Integer cdh sections on seminormal schemes

A pointwise injective map of presheaves $F\to G$ induces a pointwise injective map $F^{+}\to G^{+}$. Indeed, if two matching families from $F$ have equal images in $G^{+}(T)$, their images agree on some common covering subsieve by [F3]. Pointwise injectivity makes the original families agree there, so they represent the same element of $F^{+}(T)$. Applying the same argument again shows that sheafification preserves such injections. In particular, for every inclusion of sets $A\subseteq B$, the induced map $\underset{‾}{A}\to \underset{‾}{B}$ is pointwise injective.

For finite $A\subseteq \mathbb{Z}$, [F1] identifies $\underset{‾}{A}(X)$ with functions ${\pi }_0(X)\to A$. By naturality, whenever $A\subseteq B$ are finite subsets of $\mathbb{Z}$, the map between these section sets corresponds to inclusion of the corresponding function sets. By step 1.1, the map from each $\underset{‾}{A}(X)$ into $\underset{‾}{\mathbb{Z}}(X)$ is injective. Moreover, the images for two finite subsets $A,B$ are both contained in the image for the finite union $A\cup B$.

By [F2], the images in step 2.1 cover $\underset{‾}{\mathbb{Z}}(X)$. They therefore give a well-defined bijection from $\underset{‾}{\mathbb{Z}}(X)$ to the set of finite-image functions ${\pi }_0(X)\to \mathbb{Z}$: if a section is represented using two finite subsets, injectivity into the integer sheaf and passage to their union show that the two representations coincide there, and [F1] then gives the same function. Conversely every finite-image function is obtained from [F1] using its finite image as $A$. This also handles an empty image and an empty scheme.

The bijection respects addition. If sections are represented by finite-value functions with values in $A$ and $B$, their sum is represented by the induced map of sheaves $\underset{‾}{A}\times \underset{‾}{B}\to \underset{‾}{C}$, where $C=\{a+b:a\in A,b\in B\}$, followed by $\underset{‾}{C}\to \underset{‾}{\mathbb{Z}}$. This follows locally from the addition map of constant presheaves and the sheaf axiom; its value on each component is the sum of the two values. Thus the bijection is a group isomorphism onto the finite-image functions.

By [F4], ${\pi }_0(X)$ is finite, so every function on it has finite image. Step 4.1 is consequently the claimed isomorphism to ${\mathbb{Z}}^{{\pi }_0(X)}$. The construction uses natural sheaf maps and the natural bijections in [F1], so it is canonical. If $X$ is nonempty and connected, it has exactly one connected component and the target is $\mathbb{Z}$; if $X$ is empty, the target is the zero group.