Cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings (Q723723)

Let \((R,m)\) be commutative Noetherian local ring and \(M\) a finitely generated \(R\)-module. Let \( W= \{ \mathrm{depth} M_P +\mathrm{ht} \frac{I+P}{P} : P \in \mathrm{Spec}(R) - V(I) \}\) and \(\omega (I,M) = \max \{ i: H^i _I (M) \text{ is not weakly Laskerian}\}\). \textit{C. Huneke} [Res. Notes Math. 2, 93--108 (1992; Zbl 0782.13015)] in Problem 3.3 asked that is it true \( 0 \leq n \notin W \) if and only if \( H^n_I (M)\) is finitely generated? Bagheriyeh, Bahmanpour and A'zami [\textit{I. Bagheriyeh} et al., J. Commut. Algebra 6, No. 3, 305--321 (2014; Zbl 1299.13019)] answered a similar result to this problem when \(R\) is a complete local ring and \(I\) is the maximal ideal of \(R\). In this paper, the authors prove that this result is still true if \(R\) is a homomorphic image of a Cohen-Macaulay ring. Also, they show that for catenary ring \(R\) and finitely generated equidimensional \(R\)-module of dimension \(d\) and \(x_1, x_2, \dots , x_t\) an \(I\)-filter regular sequence on \(M\), then \((0 :_{H^{d-j}_I (\frac{M}{(x_1, \dots, x_{i-1}) M})} x_i)\) is \(I\)-cofinite if and only if \( H^{d-j} _I (\frac{M}{(x_1, \dots, x_{i-1}) M})\) is \(I\)-cofinite for all \(i=1, 2, \dots, t\) and all \(i \leq j \leq t\). Furthermore, they prove that \(\omega (I,M) = \max \{i: H^i_I (M)\text{ has no finite support}\}\) and that \(\frac{H^{\omega (I,M)} _I (M)}{I H^{\omega (I,M)}_I (M)}\) has finite support.