11.3 Homological dimensions

Homological dimensions 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. Global dimension measures a ring's homological complexity, while Krull dimension relates to its algebraic structure.

Projective and Injective Dimensions

Measuring Resolutions with Projective and Injective Dimensions

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• Projective dimension measures the length of the shortest projective resolution of a module
  • Defined as the smallest integer $n$ such that there exists a projective resolution of length $n$
  • If no finite projective resolution exists, the projective dimension is infinite
  • Projective dimension of $0$ means the module is projective itself
• Injective dimension measures the length of the shortest injective resolution of a module
  • Defined as the smallest integer $n$ such that there exists an injective resolution of length $n$
  • If no finite injective resolution exists, the injective dimension is infinite
  • Injective dimension of $0$ 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 $M$ over a local ring $R$ to the depth of $M$
  • Formula states: $\text{projective dimension}(M) + \text{depth}(M) = \text{depth}(R)$
  • 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 $0$ are called semisimple rings (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
  • Hereditary rings are rings with global dimension at most $1$ (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

• Flat dimension is another invariant that measures the length of the shortest flat resolution of a module
  • Defined as the smallest integer $n$ such that there exists a flat resolution of length $n$
  • If no finite flat resolution exists, the flat dimension is infinite
  • Flat dimension of $0$ 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

• Bass numbers 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 dimension of a module (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)