Recall we had this definition of a correspondence. (Our convention is that all schemes are separated.)
Definition 1. For $X/k$ a smooth separated scheme and $Y/k$ a separated scheme, the abelian group $\operatorname{Cor}_ k(X, Y)$ is the free abelian group generated by closed integral subschemes $Z \subseteq X \times_k Y$ that are finite surjective over a connected component of $X$.
Remark 2. Given a closed subscheme $Z \subseteq X \times_k Y$ is not integral, as long as it is finite over $X$ and the image of $Z \to X$ is open, it defines a correspondence $[Z] \in \operatorname{Cor}_ k(X, Y)$.
Étale sheaves with transfer
We’ve seen that for any separated scheme $Y$ the functor $\operatorname{Cor}_ k(-, Y)$ is a Zariski sheaf on $\mathsf{Sm}_ k$.
Proposition 3. For every separated scheme $Y/k$, the functor $\operatorname{Cor}_ k(-, Y)$ defines an étale sheaf on $\mathsf{Sm}_ k$.
Proof.
We need to show that given $U \to X$ a surjective étale map, the natural map $$ 0 \to \operatorname{Cor}_ k(X, Y) \to \operatorname{Cor}_ k(U, Y) \rightrightarrows \operatorname{Cor}_ k(U \times_X U, Y) $$ is an equalizer. Here what are the maps? These are just pulling back along the étale maps $U \times Y \to X \times Y$, where we note that étale covers of reduced schemes are reduced (Stacks 033B). We now see that this pullback map $\operatorname{Cor}_ k(X, Y) \to \operatorname{Cor}_ k(U, Y)$ sends each generator, i.e., integral $Z \subseteq X \times Y$, to linear combinations that have disjoint support. This immediately shows the injectivity, also reduces the second exactness to a statement about fields by passing to generic points. Namely, we need to show that if $E/F$ is an étale algebra, we want to show that $$ 0 \to \mathbb{Z} \to \operatorname{Hom}(\lvert \Spec E \rvert, \mathbb{Z}) \rightrightarrows \Hom(\lvert \Spec (E \otimes_F E) \rvert, \mathbb{Z}) $$ is exact, where the transition maps are compositions. This follows from the fact that $\lvert \Spec (E \otimes_F E) \rvert \to \lvert \Spec E \rvert^2$ is surjective.
Example 4. We really are using étaleness here. If we have a purely inseparable field extension $k^\prime / k$ of degree $p$, then the sequence $$ \operatorname{Cor}_ k(\Spec k, \mathbb{A}_ k^1) \to \operatorname{Cor}_ k(\Spec k^\prime, \mathbb{A}_ k^1) \rightrightarrows \operatorname{Cor}_ k(\Spec (k^\prime \otimes_k k^\prime), \mathbb{A}_ k^1) $$ is not exact, because if we take a closed point $x \in \mathbb{A}_ k^1$ with residue field $k^\prime$ then $[x] \in \operatorname{Cor}_ k(\Spec k, \mathbb{A}_ k^1)$ pulls back to $p[x] \in \operatorname{Cor}_ k(\Spec k^\prime, \mathbb{A}_ k^1)$ but $[x] \in \operatorname{Cor}_ k(\Spec k^\prime, \mathbb{A}_ k^1)$ is in the equalizer.
Recall that $\mathsf{Cor}_ k$ is the category whose objects are $\mathsf{Sm}_ k$ and morphisms are $\operatorname{Cor}_ k$. Because all schemes are separated, we have a natural functor $\mathsf{Sm}_ k \to \mathsf{Cor}_ k$.
Definition 5. An étale sheaf with transfer is a functor $\mathscr{F} \colon \mathsf{Cor}_ k^\mathrm{op} \to \mathsf{Ab}$ satisfying the property that the composition $\mathsf{Sm}_ k^\mathrm{op} \to \mathsf{Cor}_ k^\mathrm{op} \xrightarrow{\mathscr{F}} \mathsf{Ab}$ is an étale sheaf. The category of such things is denoted by $\mathsf{Sh}_ \mathrm{et}(\mathsf{Cor}_ k)$.
Here is another source of étale sheaves with transfer.
Definition 6. An étale sheaf on $\mathsf{Sm}_ k$ is locally constant when it is pulled back from an étale sheaf on $(\Spec k)_ \mathrm{et}$, i.e., it comes from a discrete $\Gal(k^\mathrm{sep}/k)$-module.
Lemma 7. Given any locally constant sheaf $\mathscr{F} \colon \mathsf{Sm}_ k^\mathrm{op} \to \mathsf{Ab}$, it uniquely lifts to a étale sheaf with transfer $\mathscr{F} \in \mathsf{Sh}_ \mathrm{et}(\mathsf{Cor}_ k)$.
Proof.
Let $X, Y \in \mathsf{Sm}_ k$ and let $Z \subseteq X \times_k Y$ be an elementary correspondence. Given $s \in \mathscr{F}(Y)$, we have to think about how to obtain an element of $\mathscr{F}(X)$.
What is an element $s \in \mathscr{F}(Y)$? Note that $\mathscr{F}$ can be represented as a scheme $M \to \Spec k$ that is separated étale (but not necessarily quasi-compact). Then we can write $s \in M(Y)$. Now the relevant fact for is us the following. If $X$ is a normal integral scheme and $K(Z^\prime) / K(X)$ is a finite normal extension, we can form its relative normalization $Z^\prime \to X$ that is finite (since finite type algebras over fields are Japanese). Now $\Aut(K(Z^\prime) / K(X))$ acts on $Z^\prime$, and the claim is that $$ M(Z^\prime)^{\Aut(K(Z^\prime)/K(X))} = M(X). $$
We find $K(Z^\prime)$ that contains $K(Z)$ so that $K(Z^\prime) / K(X)$ is normal. Now we check that maps $Z^\prime \to Z$ over $X$ correspond to embeddings $K(Z) \hookrightarrow K(Z^\prime)$. The number of such thing is the separable degree of $K(Z)/K(X)$. So what we do is to and pull back $s \in M(Y)$ to an element of $M(Z)$, and the pullbacks to $M(Z^\prime)$ along all possible $Z^\prime \to Z$. Now this is invariant under $\Aut(Z^\prime/X)$, and so it defines an element of $M(X)$. Now we just multiply this by the purely inseparable degree of $K(Z)/K(X)$.
The étale sheafification
Our goal is to prove the following theorem.
Theorem 8. Let $\mathscr{F} \colon \mathsf{Cor}_ k^\mathrm{op} \to \mathsf{Ab}$ be a presheaf with transfer. Let $\mathscr{F}_ \mathrm{et} \colon \mathsf{Sm}_ k^\mathrm{op} \to \mathsf{Ab}$ be its étale sheafification. Then there exists a unique upgrade of $\mathscr{F}_ \mathrm{et}$ to an étale sheaf with transfer, such that $\mathscr{F} \to \mathscr{F}_ \mathrm{et}$ is a morphism of presheaves with transfer.
We start with the following lemma.
Lemma 9. Let $Y \to X$ be an étale cover between separated schemes over $k$. Then the complex $$ \dotsb \to \mathbb{Z}_ \mathrm{tr}(Y \times_X Y) \to \mathbb{Z}_ \mathrm{tr}(Y) \to \mathbb{Z}_ \mathrm{tr}(X) \to 0 $$ is exact as étale sheaves on $\mathsf{Sm}_ k$.
This is true even in the Nisnevich topology, i.e., if $U \to X$ is a Nisnevich cover then this is exact in the Nisnevich topology.
Remark 10. What is this map $\mathbb{Z}_ \mathrm{tr}(Y) \to \mathbb{Z}_ \mathrm{tr}(X)$? This map actually exists for any map $Y \to X$ of separated schemes over $k$. Given any integral subscheme $Z \subseteq T \times Y$ that is finite surjective over $T$, we can look at its scheme-theoretic image $W \subseteq T \times X$. This is indeed reduced because $Z \to T \times X$ is quasi-compact and separated, and also $W \to T$ is clearly quasi-finite. We also see from something like Lemma 0AH6 that $W \to T$ is proper, and hence $W \to T$ is finite surjective.
Proof.
Because points in the étale topology are strictly Henselian local rings, it suffices to show that for a strictly Henselian local ring $S = \varprojlim S_i$, where $S_i$ are smooth affine schemes, the complex $$ \dotsb \to \varinjlim \mathbb{Z}_ \mathrm{tr}(Y \times_X Y)(S_i) \to \varinjlim \mathbb{Z}_ \mathrm{tr}(Y)(S_i) \to \varinjlim \mathbb{Z}_ \mathrm{tr}(X)(S_i) \to 0 $$ is exact. Now we can think of $\varinjlim \mathbb{Z}_ \mathrm{tr}(X)(S_i)$ as the free abelian group generated by closed integral subschemes $Z \subseteq S \times_k X$ that are finite surjective over $S$. Note that these $Z$ are also strictly Henselian local. (A local ring $R$ is Henselian if and only if every finite $R$-algebra is a product of local rings.)
Now if we look at the preimages of $Z$ in these étale covers $Y \times_X \dotsb \times_X Y \times_k S$ then they are just disjoint unions of the same local ring $S$. Then we are reduced to proving that for any set $I$ the chain complex $$ \dotsb \to \mathbb{Z}^{\oplus I^2} \to \mathbb{Z}^{\oplus I} \to \mathbb{Z} \to 0 $$ is exact. But this is the simplicial chain complex for the simplicial set on vertices indexed by $I$ and $n$-simplices given by functions $[n] \to I$. This is clearly contractible.
Corollary 11. If $U_1, \dotsc, U_n \subseteq X$ form a Zariski cover of $X$, then the complex $$ 0 \to \mathbb{Z}_ \mathrm{tr}(U_1 \cap \dotsb \cap U_n) \to \dotsb \to \bigoplus_{i=1}^n \mathbb{Z}_ \mathrm{tr}(U_i) \to \mathbb{Z}_ \mathrm{tr}(X) \to 0 $$ is exact in the étale (Nisnevich) topology.
Proof.
For $n = 2$, this follows from the previous lemma. For $n \gt 2$, we do induction using a simple spectral sequence.
Lemma 12. Let $p \colon U \to Y$ be an étale covering and $f \in \operatorname{Cor}_ k(X, Y)$, where all schemes are smooth over $k$. Then there exists an étale cover $p^\prime \colon V \to X$ so that $$ \begin{CD} V @>{f^\prime}>> U \br @V{p^\prime}VV @V{p}VV \br X @>{f}>> Y \end{CD} $$ commutes in $\mathsf{Cor}_ k$.
Proof.
Let $f$ correspond to an integral subscheme $Z \subseteq X \times Y$ finite surjective over $X$. Then we may look at the base change $Z_U \subseteq X \times U$, which is an étale cover of $Z$.
Now the claim is that if $Z_U \to Z$ is étale and $Z \to X$ is finite, then $Z_U \to Z$ splits étale locally on $X$. To see this, we can pass to wehre $X$ is strictly henselian and then $Z$ is strictly henselian as well, hence the map splits. Therefore there is an étale cover $V \to X$ such that we have a splitting $s \colon V \times_X Z \to V \times_X Z_U$.
Now the composition $f \circ p^\prime \in \operatorname{Cor}_ k(V, Y)$ is given by $V \times_X Z \subseteq V \times Y$. Therefore if we consider $s(V \times_X Z) \subseteq V \times U$ and use this for $f^\prime \in \operatorname{Cor}_ k(V, U)$ then this pushes forward to $V \times_X Z \in \operatorname{Cor}_ k(V, Y)$.
We now prove the theorem.
Theorem 13. For every $\mathscr{F} \in \mathsf{PST}(k)$, its étale sheafification $\mathscr{F}_ \mathrm{et}$ can be uniquely promoted to an étale sheaf with transfer, so that $\mathscr{F} \to \mathscr{F}_ \mathrm{et}$ is a map in $\mathsf{PST}(k)$.
Let us first check uniqueness. Given $f \in \operatorname{Cor}_ k(X, Y)$ and $u \in \mathscr{F}_ \mathrm{et}(Y)$, we need to determine what corresponding element $f^\ast(u) \in \mathscr{F}_ \mathrm{et}(X)$ is. By definition, there is an étale cover $U \to Y$ for which $u \vert_U \in \mathscr{F}(U)$. Using the above lemma, we choose an étale cover $V \to X$ together with a diagram $$ \begin{CD} V @>{f^\prime}>> U \br @VVV @VVV \br X @>{f}>> Y. \end{CD} $$ Then we would need $f^\ast(u) \vert_V = (f^\prime)^\ast (u \vert_U)$. Because restricting along $V \to X$ is injective, this determines what $f^\ast(u)$ has to be.
We now prove existence. We could try to use the above diagram to construct it, but it would be quite annoying to check independence of all the choices. So instead, we note that for every $X \in \mathsf{Sm}_ k$ there is an map $$ \mathscr{F}(X) \cong \Hom_\mathsf{PST}(\mathbb{Z}_ \mathrm{tr}(X), \mathscr{F}) \to \Hom_\mathsf{Sh}(\mathbb{Z}_ \mathrm{tr}(X), \mathscr{F}_ \mathrm{et}). $$ On the other hand, we have seen before that the right hand side is an étale sheaf in $X$, because we have a right exact sequence of étale sheaves $$ \mathbb{Z}_ \mathrm{tr}(U \times_X U) \to \mathbb{Z}_ \mathrm{tr}(U) \to \mathbb{Z}_ \mathrm{tr}(X) \to 0 $$ for $U \to X$ an étale cover. Therefore we have a factorization $$ \mathscr{F}(X) \to \mathscr{F}_ \mathrm{et}(X) \to \Hom_\mathsf{Sh}(\mathbb{Z}_ \mathrm{tr}(X), \mathscr{F}_ \mathrm{et}), $$ functorial in $X$ along maps in $\mathsf{Sm}_ k$. Now given any $f \in \operatorname{Cor}_ k(X, Y)$ and $y \in \mathscr{F}_ \mathrm{et}(Y)$, we can cake the composition $$ \mathbb{Z}_ \mathrm{tr}(X) \xrightarrow{f} \mathbb{Z}_ \mathrm{tr}(Y) \xrightarrow{[y]} \mathscr{F}_ \mathrm{et} $$ and look at the image of the identity map $[\Delta_X] \in \mathbb{Z}_ \mathrm{tr}(X)(X)$ to define $f^\ast(y) \in \mathscr{F}_ \mathrm{et}(X)$.
Some extra homological algebra
Proposition 14. The abelian category $\mathsf{Sh}_ \mathrm{et}(\mathsf{Cor}_ k)$ has enough injectives.
We verify Grothendieck’s criterion.
Theorem 15. Let $\mathcal{A}$ be an abelian category with the following properties:
- $\mathcal{A}$ has direct sums (and hence all small colimits),
- filtered colimits are exact,
- there is a set of generators $\lbrace U_i \rbrace \subseteq \mathcal{A}$, i.e., for every $N \subsetneq M$ in $\mathcal{A}$, there exists an $i \in I$ and a map $U_i \to M$ that does not factor through $N$.
Then $\mathcal{A}$ has enough injectives.
Proof.
The first observation to make is that an object $X \in \mathcal{A}$ is injective if and only if for all $i \in I$ and subobjects $V_i \subsetneq U_i$ every map $V_i \to X$ extends to $U_i \to X$. This is because we can use Zorn’s lemma (note that we are using exactness of filtered colimits).
Now for each object $X \in \mathcal{A}$, we define the pushout $$ \begin{CD} \bigoplus_{i \in I} \bigoplus_{V \subseteq U_i} \bigoplus_{\varphi \in \Hom(V, X)} V @>>> X \br @VVV @VVV \br \bigoplus_{i \in I} \bigoplus_{V \subseteq U_i} \bigoplus_{\varphi \in \Hom(V, X)} U_i @>>> \mathbb{M}(X). \end{CD} $$ Using transfinite induction we can define $\mathbb{M}^\alpha(X)$ for all ordinals $\alpha$. Now if we choose an ordinal $\alpha$ whose cofinality is greater than the cardinality of $\coprod_{i \in I} \operatorname{Sub}(U_i)$, then we claim that $\mathbb{M}^\alpha(X)$ is injective. This is because given any $V \subset U_i$ and a map $f \colon V \to \mathbb{M}^\alpha(X)$, we see that the subobjects $\lbrace f^{-1}(\mathbb{M}^\beta) : \beta \lt \alpha \rbrace$ fill $V$, and hence $f$ has to factor through some $\mathbb{M}^\beta(X)$ for $\beta \lt \alpha$. Now $f$ extends to $U_i \to \mathbb{M}^{\beta+1}(X)$.
This criterion is satisfied, where existence of colimits and exactness of filtered colimits is a general thing about a topos, and the sheaves $\mathbb{Z}_ \mathrm{tr}(X)$ for $X \in \mathsf{Sm}_ k$ form a set of generators.
There is also a version of the Godement resolution. Fix an algebraic closure $\bar{k}$. Given $\mathscr{F} \in \mathsf{Sh}_ \mathrm{et}(\mathsf{Sm}_ k)$, we can always define $$ E(\mathscr{F})(X) = \prod_{x \in X(\bar{k})} \mathscr{F}_ {\bar{x}}. $$ This has an obvious presheaf structure.
Proposition 16. This $E(\mathscr{F})$ is an acyclic étale sheaf, and moreover the natural map $\mathscr{F} \to E(\mathscr{F})$ is injective.
Proof.
It is clear that it is an étale sheaf, because the sheaf condition is a product of some obvious sheaf conditions. Why is it acyclic? I don’t know of a simple proof, but one way is to identify sheaf cohomology with Čech hypercover cohomology, namely we have $$ H^i(X, E(\mathscr{F})) = \varinjlim_{\mathfrak{K}} \check{H}^i(\mathfrak{K}, E(\mathscr{F})), $$ where $\mathfrak{K}$ ranges over hypercovers of $X$. Then we see that $\check{H}^i(\mathfrak{K}, E(\mathscr{F}))$ vanishes for every hypercover $\mathfrak{K}$, because the corresponding chain complex is $$ \prod_{x \in X(\bar{k})} C_\mathrm{simp}^\bullet(\mathfrak{K}_ x, \mathscr{F}_ x), $$ where $C_\mathrm{simp}^\bullet(\mathfrak{K}_ x, \mathscr{F}_ x)$ is the simplicial cochain complex associated to a trivial Kan complex $\mathfrak{K}_ x$. Its higher cohomology vanish because $\mathfrak{K}_ x$ is contractible.
Lemma 17. If $\mathscr{F}$ has transfers, the we can uniquely upgrade $E(\mathscr{F})$ to an étale sheaf with transfers so that $\mathscr{F} \to E(\mathscr{F})$ is a morphism in $\mathsf{PST}_ k$.
Proof.
For an integral $Z \subseteq X \times Y$ that is finite surjective over $X$, we need to define a map $$ \prod_{y \in Y(\bar{k})} \mathscr{F}_ y \to \prod_{x \in X(\bar{k})} \mathscr{F}_ x. $$ How do we do this? Given $x \in X(\bar{k})$, we consider the finite set of points $\lbrace z_i \rbrace \subseteq Z(\bar{k})$ that map to $x$, and write $z_i = (x, y_i)$. Then because we have a diagram $$ X_x^\mathrm{sh} \leftarrow X_x^\mathrm{sh} \times_X Z = \coprod_i Z_{z_i}^\mathrm{sh} \to \coprod_i Y_{y_i}^\mathrm{sh}, $$ the transfer structure on $\mathscr{F}$ actually induces a map $$ \prod_i \mathscr{F}_ {y_i} \to \mathscr{F}_ x. $$ We can now take the product of such maps to define the transfer structure on $E(\mathscr{F})$.
Corollary 18. Let $\mathscr{F}$ be an étale sheaf with transfer. Then $H_\mathrm{et}^n(-, \mathscr{F})$ is naturally a presheaf with transfer.
Proof.
We consider the canonical acyclic resolution $$ \mathscr{F} \to E(\mathscr{F}) \to E(E(\mathscr{F})/\mathscr{F}) \to \dotsb. $$ Then $H_\mathrm{et}^n(-, \mathscr{F})$ is simply the cohomology of this chain complex, as presheaves on $\mathsf{Sm}_ k$. But because this is also naturally a chain complex in $\mathsf{PST}_ k$, we see that the cohomology presheaves have transfer.