An invariant that records multiplicities of copies of indecomposable injective hulls (typically E(R/p) for prime ideals p) appearing in each spot of a minimal injective resolution of a module; often indexed by prime and degree.

Principle

Bass numbers are organized by minimal injective resolutions and Matlis‑type decompositions: the μ^{i}(p,M) Bass number counts multiplicity of the injective hull of R/p appearing in degree i of a minimal injective resolution of M, making them local and degreewise measures of injective complexity.

Demonstration

For a Noetherian local ring (R,m) and a finitely generated module M, one computes a minimal injective resolution and reads off multiplicities of E(R/m) in each degree to obtain the Bass numbers μ^{i}(m,M). In Gorenstein local rings many Bass numbers vanish except in predictable degrees, illustrating structural constraints.

Misapplication

Confusing Bass numbers with Betti numbers (projective vs injective counts), or attempting to read torsion‑free rank information from Bass numbers; another misuse is applying the standard Bass count without the minimality requirement, which changes multiplicities.

Consequence

Bass numbers give fine local information about injective structure, detect properties such as injective dimension behavior and certain duality phenomena, and are key invariants in classification problems for modules and local rings.

Reversal

The dual notion are Betti numbers: where Bass numbers measure multiplicities of injective summands in injective resolutions, Betti numbers measure multiplicities of free/projective summands in projective resolutions; comparing both yields dual perspectives on module complexity.

Boundary

Defined primarily for modules over Noetherian rings where minimal injective resolutions and decompositions into indecomposable injectives exist; outside Noetherian hypotheses or without uniqueness of minimal resolutions the raw Bass counts may not be available or well‑defined.

Semantic Tension

Bass numbers often compete conceptually with Betti numbers and with depth/width invariants: they emphasize injective decomposition data rather than free resolution data, producing tension when both types of invariants are used to classify singularities or module types.

Synthesis

Bass numbers are degreewise, prime‑indexed multiplicities of indecomposable injective hulls occurring in a minimal injective resolution of a module; they package local injective complexity and provide a dual complement to Betti numbers in homological classification.