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Because $X$ is a Noetherian scheme, choose a finite affine cover $X={\bigcup }_i{U}_i$. Every intersection $U_i\cap U_j$ is quasi-compact, so choose a finite affine cover $U_i\cap U_j={\bigcup }_k{U}_{ijk}$. If $j_i:U_i\hookrightarrow X$ and $j_{ijk}:U_{ijk}\hookrightarrow X$ are the inclusions, the sheaf axiom gives an exact sequencewhere the last arrow is the difference of the two restrictions to each overlap chart.

Applying the left-exact direct image sheaf functor identifies $f_{\ast }\mathcal{F}$ with the kernel ofEvery map $U_i\to Y$ and $U_{ijk}\to Y$ is a morphism between affine schemes. Part (a) shows that all sheaves in the two finite sums are quasi-coherent sheaves. Kernels of morphisms of quasi-coherent sheaves on the affine scheme $Y$ are quasi-coherent, because they correspond to kernels of module homomorphisms. Therefore $f_{\ast }\mathcal{F}$ is quasi-coherent. This is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism in the present Noetherian case.

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.

The two standard affine charts of the blowup restrict to an affine open cover of $X_0$. On the $x$-chart write $y=xu$. Since $xy=x^{2}u$, the fiber isOn the $y$-chart write $x=yv$, givingOn their overlap, $u$ and $v$ are invertible with $v=u^{-1}$ and $y=xu$. The squared factors $x^2$ and $y^2$ display the doubled exceptional component explicitly.

The localization of a ring $A$ at the powers of $f$ is generated over $A$ by $1/f$. The presentation $A_f\cong A[t]/(ft-1)$ follows from the universal property of localization: both sides universally receive a ring homomorphism from $A$ making $f$ invertible. This proof works with zero divisors and includes the zero ring when $f$ is nilpotent. In particular, a principal open subscheme of an affine scheme has an inclusion which is a morphism of finite type.

For $f\in A$, the principal open subset $D(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ of $\mathrm{Spec}A$ is the affine scheme $\mathrm{Spec}{A}_f$.

A scheme is quasi-compact when its underlying topological space is quasi-compact: every open cover has a finite subcover. Since affine open subsets form a base, this is equivalent to possessing a finite affine open cover. Every affine scheme is quasi-compact.

For a positive-degree homogeneous element $f\in S$, the standard openis the affine scheme $\mathrm{Spec}(S_f{)}_0$.

Every quasi-coherent sheaf on an affine scheme has zero higher sheaf cohomology. Together with the equivalence between modules and quasi-coherent sheaves on an affine scheme, this makes affine varieties the basic acyclic pieces for computing cohomology. An affine open cover of a separated scheme has affine finite intersections, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology.