# multiplication in spectral sequence

I am trying to understand this paper. Let $$M$$ be a compact Kaehler manifold of dimension $$n$$, $$X$$ is a holomorphic vector field, $$i_X$$ the contraction operator, i.e. for $$\alpha$$ a $$p$$-form, then $$i_X(\alpha) = \alpha(X, \ldots)$$ a $$(p-1)$$-forms. Let $$\mathcal{O}_M$$ be the sheaf of holomorphic functions and $$\Omega^p$$ be the sheaf of holomorphic $$p$$-form. I have a complex

$$0 \rightarrow \Omega^n \xrightarrow{i_X} \Omega^{n-1} \xrightarrow{i_X} \ldots \xrightarrow{i_X} \Omega \xrightarrow{i_X} \mathcal{O}_M \rightarrow 0$$

I then construct the Cartan-Eilenberg resolution $$\mathcal{I}^{\bullet,\bullet}$$, then apply the global section functor to get $$H^0(M, \mathcal{I}^{\bullet,\bullet})$$ and get two spectral sequences from the filtered total complex. Consider the vertical one. It has the first page $$E^{p,q}_1 = H^q(M, \Omega^p)$$, since each column $$\mathcal{I}^{p, \bullet}$$ is an injective resolution of $$\Omega^p$$. The author then claimed that the wedge product $$\Omega^a \times \Omega^b \rightarrow \Omega^{a+b}$$ induces natural product $$E^{a,b}_r \times E^{c,d}_r \rightarrow E^{a+c, b+d}_r$$ and I don't know why.

We do have the wedge product on the first page $$H^{q_1}(M, \Omega^{p_1}) \times H^{q_2}(M, \Omega^{p_2}) \rightarrow H^{q_1+q_2}(M, \Omega^{p_1+p_2})$$ thanks to the isomorphism: $$H^{q}(M, \Omega^p) = H^{p,q}_{\bar{\partial}}(M)$$ but normally to define the multiplication we need the multiplication on the total complex $$T^{\bullet}$$ which respects the filtration i.e. $$F_aT^{b} \times F_cT^{d} \rightarrow F_{a+c}T^{b+d}$$

but the complex consists of $$H^0(M, \mathcal{I}^{\bullet, \bullet})$$ and I don't know how to extend the multiplication on either $${I}^{\bullet, \bullet}$$ or $$H^0(M, \mathcal{I}^{\bullet, \bullet})$$. One attempt is to consider the resolution $$0 \rightarrow \Omega^p \rightarrow \mathcal{I}^{p, \bullet}$$ then we have a complex $$\Omega^p \times \Omega^q \rightarrow \mathcal{J}^{\bullet}$$, where $$\mathcal{J}^n = \bigoplus_{i+j=n} \mathcal{I}^{p,i} \times \mathcal{I}^{q,j}$$. I would like to induce a morphism $$\mathcal{J}^{\bullet} \rightarrow \mathcal{I}^{p+q, \bullet}$$ from $$\Omega^p \times \Omega^q \rightarrow \Omega^{p+q}$$, but there is no guarantee that the complex $$\mathcal{J}^{\bullet}$$ is exact to do that.