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 ${\displaystyle \operatorname {Ext} _{R}^{i}(k,M)}$. More generally the Bass number ${\displaystyle \mu _{i}(p,M)}$ 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 ${\displaystyle \mu _{i}(p,M)}$

References

• Bass, Hyman (1963), "On the ubiquity of Gorenstein rings", Mathematische Zeitschrift, 82: 8–28, CiteSeerX 10.1.1.152.1137, doi:10.1007/BF01112819, ISSN 0025-5874, MR 0153708, S2CID 10739225
• Helm, David; Miller, Ezra (2003), "Bass numbers of semigroup-graded local cohomology", Pacific Journal of Mathematics, 209 (1): 41–66, arXiv:math/0010003, doi:10.2140/pjm.2003.209.41, MR 1973933, S2CID 9114225
• 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