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An open immersion $j:U\to X$ is a morphism of schemes that identifies $U$, including its structure sheaf, with an open subscheme of $X$.

Let $X={\mathbb{A}}_k^2$ and $U=X\setminus \{(0,0)\}$. The inclusion is an open immersion and $X$ is an affine scheme, but $U$ is not affine. Indeed, regular functions extend across the missing codimension-two point, soWere $U$ affine, the inclusion would therefore correspond to an isomorphism of coordinate rings and would be an isomorphism $U\cong X$, a contradiction.

For an example with both schemes affine, the principal open subschemeis a nontrivial open immersion. The affine line is connected because $k[t]$ has no nontrivial idempotent elements.

An ${\mathcal{O}}_X$-module $\mathcal{F}$ is a quasi-coherent sheaf when every affine open $V=\mathrm{Spec}A\subseteq X$ has $\mathcal{F}{|}_V\cong \tilde{M}$ for some $A$-module $M$. For a morphism $f:X\to Y$, its direct image sheaf iswhile the pullback of a sheaf of modules is

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, then on an affine open $V=\mathrm{Spec}A\subseteq X$ one has $Z\cap V=\mathrm{Spec}(A/I)$. The restriction of $i_{\ast }{\mathcal{O}}_Z$ corresponds to the cyclic $A$-module $A/I$, so it is finitely generated. Hence $i_{\ast }{\mathcal{O}}_Z$ is a coherent sheaf; more generally this is the direct image of a coherent sheaf under a closed immersion.

Coherence need not survive an arbitrary pushforward. Letbe the open immersion. The sheaf ${\mathcal{O}}_{D(x)}$ is coherent, butis not a finitely generated $k[x]$-module. Therefore $j_{\ast }{\mathcal{O}}_{D(x)}$ is not coherent.