I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. We will have a look at the cohomology, the Chow groups and the algebraic K-theory of projective space -- a discussion probably interesting to non-motivic people as well. After these invariants, I will look at the motive and the A¹-homotopy type. Then I want to describe the decomposition of the motive (and the homotopy type) homotopy-theoretically, by means of cofiber sequences. (We will see that projective space is not isomorphic to a coproduct of motivic spheres with the same motive). Of course, nothing is new, I'm just working out exercises here.

In this part 1, I discuss only the cohomology of $\mathbb{P}^n$. Part 2 contains a discussion of the intersection theory and bundles and part 3 contains the motivic stuff. I intentionally left out usage of projective bundle formulas, as I will discuss them separately.

CW structure on the associated analytic space

We can compute the Betti cohomology, i.e. the singular cohomology of the associated analytic space, by showing that the manifold $\mathbb{CP}^n := (\mathbb{P}^n(\mathbb{C}))^{an}$ admits a CW-structure and computing the cellular cohomology (which also satisfies the Eilenberg-Mac Lane axioms, hence is isomorphic).
For $n=0$ this is easy, as $\mathbb{CP}^0$ is just a single point. Suppose now we already have a CW-structure on $\mathbb{CP}^n$, then we will construct $\mathbb{CP}^{n+1}$ by attaching one $2n+2$-cell $\mathbb{C}^{n+1}$ by the gluing map $\eta : \mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^n$ given by quotienting out the $\mathbb{C}^\times$-action, i.e. I claim $\mathbb{CP}^{n+1} \simeq \mathbb{CP}^{n} \cup_{\eta} \mathbb{C}^{n+1}$. The map $\eta$ is known under the name "Hopf map", and it is a surjective map.
The CW structure of $\mathbb{CP}^n$, as we have just seen, is such that there is a single $2k$-dimensional cell for each $k = 0,\dots,n$. The chain complex computing cellular cohomology thus has no differentials, so we can quickly see that

$$H^i_{cell}(\mathbb{CP}^n,\mathbb{Z}) = \begin{cases} \mathbb{Z}, & \text{ for } i=2k, k\leq n\\ 0 & \text{else.} \end{cases}$$

Smooth de Rham theory and ring structure

We can also look at the deRham cohomology of $\mathbb{CP}^n$ as a smooth $2n$-dimensional real manifold. The advantage is that we would see the cup product structure (coming from the wedge of differential forms). From what we already know, there should be a closed smooth differential $2k$-form $\omega_k$ which isn't exact for each $k = 0,\dots,n$. We know that $\omega_n$ will be a volume form and similarly $\omega_k$ can be chosen to be a volume form of $\mathbb{CP}^k$ when restricted to that subspace. Then we have $\omega_k \wedge \omega_j$ homologous to $\omega_{k+j}$ for $k+j \leq n$, so that the multiplicative structure in the cohomology can be described as $[\omega_k] = [\omega_1]^k$ and by writing $x = [\omega_k]$ (of degree $2$) we have the identity of graded rings

$$H^\bullet_{dR}(\mathbb{CP}^n,\mathbb{R}) = \mathbb{R}[x]/(x^{n+1}).$$

Some people like to describe the cohomology of projective space by choosing an arbitrary hyperplane $H \subset \mathbb{CP}^n$ (so the complement is isomorphic to $\mathbb{CP}^{n-1}$) and then $H^\bullet_{sing}(\mathbb{CP}^n,\mathbb{Z}) = \mathbb{Z}[H]/(H^{n+1}),$ with $H$ in degree $2$. This description arises from the Poincaré duality map taking singular $k$-cycles in $\mathbb{CP}^n$ to singular $2n-k$-cocycles, which takes any complex hyperplane $H \subset \mathbb{CP}^n$ to the same cohomology class of degree $2$ (since the real codimension of a complex hyperplane is $2$).

Cup product with integer coefficients

Of course, it is unsatisfying to have the multiplicative structure only with real coefficients, so we can work a little bit more (or differently) and compute the cup product via Yoneda products in sheaf cohomology of the constant sheaf $\underline{\mathbb{Z}}$. I'm too lazy now, but in the end you get

$$H^\bullet_{Betti}(\mathbb{P}^n,\mathbb{Z}) = \mathbb{H}^\bullet(\mathbb{CP}^n, \underline{\mathbb{Z}}_{\mathbb{CP}^n}) = \mathbb{Z}[x]/(x^{n+1}).$$

Yet another way to get the cup product is the most "canonical" way with cellular cohomology (in my eyes), where we use only the diagonal morphism and the Künneth formula. I'm going to do that:
The diagonal $\mathbb{P}^n \to \mathbb{P}^n \times \mathbb{P}^n$ induces a homomorphism $H^\bullet_{cell}(\mathbb{P}^n \times \mathbb{P}^n) \to H^\bullet_{cell}(\mathbb{P}^n)$ and the Künneth formula tells us $H^\bullet_{cell}(\mathbb{P}^n \times \mathbb{P}^n) \simeq H^\bullet_{cell}(\mathbb{P}^n \otimes H^\bullet_{cell}(\mathbb{P}^n))$, so we can take classes $\alpha,\beta \in H^\bullet_{cell}(\mathbb{P}^n)$, form $\alpha \otimes \beta$, move this through the Künneth isomorphism and the morphism induced by the diagonal and end up in $H^\bullet_{cell}(\mathbb{P}^n)$ again, where we call the result $\alpha \cup \beta$. Since Künneth is a graded isomorphism, this map is graded and thus we have all properties of a cup product. All other constructions (via Yoneda Ext products or wedge products of differential forms) yield the same product (otherwise we wouldn't call it cup product).

Hodge theory

There is still something left out in this discussion, namely Hodge theory. Betti cohomology of a smooth projective variety carries a Hodge structure. For projective space, there isn't much to discuss, as the Hodge structure is trivial in the sense that every class in $H^{2k}_{B}(\mathbb{P}^n)$ is of Hodge type $(k,k)$, i.e. the Hodge diamond of $\mathbb{P}^n$ has only the Hodge numbers $h^{k,k}=1$ and all other Hodge numbers vanish. The computation can be found in Voisin's book, section 7.2 (page 167 of book one of the english edition).

The structure of a homogeneous space

Now, a little bit of general theory not necessary to proceed:
The $n$-dimensional projective space parametrizes 1-dimensional linear subspaces $L$ of affine $n+1$-space, which we can consider as partial flags of linear subspaces $0 \leq L \leq \mathbb{A}^{n+1}$. We have a natural $GL_{n+1}$-action on $\mathbb{A}^{n+1}$ (hence on flags). Partial flags of linear subspaces in $\mathbb{A}^{n+1}$ are stabilized by certain parabolic subgroups $P$ of $GL_{n+1}$, for example full flags are stabilized by a Borel subgroup. The quotient $GL_{n+1}/B$ thus parametrizes precisely full flags in $\mathbb{A}^{n+1}$. The quotients $GL_{n+1}/P$ that parametrize partial flags of a certain shape (determined by $P$) are called generalized Grassmannians.
I included this material here to give an outlook on how to proceed past projective space later on, in further calculations.

More concretely:
In $GL_{n+1}$ look at the subgroup $P$ given by block matrices with a block of size $n \times n$ just all of $GL_n$ (in the upper left corner) and a block of size $1 \times 1$ (in the lower right corner) just $GL_1$, i.e. $\mathbb{G}_m$, and a block of size $n \times 1$ just $\mathbb{A}^n$ (in the upper right corner), with a block of size $1 \times n$ just zeroes (in the lower left corner). If we look at $GL_{n+1}/P$, we see that, as a variety, the big $n \times n$ block in the upper left corner of $GL_{n+1}$ is killed and what remains is $\mathbb{A}^{n+1} \setminus \{0\}$ from the right column of $GL_{n+1}$, modulo the group action from $\mathbb{G}_m$ from the lower right corner of $P$. So we really have

$$GL_{n+1}/P = (\mathbb{A}^{n+1} \setminus \{0\})/\mathbb{G}_m = \mathbb{P}^n.$$

The good thing about this description is that $P$ contains a maximal torus (the diagonal matrices) and then there is a nice general theory of Schubert calculus to be applied. Topologically, this also gives us a CW structure, isomorphic to the structure we built "manually" before.

Algebraic de Rham cohomology over the rationals

By definition, $H^\bullet_{dR}(\mathbb{P}^n) = \mathbb{H}^\bullet(\mathbb{P}^n, \Omega^\bullet_{\mathbb{P}^n/\mathbb{Q}})$, a sheaf cohomology group. A cup product structure comes from the wedge product on the de Rham complex $\Omega^\bullet_{\mathbb{P}^n/\mathbb{Q}}$. One can compute algebraic de Rham cohomology by the Hodge-to-de Rham spectral sequence, which is

$$E^{p,q}_1 := \mathbb{H}^q(\mathbb{P}^n,\Omega^p_{\mathbb{P}^n/\mathbb{Q}}) \Rightarrow H^{p+q}_{dR}(\mathbb{P}^n).$$

From $\mathbb{H}^q(\mathbb{P}^n,\Omega^p) = 0$ for $p \neq q$ and $\mathbb{H}^q(\mathbb{P}^n,\Omega^p) \simeq \mathbb{Q}$ for $0 \leq p=q \leq n$ (this is Hartshorne's Exercise III.7.3) we see that the spectral sequence degenerates hence the odd-dimensional algebraic de Rham cohomology of $\mathbb{P}^n$ vanishes and $H^{2k}_{dR}(\mathbb{P}^n) \simeq \mathbb{Q}$ for $k = 1,\dots,n$.

In particular, we have $H^\bullet_{dR}(\mathbb{P}^1) \simeq \mathbb{Q}[x]/(x^2)$. With the Künneth formula we compute for an $n$-fold product $H^\bullet_{dR}(\mathbb{P}^1\times\cdots\times\mathbb{P}^1) \simeq \mathbb{Q}[x_1,\dots,x_n]/(x_i^2)$. The symmetric group $S_n$ acts on this polynomial ring as well as on the $n$-fold product $(\mathbb{P}^1)^{\times n}$. We let the symmetric group act trivially on $\mathbb{P}^n$ and the map $(\mathbb{P}^1)^{\times n} \to \mathbb{P}^n$ becomes $S_n$-equivariant. It induces an $S_n$-equivariant injection $H^\bullet_{dR}(\mathbb{P}^n) \to H^\bullet_{dR}((\mathbb{P}^1)^{\times n})$, so after taking invariants we obtain

$$H^\bullet_{dR}(\mathbb{P}^n) \simeq \mathbb{Q}[x]/(x^{n+1})$$

, where $x = x_1 + \cdots + x_n,$
with $x$ in degree $2$. Observe just that $S_n$ is the Weyl group of $GL_n$, and its appearance here is no accident.

l-adic cohomology

I don't want to compute l-adic cohomology here, since even the definitions are a bit lengthy.

For example, $H^k_\ell(\mathbb{P}^n) \simeq 0$ for odd $k$ and $H^k_\ell(\mathbb{P}^n) \simeq \mathbb{Z}_\ell$ for even $k$. There is a Galois action on $H^k_\ell(\mathbb{P}^n)$ such that $H^2_\ell(\mathbb{P}^n)$ is equivariantly isomophic to $T_\ell(\mu)$, the limit over the roots of unity $\mu_{\ell^n} \subseteq \mathbb{C}$. Also, $H^0_\ell(\mathbb{P}^n) \simeq \mathbb{Z}_\ell$, where we equip $\mathbb{Z}_\ell$ with a trivial Galois action. Note that there are many non-equivariant isomorphisms $\mathbb{Z}_\ell \to T_\ell(\mu)$.

Similar in spirit, there is crystalline cohomology (or rigid cohomology) but I don't know enough about the subject to compute anything, so I have to leave that out.

Counting points over finite fields & Zeta function

I have written about this before (although with a different aim), so I don't want to repeat it.

The Zeta function of projective space is $Z(\mathbb{P}^n,s) = \prod_{k=0}^n{(1-p^kt)^{-1}}$,
as you can compute by hand from the decomposition of $\mathbb{F}_q$-rational points $\mathbb{P}^n(\mathbb{F}_q) = \mathbb{A}^n(\mathbb{F}_q) \cup \mathbb{A}^{n-1}(\mathbb{F}_q) \cup \cdots \cup \mathbb{A}^0(\mathbb{F}_q)$.

So far this is what I had to tell about the cohomology of projective space.