Let $(A,\fm)$ be a commutative quasi-local ring with non-zero identity with infinite residue field and let $I$ be an ideal of $A$. Let $M$ be an Artinian $A$-module and $G(I,M)$ be a dual of associated graded module and we denote by $s(I,M)$ the analytic spread of $I$ with respect to $M$. The dual of Burch's inequality says that $s(I,M)+\inf\{\width(0:_MI^n): n\geq 1\}\leq\dim M$, and it is well known that equality holds if $G(I,M)$ is co-Cohen-Macaulay. Thus, in that case one can compute the width of dual of the associated graded module $I$ as $\width G(I,M)=s(I,M)+\inf\{\width(0:_MI^n): n\geq 1\}$. We study when such an equality is also valid when $G(I,M)$ is not necessarily co-Cohen-Macaulay.
