Behaviour of Finiteness of the Set of Associated Primes under Ring Extensions

We study the behaviour of the finiteness of the set of associated primes of local cohomology modules, more generally of Lyubeznik functors, under various ring extensions. At first, we review the results for flat and faithfully flat extensions and we present new applications of them. Then, we focus how the finiteness property of the set of associated primes of local cohomology modules and Lyubeznik functors is transferred from extended ring to the base ring of pure and cyclically pure ring extensions. We show that finiteness property can be transferred from a ring to its pure local subring and this extends the result of Theorem 1.1 of \cite{Nu}. Further, we observed that under mild conditions on the rings, finiteness property comes down from cyclically pure ring extensions to its local base ring. In particular, we observe that the set of associated primes of Lyubeznik functors of a cyclically pure local subring (which turns out to be Cohen-Macaulay) of equicharacteristic or unramified regular local ring, is finite. There is an appendix on behaviour of the Bass numbers under pure and cyclically pure ring extensions.