1

I'm still very new to homological algebra.

I would like to know what are the groups cohomology derived from the functor $Hom_R(\_, D)$ of the $R$-module $A$ (i.e. compute $Ext^n(A,D)$ ),

-in the particular case where $R$ is a principal ideal domain and $A$ is a finitely generated module

-in the case where $R$ is noetherian ring and $A$ is finitely generated

I think I must specify that the way I construct the cohomology groups is using a projective resolution : $$...\to P_n \to P_{n-1} \to ... \to P_0 \to A \to 0 $$ then applying functor $Hom_R(\_, D)$ to the sequence and taking the cohomology group of the new sequence.

Feel free to add some hypothesis if you need (as I am not sure if what I said makes perfect sense), or even come up with other particular cases where it is relatively easy to compute those cohomology groups.

KiwiKiwi
  • 159
  • When R is a PID you can use the structure theorem that says any of module over a PID is torsion + free. So the calculation essentially boils down to torsion modules which have a length one resolution $Z \rightarrow Z$ . In the Noetherian case calculating the resolution in specific cases isn't difficult (Koszul complex) but you will need nontrivial theorem to get finite resolution. – DBS May 30 '16 at 08:09
  • Of course it also depends on what D is and sometimes you can use its injective resolution instead. – DBS May 30 '16 at 08:10

0 Answers0