these facts we obtain isomorphisms

$$R\Gamma_Y\underline{{\mathbb{C}}}_X\simeq R\Gamma_Y R{\varepsilon}_*\underline{{\mathbb{C}}}_{X_{{\bullet}}}\simeq R\Gamma_Y R{\varepsilon}_*\Omega_{X_{{\bullet}}}^{{\bullet}}=R\Gamma_Y DR^{{\bullet}}_X$$

that have counterparts for all the $(m-2)$-cubical hyperresolution in $H(X)_{\square}$. Thus we obtain the short exact sequence of objects of $D_+(Sh(X))$

$$\label{eq:SESforX}
0\rightarrow R(g\circ a)_*R\Gamma_{D_Y} DR_{D_X}^{{\bullet}}[-1]\rightarrow\Gamma_Y DR_X^{{\bullet}}\rightarrow Rb_*\Gamma_{\tilde{Y}}DR_{\tilde{X}}^{{\bullet}}\oplus Rg_*\Gamma_{\Sigma_Y}DR_{\Sigma_X}^{{\bullet}}\rightarrow 0$$

which yields the long exact sequence of algebraic de Rham cohomology groups with supports

$$\label{eq:LECSX}
\dots\rightarrow H_{DR,\Sigma_Y}^{{\bullet}}(\Sigma_X)\oplus H_{DR,\tilde{Y}}^{{\bullet}}(\tilde{X})\rightarrow H_{DR,D_Y}^{{\bullet}}(D_X)\rightarrow H_{DR,Y}^{{\bullet}+1}(X)\rightarrow\cdots.$$

Consider now the $m_1$-cubical hyperresolutions ${D_Y}_{{\square}}$ of $D_Y$ and ${D_X}_{{\square}}$ of $D_X$ provided by hypothesis (ii); we want to use the following result:

\[lem:tracemaps\] [@H1 Lemma II.3.1] Let ${\mathbb{K}}$ be a field of characteristic zero. Let $f:X\rightarrow Y$ be either a smooth morphism or a closed immersion of smooth schemes of finite type over ${\mathbb{K}}$. Let $Z$ be a closed subscheme of $X$ such that the induced map $f:Z\rightarrow Y$ is a closed immersion. Then the trace map gives an isomorphism of complexes in $D_+(Sh(Y))$

$$\text{Tr}_f:f_*R\Gamma_Z\Omega_X^{{\bullet}}[2n]\rightarrow R\Gamma_Z\Omega_Y^{{\bullet}}$$

where $n=\dim(X)-\dim(Y)$ and $\Gamma_Z$ denotes the restriction of sections functor.

Fix any $I\in{\square}_{m_1}^*$ and consider any irreducible component $V$ of ${D_X}_I$; two things can happen:

(I) To $V$ there corresponds a unique irreducible component $U$ of ${D_Y}_I$ that admits a closed immersion $i:U\hookrightarrow V$ of codimension $1$.

(II) To $V$ there corresponds 