Bass number

In mathematics, the ith Bass number of a module M over a local ring R with residue field k is the k-dimension of Extⁱ
_R(k,M). More generally the Bass number μ_i(p,M) of a module M over a ring R at a prime ideal p is the Bass number of the localization of M for the localization of R (with respect to the prime p). Bass numbers were introduced by Hyman Bass (1963, p.11).

The Bass numbers describe the minimal injective resolution of a finitely-generated module M over a Noetherian ring: for each prime ideal p there is a corresponding indecomposable injective module, and the number of times this occurs in the ith term of a minimal resolution of M is the Bass number μ_i(p,M).

References

• Bass, Hyman (1963), "On the ubiquity of Gorenstein rings", Mathematische Zeitschrift, 82: 8–28, doi:10.1007/BF01112819, ISSN 0025-5874, MR 0153708
• http://arxiv.org/abs/math/0010003
• Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956