I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. The discussion started with cohomology (part 1), continued with bundles and cycles (part 2) and in this part 3, we discuss motivic stuff.

Motives

Chow motive of projective space

One can compute the Chow motive of projective space by guessing it and using Manin's identity principle, a variant of the Yoneda lemma. To do such a "calculation" in general, the guessing part will be a problem. We can try to do systematic guessing. In fact, the previous two posts on projective space have prepared this. We expect, from the Weil cohomology computations, to have a motive $h(\mathbb{P}^n) = \bigoplus_{s=0}^n 1(-s)[-2s]$ (we can forget about the grading and the $[-2s]$ for now).

Manin's identity principle states that the functor that maps a motive $M$ to the functor
$\omega_M : \mathcal{P}(k) \ni Y \mapsto M_\sim(h(Y),M(\ast)) := \bigoplus_{r \in \mathbb{Z}} M_\sim(h(Y),M(r))$
is fully faithful (where the grading on the right hand side is taken to be the grading in intersection groups). If you know the Yoneda lemma, this is an easy consequence. If you don't know the Yoneda lemma, you should be sitting at your desk, trying to prove it!

We compare the motives $h(\mathbb{P}^n)$ and $\bigoplus_{s=0}^n 1(-s)$ by looking at their corresponding functors $\omega_M$. The only input we need from intersection theory is a projective bundle formula
$A^\bullet(X \times \mathbb{P}^n) \simeq A^\bullet(X)_F[H]/(H^{n+1}) = \bigoplus_{s=0}^n A^\bullet(X) \cdot H^s.$
We compute for any $Y \in \mathcal{P}(k)$:
$M_\sim(h(Y),h(\mathbb{P}^n)(r)) = Z_\sim^{d_Y+r}(Y \times \mathbb{P}^n) \simeq \bigoplus_{s=0}^n Z_\sim^{d_Y+r-s}(Y),$
$M_\sim(h(Y),\left(\bigoplus_{s=0}^n 1(-s)\right)(r)) = \bigoplus_{s=0}^n Z_\sim^{d_Y+r-s}(Y).$
Using the first line with $Y=\mathbb{P}^n$ and $r=0$ we can take the identity on $\mathbb{P}^n$ to yield a canonical morphism $h(\mathbb{P}^n) \to \bigoplus_{s=0}^n 1(-s)$, which is an isomorphism, since it induces an isomorphism of corresponding functors.

We can use this now to compute realizations of the Chow motive. As we have seen, we just need to know how the Lefschetz motive $1(-1)[-2]$ realizes, i.e. what a Weil cohomology does on $\mathbb{P}^1$. For example, Betti realization $\omega_B$ gives us a one-dimensional vector space $\omega_B(1(-1)) = V \simeq \mathbb{Q}$ with Hodge decomposition $V = V^{1,1}$. The $\ell$-adic realization gives us a one-dimensional vector space $\omega_\ell(1(-1)) = \mathbb{Q}_\ell(1) = T_\ell(\mu) \otimes \mathbb{Q}_\ell$, where $T_\ell(\mu)$ is the $\ell$-adic Tate module of roots of unity $\mu$, which has a natural Galois action (which is part of the realization). Consequently, $H_B^{2k}(\mathbb{P}^n) = \left(H_B^{1}(\mathbb{P}^1)\right)^{\otimes k} = V^{\otimes k}$ with Hodge decomposition $V^{\otimes k} = (V^{\otimes k})^{(k,k)}$ and $H_\ell^{k}(\mathbb{P}^n) = \left(H_\ell^{1}(\mathbb{P}^1)\right)^{\otimes k} = \mathbb{Q}_\ell(k) = T_\ell(\mu)^{\otimes k}$.

We could have skipped the Weil cohomology computations for $\mathbb{P}^n$, it would have sufficed to compute the motive and the realizations of $\mathbb{P}^1$. But how would we have guessed the motive then? Well, using geometry. I will come to that later in this article.

Voevodsky motives

Now I want to describe the motive of projective space (and its decomposition) in Voevodsky's framework of the derived category of mixed motives (which isn't constructed as the derived category of an abelian category, however it looks like that). By a general theorem (which is not too hard), the motive of a smooth projective variety is in the image of a (contravariant!) functor from Chow motives, so we don't need to work any longer for projective space. The following is for educational purposes only. I want to consider only perfect fields $k$, since I don't know what's possible for non-perfect fields.

The triangulated category of effective geometrical motives over $k$, denoted $DM_{gm}^{eff}(k)$ is defined as the pseudo-abelian envelope of a localization (at the minimal thick subcategory containing $X\times \mathbb{A}^1 \to X$ and Mayer-Vietoris sequences) of the homotopy category of bounded complexes over $SmCor(k)$, the category of finite correspondences of smooth schemes over $k$. We denote the image of a smooth scheme $X$ in this category by $M_{gm}(X)$ (following Voevodsky).

A few easy calculations:
Since we have the pseudo-abelian property, we can do at least the usual splitting $M_{gm}(\mathbb{P}^1) = \mathbb{Z} \oplus \mathbb{L}$, where $\mathbb{Z} := M_{gm}(Spec k)$ is the unit object for the tensor structure and $\mathbb{L}$ is the reduced motive of $\mathbb{P}^1$. The Tate object is defined as $\mathbb{Z}(1) := \mathbb{L}[-2]$ (warning: Voevodsky motives are covariant, while Chow motives are contravariant, hence some formula look different; this is such a formula).
For Mayer-Vietoris, we can take the usual two charts $U,V : \mathbb{A}^1 \to \mathbb{P}^1$ which are a Zariski open covering of $X := \mathbb{P}^1$, so there is a distinguished triangle
$M_{gm}(U\cap V) \to M_{gm}(U) \oplus M_{gm}(V) \to M_{gm}(X) \to M_{gm}(U \cap V)[1]$
which in our case looks like
$M_{gm}(\mathbb{G}_m) \to 0 \oplus 0 \to M_{gm}(\mathbb{P}^1) \to M_{gm}(\mathbb{G}_m)[1]$
so that $M_{gm}(\mathbb{P}^1) \to M_{gm}(\mathbb{G}_m)[1]$ is an isomorphism.

To work with these motives, it is better to look at another category, which is denoted by $DM_{-}^{eff}(k)$. A presheaf with transfers on $Sm_k$ is an additive contravariant functor from $SmCor(k)$ to abelian groups. Such a presheaf with transfers can be seen as a presheaf on $Sm_k$ with additional restriction maps for each correspondence which isn't the graph of a morphism; in particular for the transposed graphs, which give restriction maps "in the other direction", hence the name "with transfers". A presheaf with transfers on $Sm_k$ is called Nisnevich sheaf if the corresponding presheaf of abelian groups on $Sm_k$ is a Nisnevich sheaf. A (pre)sheaf with transfers is called homotopy invariant if every projection map $X \times \mathbb{A}^1 \to X$ induces an isomorphism of sections. Now $DM_{-}^{eff}(k)$ is the full subcategory of $D^{-1}(Shv_{Nis}(SmCor(k)))$ of complexes with homotopy invariant cohomology sheaves. This category is a triangulated pseudo-abelian category. One can show that $DM_{gm}^{eff}(k)$ admits a full embedding (as tensor triangulated category) into $DM_{-}^{eff}(k)$.

Both categories of effective motives yield larger categories of motives by inverting the Tate twist operation $\otimes 1(1)$, thus one has $DM_{gm}(k)$ a full tensor triangulated subcategory of $DM_{-}(k)$. Now one could write down a Gysin sequence and a projective bundle theorem which can be used to compute the motive of $\mathbb{P}^n$, entirely in terms of $DM_{gm}^{eff}(k)$; the problem is that the proof I know of goes through the computation of the motive of $\mathbb{P}^n$, in terms of $DM_{-}(k)$.

Denote by $\mathbb{Z}_{tr}(X) := Cor(-,X)$ the presheaf with transfers associated to a smooth scheme $X$ and by $C_\ast\mathbb{Z}_{tr}(X)$ the complex obtained from the simplicial object $Cor(- \times \Delta^\bullet,X)$.

Voevodsky motive of projective space

We use $\mathbb{Z}(q) := C_\ast\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge q})[-q]$, which is called a motivic complex. We have already discussed $M_{gm}(\mathbb{P}^1) = M_{gm}(\mathbb{G}_m)[1]$. In a similar spirit, we have $C_\ast\mathbb{Z}_{tr}(\mathbb{P}^1) = \mathbb{Z}(1)[2]$, using the following lemma: for $\mathcal{U}$ a Zariski covering of $X$ the Cech resolution $Tot C_\ast \mathbb{Z}_{tr}(\mathcal{U}) \to C_\ast \mathbb{Z}_{tr}(X)$ is a quasi-isomorphism in the Zariski topology.

Let's denote $0 := [1:0:\cdots:0] \in \mathbb{P}^n$ and look at the map $f : \mathbb{P}^n \setminus 0 \to \mathbb{P}^{n-1}$ given by $[x_0:\cdots:x_n] \mapsto [x_1:\cdots:x_n]$. The fibers of this map are just $\mathbb{A}^1$. There is an $\mathbb{A}^1$-homotopy inverse $g : \mathbb{P}^{n-1} \to \mathbb{P}^n \setminus 0$ given by the section $[x_1:\cdots:x_n] \mapsto [0:x_1:\cdots:x_n]$, and the $\mathbb{A}^1$-homotopy of $g \circ f$ to $id$ is given by multiplication of $x_0$ with $\lambda \in \mathbb{A}^1$. Such a pair of $\mathbb{A}^1$-homotopy inverse maps yield a chain homotopy equivalence $C_\ast \mathbb{Z}_{tr}(\mathbb{P}^n \setminus 0) \to C_\ast \mathbb{Z}_{tr}(\mathbb{P}^{n-1})$.

Theorem: For each $n$, there are quasi-isomorphisms of Zariski sheaves:
$C_\ast\left( \mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \right) \simeq C_\ast \mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge n})[n] = \mathbb{Z}(n)[2n]$.

The proof of this theorem (which I got from the Mazza-Voevodsky-Weibel book, Chapter 15) uses a few facts about presheaves with transfers, for example the Nisnevich sheafification $F_{Nis}$ of a homotopy invariant presheaf with transfers $F$ is again homotopy invariant; and more is true: all the presheaves $H^n_{Nis}(-,F_{Nis})$ are homotopy invariant. Using this, one can show that a presheaf with transfers $F$ that satisfies $F_{Nis} =0$ also satisfies $(C_\ast F)_{Nis} \simeq 0$ and $(C_\ast F)_{Zar} \simeq 0$.

Proof sketch: Let $\mathcal{U}$ be the usual cover of $\mathbb{P}^n$ by $n+1$ charts $\mathbb{A}^n$ and let $\mathcal{V}$ be the cover by $n$ charts of $\mathbb{P}^{n} \setminus 0$. Intersecting $i+1$ charts gives $\mathbb{A}^{n-i} \times (\mathbb{A}^1 \setminus 0)^i$. There are quasi-isomorphisms (of complexes of Nisnevich sheaves with transfers) $\mathbb{Z}_{tr}(\mathcal{U}) \to \mathbb{Z}_{tr}(\mathbb{P}^n)$ and $\mathbb{Z}_{tr}(\mathcal{V}) \to \mathbb{Z}_{tr}(\mathbb{P}^n\setminus 0)$, so $Q_\ast := \mathbb{Z}_{tr}(\mathcal{U})/\mathbb{Z}_{tr}(\mathcal{V})$ is a resolution of $\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n \setminus 0)$ (as Nisnevich sheaf). Now $Tot C_\ast Q_\ast$ is quasi-isomorphic to $C_\ast\left(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n \setminus 0)\right)$, hence to $C_\ast\left(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^{n-1})\right)$ for the Zariski topology.
One can write down a resolution $R_\ast$ of $\mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge n})[n]$ such that one gets a map $Q_\ast \to R_\ast$ whose terms are direct sums of $\mathbb{A}^1$-homotopy equivalences, so $C_\ast Q_\ast \to C_\ast R_\ast$ is a quasi-isomorphism. Applying $Tot$ gives us
$C_\ast\left(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^{n-1})\right) \simeq Tot C_\ast Q_\ast \simeq Tot C_\ast R_\ast \simeq C_\ast \mathbb{Z}_{tr}(\mathbb{G}_m^{\wedge n})[n]$.

One can show that the isomorphism in this theorem factors through every inclusion $C_\ast \mathbb{Z}_{tr}(\mathbb{P}^i) \to C_\ast \mathbb{Z}_{tr}(\mathbb{P}^n)$ with $n > i$.

Corollary: There is a quasi-isomorphism $M(\mathbb{P}^n)=C_\ast\mathbb{Z}_{tr}(\mathbb{P}^n) \to \bigoplus_{s=0}^n \mathbb{Z}(s)[2s]$.

Proof: by induction, where the case $n=1$ is already done. The map $\mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \to \mathbb{Z}_{tr}(\mathbb{P}^{n})$ is split injective in $DM_{-}^{eff}(k)$, since the quasi-isomorphism $\mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \to \bigoplus_{s=0}^{n-1} \mathbb{Z}(s)[2s]$ (from the induction hypothesis) factors through it. Hence the distinguished triangle
$C_\ast \mathbb{Z}_{tr}(\mathbb{P}^{n-1}) \to C_\ast \mathbb{Z}_{tr}(\mathbb{P}^{n}) \to \mathbb{Z}(n)[n] \to [1]$
splits.

Computations from the motive: motivic cohomology

Now that we have the motive, we can (in principle) compute motivic cohomology
$H^{p,q}_{mot}(\mathbb{P}^n,\mathbb{Z}) := H^p_{Zar}(\mathbb{P}^n,\mathbb{Z}(q)) = Ext^p(\mathbb{Z}_{tr}(X),\mathbb{Z}(q))$
where the Ext is in the category of Nisnevich sheaves with transfer.
So we have to understand $Hom_{DM_{-}}(\mathbb{Z}(s)[2s],\mathbb{Z}(q)[p])$. This can be simplified to $Hom(\mathbb{Z},\mathbb{Z}(q-s)[p-2s]) = H^{p-2s,q-s}_{mot}(Spec(k))$, so we are reduced to understand the motivic cohomology of a point.

From the motivic cycle class isomorphism $CH^q(X,2q-p) \simeq H^{p,q}(X)$ one can recover the Chow groups of $\mathbb{P}^n$. The motivic Chern character yields an isomorphism computing the algebraic K-Theory of $\mathbb{P}^n$ (see this article about the divisorial jungle, where I discuss this briefly).

 

A¹-homotopy type and motivic cell structure

We can take one step backwards and look at $\mathbb{P}^n$ not with (co)homological eyes, but with homotopical ones. The functor $M : Sm_k \to DM_{-}(k)$ factors through a model category $\mathcal{M}$ of simplicial Nisnevich sheaves on $Sm_k$, where one can do homotopy theory.

In this model category $\mathcal{M}$ one can write down $X := \bigvee_{s=0}^n \mathbb{G}_m^{\wedge s} \wedge S^s_s$, a space (=simplicial Nisnevich sheaf) which has the same motive as $\mathbb{P}^n$, since wedging with the $s$-dimensional simplicial sphere $S_s^s$ induces a shift by $s$, hence $M(\mathbb{G}_m^{\wedge s} \wedge S^s_s) = \mathbb{Z}(s)[2s]$. One could now try to write down (or prove existence of) an $\mathbb{A}^1$-homotopy equivalence of $X$ with $\mathbb{P}^n$.

This turns out to be impossible in general, since the homotopy types are different (but they become isomorphic over a quadratically closed base field). We can already see that in the real realization, which is a functor that assigns to a homotopy type a topological space which acts like the $\mathbb{R}$-points. The projective space has non-orientable real realization, while $X$ is orientable. This is like the difference between $\mathbb{RP}^2$ and $S^2$.

One can write down a motivic cell structure for $\mathbb{P}^n$, where the attaching maps split over a quadratically closed field. One can say that motives don't distinguish between $X$ and $\mathbb{P}^n$, but the homotopy type does (even the stable homotopy type).

Such a motivic cell structure can be constructed like in topology: start with a point $\mathbb{P}^0$ and attach a 1-cell $\mathbb{A}^1$ along the attaching map $\eta_1 : \mathbb{A}^1 \setminus 0 \to \mathbb{P}^0$ which is the quotient map after the $\mathbb{G}_m$-action. You get a $\mathbb{P}^1$. Then attach a 2-cell $\mathbb{A}^2$ along $\eta_2 : \mathbb{A}^2 \setminus 0 \to \mathbb{P}^1$, you get $\mathbb{P}^n$ and so on.

We get the motive out of a motivic cell structure, since a cofiber sequence $\mathbb{A}^n \setminus 0 \to \mathbb{P}^{n-1} \to \mathbb{P}^n$ yields a distinguished triangle and $\mathbb{A}^n \setminus 0 \simeq S^{2n-1,n}$ has the motive $\mathbb{Z}(n)[2n-1]$, so we can write the distinguished triangle as
$M(\mathbb{P}^{n-1}) \to M(\mathbb{P}^n) \to \mathbb{Z}(n)[2n] \to$
which splits, i.e. $M(\mathbb{P}^n) = M(\mathbb{P}^{n-1}) \oplus \mathbb{Z}(n)[2n]$, since the morphism $\mathbb{Z}(n)[2n-1] \to M(\mathbb{P}^{n-1})$ is trivial.

This is the calculation of the motive, thus of higher Chow groups, algebraic K-Theory and all Weil cohomology theories, that I like most.