2015/07/30 by Tony Se, Se, Tony
Mathematics · #13A02 #13A15 #13A30 #13E05 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13A02 #msc:13A15 #msc:13A30 #msc:13E05
paper · pdf · doi:10.48550/arxiv.1507.08389
12 pages
arxiv created 2015/07/30 · arxiv updated 2015/07/31
Let R be a commutative Noetherian ring, I and J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then the sets of associated primes of F(M/In M) and F(In-1 M / In M) and the J-depths of F(M/In M) and F(In-1 M / In M) become independent of n for large n. Next, we consider several examples in which F is a rather familiar functor, but is not coherent or not even finitely generated in general. In these cases, the set of associated primes of F(M/In M) still becomes independent of n for large n. We then show one negative result where F is not finitely generated. Finally, we give a positive result where F belongs to a special class of functors which are not finitely generated in general, an example of which is the zeroth local cohomology functor.