Invariants of projective space III: Motives

Monday, December 10th, 2012 | Author:

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.


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]

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.

Category: English, Mathematics

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