Presheaf with transfers

In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences (defined below) to the category of abelian groups (in category theory, “presheaf” is another term for a contravariant functor).

When a presheaf F with transfers is restricted to the subcategory of smooth separated schemes, it can be viewed as a presheaf on the category with extra maps $$F(Y) \to F(X)$$, not coming from morphisms of schemes but also from finite correspondences from X to Y

A presheaf F with transfers is said to be $\mathbb{A}^1$-homotopy invariant if $$F(X) \simeq F(X \times \mathbb{A}^1)$$ for every X.

For example, Chow groups as well as motivic cohomology groups form presheaves with transfers.

Finite correspondence
Let $$X, Y$$ be algebraic schemes (i.e., separated and of finite type over a field) and suppose $$X$$ is smooth. Then an elementary correspondence is an irreducible closed subscheme $$W \subset X_i \times Y$$, $$X_i$$ some connected component of X, such that the projection $$\operatorname{Supp}(W) \to X_i$$ is finite and surjective. Let $$\operatorname{Cor}(X, Y)$$ be the free abelian group generated by elementary correspondences from X to Y; elements of $$\operatorname{Cor}(X, Y)$$ are then called finite correspondences.

The category of finite correspondences, denoted by $$Cor$$, is the category where the objects are smooth algebraic schemes over a field; where a Hom set is given as: $$\operatorname{Hom}(X, Y) = \operatorname{Cor}(X, Y)$$ and where the composition is defined as in intersection theory: given elementary correspondences $$\alpha$$ from $$X$$ to $$Y$$ and $$\beta$$ from $$Y$$ to $$Z$$, their composition is:
 * $$\beta \circ \alpha = p_{{13}, *}(p^*_{12} \alpha \cdot p^*_{23} \beta)$$

where $$\cdot$$ denotes the intersection product and $$p_{12}: X \times Y \times Z \to X \times Y$$, etc. Note that the category $$Cor$$ is an additive category since each Hom set $$\operatorname{Cor}(X, Y)$$ is an abelian group.

This category contains the category $$\textbf{Sm}$$ of smooth algebraic schemes as a subcategory in the following sense: there is a faithful functor $$\textbf{Sm} \to Cor$$ that sends an object to itself and a morphism $$f: X \to Y$$ to the graph of $$f$$.

With the product of schemes taken as the monoid operation, the category $$Cor$$ is a symmetric monoidal category.

Sheaves with transfers
The basic notion underlying all of the different theories are presheaves with transfers. These are contravariant additive functors"$F:\text{Cor}_k \to \text{Ab}$"and their associated category is typically denoted $$\mathbf{PST}(k)$$, or just $$\mathbf{PST}$$ if the underlying field is understood. Each of the categories in this section are abelian categories, hence they are suitable for doing homological algebra.

Etale sheaves with transfers
These are defined as presheaves with transfers such that the restriction to any scheme $$X$$ is an etale sheaf. That is, if $$U \to X$$ is an etale cover, and $$F$$ is a presheaf with transfers, it is an Etale sheaf with transfers if the sequence"$0 \to F(X) \xrightarrow{\text{diag}} F(U) \xrightarrow{(+,-)} F(U\times_XU)$|undefined"is exact and there is an isomorphism"$F(X\coprod Y) = F(X)\oplus F(Y)$"for any fixed smooth schemes $$X,Y$$.

Nisnevich sheaves with transfers
There is a similar definition for Nisnevich sheaf with transfers, where the Etale topology is switched with the Nisnevich topology.

Units
The sheaf of units $$\mathcal{O}^*$$ is a presheaf with transfers. Any correspondence $$W \subset X \times Y$$ induces a finite map of degree $$N$$ over $$X$$, hence there is the induced morphism"$\mathcal{O}^*(Y) \to \mathcal{O}^*(W) \xrightarrow{N} \mathcal{O}^*(X)$"showing it is a presheaf with transfers.

Representable functors
One of the basic examples of presheaves with transfers are given by representable functors. Given a smooth scheme $$X$$ there is a presheaf with transfers $$\mathbb{Z}_{tr}(X)$$ sending $$U \mapsto \text{Hom}_{Cor}(U,X)$$.

Representable functor associated to a point
The associated presheaf with transfers of $$\text{Spec}(k)$$ is denoted $$\mathbb{Z}$$.

Pointed schemes
Another class of elementary examples comes from pointed schemes $$(X,x)$$ with $$x: \text{Spec}(k) \to X$$. This morphism induces a morphism $$x_*:\mathbb{Z} \to \mathbb{Z}_{tr}(X)$$ whose cokernel is denoted $$\mathbb{Z}_{tr}(X,x)$$. There is a splitting coming from the structure morphism $$X \to \text{Spec}(k)$$, so there is an induced map $$\mathbb{Z}_{tr}(X) \to \mathbb{Z}$$, hence $$\mathbb{Z}_{tr}(X) \cong \mathbb{Z}\oplus\mathbb{Z}_{tr}(X,x)$$.

Representable functor associated to A1-0
There is a representable functor associated to the pointed scheme $$\mathbb{G}_m = (\mathbb{A}^1-\{0\},1)$$ denoted $$\mathbb{Z}_{tr}(\mathbb{G}_m)$$.

Smash product of pointed schemes
Given a finite family of pointed schemes $$(X_i, x_i)$$ there is an associated presheaf with transfers $$\mathbb{Z}_{tr}((X_1,x_1)\wedge\cdots\wedge(X_n,x_n))$$, also denoted $$\mathbb{Z}_{tr}(X_1\wedge\cdots\wedge X_n)$$ from their Smash product. This is defined as the cokernel of"$\text{coker}\left( \bigoplus_i \mathbb{Z}_{tr}(X_1\times \cdots \times \hat{X}_i \times \cdots \times X_n) \xrightarrow{id\times \cdots \times x_i \times \cdots \times id} \mathbb{Z}_{tr}(X_1\times\cdots\times X_n) \right)$"For example, given two pointed schemes $$(X,x),(Y,y)$$, there is the associated presheaf with transfers $$\mathbb{Z}_{tr}(X\wedge Y)$$ equal to the cokernel of"$\mathbb{Z}_{tr}(X)\oplus \mathbb{Z}_{tr}(Y) \xrightarrow{ \begin{bmatrix}1\times y & x\times 1 \end{bmatrix}} \mathbb{Z}_{tr}(X\times Y)$ |undefined"This is analogous to the smash product in topology since $$X\wedge Y = (X \times Y) / (X \vee Y)$$ where the equivalence relation mods out $$X\times \{y\} \cup \{x\}\times Y$$.

Wedge of single space
A finite wedge of a pointed space $$(X,x)$$ is denoted $$\mathbb{Z}_{tr}(X^{\wedge q}) = \mathbb{Z}_{tr}(X\wedge \cdots \wedge X)$$. One example of this construction is $$\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge q})$$, which is used in the definition of the motivic complexes $$\mathbb{Z}(q)$$ used in Motivic cohomology.

Homotopy invariant sheaves
A presheaf with transfers $$F$$ is homotopy invariant if the projection morphism $$p:X\times\mathbb{A}^1 \to X$$ induces an isomorphism $$p^*:F(X) \to F(X\times \mathbb{A}^1)$$ for every smooth scheme $$X$$. There is a construction associating a homotopy invariant sheaf  for every presheaf with transfers $$F$$ using an analogue of simplicial homology.

Simplicial homology
There is a scheme"$\Delta^n = \text{Spec}\left( \frac{k[x_0,\ldots,x_n]}{\sum_{0 \leq i \leq n} x_i - 1} \right)$"giving a cosimplicial scheme $$\Delta^*$$, where the morphisms $$\partial_j:\Delta^n \to \Delta^{n+1}$$ are given by $$x_j = 0$$. That is,"$\frac{k[x_0,\ldots,x_{n+1}]}{(\sum_{0 \leq i \leq n} x_i - 1)} \to \frac{k[x_0,\ldots,x_{n+1}]}{(\sum_{0 \leq i \leq n} x_i - 1, x_j)} $"gives the induced morphism $$\partial_j$$. Then, to a presheaf with transfers $$F$$, there is an associated complex of presheaves with transfers $$C_*F$$ sending"$C_iF: U \mapsto F(U \times \Delta^i)$"and has the induced chain morphisms"$\sum_{i=0}^j (-1)^i \partial_i^*: C_jF \to C_{j-1}F$"giving a complex of presheaves with transfers. The homology invaritant presheaves with transfers $$H_i(C_*F)$$ are homotopy invariant. In particular, $$H_0(C_*F)$$ is the universal homotopy invariant presheaf with transfers associated to $$F$$.

Relation with Chow group of zero cycles
Denote $$H_0^{sing}(X/k) := H_0(C_*\mathbb{Z}_{tr}(X))(\text{Spec}(k))$$. There is an induced surjection $$H_0^{sing}(X/k) \to \text{CH}_0(X)$$ which is an isomorphism for $$X$$ projective.

Zeroth homology of Ztr(X)
The zeroth homology of $$H_0(C_*\mathbb{Z}_{tr}(Y))(X) $$ is $$\text{Hom}_{Cor}(X,Y)/\mathbb{A}^1 \text{ homotopy}$$ where homotopy equivalence is given as follows. Two finite correspondences $$f,g:X \to Y$$ are $$\mathbb{A}^1$$-homotopy equivalent if there is a morphism $$h:X\times\mathbb{A}^1 \to X$$ such that $$h|_{X\times 0} = f$$ and $$h|_{X \times 1} = g$$.

Motivic complexes
For Voevodsky's category of mixed motives, the motive $$M(X)$$ associated to $$X$$, is the class of $$C_*\mathbb{Z}_{tr}(X)$$ in $$DM_{Nis}^{eff,-}(k,R)$$. One of the elementary motivic complexes are $$\mathbb{Z}(q)$$ for $$q \geq 1$$, defined by the class of"$\mathbb{Z}(q) = C_*\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge q})[-q]$"For an abelian group $$A$$, such as $$\mathbb{Z}/\ell$$, there is a motivic complex $$A(q) = \mathbb{Z}(q) \otimes A$$. These give the motivic cohomology groups defined by"$H^{p,q}(X,\mathbb{Z}) = \mathbb{H}_{Zar}^p(X,\mathbb{Z}(q))$"since the motivic complexes $$\mathbb{Z}(q)$$ restrict to a complex of Zariksi sheaves of $$X$$. These are called the $$p$$-th motivic cohomology groups of weight $$q$$. They can also be extended to any abelian group $$A$$,"$H^{p,q}(X,A) = \mathbb{H}_{Zar}^p(X,A(q))$"giving motivic cohomology with coefficients in $$A$$ of weight $$q$$.

Special cases
There are a few special cases which can be analyzed explicitly. Namely, when $$q = 0,1$$. These results can be found in the fourth lecture of the Clay Math book.

Z(0)
In this case, $$\mathbb{Z}(0) \cong \mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge 0})$$ which is quasi-isomorphic to $$\mathbb{Z}$$ (top of page 17), hence the weight $$0$$ cohomology groups are isomorphic to $$H^{p,0}(X,\mathbb{Z}) = \begin{cases} \mathbb{Z}(X) & \text{if } p = 0 \\ 0 & \text{otherwise} \end{cases}$$ where $$\mathbb{Z}(X) = \text{Hom}_{Cor}(X,\text{Spec}(k))$$. Since an open cover

Z(1)
This case requires more work, but the end result is a quasi-isomorphism between $$\mathbb{Z}(1)$$ and $$\mathcal{O}^*[-1]$$. This gives the two motivic cohomology groups $$\begin{align} H^{1,1}(X,\mathbb{Z}) &= H^0_{Zar}(X,\mathcal{O}^*) = \mathcal{O}^*(X) \\ H^{2,1}(X,\mathbb{Z}) &= H^1_{Zar}(X,\mathcal{O}^*) = \text{Pic}(X) \end{align}$$ where the middle cohomology groups are Zariski cohomology.

General case: Z(n)
In general, over a perfect field $$k$$, there is a nice description of $$\mathbb{Z}(n)$$ in terms of presheaves with transfer $$\mathbb{Z}_{tr}(\mathbb{P}^n)$$. There is a quasi-ismorphism $$C_*(\mathbb{Z}_{tr}(\mathbb{P}^n) / \mathbb{Z}_{tr}(\mathbb{P}^{n-1})) \simeq C_*\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge q})[n] $$ hence"$\mathbb{Z}(n) \simeq C_{*}(\mathbb {Z} _{tr}(\mathbb {P} ^{n})/\mathbb {Z} _{tr}(\mathbb {P} ^{n-1}))[-2n] $"which is found using splitting techniques along with a series of quasi-isomorphisms. The details are in lecture 15 of the Clay Math book.