On the Associated Primes of Matlis Duals of Top Local Cohomology Modules

After motivating the question, we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases, we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.     

Numerical semigroups generated by primes

Semigroup Forum - Let $$p_1=2, p_2=3, p_3=5, \ldots$$ be the consecutive prime numbers, $$S_n$$ the numerical semigroup generated by the primes not less than $$p_n$$ and $$u_n$$ the largest...     

Correction to: Numerical semigroups generated by primes

Semigroup Forum - A correction to this paper has been published: https://doi.org/10.1007/s00233-021-10198-7     

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of , where is a -dimensional local ring and an ideal such that and .

     

On endomorphism rings of local cohomology modules

Let be a local complete ring. For an -module the canonical ring map is in general neither injective nor surjective; we show that it is bijective for every local cohomology module if for every ( an ideal of ); furthermore the same holds for the Matlis dual of such a module. As an application we prove new criteria for an ideal to be a set-theoretic complete intersection.

     

Matlis Duals of Top Local Cohomology Modules and the Arithmetic Rank of an Ideal

Let I be an ideal of a local ring (R, 𝔪). Using local cohomology, we present new criteria (see 1.4, respectively 1.5) for the conditions ara (I) ≤ 1 respectively ara (I) ≤ 2, where ara (I) stands for the number of generators of I up to radical. Though this works equally well for the local and for the graded case, we show some subtle differences between the local and the graded situation in Section 2. Finally, in Section 3, we show that the Matlis dual of certain local cohomology modules, though not finite, is well behaved in some sense.     

Finiteness Properties of Duals of Local Cohomology Modules

We investigate Matlis duals of local cohomology modules and prove that, in general, their zeroth Bass number with respect to the zero ideal is not finite. We also prove that, somewhat surprisingly, if we apply local cohomology again (i.e., to the Matlis dual of the local cohomology module), we get (under certain hypotheses) either zero or E, an R-injective hull of the residue field of the local ring R.     

On the Associated Primes of Matlis Duals of Local Cohomology Modules II

In continuation of [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zöschinger on coassociated primes of arbitrary modules with results from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] 4-6 Hellus , M. , Stückrad , J. ( 2008 ). On endomorphism rings of local cohomology modules . Proceedings of the American Mathematical Society 136 : 2333 – 2341 . Hellus , M. , Stückrad , J. ( 2008 ). Matlis duals of top local cohomology modules . Proceedings of the American Mathematical Society 136 : 489 – 498 . Hellus , M. , Stückrad , J. ( 2009 ). Artinianness of local cohomology . Journal of Commutative Algebra 1 : 269 – 274 . ], and obtain partial answers to questions which were left open in [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. These partial answers give further support for conjecture (*) from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zöschinger, we prove some non-finiteness results of local cohomology.