I'm studying this paper "Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary".
At the bottom of page 4, it says "$CcC^p_N$ is the orthogonal complement of the exact $p$-forms within the closed ones, so $CcC^p_N \cong H^p(M;\mathbb{R})$". This is straightforward by the definition of cohomology.
Then it says "$CcC^p_D$ is the orthogonal complement of the co-exact $p$-forms within the co-closed ones, so $CcC^p_D \cong H^p(M, \partial M;\mathbb{R})$". My understanding is that the co-closed $p$-forms modulo the co-exact $p$-forms gives $H^p(M;\mathbb{R})$, which is isomorphic to $H^p(M, \partial M;\mathbb{R})$ by the Poincaré–Lefschetz duality, so we have $CcC^p_D \cong H^p(M, \partial M;\mathbb{R})$.
Is my understanding correct? If so, why not just saying $CcC^p_D \cong H^p(M;\mathbb{R})$? In other words, what's the point of bringing up the relative cohomology $H^p(M, \partial M;\mathbb{R})$?