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Associated primes of the local cohomology modules. (English) Zbl 1124.13009
Goeters, Pat (ed.) et al., Abelian groups, rings, modules, and homological algebra. Selected papers of a conference on the occasion of Edgar Earle Enochs’ 72nd birthday, Auburn, AL, USA, September 9–11, 2004. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-552-1/hbk). Lecture Notes in Pure and Applied Mathematics 249, 51-58 (2006).
This is a short survey on the question whether local cohomology modules of modules over a noetherian ring have only finitely many associated primes. It also contains a few new results. Section 6.2 deals with the question whether all local cohomology modules of all finitely generated modules have only finitely many associated primes. There are rings for which this is not true but the answer is yes over regular local rings of positive characteristic. Section 6.3 concerns a fixed local cohomology module of a fixed module. Many results are stated of the following type: If certain modules are finitely generated and the local cohomology modules of dimension less than \(t\) have only finitely many associated primes and so does the local cohomology module of dimension \(t\). An interesting application is Theorem 6.3.9 claiming that if \(\text{Ext}^{t-j}_R(R/a, H^j_a(M))\) is finitely generated for all \(j < s\) and \(t = s, s+1\) then \(\text{Ext}^s_R(R/a, M)\) is finitely generated if and only if so is \(\text{Hom}_R(R/a, H^s_a(M))\). Section 6.4 generalizes many results from Section 6.3 to generalized local cohomology modules.
For the entire collection see [Zbl 1085.20500].

13D45 Local cohomology and commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13E05 Commutative Noetherian rings and modules