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 $\operatorname{Ext}^i_R(k,M)$ . More generally the Bass number $\mu_i(p,M)$ 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 {{harvs|txt|authorlink= Hyman Bass|last=Bass|first=Hyman|year=1963|loc=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 $\mu_i(p,M)$ .

References

{{Citation | last=Bass | first=Hyman |authorlink= Hyman Bass| title=On the ubiquity of Gorenstein rings | doi=10.1007/BF01112819 |mr=0153708 | year=1963 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=82 | pages=8–28| citeseerx=10.1.1.152.1137 | s2cid=10739225 }}
{{Citation| last1=Helm| first1= David|

last2=Miller| first2= Ezra|

title=Bass numbers of semigroup-graded local cohomology|

journal=Pacific Journal of Mathematics |

volume=209|

date=2003|

issue=1|

pages=41–66|

mr=1973933|

doi=10.2140/pjm.2003.209.41|

arxiv=math/0010003| s2cid= 9114225}}

{{Citation | last1=Bruns | first1=Winfried | last2=Herzog | first2=Jürgen | title=Cohen-Macaulay rings | url=https://books.google.com/books?id=LF6CbQk9uScC | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-41068-7 |mr=1251956 | year=1993 | volume=39}}

Category:Commutative algebra

Category:Homological algebra