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An open cover is acyclic for a specified sheaf if its restriction to every nonempty finite intersection has zero higher sheaf cohomology. The acyclic cover theorem then identifies the cover's Čech cohomology with sheaf cohomology. On a separated scheme, an affine open cover is acyclic for every quasi-coherent sheaf.

Let $j:{\mathbb{A}}_k^1\hookrightarrow {\mathbb{P}}_k^1$ be a standard affine chart and $\mathcal{G}=\tilde{M}$ a quasi-coherent sheaf there. Its direct image is quasi-coherent because $j$ is an affine morphism. The standard two-chart acyclic cover has Čech cochain complex $M\oplus M_x\to M_x$ in degrees zero and one, with differential $(m,n)\mapsto n-m/1$. Surjectivity gives $H^i({\mathbb{P}}_k^1,j_{\ast }\mathcal{G})=0$ for $i>0$. This does not assert that the direct image itself is flasque.

An open subset of an algebraic variety equipped with an identification with an affine variety. Such charts make quasi-coherent sheaves and regular maps accessible through their local coordinate rings.

For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

On an affine variety or affine scheme with coordinate ring $A$, the global section functor and $M\mapsto \tilde{M}$ are inverse equivalences between modules and quasi-coherent sheaves. On every principal open subset, sections are the appropriate localization of a module. Localization detects exactness at maximal ideals, so these functors are exact even for modules which are not finitely generated.

A morphism $f:X\to Y$ is affine when the inverse image of every affine open subscheme of $Y$ is affine. For a quasi-coherent sheaf $\mathcal{F}$ on $X$, one has $R^q{f}_{\ast }\mathcal{F}=0$ for $q>0$.

For a finite affine cover of a semi-separated scheme, every finite intersection is affine. The augmented sheaf Čech cochain complex is exact: near each point, an open of the cover contains the neighbourhood under consideration, and inserting its index contracts the complex. Its terms are acyclic for a quasi-coherent sheaf, by affine cohomology vanishing and the affine-intersection property. Alternatively, apply the section-level Čech cochain complex to a flasque resolution; the two computations of the resulting double complex identify Čech cohomology with sheaf cohomology. Stalks of direct images from opens outside those opens need not vanish, so exactness must not be justified by treating these direct images as extensions by zero.

For a finite morphism between Noetherian schemes, a quasi-coherent sheaf is coherent exactly when its direct image sheaf is coherent. Over an affine target chart, let $B$ be the finite algebra over the target ring $A$, and let $M$ represent the sheaf on $\mathrm{Spec}B$. Finite generation of $M$ over $A$ implies finite generation over $B$, using the same generators; finite generation over $B$ implies finite generation over $A$ because $B$ is finite over $A$.

Coherent cohomology means sheaf cohomology with coefficients in a coherent sheaf. On a separated Noetherian scheme, a finite affine cover computes it using the Čech cochain complex, because intersections of affine opens are affine and positive-degree quasi-coherent sheaf cohomology vanishes there. Short exact sequences give long exact sequences in sheaf cohomology.

On a locally Noetherian scheme, a coherent sheaf is a quasi-coherent sheaf that is locally associated with a finitely generated module. Kernels, cokernels, and finite direct sums of coherent sheaves remain coherent.

If a separated scheme has an acyclic affine cover by $r$ open sets, then every quasi-coherent sheaf on it has vanishing sheaf cohomology in degrees at least $r$. The corresponding Čech cochain complex has no terms in those degrees.

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, $X$ is separated, and $\mathcal{F}$ is quasi-coherent on $Z$, thenIndeed, an affine cover of $X$ induces an affine cover of $Z$, and the two associated Čech cochain complexes are termwise identical.

For an affine morphism $f:X\to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$, the direct image sheaf has the same sheaf cohomology on $Y$ as $\mathcal{F}$ on $X$. To prove this, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Its direct images are flasque sheaves. On each affine open $V\subset Y$, the complex of sections of the direct images is $\Gamma (f^{-1}V,{\mathcal{I}}^{\bullet })$. By vanishing of quasi-coherent cohomology on an affine scheme, this augmented complex is exact. Checking on the affine basis proves that $f_{\ast }{\mathcal{I}}^{\bullet }$ resolves $f_{\ast }\mathcal{F}$. Its global sections are identical to those of the original resolution.

For a closed immersion, this quasi-coherent sheaf measures the first-order equations of the subvariety. Its local module is $I/I^2$, and its map to the restricted Kähler differential sheaf sends the class of $f$ to $df$. This is the definition in Stacks Project, Section 29.32.

A quasi-coherent sheaf of ideals $\mathcal{I}\subseteq {\mathcal{O}}_X$ gives a closed subscheme by gluing $\mathrm{Spec}(A/J)$ on affine open subschemes where $\mathcal{I}=\tilde{J}$. Compatibility follows from exactness of localization. On the closed set $Z=\{x:{\mathcal{I}}_x\ne {\mathcal{O}}_{X,x}\}$, the structure sheaf is $i^{-1}({\mathcal{O}}_X/\mathcal{I})$, where $i:Z\hookrightarrow X$ is the inclusion. Its pushforward is ${\mathcal{O}}_X/\mathcal{I}$.

For an affine morphism $f:X\to Y$, the direct image of every quasi-coherent sheaf on $X$ is quasi-coherent. On affine charts this is the fact that restriction of scalars turns the module sheaf $\tilde{M}$ into the module sheaf associated with the same underlying module over the target ring.

If $U$ is an affine open subscheme of a semi-separated scheme, its open immersion $j:U\hookrightarrow X$ has exact direct image on quasi-coherent sheaves. Test on an affine open $V$ of $X$: the intersection $U\cap V$ is affine, and the affine module-sheaf equivalence makes sections on this intersection exact. The direct image is also quasi-coherent, since this open immersion is an affine morphism.

On the projective line, an infinite direct sum of copies of $\mathcal{O}(-1)$ is a quasi-coherent sheaf but not a coherent sheaf: at any point its residue-vector-space dimension is infinite. Its global sections vanish because each summand has no sections, and sections commute with direct sums for the finite two-chart equalizer. Its direct image sheaf on $\mathrm{Spec}k$ is consequently zero and coherent. This shows that a nonaffine direct image can lose information needed to detect coherence.

On an affine scheme, a morphism between quasi-coherent sheaves is determined by the induced map on global sections, because its maps on principal open subsets are their localizations. Exactness of localization gives the displayed identity. Thus the kernel sheaf of a morphism between quasi-coherent sheaves is quasi-coherent on an arbitrary scheme, by applying the affine argument locally. No finite generation is needed.