Presheaf with transfers

These are defined as presheaves with transfers such that the restriction to any scheme ${\displaystyle X}$ is an etale sheaf. That is, if ${\displaystyle U\to X}$ is an etale cover, and ${\displaystyle F}$ is a presheaf with transfers, it is an Etale sheaf with transfers if the sequence

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

is exact and there is an isomorphism

${\displaystyle F(X\coprod Y)=F(X)\oplus F(Y)}$

for any fixed smooth schemes ${\displaystyle X,Y}$.

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

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

${\displaystyle {\mathcal {O}}^{*}(Y)\to {\mathcal {O}}^{*}(W){\xrightarrow {N}}{\mathcal {O}}^{*}(X)}$[2]

showing it is a presheaf with transfers.

One of the basic examples of presheaves with transfers are given by representable functors. Given a smooth scheme ${\displaystyle X}$ there is a presheaf with transfers ${\displaystyle \mathbb {Z} _{tr}(X)}$ sending ${\displaystyle U\mapsto {\text{Hom}}_{Cor}(U,X)}$.[2]

The associated presheaf with transfers of ${\displaystyle {\text{Spec}}(k)}$ is denoted ${\displaystyle \mathbb {Z} }$.

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

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

Given a finite family of pointed schemes ${\displaystyle (X_{i},x_{i})}$ there is an associated presheaf with transfers ${\displaystyle \mathbb {Z} _{tr}((X_{1},x_{1})\wedge \cdots \wedge (X_{n},x_{n}))}$, also denoted ${\displaystyle \mathbb {Z} _{tr}(X_{1}\wedge \cdots \wedge X_{n})}$[2] from their Smash product. This is defined as the cokernel of

${\displaystyle {\text{coker}}\left(\bigoplus _{i}\mathbb {Z} _{tr}(X_{1}\times \cdots \times {\hat {X}}_{i}\times \cdots \times X_{n}){\xrightarrow {id\times \cdots \times x_{i}\times \cdots \times id}}\mathbb {Z} _{tr}(X_{1}\times \cdots \times X_{n})\right)}$

For example, given two pointed schemes ${\displaystyle (X,x),(Y,y)}$, there is the associated presheaf with transfers ${\displaystyle \mathbb {Z} _{tr}(X\wedge Y)}$ equal to the cokernel of

${\displaystyle \mathbb {Z} _{tr}(X)\oplus \mathbb {Z} _{tr}(Y){\xrightarrow {\begin{bmatrix}1\times y&x\times 1\end{bmatrix}}}\mathbb {Z} _{tr}(X\times Y)}$[3]

This is analogous to the smash product in topology since ${\displaystyle X\wedge Y=(X\times Y)/(X\vee Y)}$ where the equivalence relation mods out ${\displaystyle X\times \{y\}\cup \{x\}\times Y}$.

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

There is a scheme

${\displaystyle \Delta ^{n}={\text{Spec}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{\sum _{0\leq i\leq n}x_{i}-1}}\right)}$

giving a cosimplicial scheme ${\displaystyle \Delta ^{*}}$, where the morphisms ${\displaystyle \partial _{j}:\Delta ^{n}\to \Delta ^{n+1}}$ are given by ${\displaystyle x_{j}=0}$. That is,

${\displaystyle {\frac {k[x_{0},\ldots ,x_{n+1}]}{(\sum _{0\leq i\leq n}x_{i}-1)}}\to {\frac {k[x_{0},\ldots ,x_{n+1}]}{(\sum _{0\leq i\leq n}x_{i}-1,x_{j})}}}$

gives the induced morphism ${\displaystyle \partial _{j}}$. Then, to a presheaf with transfers ${\displaystyle F}$, there is an associated complex of presheaves with transfers ${\displaystyle C_{*}F}$ sending

${\displaystyle C_{i}F:U\mapsto F(U\times \Delta ^{i})}$

and has the induced chain morphisms

${\displaystyle \sum _{i=0}^{j}(-1)^{i}\partial _{i}^{*}:C_{j}F\to C_{j-1}F}$

giving a complex of presheaves with transfers. The homology invaritant presheaves with transfers ${\displaystyle H_{i}(C_{*}F)}$ are homotopy invariant. In particular, ${\displaystyle H_{0}(C_{*}F)}$ is the universal homotopy invariant presheaf with transfers associated to ${\displaystyle F}$.

Denote ${\displaystyle H_{0}^{sing}(X/k):=H_{0}(C_{*}\mathbb {Z} _{tr}(X))({\text{Spec}}(k))}$. There is an induced surjection ${\displaystyle H_{0}^{sing}(X/k)\to {\text{CH}}_{0}(X)}$ which is an isomorphism for ${\displaystyle X}$ projective.

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

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

In this case, ${\displaystyle \mathbb {Z} (0)\cong \mathbb {Z} _{tr}(\mathbb {G} _{m}^{\wedge 0})}$ which is quasi-isomorphic to ${\displaystyle \mathbb {Z} }$ (top of page 17),[2] hence the weight ${\displaystyle 0}$ cohomology groups are isomorphic to

${\displaystyle H^{p,0}(X,\mathbb {Z} )={\begin{cases}\mathbb {Z} (X)&{\text{if }}p=0\\0&{\text{otherwise}}\end{cases}}}$

where ${\displaystyle \mathbb {Z} (X)={\text{Hom}}_{Cor}(X,{\text{Spec}}(k))}$. Since an open cover

This case requires more work, but the end result is a quasi-isomorphism between ${\displaystyle \mathbb {Z} (1)}$ and ${\displaystyle {\mathcal {O}}^{*}[-1]}$. This gives the two motivic cohomology groups

${\displaystyle {\begin{aligned}H^{1,1}(X,\mathbb {Z} )&=H_{Zar}^{0}(X,{\mathcal {O}}^{*})={\mathcal {O}}^{*}(X)\\H^{2,1}(X,\mathbb {Z} )&=H_{Zar}^{1}(X,{\mathcal {O}}^{*})={\text{Pic}}(X)\end{aligned}}}$

where the middle cohomology groups are Zariski cohomology.

In general, over a perfect field ${\displaystyle k}$, there is a nice description of ${\displaystyle \mathbb {Z} (n)}$ in terms of presheaves with transfer ${\displaystyle \mathbb {Z} _{tr}(\mathbb {P} ^{n})}$. There is a quasi-ismorphism

${\displaystyle C_{*}(\mathbb {Z} _{tr}(\mathbb {P} ^{n})/\mathbb {Z} _{tr}(\mathbb {P} ^{n-1}))\simeq C_{*}\mathbb {Z} _{tr}(\mathbb {G} _{m}^{\wedge q})[n]}$

hence

${\displaystyle \mathbb {Z} (n)\simeq C_{*}(\mathbb {Z} _{tr}(\mathbb {P} ^{n})/\mathbb {Z} _{tr}(\mathbb {P} ^{n-1}))[-2n]}$

which is found using splitting techniques along with a series of quasi-isomorphisms. The details are in lecture 15 of the Clay Math book.