Sheaf Cohomology Cech Cohomology and Comparison — Examples

The sequence $0\to \underset{‾}{\mathbb{Z}}\to \mathcal{C}\to \mathcal{Q}\to 0$ is short exact, where the first map is the inclusion of locally constant integer-valued functions and the second is induced by the presheaf quotient maps $\mathcal{C}(V)\to \mathcal{C}(V)\mathrm{/}\underset{‾}{\mathbb{Z}}(V)$ followed by sheafification. The kernel of the second map is $\underset{‾}{\mathbb{Z}}$: its stalk at $x$ is computed from the stalk of $\mathcal{Q}$, which is the colimit of the groups $\mathcal{C}(V)\mathrm{/}\underset{‾}{\mathbb{Z}}(V)$ over the filtered neighbourhood system of $x$ by [F6], and this colimit is ${\mathcal{C}}_x\mathrm{/}{\underset{‾}{\mathbb{Z}}}_x$ because filtered colimits of abelian groups are exact [F7]; hence the stalk of the kernel is ${\underset{‾}{\mathbb{Z}}}_x$ for every $x$, and kernels of sheaf morphisms are computed stalkwise [F12]. The second map is an epimorphism with the same stalkwise computation, its stalk at $x$ being the surjection ${\mathcal{C}}_x\to {\mathcal{C}}_x\mathrm{/}{\underset{‾}{\mathbb{Z}}}_x$; by [F5] and [F4] the displayed sequence is short exact.

Put $A:=(-1\mathrm{/}2,1\mathrm{/}2)$ and $B:=(0,1)$ and $U_0:=p(A)$, $U_1:=p(B)$. Both are open in $S^1$, because $p^{-1}(U_0)=A+\mathbb{Z}$ and $p^{-1}(U_1)=B+\mathbb{Z}$ are unions of open intervals, hence open in $\mathbb{R}$ [F9]. Both are connected, being continuous images under $p$ of the intervals $A$ and $B$ [F10]. Their union is $S^1$: every real number is congruent modulo $\mathbb{Z}$ to a point of $A\cup B=(-1\mathrm{/}2,1)$, so every class in $S^1$ lies in $U_0\cup U_1$. Their intersection is $p(A\cap (B+\mathbb{Z}))=p((-1\mathrm{/}2,0)\cup (0,1\mathrm{/}2))=p((1\mathrm{/}2,1))\cup p((0,1\mathrm{/}2))$, a union of two disjoint nonempty open connected subsets, so $U_0\cap U_1$ has exactly these two connected components; neither of them meets the other since they are disjoint and open, so every connected subset of the intersection lies in one of them [F11].

On a connected open subset of $S^1$ every locally constant $\mathbb{Z}$-valued function is constant: its fibres are open, pairwise disjoint and cover the set, and two nonempty fibres would exhibit a separation [F11]; so by [F8] the sections of $\underset{‾}{\mathbb{Z}}$ over $U_0$ and over $U_1$ are the constant integer functions, $\underset{‾}{\mathbb{Z}}(U_0)\cong \mathbb{Z}$ and $\underset{‾}{\mathbb{Z}}(U_1)\cong \mathbb{Z}$, while over the two-component intersection $\underset{‾}{\mathbb{Z}}(U_0\cap U_1)\cong \mathbb{Z}\times \mathbb{Z}$, a locally constant function on the intersection being determined by, and arbitrary on, the two components of [step 1.2].

Let ${\theta }_0:U_0\to A$ and ${\theta }_1:U_1\to B$ be the inverses of the bijections induced by $p$ on $A$ and $B$; they are continuous because $p$ is a quotient map and these are homeomorphisms onto their images [F9]. On $U_0\cap U_1$ the difference ${\theta }_1-{\theta }_0$ is a continuous integer-valued function: the two lifts of a point of the intersection differ by an integer, and on $p((0,1\mathrm{/}2))$ both lifts lie in $(0,1\mathrm{/}2)$, giving the value $0$, while on $p((1\mathrm{/}2,1))$ the lift in $B$ is the lift in $A$ shifted by $1$, giving the value $1$. Hence ${\theta }_0$ and ${\theta }_1$ have the same image in $\mathcal{Q}(U_0\cap U_1)$, since their difference lies in $\underset{‾}{\mathbb{Z}}(U_0\cap U_1)$, and the sheaf axiom for $\mathcal{Q}$ over the cover $S^1=U_0\cup U_1$ glues them to a global section $q\in \mathrm{\Gamma }(S^1,\mathcal{Q}),q{\mathrm{\mid }}_{U_0}=[{\theta }_0],q{\mathrm{\mid }}_{U_1}=[{\theta }_1]\mathrm{.}$

There is no $g\in \mathrm{\Gamma }(S^1,\mathcal{C})$ with image $q$. Suppose there were. Then on each $U_i$ the difference $g{\mathrm{\mid }}_{U_i}-{\theta }_i$ has zero image in $\mathcal{Q}(U_i)$, hence has locally constant integer values; being continuous with values in the discrete set $\mathbb{Z}$, it is locally constant and therefore constant on the connected set $U_i$ [step 1.2], say $g{\mathrm{\mid }}_{U_i}-{\theta }_i=n_i$ with $n_i\in \mathbb{Z}$ by [step 2.1]. On the first component $p((0,1\mathrm{/}2))$ of the intersection the difference ${\theta }_1-{\theta }_0$ vanishes, so $n_1-n_0=({\theta }_0-{\theta }_1)\mathrm{\mid }=0$; on the second component $p((1\mathrm{/}2,1))$ the same difference equals $1$, so $n_1-n_0=-1$. This is impossible, so $q$ is not in the image of $\mathrm{\Gamma }(S^1,\mathcal{C})\to \mathrm{\Gamma }(S^1,\mathcal{Q})$.

By [F2] applied to the short exact sequence of [step 1.1] the sequence $\mathrm{\Gamma }(S^1,\mathcal{C})⟶\mathrm{\Gamma }(S^1,\mathcal{Q})\stackrel{\mathrm{\partial }}{\to }{H}^1(S^1,\underset{‾}{\mathbb{Z}})$ is exact, the first group being $H^0(S^1,\mathcal{C})$ and the middle one $H^0(S^1,\mathcal{Q})$ by [F3]; exactness at the middle group [F4] says that the image of the first map is the kernel of $\mathrm{\partial }$. By [step 3.1] the element $q$ is not in that image, so $\mathrm{\partial }(q)\mathrm{\ne }0$ and $H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{\ne }0$; the section $q$ is thus an explicit witness for the failure of surjectivity asserted in the statement, and its obstruction is detected by the degree one connecting class. The Axiom of Choice enters exactly through the long exact sequence [F2], whose construction uses the supplied injective resolution datum and the Dependent Choice it requires [F13]; the computations of [step 1.2], [step 2.1], [step 2.2] and [step 3.1] use no choice principle. ∎

$p^{-1}(U_0)=(-1\mathrm{/}2,1\mathrm{/}2)+\mathbb{Z}$ and $p^{-1}(U_1)=(0,1)+\mathbb{Z}$ are unions of open intervals, hence open in $\mathbb{R}$, so $U_0$ and $U_1$ are open in $S^1$ [F7]. Both are nonempty and connected, being the images under the continuous map $p$ of the connected intervals $(-1\mathrm{/}2,1\mathrm{/}2)$ and $(0,1)$ [F8, F9]. Their union is all of $S^1$, because every real number differs by an integer from a point of $(-1\mathrm{/}2,1\mathrm{/}2)\cup (0,1)=(-1\mathrm{/}2,1)$. For the intersection, $p^{-1}(U_0\cap U_1)=((-1\mathrm{/}2,1\mathrm{/}2)+\mathbb{Z})\cap ((0,1)+\mathbb{Z})=((-1\mathrm{/}2,0)\cup (0,1\mathrm{/}2))+\mathbb{Z}$, so $U_0\cap U_1=p((-1\mathrm{/}2,0)\cup (0,1\mathrm{/}2))=p((1\mathrm{/}2,1))\cup p((0,1\mathrm{/}2))=C_2\cup C_1$, where $p(x+1)=p(x)$ was used [F6]. Each of $C_1,C_2$ is open, since $p^{-1}(C_1)=(0,1\mathrm{/}2)+\mathbb{Z}$ and $p^{-1}(C_2)=(1\mathrm{/}2,1)+\mathbb{Z}$ are open; each is nonempty and connected as the image of an interval [F8, F9]; and $C_1\cap C_2=\mathrm{\varnothing }$ because $(0,1\mathrm{/}2)+\mathbb{Z}$ and $(1\mathrm{/}2,1)+\mathbb{Z}$ are disjoint. So $(C_1,C_2)$ is a separation of $U_0\cap U_1$ in the sense of [F10], and every connected subset $T\subseteq U_0\cap U_1$ lies in $C_1$ or in $C_2$: if $T$ met both, then $T\cap C_1$ and $T\cap C_2$ would be nonempty disjoint relatively open subsets covering $T$, a separation of $T$ [F10]. Hence $C_1$ and $C_2$ are exactly the connected components of $U_0\cap U_1$.

By [F3] the sections of $\underset{‾}{\mathbb{Z}}$ over an open set are its locally constant $\mathbb{Z}$-valued functions. On each of the connected sets $U_0$, $U_1$, $C_1$, $C_2$ of [step 1.1] such a function is constant: its fibres are open, pairwise disjoint and cover the set, so if two distinct fibres were nonempty, one nonempty fibre and the union of all the other fibres would form a separation [F10]. Consequently the constant value of a section over $U_0$, over $U_1$, over $C_1$ and over $C_2$ is well defined, and the maps $f⟼f{\mathrm{\mid }}_{C_1}\text{’s value},f⟼f{\mathrm{\mid }}_{C_2}\text{’s value}$ give a bijection $\underset{‾}{\mathbb{Z}}(U_0\cap U_1)\to \mathbb{Z}\oplus \mathbb{Z}$: it is injective because a locally constant function on $U_0\cap U_1=C_1\cup C_2$ [step 1.1] is determined by its two constant values, and it is surjective because for $(m,n)\in \mathbb{Z}\oplus \mathbb{Z}$ the function equal to $m$ on $C_1$ and to $n$ on $C_2$ is locally constant, the two sets being open and disjoint [step 1.1]. The same argument with one connected set gives bijections $\underset{‾}{\mathbb{Z}}(U_0)\to \mathbb{Z}$ and $\underset{‾}{\mathbb{Z}}(U_1)\to \mathbb{Z}$ by constant value. All these bijections are group isomorphisms for the pointwise group structure of [F3], addition of locally constant functions being computed valuewise.

By [F1] applied to the two-member cover $\mathcal{U}=(U_0,U_1)$ the groups are $C^0(\mathcal{U},\underset{‾}{\mathbb{Z}})=\underset{‾}{\mathbb{Z}}(U_0)\oplus \underset{‾}{\mathbb{Z}}(U_1)$ and $C^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=\underset{‾}{\mathbb{Z}}(U_0\cap U_1)$, and $C^p(\mathcal{U},\underset{‾}{\mathbb{Z}})=0$ for $p\ge 2$. Under the identifications of [step 2.1] the pair $(a,b)\in \mathbb{Z}\oplus \mathbb{Z}$ corresponds to the $0$-cochain whose components are the functions constantly equal to $a$ on $U_0$ and to $b$ on $U_1$; the difference $s_1{\mathrm{\mid }}_{U_0\cap U_1}-s_0{\mathrm{\mid }}_{U_0\cap U_1}$ of [F2] is then the function constantly equal to $b-a$ on $U_0\cap U_1$, hence restricts to the constant value $b-a$ on each of $C_1$ and $C_2$ and corresponds to $(b-a,b-a)\in \mathbb{Z}\oplus \mathbb{Z}$. Therefore ${\delta }^0(a,b)=(b-a,b-a)$ under the identifications, and ${\delta }^1=0$ since $C^2(\mathcal{U},\underset{‾}{\mathbb{Z}})=0$.

By [step 3.1] an element $(a,b)$ is a Čech $0$-cocycle exactly when $b-a=0$, that is exactly when $a=b$; hence the group of $0$-cocycles is the diagonal $\mathrm{\Delta }=\{(n,n):n\in \mathbb{Z}\}\cong \mathbb{Z}$ under $n\mapsto (n,n)$. The image of ${\delta }^0$ is $\{(b-a,b-a):a,b\in \mathbb{Z}\}=\mathrm{\Delta }$ as well. By [F4] applied to the complex of [step 3.1], whose terms $C^0,C^1$ are displayed there and whose terms in degrees $p\ge 2$ vanish, ${\check{H}}^0(\mathcal{U},\underset{‾}{\mathbb{Z}})=\ker {\delta }^0=\mathrm{\Delta }\cong \mathbb{Z},{\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=(\mathbb{Z}\oplus \mathbb{Z})\mathrm{/}\mathrm{\Delta },{\check{H}}^p(\mathcal{U},\underset{‾}{\mathbb{Z}})=0(p\ge 2)\mathrm{.}$ The homomorphism $\phi :\mathbb{Z}\oplus \mathbb{Z}\to \mathbb{Z}$, $\phi (x,y)=x-y$, is surjective because $\phi (x,0)=x$, and $\ker \phi =\mathrm{\Delta }$; by the first isomorphism theorem [F11] it induces an isomorphism $(\mathbb{Z}\oplus \mathbb{Z})\mathrm{/}\mathrm{\Delta }\to \mathbb{Z}$, so ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})\cong \mathbb{Z}$.

Let $\kappa \in \underset{‾}{\mathbb{Z}}(U_0\cap U_1)$ be the locally constant function equal to $1$ on $C_1$ and to $0$ on $C_2$; it is a well-defined section by [step 2.1], and it corresponds to $(1,0)\in \mathbb{Z}\oplus \mathbb{Z}$ there. Every $1$-cochain is a cocycle because ${\delta }^1=0$ [step 3.1], so $\kappa$ has a class in ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=(\mathbb{Z}\oplus \mathbb{Z})\mathrm{/}\mathrm{\Delta }$ of [step 4.1], namely the class of $(1,0)$; under the isomorphism induced by $\phi (x,y)=x-y$ this class corresponds to $\phi (1,0)=1$. Since $1$ generates $\mathbb{Z}$, the class $[\kappa ]$ generates ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})\cong \mathbb{Z}$.

By [F5] the restriction map $\mathrm{\Gamma }(S^1,\underset{‾}{\mathbb{Z}})\to {\check{H}}^0(\mathcal{U},\underset{‾}{\mathbb{Z}})$ is an isomorphism. The circle is connected: $S^1=p(\mathbb{R})$ is the image of the connected interval $\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$ [F8] under the continuous map $p$ [F9]. A locally constant $\mathbb{Z}$-valued function on the connected space $S^1$ is constant, by the argument of [step 2.1] applied to the whole circle: its fibres are open and partition $S^1$, so two distinct nonempty fibres would be a separation [F10]. Hence $\mathrm{\Gamma }(S^1,\underset{‾}{\mathbb{Z}})\cong \mathbb{Z}$ and ${\check{H}}^0(\mathcal{U},\underset{‾}{\mathbb{Z}})\cong \mathbb{Z}$, in agreement with the computation of [step 4.1]. This is the promised calculation for the two-arc cover: ${\check{H}}^0$ and ${\check{H}}^1$ are both isomorphic to $\mathbb{Z}$, with ${\check{H}}^1$ generated by the class of the overlap cocycle $\kappa$ of [step 5.1], and all higher Čech groups of the cover vanish. ∎

Take $X=S^1$ the circle with quotient map $p$ [F9], the index set $I=\{0\}$ with its unique linear order and $U_0:=S^1$; the circle is open in itself, so $\mathcal{U}=(U_0)$ is a cover of $X=S^1$ by open subsets. The increasing tuples in $I$ are the single $0$-tuple $(0)$ in degree $0$, and there is no increasing $(p+1)$-tuple for $p\ge 1$; hence by [F1] $C^0(\mathcal{U},\underset{‾}{\mathbb{Z}})=\underset{‾}{\mathbb{Z}}(U_0)=\underset{‾}{\mathbb{Z}}(S^1),C^p(\mathcal{U},\underset{‾}{\mathbb{Z}})=0(p\ge 1)\mathrm{.}$ So the differential ${\delta }^0:C^0\to C^1$ has target $0$ and is the zero map, and all higher differentials vanish as well. By [F2] the cohomology of the complex $C^0\to 0\to \cdots$ is ${\check{H}}^0(\mathcal{U},\underset{‾}{\mathbb{Z}})=\ker {\delta }^0=C^0=\underset{‾}{\mathbb{Z}}(S^1)$ and ${\check{H}}^p(\mathcal{U},\underset{‾}{\mathbb{Z}})=\ker {\delta }^p\mathrm{/}\mathrm{im}{\delta }^{p-1}=0\mathrm{/}0=0$ for every $p\ge 1$; in particular ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=0$.

By [F4] there is a global section $q$ of the quotient sheaf $\mathcal{Q}$ on $S^1$ — the sheafification of the presheaf quotient of the continuous real-valued functions by the locally constant integer-valued functions — whose connecting class $\mathrm{\partial }(q)\in H^1(S^1,\underset{‾}{\mathbb{Z}})$ is nonzero; hence $H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{\ne }0$. The coefficient sheaf there is the constant sheaf with value $\mathbb{Z}$ on $S^1$, identified with the subsheaf of locally constant integer-valued functions [F8], which is the same sheaf $\underset{‾}{\mathbb{Z}}$ used in [step 1.1]; the two occurrences of $H^1(S^1,\underset{‾}{\mathbb{Z}})$ denote the same group, formed from the same fixed injective resolution datum.

By [F3], under the Axiom of Choice, the comparison map for the cover $\mathcal{U}$ of [step 1.1] is defined: ${\phi }_{\mathcal{U}}^1:{\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})⟶H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{.}$ Its source is the zero group by [step 1.1], so ${\phi }_{\mathcal{U}}^1$ is the zero homomorphism, while its target is nonzero by [step 2.1]. A homomorphism whose target is nonzero and whose source is the zero group has image $0\mathrm{\ne }{H}^1(S^1,\underset{‾}{\mathbb{Z}})$, so it is not surjective and in particular not an isomorphism.

The cover $\mathcal{U}$ is not $\underset{‾}{\mathbb{Z}}$-acyclic. Indeed, by [F6] $\mathcal{U}$ is $\underset{‾}{\mathbb{Z}}$-acyclic exactly when $H^q(W,\underset{‾}{\mathbb{Z}}{\mathrm{\mid }}_W)=0$ for every $q>0$ and every nonempty finite intersection $W$ of members of $\mathcal{U}$; the only member is $U_0=S^1$, so the only such $W$ is $S^1$ itself, whose restriction of $\underset{‾}{\mathbb{Z}}$ is $\underset{‾}{\mathbb{Z}}$, and $H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{\ne }0$ by [step 2.1]. Hence the hypothesis of the Leray comparison theorem [F7] fails for this cover, and no contradiction with that theorem arises from [step 3.1]: the theorem gives an isomorphism only for $\mathcal{F}$-acyclic covers, and the one-member cover of the circle is not one.

Assembling the pieces for the circle, the constant sheaf $\underset{‾}{\mathbb{Z}}$ and the one-member cover $\{S^1\}$: ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=0$ by [step 1.1], the comparison map ${\phi }_{\mathcal{U}}^1$ is the zero map out of the zero group by [step 3.1], the sheaf cohomology $H^1(S^1,\underset{‾}{\mathbb{Z}})$ is nonzero by [step 2.1], and the cover fails acyclicity by [step 4.1]. Therefore the assertion that the comparison map ${\phi }_{\mathcal{U}}^p$ is an isomorphism in every degree for every cover is false, with degree $p=1$ and this cover as the witness; on the other hand ${\phi }_{\mathcal{U}}^0$ is an isomorphism for every cover, since it is the identity of the global sections up to the canonical identifications, so degree zero provides no obstruction. The Axiom of Choice of [F5] enters exactly twice: in [step 3.1] through the construction of the comparison map, which uses the Godement resolution of [F3], and in [step 2.1] through the long exact sequence in sheaf cohomology of [F4], whose connecting homomorphism is formed from injective resolutions; no other selection is used, the cochains of [step 1.1] being a single section group. ∎

Let $U\subseteq V\subseteq X$ be open and consider the restriction map ${\rho }_U^V$ of $i_{x,\ast }A$. If $x\in U$ then $x\in V$, so by [F1] both groups are $A$ and ${\rho }_U^V$ is the identity of $A$, which is surjective. If $x\mathrm{\notin }U$ then by [F1] the group $(i_{x,\ast }A)(U)$ is the zero group $0$, and any map into the zero group is surjective, indeed the only such map is the zero homomorphism; no hypothesis on $V$ is needed for this case. Since $x\in U$ or $x\mathrm{\notin }U$, these two cases exhaust all pairs $U\subseteq V$ of open subsets, so every restriction map of $i_{x,\ast }A$ is surjective; by [F2] the sheaf $i_{x,\ast }A$ is flasque.

By [step 1.1] the sheaf $i_{x,\ast }A$ on $X$ is flasque, so the vanishing theorem [F3] applies to it: for every open subspace $U\subseteq X$ and every integer $q>0$ the cohomology of the restriction vanishes, $H^q(U,i_{x,\ast }A{\mathrm{\mid }}_U)=0.$ Taking $U=X$, where the restriction of $i_{x,\ast }A$ to $X$ is $i_{x,\ast }A$ itself, gives $H^q(X,i_{x,\ast }A)=0$ for every $q>0$.

The two conclusions are the flasqueness of $i_{x,\ast }A$ from [step 1.1] and the vanishing $H^q(U,i_{x,\ast }A{\mathrm{\mid }}_U)=0$ for all open $U\subseteq X$ and all $q>0$ from [step 2.1], in particular $H^q(X,i_{x,\ast }A)=0$ for $q>0$; note that the flasqueness gives no information in degree zero, where $H^0(X,i_{x,\ast }A)=\mathrm{\Gamma }(X,i_{x,\ast }A)=(i_{x,\ast }A)(X)=A$ because $x\in X$. The Axiom of Choice of [F4] enters exactly once, in [step 2.1]: the vanishing theorem [F3] is proved by applying a supplied functorial injective resolution datum, whose availability is obtained from the Axiom of Choice through the implication to Dependent Choice recorded in [F5]. No further choice is made in this example — the point $x$ is part of the given data, not selected — and the computations of [step 1.1] are case distinctions on whether $x\in U$. ∎

For open $U\subseteq X$ let $\mathcal{F}(U)=\mathrm{Map}(U,A)$ be the set of all functions $U\to A$, and for open $U\subseteq V$ let ${\rho }_U^V(f):=f{\mathrm{\mid }}_U$; both $U$ and $V$ are open sets of the topology of [F1]. Restriction of functions satisfies ${\rho }_U^U(f)=f$ and ${\rho }_W^U(f)={\rho }_W^V({\rho }_V^U(f))$ for $W\subseteq V\subseteq U$, since both sides send $x\in W$ to $f(x)$; by [F2] this makes $\mathcal{F}$ a presheaf of sets on $X$. It is a presheaf of abelian groups under pointwise addition $(f+g)(x):=f(x)+g(x)$: the pointwise sum of two functions $U\to A$ is a function $U\to A$ [F5], addition is associative and commutative and has the constant zero function as identity because $A$ is an abelian group, and each ${\rho }_U^V$ is a group homomorphism since restrictions are computed valuewise.