We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\xrightarrow[]{d_{n-3}} V_{n-2}\xrightarrow []{d_{n-2}}V_{n-1}\xrightarrow[]{d_{n-1}} V_{n}\rightarrow 0$$ Furthermore $V=\bigoplus_{j=1}^{3}X_{j}$ and $V_{n}\cap X_{3}=\{0\}.$ The differentials have following properties:
1) the differentials $d_{i},\,\, 0\leq i\leq n-1$ acts on bases elements non-trivially.
2) $d_{i}(X_{j}\cap V_{i})\subseteq X_{j}\cap V_{i+1}\quad\mbox{for}\,\, j=1,2. $
The action of differentials on the bases elements of $X_{3}$ in the following way:
$d_{n-1}(x_{3})=\alpha_{1}x_{1}+\alpha_{2}x_{2}$ where $\alpha_{1},\alpha_{2}\neq0$ and $x_{j}\in Bases(X_{j}).$
$d_{i<n-1}(x_{3})=\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}$ where $\alpha_{1},\alpha_{2},\alpha_{3}\neq0$ and $x_{j}\in Bases(X_{j}).$
From property 2) we get subcochain complex: $$0\rightarrow V_{0}\cap X_{1}\xrightarrow[]{d_{0}} V_{1}\cap X_{1}\xrightarrow[]{d_{1}}\ldots\xrightarrow[]{d_{n-3}} V_{n-2}\cap X_{1}\xrightarrow []{d_{n-2}}V_{n-1}\cap X_{1}\xrightarrow[]{d_{n-1}} V_{n}\cap X_{1}\rightarrow 0.$$ Question 1) is it true $dim(Img(d_{i-1})\cap Ker(d_{i}\mid_{X_{1}}))=dim(Img(d_{i-1}\mid_{X_{1}}))?$
Question 2) is it true $dim(H^{i}(V);\mathbb{Q})\geq dim(H^{i}(V\cap X_{1});\mathbb{Q})?$
