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 [math]\displaystyle{ \operatorname{Ext}^i_R(k,M) }[/math]. More generally the Bass number [math]\displaystyle{ \mu_i(p,M) }[/math] 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 [math]\displaystyle{ \mu_i(p,M) }[/math]
References
- Bass, Hyman (1963), "On the ubiquity of Gorenstein rings", Mathematische Zeitschrift 82: 8–28, doi:10.1007/BF01112819, ISSN 0025-5874
- Helm, David; Miller, Ezra (2003), "Bass numbers of semigroup-graded local cohomology", Pacific Journal of Mathematics 209 (1): 41–66, doi:10.2140/pjm.2003.209.41
- Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, ISBN 978-0-521-41068-7, https://books.google.com/books?id=LF6CbQk9uScC
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