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
, not coming from morphisms of schemes but also from finite correspondences from X to Y
A presheaf F with transfers is said to be
-homotopy invariant if
for every X.
For example, Chow groups as well as motivic cohomology groups form presheaves with transfers.
Finite correspondence
Let
be algebraic schemes (i.e., separated and of finite type over a field) and suppose
is smooth. Then an elementary correspondence is an irreducible closed subscheme
,
some connected component of X, such that the projection
is finite and surjective.[1] Let
be the free abelian group generated by elementary correspondences from X to Y; elements of
are then called finite correspondences.
The category of finite correspondences, denoted by
, is the category where the objects are smooth algebraic schemes over a field; where a Hom set is given as:
and where the composition is defined as in intersection theory: given elementary correspondences
from
to
and
from
to
, their composition is:
![{\displaystyle \beta \circ \alpha =p_{{13},*}(p_{12}^{*}\alpha \cdot p_{23}^{*}\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8de34ac78f7b422ca538f55112e8163964f39ef7)
where
denotes the intersection product and
, etc. Note that the category
is an additive category since each Hom set
is an abelian group.
This category contains the category
of smooth algebraic schemes as a subcategory in the following sense: there is a faithful functor
that sends an object to itself and a morphism
to the graph of
.
With the product of schemes taken as the monoid operation, the category
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
![{\displaystyle F:{\text{Cor}}_{k}\to {\text{Ab}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a2fc920cd1c9fcecb29d3001da262262964d6)
and their associated category is typically denoted
, or just
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
is an etale sheaf. That is, if
is an etale cover, and
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb65705fad7acf6ecc8d7e6421dea798426ddfff)
is exact and there is an isomorphism
![{\displaystyle F(X\coprod Y)=F(X)\oplus F(Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/337f3d74e37f7065cbf8a86d90a222269f6fb2e1)
for any fixed smooth schemes
.
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
is a presheaf with transfers. Any correspondence
induces a finite map of degree
over
, hence there is the induced morphism
[2]
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
there is a presheaf with transfers
sending
.[2]
Representable functor associated to a point
The associated presheaf with transfers of
is denoted
.
Pointed schemes
Another class of elementary examples comes from pointed schemes
with
. This morphism induces a morphism
whose cokernel is denoted
. There is a splitting coming from the structure morphism
, so there is an induced map
, hence
.
Representable functor associated to A1-0
There is a representable functor associated to the pointed scheme
denoted
.
Smash product of pointed schemes
Given a finite family of pointed schemes
there is an associated presheaf with transfers
, also denoted
[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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d23a1827e547176f965f9cdd8a88f4a162a424e)
For example, given two pointed schemes
, there is the associated presheaf with transfers
equal to the cokernel of
[3]
This is analogous to the smash product in topology since
where the equivalence relation mods out
.
Wedge of single space
A finite wedge of a pointed space
is denoted
. One example of this construction is
, which is used in the definition of the motivic complexes
used in Motivic cohomology.
Homotopy invariant sheaves
A presheaf with transfers
is homotopy invariant if the projection morphism
induces an isomorphism
for every smooth scheme
. There is a construction associating a homotopy invariant sheaf[2] for every presheaf with transfers
using an analogue of simplicial homology.
Simplicial homology
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e75e6d2dfcc171c442aa7986e437e2a7511b9c31)
giving a cosimplicial scheme
, where the morphisms
are given by
. 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})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54343dd4b4c568fa402bb2b4e3cdc86cbc93dbf0)
gives the induced morphism
. Then, to a presheaf with transfers
, there is an associated complex of presheaves with transfers
sending
![{\displaystyle C_{i}F:U\mapsto F(U\times \Delta ^{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/607bcf0d218cbc8ff4ddf8dd1c464d123bf94aeb)
and has the induced chain morphisms
![{\displaystyle \sum _{i=0}^{j}(-1)^{i}\partial _{i}^{*}:C_{j}F\to C_{j-1}F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2c294c6238bf0688ab83322912ad6376064480)
giving a complex of presheaves with transfers. The homology invaritant presheaves with transfers
are homotopy invariant. In particular,
is the universal homotopy invariant presheaf with transfers associated to
.
Relation with Chow group of zero cycles
Denote
. There is an induced surjection
which is an isomorphism for
projective.
Zeroth homology of Ztr(X)
The zeroth homology of
is
where homotopy equivalence is given as follows. Two finite correspondences
are
-homotopy equivalent if there is a morphism
such that
and
.
Motivic complexes
For Voevodsky's category of mixed motives, the motive
associated to
, is the class of
in
. One of the elementary motivic complexes are
for
, defined by the class of
[2]
For an abelian group
, such as
, there is a motivic complex
. These give the motivic cohomology groups defined by
![{\displaystyle H^{p,q}(X,\mathbb {Z} )=\mathbb {H} _{Zar}^{p}(X,\mathbb {Z} (q))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d4a3d459e0c490f48a6357c505c24b3b59720d)
since the motivic complexes
restrict to a complex of Zariksi sheaves of
.[2] These are called the
-th motivic cohomology groups of weight
. They can also be extended to any abelian group
,
![{\displaystyle H^{p,q}(X,A)=\mathbb {H} _{Zar}^{p}(X,A(q))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40b5bea460cc542100abe173960e61b0090970ea)
giving motivic cohomology with coefficients in
of weight
.
Special cases
There are a few special cases which can be analyzed explicitly. Namely, when
. These results can be found in the fourth lecture of the Clay Math book.
Z(0)
In this case,
which is quasi-isomorphic to
(top of page 17),[2] hence the weight
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb7e09dcfe76fb1132e5ba19ab0c3cb4c20ce7d)
where
. Since an open cover
Z(1)
This case requires more work, but the end result is a quasi-isomorphism between
and
. 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0b1373649cf8a44228d0c383deceb18afc1174)
where the middle cohomology groups are Zariski cohomology.
General case: Z(n)
In general, over a perfect field
, there is a nice description of
in terms of presheaves with transfer
. 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26493e1112945c9674c6b16d8e40bc7a1754ea8)
hence
![{\displaystyle \mathbb {Z} (n)\simeq C_{*}(\mathbb {Z} _{tr}(\mathbb {P} ^{n})/\mathbb {Z} _{tr}(\mathbb {P} ^{n-1}))[-2n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f63855200ac11207905ec52da572505008e91a6)
which is found using splitting techniques along with a series of quasi-isomorphisms. The details are in lecture 15 of the Clay Math book.
See also
- Relative cycle
- Motivic cohomology
- Mixed motives (math)
- Étale topology
- Nisnevich topology
References
- ^ Mazza, Voevodsky & Weibel 2006, Definition 1.1.
- ^ a b c d e f g Lecture Notes on Motivic Cohomology (PDF). Clay Math. pp. 13, 15–16, 17, 21, 22.
- ^ Note
giving
- Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
External links
- https://ncatlab.org/nlab/show/sheaf+with+transfer