Dolbeault cohomology of a domain

By [F2], every image $\bar{\mathrm{\partial }}\gamma$ with $\gamma \in {\mathrm{\Omega }}^{p,q-1}(U)$ is killed by $\bar{\mathrm{\partial }}$, so $B_{\bar{\mathrm{\partial }}}^{p,q}(U)\subseteq Z_{\bar{\mathrm{\partial }}}^{p,q}(U)$ and the quotient in the Definition is well-defined. For $q=0$ the image is zero by the stated convention; for $q=n$ the target of $\bar{\mathrm{\partial }}$ is zero.

For an inclusion $j:U^{\mathrm{\prime }}\hookrightarrow U$, restriction commutes with coordinate differentiation: the coefficient formula for $\bar{\mathrm{\partial }}$ gives $j^{\ast }(\bar{\mathrm{\partial }}\eta )=\bar{\mathrm{\partial }}(j^{\ast }\eta )$ term by term. Hence closed forms restrict to closed forms and exact forms restrict to exact forms.

Define $j^{\ast }:H_{\bar{\mathrm{\partial }}}^{p,q}(U)\to H_{\bar{\mathrm{\partial }}}^{p,q}(U^{\mathrm{\prime }})$ by $[\eta ]\mapsto [j^{\ast }\eta ]$ for closed $\eta$. If $[\eta ]=[{\eta }^{\mathrm{\prime }}]$, then $\eta -{\eta }^{\mathrm{\prime }}=\bar{\mathrm{\partial }}\gamma$; step 1.2 gives $j^{\ast }\eta -j^{\ast }{\eta }^{\mathrm{\prime }}=\bar{\mathrm{\partial }}(j^{\ast }\gamma )$, so the class is independent of the representative.

Restricting a form to itself is the identity, and for open inclusions $U^{\mathrm{\prime }\mathrm{\prime }}\subseteq U^{\mathrm{\prime }}\subseteq U$, $(\eta {\mathrm{\mid }}_{U^{\mathrm{\prime }}}){\mathrm{\mid }}_{U^{\mathrm{\prime }\mathrm{\prime }}}=\eta {\mathrm{\mid }}_{U^{\mathrm{\prime }\mathrm{\prime }}}$. Therefore the induced cohomology maps satisfy the same identity and composition laws.