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 $\mathcal{U}$.

For a commutative ring $R$, remove the section $[1:0:0]$ from ${\mathbb{P}}_R^2$. The two remaining standard charts have coordinate rings $R[a,b]$ and $R[a/b,b^{-1}]$ inside the Laurent polynomial ring $R[a,b,b^{-1}]$. Their intersection consists of constants: their monomial exponent regions are $i\ge 0,j\ge 0$ and $i\ge 0,j\le -i$, respectively. The Čech cohomology in degree one is free over $R$ on the missing monomials $a^i{b}^{-k}$ with $1\le k<i$. Equivalently its basis consists of the classes of $x_0^{m+n}/(x_1^m{x}_2^n)$ for $m,n\ge 1$. Degree zero is $R$ and every degree at least two is zero. The charts and their intersection are affine schemes, so this computes sheaf cohomology. If $R$ is nonzero, the first sheaf cohomology is nonzero and this scheme is not an affine scheme.

For $r\ge 1$, a twisting sheaf on projective space has $H^0({\mathbf{P}}_k^r,\mathcal{O}(n))=k[X_0,\dots ,X_r{]}_n$ when $n\ge 0$, and top cohomology with basis the Laurent monomials of total degree $n$ whose exponents are all negative. All intermediate sheaf cohomology vanishes. Their dimensions are $\left(\genfrac{}{}{0ex}{}{n+r}{r}\right)$ and $\left(\genfrac{}{}{0ex}{}{-n-1}{r}\right)$ respectively, with the top group nonzero only for $n\le -r-1$. Decompose the standard Čech cochain complex by exponent vector: no negative exponents gives a simplex with cohomology in degree zero; a nonempty proper negative-support set admits a contracting homotopy inserting a vertex outside that set; all negative exponents give only the top term. For $r=0$, every twist is trivial on $\mathrm{Spec}k$ and has $H^0=k$.

The two affine opens $D(x),D(y)$ cover the punctured affine plane. Their intersection has ring $k[x^{\pm 1},y^{\pm 1}]$. The Čech cohomology in degree one is the quotient by $k[x^{\pm 1},y]+k[x,y^{\pm 1}]$. Its basis consists precisely of the Laurent monomials whose two exponents are negative, so this vector space is infinite-dimensional. Global functions, by contrast, form $k[x,y]$.

On the standard two-affine cover of the projective line, with coordinate $t$, a frame for the twisting sheaf on projective space $\mathcal{O}(n)$ on the second chart equals $t^n$ times the frame on the first. Its Čech cochain complex is $k[t]\oplus k[t^{-1}]\to k[t,t^{-1}]$, with map $(a,b)\mapsto t^{n}b-a$. The kernel has basis $1,t,\dots ,t^n$ for $n\ge 0$. The cokernel has basis $t^{n+1},\dots ,t^{-1}$ for $n\le -2$. The two-chart complex also proves vanishing in degrees at least two.

For analytic holomorphic sections, the two-chart Čech cohomology uses convergent Laurent series, rather than only Laurent polynomials. A frame on the infinity chart equals $z^n$ times a finite-chart frame. Entire finite-chart functions remove Laurent powers $k\ge 0$, and infinity-chart functions remove powers $k\le n$. The positive and negative tails converge to entire chart functions, so only the finite gap $n<k<0$ remains. Thus the first cohomology has basis $z^{n+1},\dots ,z^{-1}$ when $n\le -2$ and is zero when $n\ge -1$. The same transition relation gives global sections of dimension $\max (n+1,0)$.

On $X={\mathbb{A}}_k^1$, let $K=k(t)$ and let $\mathcal{K}$ be its constant sheaf. For the two closed rational points $0,1$, take skyscraper sheaves $S_0,S_1$ with value $K$, and put $\mathcal{F}=\ker (\mathcal{K}\to S_0\oplus S_1)$. The maps on the two stalks are identities, so this is a sheaf surjection. All three displayed outer sheaves are flasque. The induced global map is the diagonal $K\to K^2$, so $H^1(X,\mathcal{F})=K^2/\Delta K\simeq K$. The one-member affine cover has zero positive Čech cochains. This even gives an ${\mathcal{O}}_X$-module example: let regular functions act on all copies of $K$ by multiplication as rational functions, not by evaluation at the closed point. The kernel is not quasi-coherent.

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 well-order on the indices of an open cover, the ordered Čech cochain complex uses only strictly increasing tuples, with the alternating restriction Čech differential. It computes the same Čech cohomology as the complex using all tuples. If every finite intersection is nonempty and the coefficient constant sheaf has the same abelian group $A$ on each intersection, inserting the least index $\alpha$ gives a cochain homotopy: set $(hs{)}_I=s_{\alpha ,I}$ when $\alpha$ is not in $I$, and $(hs{)}_I=0$ otherwise. Expanding the alternating sums gives $\delta h+h\delta =\mathrm{id}$ in every positive degree, since the terms deleting an index of $I$ cancel and the term deleting $\alpha$ is $s_I$. Hence positive Čech cohomology vanishes in this situation; degree zero remains $A$.

The Čech differential sends a cochain on $(p+1)$-fold intersections to the alternating sum of its restrictions to $(p+2)$-fold intersections. Each pair of omitted indices occurs twice with opposite signs in the composite, so ${\delta }^2=0$. It defines the Čech cohomology of the cover.

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 every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for $\mathcal{F}$, then the cover's Čech cohomology computes $H^p(X,\mathcal{F})$. An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.

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.