Transfers on commutative group schemes

Voevodsky’s theory of presheaves with transfers (see for example [MVW06, Part 1]) plays a central role in his theory of motives over a field. Roughly speaking, they are presheaves on the category of smooth schemes equipped with covariant “transfer maps” for finite surjective morphisms, functorial in an appropriate sense. One important example is commutative group schemes, e.g. $\mathbb{G}_{a}$, $\mathbb{G}_{m}$, the Witt ring scheme and abelian varieties. In particular, smooth commutative group schemes over a field equipped with transfers are known to give examples of reciprocity sheaves (see [KSY16], [KSY]), for which a nice motivic theory can be applied.

On the other hand, transfer structures on group schemes over a general base scheme has not been studied much yet. In this paper we prove the following general existence result.

One way to prove that a presheaf has a transfer structure is to show that it is a qfh sheaf. In this way, Ancona-Huber-Lehalleur [AHL14] proved that for any smooth commutative group scheme $G$ over a noetherian excellent scheme $S$, the presheaf $G_{\mathbb{Q}}=G\otimes_{\mathbb{Z}}\mathbb{Q}$ admits a unique transfer structure. However, being a qfh sheaf is so strong a condition that we cannot expect in general that $G$ itself should be so. Also, transfers constructed in this way are comparatively inexplicit and difficult to compute.

Another way is to use the symmetric product and construct transfers geometrically. In this way, Spieß-Szamuely [SS03] and Barbieri-Viale-Kahn [BVK16, Lemma 1.4.4] proved that any commutative group scheme locally of finite type over a field admits a canonical transfer structure. Our proof of Theorem 0.1 is based on this idea. In section 2, we imitate the construction of locally free transfers due to Spieß-Szamuely in a more general setting. In section 3, we define the canonical transfer on group schemes and prove the functoriality to establish Theorem 0.1. In section 4, we characterize the canonical transfer by a simple condition on radicial transfers.

In this section we recall the theory of relative cycles in the style of Cisinski-Déglise [CD19]; nothing in this section is our original. Let $X$ be a noetherian scheme. A cycle on $X$ is a formal $\mathbb{Z}$-linear combination of integral closed subschemes of $X$. A component of a cycle $\alpha$ is an integral closed subscheme whose coefficient in $\alpha$ is non-zero. For a closed subscheme $W$ of $X$, we define the associated cycle $[W]$ by

where $W_{1},\dots,W_{n}$ are the irreducible components of $W$. If $f\colon X\to Y$ is a morphism between noetherian schemes and $\alpha$ is a cycle on $X$, then the cycle $f_{*}\alpha$ on $Y$ is defined by linearly extending

where $\xi$ is the generic point of $V$.

Until the end of this section, we fix a noetherian base scheme $S$ and consider only noetherian schemes over $S$.

Consider the following diagram of noetherian schemes.

Let $\alpha$ (resp. $\beta$) be a cycle on $X$ (resp. $Y$). If $\alpha$ is $\mathbb{Z}$-universal over $S$ in the sense of [CD19], then a cycle $\alpha\otimes_{S}\beta$ on $X\times_{S}Y$ called the pullback of $\alpha$ by $\beta$ is defined. The operation ${-}\otimes_{S}{-}$ is bilinear.

Let $X,Y,Z$ be noetherian $S$-schemes. We set $c_{S}(X,Y):=c_{0}(X\times_{S}Y/X)$ and call its elements finite correspondences from $X$ to $Y$ over $S$. For example, the graph $\Gamma_{f}$ of a morphism $f\colon X\to Y$ over $S$ gives an element $[\Gamma_{f}]\in c_{S}(X,Y)$, for which we simply write $f$. For $\alpha\in c_{S}(X,Y)$ and $\beta\in c_{S}(Y,Z)$, we define a cycle $\beta\circ\alpha$ on $X\times_{S}Z$ by the formula

where ${{\mathrm{pr}}_{13}}_{*}\colon X\times_{S}Y\times_{S}Z\to X\times_{S}Z$ is the canonical projection. By the definition of pullback (cf. [CD19, Theorem 8.1.39]) and [CD19, Corollary 8.2.6], we have $\beta\otimes_{Y}\alpha\in c_{0}(X\times_{S}Y\times_{S}Z/X)$. If $Z$ is separated and of finite type over $S$, then it follows from [CD19, Section 9.1.1] that $\beta\circ\alpha\in c_{S}(X,Z)$. This gives a bilinear pairing

Let ${\mathrm{Sm}}_{S}$ denote the category of smooth separated $S$-schemes of finite type. Using the pairing above as composition, we can define an additive category ${\mathrm{Cor}}_{S}$ whose objects are the same as ${\mathrm{Sm}}_{S}$ and the morphisms are finite correspondences. The category ${\mathrm{Sm}}_{S}$ can be embedded into ${\mathrm{Cor}}_{S}$ by the graph construction. A presheaf ${\mathrm{Cor}}_{S}^{\mathrm{op}}\to{\mathrm{Ab}}$ is called a presheaf with transfers over $S$.

In this section we construct transfers for finite locally free morphisms.

Let $A$ be a ring and $B$ a finite $A$-algebra. We write $B^{\odot d}$ for the $A$-algebra $(B^{\otimes d})^{S_{d}}$, i.e. the subalgebra of $B^{\otimes d}$ fixed under the canonical action of the symmetric group $S_{d}$.

Suppose that $B$ is free of rank $d$ as an $A$-module. Let $e_{1},\dots,e_{d}$ be a basis of $B$ over $A$ and let $I_{[m,n]}^{d}$ denote the set of $S_{d}$-orbits in $\{m,m+1,\dots,n\}^{d}$. For an orbit $\Gamma\in I_{[1,d]}^{d}$, we set

Then $\{e_{\Gamma}\}_{\Gamma\in I_{[1,d]}^{d}}$ is a basis of $B^{\odot d}$ over $A$. We define an action of $B^{\odot d}$ on $\wedge^{d}B$ by

This is well-defined since we are taking a sum over all possible permutations of $(i_{1},\dots,i_{d})$. In this way we obtain a morphism of $A$-algebras

One can easily verify that this does not depend on the choice of the basis.

Let $S$ be a scheme. For a finite morphism $f\colon Y\to X$ between $S$-schemes, the symmetric product $\operatorname{Sym}^{d}_{X}(Y)$ is defined to be the quotient of $Y\times_{X}\dots\times_{X}Y$ ($d$ times) by the canonical action of the symmetric group $S_{d}$, i.e.

(see [Gro71, Exposé V, 1] for basic facts about quotients of schemes by finite groups).

Suppose that $f$ is a finite locally free morphism of constant rank $d$. The morphism of $\mathcal{O}_{X}$-algebras $u\colon(f_{*}\mathcal{O}_{Y})^{\odot d}\to\mathcal{O}_{X}$ (see the preamble to Lemma 2.1) gives rise to a morphism $f^{\sharp}\colon X\to\operatorname{Sym}^{d}_{X}(Y)$ (note that this morphism is considered also in [SV96, Section 6]). Now let $G$ be a commutative group scheme over $S$. For any $g\in G(Y)$, the morphism

is $S_{d}$-invariant, so it descends to a morphism $\sigma(g)\colon\operatorname{Sym}_{X}^{d}(Y)\to G$. We define $f_{*}g\in G(X)$ to be the composition

In this section we fix a noetherian base scheme $S$ and a separated commutative group scheme $G$ over $S$. First we note the following fact.

Let $X,Y$ be noetherian integral schemes over $S$ and $\alpha\in c_{S}(X,Y)$. We construct the canonical transfer $\alpha^{*}\colon G(Y)\to G(X)$ in the following two cases:

1. (1)

   $\alpha$ is flat over $X$.

2. (2)

   $X$ is normal.

First suppose that $\alpha$ is flat over $X$. Write $\alpha=\sum_{i=1}^{n}m_{i}[V_{i}]$. Then each $V_{i}$ is finite locally free of constant rank over $X$. Let $p_{i}\colon V_{i}\to X$ and $q_{i}\colon V_{i}\to Y$ be canonical projections. We define

Note that for an $S$-morphism $f\colon X\to Y$, this definition of $f^{*}$ coincides with the usual pullback.

Next suppose that $X$ is normal. Take a dense open subset $U\subset X$ such that $\alpha|_{U}$ is flat over $U$. By the next lemma, for any $g\in G(Y)$ the element $(\alpha|_{U})^{*}g\in G(U)$ lies in the image of $G(X)\to G(U)$. We define $\alpha^{*}\colon G(Y)\to G(X)$ by setting $\alpha^{*}g:=(\alpha|_{U})^{*}g$; this does not depend on the choice of $U$.

In the following, we will prove that the canonical transfer defined above is functorial, i.e. $\alpha^{*}\beta^{*}=(\beta\circ\alpha)^{*}$. We start with a special case.