Presheaves with transfer

The norm residue isomorphism theorem

The goal is to learn Voevodsky’s theory of motives, developed by Voevodsky and Suslin in the 1990s. A remarkable achievement of this theory is the proof of the Bloch–Kato conjecture. What does this say? There is the Brauer group $\mathrm{Br}(F)$ that classifies central simple algebras. For $\operatorname{char}(F) \neq 2$ and $a, b \in F^\times$, we have the quaternion algebras $$ H(a, b) = \langle i, j \mid i^2 = a, j^2 = b, ij = -ji \rangle \in \mathrm{Br}(F)[2], $$ and more generally, if $n$ is invertible in $F$ and $\mu_n \subseteq F^\times$ we have $$ A_\zeta(a, b) = \langle i, j \mid i^n = a, j^n = b, ij = \zeta ji \rangle \in \mathrm{Br}(F)[n]. $$ Now a natural question is whether these generate the torsion in the Brauer group.

Definition 1. We define the abelian group $K_2^\mathrm{M}(F)$ generated by symbols $\lbrace a, b \rbrace$ for $a, b \in F^\times$ and with relations
$\lbrace ab, c \rbrace = \lbrace a, c \rbrace + \lbrace b, c \rbrace$,
$\lbrace a, bc \rbrace = \lbrace a, b \rbrace + \lbrace a, c \rbrace$,
$\lbrace a, 1-a \rbrace = 0$ for $a \neq 0, 1$.

There is then a norm residue map $$ K_2^\mathrm{M}(F) \to H^2(F, \mu_n^{\otimes 2}); \quad \lbrace a, b \rbrace \mapsto [a] \cup [b]. $$ Here, we are using Kummer theory to identify $H^1(F, \mu_n) = F^\times / (F^\times)^n$.

Theorem 2 (Merkurjev–Suslin). We have an isomorphism $K_2^\mathrm{M}(F) / nK_2^\mathrm{M}(F) \cong H^2(F, \mu_n^{\otimes 2})$.

We can similarly define a higher version of these Milnor K-groups, generated by these symbols $\lbrace a_1, \dotsc, a_m \rbrace$ and with the multiplication relations and $\lbrace \cdots, a, \cdots, 1-a, \cdots \rbrace = 0$.

Theorem 3 (Voevodsky). We have an isomorphism $$ K_m^\mathrm{M}(F) / nK_m^\mathrm{M}(F) \cong H^m(F, \mu_n^{\otimes m}), $$ similarly given by $\lbrace a_1, \dotsc, a_m \rbrace \mapsto [a_1] \cup \dotsb \cup [a_m]$.

The way Voevodsky proved this was by inducing a complex of sheaves $\mathbb{Z}(m)$ on the big étale site over $F$ such that its cohomology recovers Chow groups, Milnor K-theory, and so on. In particular, we will prove that $$ K_m^\mathrm{M}(F) = H_\mathrm{Zar}^m(\Spec F, \mathbb{Z}(m)). $$ On the other hand, we have $$ H_\mathrm{et}^m(\Spec F, \mathbb{Z}(m) \otimes \mathbb{Z}/n\mathbb{Z}) = H^m(F, \mu_n^{\otimes m}). $$ Then the norm residue map is simply the change of topology map. So to compare the two, what we need to show is that if we denote by $\pi \colon (\Spec F)_ \mathrm{et} \to (\Spec F)_ \mathrm{Zar}$ the morphism of topoi, that $$ R\pi_\ast(\mathbb{Z}(m) \otimes \mathbb{Z}/n\mathbb{Z}) $$ is concentrated in degree at least $m$.

Multi-valued maps

We want to allow multi-valued maps in this theory. Let $F$ be a field with no other assumptions.

Definition 4. Let $X, Y$ be algebraic varieties over $F$ and assume that $X$ is smooth. An elementary multi-valued map is an irreducible subvariety $W \subseteq X \times Y$ such that $W \to X$ is finite and surjective. A multi-valued map is a $\mathbb{Z}$-linear combination of multi-valued maps. The group of multi-valued maps is denoted by $\operatorname{Cor}(X, Y)$.

Example 5. There is the square-root map $\mathbb{A}_ x^1 \to \mathbb{A}_ y^1$, which is just the variety cut out by $y^2 = x$.

The way we interpret it as a multi-valued map is that $X \xleftarrow{\pi_X} W \xrightarrow{\pi_Y} Y$ induces a map $$ Z_0(X) \to Z_0(Y); \quad [x] \mapsto \sum n_i [y_i] $$ where the fiber over $x$ is $\pi_X^{-1} = \lbrace (x, y_i) \rbrace$ with $(x, y_i)$ of multiplicity $y$.

Example 6. A regular map can be interpreted as a multi-valued map. For a finite surjection $f \colon X \to Y$, there is a corresponding inverse $f^{-1} \in \mathrm{Cor}(Y, X)$.

Proposition 7. Let $X, Y, Z$ be algebraic varieties, where $X, Y$ are smooth. Let $V \subseteq X \times Y$ and $W \subseteq Y \times W$ be multi-valued maps. Then $\pi_Z^{-1}(V)$ and $\pi_X^{-1}(W)$ intersect properly and the intersection $\pi_Z^{-1}(V) \cap \pi_X^{-1}(W)$ defines a multi-valued map.

Proof.

Because $X$ and $Y$ are smooth, finite surjective is equivalent to finite flat. So now $V \times_Y W \to X$ is also finite flat. This already shows that the intersection is proper. Now for each component $T_i \subseteq V \times_Y W$, we look at the image of $T_i \to X \times Z$ and take their linear combinations.

Definition 8. We now define the category $\mathsf{Cor}_ F$ where objects are smooth algebraic varieties and morphisms are multi-valued maps.

Presheaves with transfer

Definition 9. A presheaf with transfers is a contravariant additive functor $\mathscr{F} \colon \mathsf{Cor}_ F \to \mathsf{Ab}$. The category of presheaves with transfers is denoted by $\mathsf{PST}_ \mathrm{F}$.

Roughly this means that we have pullbacks $f^\ast$ as well as pushfowards $f_\ast$ along finite surjective maps.

Example 10. The sheaves $\mathscr{O}^\ast$ and $\mathscr{O}$ have transfers, namely the norm and the trace. The Chow groups $\mathrm{CH}^i$ also have transfers.

Example 11. For every algebraic variety $X$ (not necessarily smooth) we have the presheaf $\mathbb{Z}_ \mathrm{tr}(X)$ defined by $\mathbb{Z}_ \mathrm{tr}(X)(U) = \operatorname{Cor}(U, X)$. We have $\mathbb{Z}_ \mathrm{tr}(\Spec F) = \mathbb{Z}$.

Definition 12. If $(X, x)$ is a pointed algebraic variety, we can define $$ \mathbb{Z}_ \mathrm{tr}(X, x) = \coker(\mathbb{Z} \to \mathbb{Z}_ \mathrm{tr}(X)). $$ If $(X, x)$ and $(Y, y)$ are two pointed algebraic varieties, we define $$ \mathbb{Z}_ \mathrm{tr}(X \wedge y) = \coker(\mathbb{Z}_ \mathrm{tr}(X \times \lbrace y \rbrace \cup \lbrace x \rbrace \cup Y) \to \mathbb{Z}_ \mathrm{tr}(X \times Y)). $$

$\mathbb{A}^1$-homotopy

We can also now to singular homology of algebraic varieties. We define $\Delta_F^n = \Spec F[x_0, \dotsc, x_n] / (x_0 + \dotsb + x_n - 1)$ and face maps $\partial_i \colon \Delta_F^n \to \Delta_F^{n+1}$ as usual.

Definition 13. For $\mathscr{F} \in \mathsf{PST}_ F$, we define a chain complex $C_\bullet \mathscr{F}$ as $$ C_n(\mathscr{F})(U) = \mathscr{F}(\Delta_F^n \times U), \quad d_n = \sum (-1)^i (\partial_i \times \id)^\ast. $$ We also write $C_\bullet(X) = C_\bullet(\mathbb{Z}_ \mathrm{tr}(X))$.

Definition 14. We say that a presheaf $\mathscr{F}$ is $\mathbb{A}^1$-invariant if for all smooth $X$ the projection $\pi \colon X \times \mathbb{A}^1 \to X$ induces an isomorphism $\mathscr{F}(X) \to \mathscr{F}(X \times \mathbb{A}^1)$.

Definition 15. Let $X, Y$ be smooth algebraic varieties, and let $f, g \in \operatorname{Cor}(X, Y)$. We say that $f, g$ are $\mathbb{A}^1$-homotopic if there exists a multi-valued map $h \colon \operatorname{Cor}(X \times \mathbb{A}^1, Y)$ such that $h \vert_{X \times \lbrace 0 \rbrace} = f$ and $h \vert_{X \times \lbrace 1 \rbrace} = g$. We say that $f$ is an $\mathbb{A}^1$-homotopy equivalence when there exists an inverse $g$ such that both $fg$ and $gf$ are $\mathbb{A}^1$-homotopic to identity maps.

This is subtly stronger than just being rationally equivalent, because we really need entire family to be finite flat over $X \times \mathbb{A}^1$.

Remark 16. This is in fact an equivalence relation. Given $f_0 \sim f_1$ and $f_1 \sim f_2$, we can simply add them and get $f_0 + f_1 \sim f_1 + f_2$, and then subtract off $f_1$ from both sides.

The goal for the rest of today is to show that an $\mathbb{A}^1$-homotopy equivalence induces a chain homotopy between $C_\bullet$.

Lemma 17. A presheaf with transfer $\mathscr{F}$ is $\mathbb{A}^1$-homotopy invariant if and only if for every $X$ the maps $i_0, i_1 \colon X \hookrightarrow X \times \mathbb{A}^1$ induce the same maps $\mathscr{F}(X) \to \mathscr{F}(X \times \mathbb{A}^1)$.

Proof.

We look at $i_0 \circ \pi$ and $i_1 \circ \pi$ as $X \times \mathbb{A}^1 \to X \times \mathbb{A}^1$. Now we apply the assumption to $X \times \mathbb{A}^1$ to see that $(x, t) \mapsto (x, t, 0)$ and $(x, t) \mapsto (x, t, 1)$ give the same maps, and then compose it.

Lemma 18. The maps $i_0^\ast, i_1^\ast \colon C_\bullet(X \times \mathbb{A}^1) \to C_\bullet(X)$ are chain homotopic.

Proof.

You do it in the same way you do it for singular homology.

Corollary 19. For every $\mathscr{F} \in \mathsf{PST}_ F$, the homology groups $H_n(C_\bullet(F))$ are $\mathbb{A}^1$-homotopy invariant. Moreover, if there exists a natural transformation $$ h_X \colon \mathscr{F}(X) \to \mathscr{F}(X \times \mathbb{A}^1) $$ such that $i_0^\ast \circ h_X = 0$ and $i_1^\ast \circ h_X = \id$, then $C_\bullet \mathscr{F}$ is chain contractible.

Theorem 20. The map $\pi^\ast \colon C_\bullet(X) \to C_\bullet(X \times \mathbb{A}^1)$ is a chain homotopy equivalence.

Proof.

Because we have a left inverse $i_0^\ast \colon C_\bullet(X \times \mathbb{A}^1) \to C_\bullet(X)$, we can split off $$ C_\bullet(X \times \mathbb{A}^1) = C_\bullet(X) \oplus C_\bullet(\mathscr{F}), \quad \mathscr{F} = \operatorname{coker}(\mathbb{Z}_ \mathrm{tr}(X) \xrightarrow{i_0^\ast} \mathbb{Z}_ \mathrm{tr}(X \times \mathbb{A}^1)). $$ By the above criterion, it suffices to find a natural transformation $$ h \colon \frac{\operatorname{Cor}(U, \mathbb{X} \times \mathbb{A}^1)}{i_0^\ast \operatorname{Cor}(U, X)} \to \frac{\operatorname{Cor}(U \times \mathbb{A}^1, X \times \mathbb{A}^1)}{i_0^\ast \operatorname{Cor}(U \times \mathbb{A}^1, X)}. $$ We can construct this by sending $f \in \operatorname{Cor}(U, X \times \mathbb{A}^1)$ to $m \circ (f \times \id) \in \operatorname{Cor}(U \times \mathbb{A}^1, X \times \mathbb{A}^1)$, where $m \colon \mathbb{A}^1 \times \mathbb{A}^1 \to \mathbb{A}^1$ is the multiplication map. We can check that $i_0^\ast \circ h = 0$ and $i_1^\ast \circ h = \id$.