# 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
   $0\to F(X){\xrightarrow {\text{diag}}}F(U){\xrightarrow {(+,-)}}F(U\times _{X}U)$
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.

## Examples
### 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 A$^1$-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
   $\mathbb {Z} _{tr}(X)\oplus \mathbb {Z} _{tr}(Y){\xrightarrow {\begin{bmatrix}1\times y&x\times 1\end{bmatrix}}}\mathbb {Z} _{tr}(X\times Y)$
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
   $C_{i}F:U\mapsto F(U\times \Delta ^{i})$
and has the induced chain morphisms
   $\sum _{i=0}^{j}(-1)^{i}\partial _{i}^{*}:C_{j}F\to C_{j-1}F$
giving a complex of presheaves with transfers. The homology invariant 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 Z$_{tr}$(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
   ${\begin{aligned}H^{1,1}(X,\mathbb {Z} )&=H_{Zar}^{0}(X,{\mathcal {O}}^{*})={\mathcal {O}}^{*}(X)\\H^{2,1}(X,\mathbb {Z} )&=H_{Zar}^{1}(X,{\mathcal {O}}^{*})={\text{Pic}}(X)\end{aligned}}$
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.