are a set of invariants associated with a module that provide insight into its homological dimensions, particularly in relation to projective and injective resolutions. They capture the number of generators required to construct certain projective modules and reflect the complexity of a module's structure. Bass numbers help in determining various homological properties and are fundamental when studying extensions and torsion theories within algebra.
Cohomological Dimension: Cohomological dimension is a measure of the 'size' of the cohomology groups of a module or a sheaf, indicating the highest degree in which non-trivial cohomology occurs. This concept helps in understanding how different algebraic structures behave with respect to cohomological techniques, serving as a critical tool in both algebraic geometry and group theory, as well as in analyzing the behavior of modules over rings.
David Auslander: David Auslander is a significant figure in the field of homological algebra, particularly known for his contributions to the understanding of homological dimensions and their implications in various mathematical contexts. His work has focused on defining and exploring concepts such as projective and injective dimensions, which are critical for understanding the structure of modules over rings. Auslander's insights have paved the way for further developments in both algebra and geometry, linking these areas with homological methods.
Derived Functors: Derived functors are a way to extend the concept of functors in category theory to measure how much a given functor fails to be exact. They provide a systematic way to derive additional information from a functor by analyzing its relationship with exact sequences and chain complexes. Derived functors are particularly useful in homological algebra as they connect various algebraic structures, allowing us to study properties like the existence of certain modules and their relationships.
Dimension Formula: The dimension formula is an essential concept in homological algebra that relates the projective dimensions, injective dimensions, and flat dimensions of modules. It provides a way to understand how these dimensions interact and can be computed for various algebraic structures. This formula highlights the relationships between different homological dimensions, allowing mathematicians to analyze the properties of modules more effectively.
Dimension of a module: The dimension of a module is a measure of its size and complexity, specifically in terms of the number of generators needed to express the module. It can indicate how 'large' or 'complicated' a module is, often reflecting the relationships it has with other modules and structures within a mathematical framework. Understanding the dimension helps to classify modules into different types, which can be essential for analyzing their properties and behavior under various operations.
Embedding Dimension: Embedding dimension refers to the smallest integer 'n' such that a given algebraic object can be realized as a subset of a Euclidean space of dimension 'n'. It provides insight into the geometric properties of algebraic varieties, particularly in relation to their homological dimensions. The concept is essential for understanding how algebraic structures can be represented and manipulated within different dimensional contexts.
Ext and tor functors: Ext and Tor functors are fundamental tools in homological algebra used to study the properties of modules over a ring. Ext measures the extent to which a module fails to be projective, while Tor quantifies how far a module is from being flat. Together, they provide critical insights into the relationships between modules, their resolutions, and various homological dimensions.
Flat Dimension: Flat dimension refers to a homological invariant that measures the 'flatness' of a module over a ring, specifically indicating the smallest number of flat modules needed to resolve that module. Understanding flat dimension is crucial as it relates to various properties of modules, such as projective dimension and injective dimension, providing insight into their structure and behavior in homological algebra.
Global dimension: Global dimension refers to the highest level of homological dimension for a given module or ring, indicating the complexity of its projective resolutions. It measures how far a module can be from being projective and is crucial for understanding the relationships between modules in terms of their homological properties. This concept provides insight into the structure of modules over rings and helps categorize them based on their projective and injective characteristics.
Hereditary Rings: Hereditary rings are a special class of rings where every left (or right) module is projective, which means they can be represented as direct summands of free modules. This property significantly influences the homological dimensions of modules over these rings, as the structure allows for the simplification of projective resolutions and the calculation of various homological invariants. The concept is crucial in understanding how these rings relate to more general algebraic structures and their implications in homological algebra.
Homological dimensions: Homological dimensions refer to a set of numerical invariants that provide information about the complexity of modules and their relationships to projective, injective, or flat resolutions. These dimensions, such as projective dimension, injective dimension, and flat dimension, help us understand how far a given module is from being projective or injective. They play a crucial role in determining properties of functors, particularly when discussing the Tor functor and its applications.
Injective Dimension: Injective dimension is a measure of how far a module is from being injective, specifically defined as the shortest length of an injective resolution of that module. It plays a vital role in homological algebra, connecting the notions of injective modules, projective resolutions, and derived functors, while also influencing computations involving Tor and Ext.
Injective resolution: An injective resolution is a type of exact sequence of injective modules that allows one to represent a module as an extension by injective modules. This concept is crucial for understanding how injective modules can be used to study other modules and their homological properties. The construction of injective resolutions provides a way to compute derived functors, including Ext, and plays an important role in various homological contexts, such as sheaf cohomology and the determination of homological dimensions.
Krull Dimension: Krull dimension is a concept in commutative algebra that measures the 'height' of a ring in terms of its prime ideals. It is defined as the supremum of the lengths of all chains of prime ideals within the ring. This dimension provides insight into the structure of rings and their modules, linking to important homological dimensions that reveal properties like projective and injective dimensions.
Lars Østerlund: Lars Østerlund is a mathematician known for his contributions to the field of homological algebra, particularly in the study of homological dimensions. His work has influenced various aspects of category theory and module theory, helping to understand the properties and relationships between different algebraic structures.
Projective Dimension: Projective dimension is a homological invariant that measures the complexity of a module by determining the length of the shortest projective resolution of that module. It connects deeply with various concepts in homological algebra, such as resolutions, derived functors, and cohomology theories, showcasing how modules relate to projective modules and their role in understanding other homological properties.
Projective Resolution: A projective resolution is a specific type of exact sequence that helps to approximate modules using projective modules. It consists of a chain of projective modules connected by homomorphisms that leads to a given module, allowing the study of properties like homological dimensions and derived functors, which are essential in understanding the structure and classification of modules.
Semisimple Rings: Semisimple rings are rings that can be expressed as a direct sum of simple modules. These rings exhibit a structure that allows them to be analyzed using the theory of modules and homological algebra, which is crucial for understanding their properties. One of the most notable features of semisimple rings is that every module over such a ring is semisimple, meaning it can be decomposed into a direct sum of simple modules.