Motivic Cohomology II: A1 algebraic topology

Where we’re at, and where we’re going

In the last post (some time ago now, as I have been delayed by responsibilities) we looked at general categorical constructions. In the coming weeks we’ll use these to construct categories of motives.

In this post, we’ll work two main angles.

For starters, for any presheaf with transfers F we’ll define a complex of presheaves with transfers depending on F which is analogous to the singular homology complex of algebraic topology. In fact, we’ll define algebraic singular homology of a smooth scheme X to be the homology of the complex associated to our representable presheaves, \mathbb{Z}_{tr} X, evaluated at a point.

Next, we’ll explore the analog of homotopy in an algebraic setting. That is, we define what it means for two morphisms between schemes to be homotopy equivalent, also said \mathbb{A}^1-homotopic. We give an extension of this definition to sheaves with transfers and show that it coincides with our previous one on representable objects.

Finally, we conclude with a definition of \mathbb{A}^1 weak equivalence. This will be the last definition we need to fully construct our category of motives. In the next post, we’ll carry out this construction for étale motives and investigate the structure of this category.

Remark: quite a lot of the proofs have been differed to [MVW]. I have no idea if I will ever complete them.

Algebraic Singular Homology

Our first problem when defining singular homology is that we can no longer use the interval, and consequently simplices, in the algebraic setting. That is to say, there is no natural analog of the interval for arbitrary fields. Even if we specialized to \mathbb{R}, there’s no immediately obvious way to describe the interval as a scheme. To circumvent this, we instead use the affine line \mathbb{A}^1 as our analog of the interval. Similarly, our notion of simplex will have to be changed. In algebraic topology the n-dimensional simplex is always isomorphic (i.e. homeomorphic) to the nth power of the interval. In our analog this means our simplex will be isomorphic to \mathbb{A}^n.

(4.1) Definition: Fix a field k. We define the algebraic n-simplex to be the scheme

\Delta^n:=\text{Spec}(k[x_0,...,x_n]/(\sum_{i=0}^{n} x_i=1).

We define the jth face maps \partial_j:\Delta^{n-1}\rightarrow \Delta^n via the morphism k[x_0,...,x_n]/(\sum x_i=1)\rightarrow k[x_0,...,x_{n-1}]/(\sum x_i=1) defined by x_i\mapsto 0.

In the case k=\mathbb{R}, and we only consider real valued points, we recover our intuition from topology in the sense that \Delta^1_\mathbb{R}(\mathbb{R}) is the line which contains the topological 1-simplex, and \Delta^2_\mathbb{R}(\mathbb{R}) is the plane which contains the topological 2-simplex.

(4.2) Definition: If F is a presheaf with transfers, we can define a new presheaf with transfers F^{\Delta^n}

F^{\Delta^n}(U):=F(U\times \Delta^n); F^{\Delta^n}(\gamma):= F(\gamma\times \Gamma_{\text{id}_{\Delta^n}})

for any smooth scheme U and any elementary correspondence \gamma \in Cor_k(U,V). Here \gamma\times \Gamma_{\text{id}_{\Delta^n}} is considered as an element of \text{Cor}_k(U\times \Delta^n, V\times \Delta^n). This extends additively to any finite correspondence.

From any presheaf with transfers, constructing F^{\Delta^n} allows us to define a chain complex of presheaves with transfers as so:

C_\ast F:\,\,\,\,\, \cdots\rightarrow F^{\Delta^2}\rightarrow F^{\Delta^1}\rightarrow F^{\Delta^0}\rightarrow 0

with morphisms F^{\Delta^n}(X)\rightarrow F^{\Delta^{n-1}}(X), that is to say evaluated on smooth schemes X, the maps F(\sum_{i=0}^n(-1)^i\Gamma_{\text{id}_X\times \partial_i}). Said differently, our morphisms are applications of F to the alternating sum of correspondences which are the graphs of the morphisms of schemes \text{id}_X\times \partial_i:X\times\Delta^{n-1}\rightarrow X\times \Delta^n. To see this is a complex we can use the functorality of F

F(\sum_{i=0}^n (-1)^i \Gamma_{\text{id}\times \partial_i}) \circ F(\sum_{i=0}^{n-1} (-1)^{n-1} \Gamma_{\text{id}\times \partial_i})= F((\sum_{i=0}^{n-1} (-1)^i \Gamma_{\text{id}\times \partial_i})\circ (\sum_{i=0}^n (-1)^i \Gamma_{\text{id}\times \partial_i}))

= F(0)=0.

where the last two equalities ( =F(0)=0) follow from a computation similar to checking the boundary operator in singular homology is a differential (I said the same thing in the second post I wrote in this series for a different complex).

(4.3) Definition: The algebraic singular homology of a scheme X with coefficients in R is defined to be the homology of the complex C_\ast R_{tr}X evaluated at a point \text{Spec}(k),

H^{sing}_p(X,R):=H_p(C_\ast R_{tr}X(\text{Spec} k)).

If we unravel all the notation, the complex for \mathbb{Z}_{tr}X(\text{Spec} k) looks like:

 \cdots\rightarrow \mathbb{Z}_{tr}X^{\Delta^2}(\text{Spec} k)\rightarrow \mathbb{Z}_{tr}X^{\Delta^1}(\text{Spec} k)\rightarrow \mathbb{Z}_{tr}X^{\Delta^0}(\text{Spec} k)\rightarrow 0

equivalently,

\cdots \rightarrow \mathbb{Z}_{tr}X(k \times_k \Delta^2)\rightarrow \mathbb{Z}_{tr}X(k \times_k\Delta^1)\rightarrow \mathbb{Z}_{tr}X( k\times_k\Delta^0)\rightarrow 0

equivalently,

\cdots \rightarrow \text{Cor}(k \times_k \Delta^2, X)\rightarrow \text{Cor}(k \times_k \Delta^1, X)\rightarrow \text{Cor}(k \times_k \Delta^0, X)\rightarrow 0

canonically identifying k\times_k \Delta^n:=\text{Spec} k \times_k \Delta^n \simeq \Delta^n,

C_\ast \mathbb{Z}_{tr}X(\text{Spec} k)\simeq \cdots \rightarrow \text{Cor}(\Delta^2, X)\rightarrow \text{Cor}(\Delta^1, X)\rightarrow \text{Cor}(\Delta^0, X)\rightarrow 0.

The last complex shows the similarity to the construction of topological singular homology; it is exactly the complex of the free abelian group on all morphisms from the simplex into our space, with the analogous boundary maps.

(4.4) Example: Fix a field k. Let’s compute H^{sing}_n(\text{Spec}(k), \mathbb{Z}) for all n. If we remember the computation from topological singular homology, and we hope our computation here is similar, this should be

H_n^{sing}(\text{Spec}(k), \mathbb{Z})=\begin{cases} \mathbb{Z} & \mbox{if} \quad n=0\\ 0 & \mbox{if}\quad n\geq 1 \end{cases}.

First, notice that \text{Cor}_k(\Delta^n, \text{Spec}(k)) is by definition the set of all \mathbb{Z}-linear combinations of closed subschemes of \Delta^n \times_k \text{Spec}(k)\simeq \Delta^n finite and surjective over \Delta^n. But the only subscheme of \Delta^n which is surjective over \Delta^n is \Delta^n. That is to say, \text{Cor}_k(\Delta^n, \text{Spec}(k))= \mathbb{Z}\cdot \Delta^n. Our complex is now

\cdots\rightarrow\mathbb{Z}\cdot \Delta^3 \rightarrow \mathbb{Z}\cdot \Delta^2 \rightarrow \mathbb{Z}\cdot \Delta^1\rightarrow \mathbb{Z}\cdot\Delta^0\rightarrow 0

and it remains to determine all of the maps. But, by definition the map \text{Cor}(\Delta^n, \text{Spec}(k))\rightarrow \text{Cor}(\Delta^n,\text{Spec}(k)) is \Delta^n\mapsto (\sum_{i=0}^n (-1)^i \Gamma_{\partial_i})\circ \Delta^n = \sum_{i=0}^n (-1)^ i \Delta^{n-1}. For n odd, the sum on the right is 0 and so the map is 0. For n even, the sum on the right is \Delta^{n-1} and so the map is an isomorphism. Our chain complex

\cdots\rightarrow\mathbb{Z}\cdot \Delta^3 \xrightarrow{0} \mathbb{Z}\cdot \Delta^2 \xrightarrow{\sim} \mathbb{Z}\cdot \Delta^1\xrightarrow{0} \mathbb{Z}\cdot\Delta^0\rightarrow 0

shows that we do indeed have H_0^{sing}(\text{Spec}(k),\mathbb{Z})=\mathbb{Z} and all higher homology vanishes. Interestingly, this proof is even essentially the same as the one we get from topology.

A1 Homotopy

We’re going to define a notion of homotopy equivalence for correspondences (a.k.a. morphisms in the category \mathsf{Cor}_k). We will also define a notion of homotopy equivalence for morphisms of sheaves on \mathsf{Cor}_k in an arbitrary Grothendieck topology. I’ll continue using the notation C_\tau for a Grothendieck topology for which it makes sense to consider sheaves on \mathsf{Cor}_k (i.e. a topology satisfying condition T1). If C_{\tau} is the étale, Nisnevich, or Zariski topology then we have a full embedding of categories \mathsf{Cor}_k\hookrightarrow \mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k) (given by X\mapsto \mathbb{Z}_{tr}X – this is due to lemma 2.7). For these topologies, our definition of \mathbb{A}^1-homotopic morphisms will of sheaves will agree with our definition of \mathbb{A}^1-homotopic correspondences when the sheaf is representable.

Defining when two morphisms f,g:X\rightarrow Y between smooth schemes should be called homotopic is a subtle issue again. We would like our definition of homotopic maps to be an equivalence relation, as it is in topology. However, the way one shows this is transitive in the topological setting requires gluing two copies of the interval together and then shrinking it to the same size as the original interval. This isn’t going to work for us in the algebraic setting because we can’t glue two copies of \mathbb{A}^1 in a meaningful way (this could probably be made precise, in the sense that there are no isomorphisms \mathbb{A}^1\coprod \mathbb{A}^1\rightarrow \mathbb{A}^1 but I won’t pursue this). However, extending the definition to correspondences does give an equivalence relation.

(4.5) Definition: two correspondences \gamma,\omega \in \text{Cor}(X,Y) are called \mathbb{A}^1-homotopic if there is a correspondence h\in \text{Cor}(X\times \mathbb{A}^1, Y) so that h\circ i_0=\gamma and h\circ i_0 = \omega (here the correspondences i_\alpha are the graphs of the compositions X\simeq X\times \text{spec}(k)\xrightarrow{\text{id}_X\times \alpha} X\times \mathbb{A}^1). I’ll write \gamma\simeq \omega if \gamma is homotopic with \omega.

(4.6) Lemma: this is an equivalence relation on the set \text{Cor}(X, Y). Moreover, the correspondences \mathbb{A}^1-homotopic to 0 are closed under addition and composition (hence form a subgroup of \text{Cor}(X,Y) among other things).

Proof. Reflexivity: For an elementary correspondence \gamma, using \gamma\times \mathbb{A}^1 gives \gamma\times \mathbb{A}^1 \circ i_0= \gamma, and \gamma\times \mathbb{A}^1 \circ i_1= \gamma. We extend by additivity to finite correspondences.

Symmetry: If h\in \text{Cor}(X\times \mathbb{A}^1, Y) is such that h\circ i_0 = \gamma and h\circ i_1= \omega, then taking the automorphism T:\mathbb{A}^1\xrightarrow{*(-1)}\mathbb{A}^1\xrightarrow{+1}\mathbb{A}^1 gives (h\circ \Gamma_{\text{id}_X\times T})\circ i_0= \omega and (h\circ \Gamma_{\text{id}_X \times T}) \circ i_1=\gamma.

Transitivity: Let h\circ i_0 = \gamma, h\circ i_1=\omega, g\circ i_0 = \omega, g\circ i_1 = \zeta. This follows using (h-(g\circ\Gamma_{\text{id}_X \times T})+ \zeta\times \mathbb{A}^1)\circ i_0 = \gamma-\zeta+\zeta =\gamma and (h- (g\circ\Gamma_{\text{id}_X\times T})+\zeta\times \mathbb{A}^1)\circ i_1 = \omega - \omega+\zeta = \zeta.

Being closed under addition is simple: say h\circ i_0 =\gamma, h\circ i_1=0, g\circ i_0=\omega, g\circ i_1 = 0, then (h+g)\circ i_0 = \gamma+\omega and (h+g)\circ i_1 = 0.

Being closed under composition is equally simple: if \gamma \in \text{Cor}(X,Y) and \omega\in \text{Cor}(Y,Z) with both \mathbb{A}^1-homotopic to 0 (say h\circ i_0 =\gamma, h\circ i_1=0, g\circ i_0=\omega, g\circ i_1 = 0), then take the correspondence g\circ (h\times \Gamma_{\text{pr}_{\mathbb{A}^1}}) as the homotopy in \text{Cor}(X\times \mathbb{A}^1, Z) giving \omega\circ \gamma\simeq 0.\square

An immediate extension of our new definition is the notion of homotopy equivalent spaces.

(4.7) Definition: A correspondence f:X\rightarrow Y is called a homotopy equivalence (between the schemes X and Y) if there exists a correspondence g:Y\rightarrow X such that g\circ f\simeq \text{id}_X and f\circ g\simeq \text{id}_Y.

We can also generalize definition (4.5) to the category \mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R)).

(4.8) Definition: two morphisms of sheaves f,g:F\rightarrow G of R-modules with transfers are called \mathbb{A}^1-homotpic if there exists a morphism h:F\otimes^{tr} R_{tr}\mathbb{A}^1\rightarrow G so that h\circ i_1= f and h\circ i_0=g. (Here i_\alpha is the morphism of sheaves R_{tr}\text{Spec}(k)\rightarrow R_{tr}\mathbb{A}^1 induced by the map \text{Spec}(k)\rightarrow \mathbb{A}^1 x\mapsto \alpha). I’ll write f\simeq g if f is homotopic with g.

Note: The tensor product in the above is the one defined construction (2.13).

As mentioned earlier, definition (4.8) recovers definition (4.5) in the case the topology is one of the Zariski, Nisnevich, or étale topologies. This is because if f,g:X\rightarrow Y are correspondences of smooth shcemes, they induce morphisms \tilde{f},\tilde{g}:\mathbb{Z}_{tr}X\rightarrow \mathbb{Z}_{tr}Y of sheaves. If \tilde{f},\tilde{g} are \mathbb{A}^1-homotopic in the sense of definition (4.8), that is to say there is an \tilde{h}:\mathbb{Z}_{tr}(X\times \mathbb{A}^1)\rightarrow \mathbb{Z}_{tr}Y such that it agrees on the restrictions to 0,1, then the Yoneda embedding followed by the sheafification functor (which acts trivially as these are already sheaves by lemma 2.7) shows this comes from a correspondence h\in \text{Cor}(X\times \mathbb{A}^1,Y) and f,g are homotopic in the sense of definition (4.5).

Homotopy Invariant Presheaves

In this section we study the effect homotopy equivalence has on functors defined over smooth schemes. Our primary interest will be: when does a functor give the same object for any two homotopy equivalent schemes? It will turn out that, generalizing our definition a little bit (in the sense we don’t actually need to consider homotopy equivalent schemes on the nose – it’s enough to consider when a functor gives the same values whether or not we have multiplied by \mathbb{A}^1), what we will call homotopy invariance is enough to guarantee homotopy equivalent schemes have isomorphic motives.

(4.9) Definition: A presheaf with transfers F is called homotopy invariant if, for every smooth scheme X, the morphism

F(X)\xrightarrow{F(\Gamma_{\text{pr}_X})} F(X\times \mathbb{A}^1)

is an isomorphism.

Remark: Since the map X\rightarrow X\times \mathbb{A}^1 given by the zero section is a splitting of the above map, we know {F(\Gamma_{\text{pr}_X})} is always injective.

(4.10) Lemma: The homotopy invariant presheaves with transfers form a Serre subcategory of \mathsf{PreSh}(\mathsf{Cor}_k(R)).

Proof. Given an exact sequence of presheaves with transfers with F,H homotopy invariant

0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0

we have the following commutative diagram, induced by any smooth scheme X,

\begin{matrix} 0 & \rightarrow & F(X) & \rightarrow & G(X) & \rightarrow & H(X) & \rightarrow & 0  \\ & & \downarrow & & \downarrow & & \downarrow & \\ 0 & \rightarrow & F(X\times \mathbb{A}^1) & \rightarrow & G(X\times \mathbb{A}^1) & \rightarrow & H(X\times \mathbb{A}^1) & \rightarrow & 0 \end{matrix}

with the outer two vertical arrows isomorphisms. Thus, the five lemma gives that the middle is also an isomorphism. \square

With the notation i_\alpha as in definition (4.5), the following lemma provides an alternative characterization of homotopy invariant presheaves with transfers.

(4.11) Lemma: A presheaf with transfers F is homotopy invariant if and only if for every smooth scheme X we have equality of morphisms F(i_0)=F(i_1):F(X\times \mathbb{A}^1)\rightarrow F(X).

Proof. Lemma 2.16 in [MVW].

(4.12) Lemma: With F as above, the morphisms of chain complexes C_\ast F(i_0),C_\ast F(i_1): C_\ast F(X\times \mathbb{A}^1)\rightarrow C_\ast F(X) induced by i_0 and i_1 are chain homotopic.

Proof. This is by the construction of an explicit homotopy. Lemma 2.18 in [MVW].

Lemma 4.12 has the following application:

(4.13) Corollary: If F is a presheaf with transfers, the homology presheaves H_n C_\ast F : X\mapsto H_n C_\ast F(X) are homotopy invariant for all n.

Proof. Lemma (4.12) says C_\ast F(i_0) and C_\ast F(i_1) induce the same map H_n C_\ast F(X)\rightarrow H_n C_\ast F(X\times \mathbb{A}^1) for all smooth schemes X and all n. By (4.11) this implies they are homotopy invariant.\square

As an immediate application, we find that H_p^{sing}(X, R)\cong H_p^{sing}(X\times \mathbb{A}^1, R) for all p. Hence, from example (4.4) we have computed:

(4.14) Example:

H_p^{sing}(\mathbb{A}^n, \mathbb{Z})=\begin{cases} \mathbb{Z} & \mbox{if} \quad p=0\\ 0 & \mbox{if}\quad p\geq 1 \end{cases}

(4.15) Lemma: there is a chain homotopy equivalence C_\ast \mathbb{Z}_{tr}(X\times \mathbb{A}^1)\rightarrow C_\ast \mathbb{Z}_{tr}X.

Proof. [MVW] Corollary 2.24.

Finally, all of the lemmas here allow us to prove

(4.16) Proposition: If f:X\rightarrow Y is an \mathbb{A}^1-homotopy equivalence with \mathbb{A}^1-homotopy inverse g, then the chain morphism induced by f, C_\ast\mathbb{Z}_{tr}X\rightarrow C_\ast\mathbb{Z}_{tr}Y, is a chain homotopy equivalence with chain homotopy inverse induced by g.

Proof. [MVW]  Lemma 2.26.


References:

[MVW] Motivic Cohomology – Mazza, Voevodsky, Weibel. Link. (Try Google if the link expires).

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