Čech cohomology (${\check{H}}^p(\mathcal{U},\mathcal{F})$)

For an open cover $\mathcal{U}=(U_i)$, the Čech cochain group iswith the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of $\mathcal{F}$ with respect to $\mathcal{U}$.

On a good cover, combine Čech degree $p$ and differential-form degree $q$ in the double complex of forms on intersections. The Čech differential and exterior derivative commute, so the total differential $\delta +(-1{)}^{p}d$ squares to zero. A partition of unity and the Poincaré lemma identify its cohomology with both constant-sheaf and de Rham cohomology. The sign convention makes Čech-de Rham curvature descent explicit.

For a finite cover whose nonempty intersections have zero sheaf cohomology in degrees $1,\dots ,i-1$, the displayed sequence is exact for $i>0$. Apply the Čech cochain complex to a flasque resolution. Local primitives and successive primitives of their overlap differences reduce a class with zero restrictions to a Čech cocycle with values in the original sheaf. Vanishing in degree $i$ on all intersections is not needed.

The Čech cochain complex places sections on $(p+1)$-fold intersections in degree $p$ and uses the alternating sum of restriction maps as its differential.

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 family generating the unit ideal in a commutative ring, this augmented Čech cochain complex of any module is exact. One proof clears the finitely many denominators and cocycle relations by powers $f_i^N$, writes $1={\sum }_i{a}_i{f}_i^N$, and contracts an alternating cocycle by the weighted insertions ${\sum }_i{a}_i{f}_i^N{c}_{iI}$. The argument uses vanishing criterion in a module localization and does not require the module to be finitely generated.

The degree-$p$ Čech cochain group for an ordered cover is ${\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$.

A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain $(g_{ij})$, this says $g_{ij}{g}_{jk}=g_{ik}$ on triple intersections.

A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain $(h_i)$ changes a one-cocycle by $g_{ij}\mapsto h_i^{-1}{g}_{ij}{h}_j$.

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.

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.