Given a vector bundle E-->X of rank r+1 one can take the projective space of lines in each fiber, which results in a projective bundle P(E)-->X. A projective bundle formula for a functor F from spaces to rings tells us that F(P(E)) is a free F(X)-module of rank r.

In this post I look at some computations around projective bundle formulae for the Chow ring, the algebraic K-Theory and the (Chow) motive of some spaces, in particular flag varieties. We recover some results from the previous posts on cohomology, cycles & bundles and motive of projective space.

 

Motivation and History

In the first section, I want to talk about (nice) topological spaces $X$ and their real vector bundles.

Classical topology

The idea of a projective bundle formula comes from classical topology.
Theorem (Projective Bundle formula for singular cohomology):
Given a vector bundle $\mathcal{E} \to X$ of rank $r+1$ the singular cohomology of its projectivization $\mathbb{P}(\mathcal{E}) \to X$ is a module over the singular cohomology of the base $X$, and there is a module homomorphism $H^\bullet(\mathbb{P}(\mathcal{E}),\mathbb{Z}) \simeq H^\bullet(X)[H]/(H^{n+1})$ where the $H$ stands for "hyperplane" and is in degree $2$.

A special case of the projective bundle formula is projective space $\mathbb{P}^n$ itself, the projectivization of a vector space $V = k^{n+1}$, considered as a vector bundle over a point. Since the cohomology of a point is concentrated in degree $0$ and there just $\mathbb{Z}$, we get $H^\bullet(\mathbb{P}^n)=\mathbb{Z}[H]/(H^{n+1})$.

One should pay attention to the fact that not every bundle with fibers projective spaces are projectivizations of vector bundles. The obstruction to this is a class in $H^2$ with values in $GL_1$, as one can see explicitly by a Cech resolution.

The projective bundle formula for singular cohomology can be seen as a special case of the Leray-Hirsch theorem, which states that a fiber bundle $F \to E \to B$ which has the property that $F \to E$ induces a surjection $H^\bullet(E) \to H^\bullet(F)$ has cohomology $H^\bullet(E) \simeq H^\bullet(B) \otimes H^\bullet(F)$, where the isomorphism is an isomorphism of $H^\bullet(B)$-modules and the tensor product is taken in the graded sense.
If the basis $B$ is a point (or just contractible, i.e. a point from the homotopy point of view) the theorem is trivial. If you take $E \to B$ to be a projective bundle of rank $r$, the fiber over any point is $F = \mathbb{P}^r$ and one can show that the classes $H^k$ that generate the cohomology of $\mathbb{P}^r$ are in the image of $H^\bullet(E) \to H^\bullet(F)$, hence Leray-Hirsch can be applied.
Leray-Hirsch follows from the Leray-Serre spectral sequence (which is, of course, just a special case of the Grothendieck spectral sequence for the composition of two functors and derivation), which is $H^p(B,\mathcal{H}^q(F)) \Rightarrow H^{p+q}(E)$.

Singular cohomology also satisfies a Künneth formula, which is $H^\bullet(X,\mathbb{Q}) \otimes H^\bullet(Y,\mathbb{Q}) \simeq H^\bullet(X \times Y,\mathbb{Q})$. If we try to do this with integral coefficients, there's not an isomorphism but a short exact sequence with a Tor-term which doesn't vanish in general. A Künneth formula would also give us a projective bundle formula for trivial projective bundles $\mathbb{P}^n \times X \to X$. In some sense, I like to think of bundle formulas as generalizations or versions of the Künneth formula (since bundles are locally products).

 

The Formulas in Algebraic Geometry

Now we change our focus and switch to algebraic geometry. I've written about definitions and relations between Chow groups and algebraic K-Theory before.

Chow ring

Let $X$ be an algebraic scheme over a field for the rest of the article.

Chow groups don't satisfy a Künneth formula! They do satisfy a projective bundle formula. I want to start with a baby version, which I'd like to call projective Künneth formula:
Theorem: $CH^\bullet(X \times \mathbb{P}^n) \simeq CH^\bullet(X)[H]/(H^{n+1})$, where $H$ is a hyperplane class in degree $1$.

This formula, together with the Yoneda lemma (in a variant known as Manin Identity Principle), already gives a decomposition of the Chow motive of projective space into irreducibles.

The full projective bundle theorem can be deduced from a localization sequence for higher Chow groups.

Theorem (localization sequence):
For $Z \to X$ a closed immersion of pure codimension $c$ with open complement $U$ there is a long exact sequence
$\cdots \to CH^{i-c}(Z,n) \to CH^{i}(X,n) \to CH^{i}(U,n) \to CH^{i-c}(Z,n-1) \to \cdots$

Theorem (projective bundle formula):
Let $\mathcal{E} \to X$ be a vector bundle of rank $r+1$ and $\mathbb{P}(\mathcal{E}) \to X$ its projectivization. Then $CH^\bullet(\mathbb{P}(\mathcal{E}),\bullet)$ is a $CH^\bullet(X,\bullet)$-module isomorphic to $CH^\bullet(X,\bullet)[H]/(H^{n+1})$, with $H$ in degree $1$.

To prove this by induction on the dimension on $X$, we can just take an open subset $U \subset X$ such that $\mathcal{E}$ is trivial over $U$, then for $U$ we have a projective Künneth formula and for $Z$ we use the induction hypothesis.

If you take only the $n=0$ part of the localization sequence (which I think was known for a longer time), you can prove the projective bundle formula for ordinary Chow groups with this method.

Algebraic K-Theory

We start with a projective bundle formula for $K_0$, i.e. the K-Theory of coherent sheaves on an algebraic scheme.
Theorem
For a vector bundle $E \to X$ of rank $n+1$, $K_0(\mathbb{P}(E))$ is a $K_0(X)$-module which is module-isomorphic to $K_0(X)[H]/(H^{n+1})$.

For trivial bundles, this is quite easy to see:
By a Theorem of Serre, any coherent sheaf on $\mathbb{P}^n$ is a quotient of some $\mathcal{O}(k)^{\oplus m}$ (with $k,m \geq 0$), hence the $\mathcal{O}(k)$ generate $K_0(\mathbb{P}^n)$.
The Koszul complex on $\mathbb{A}^{n+1}$ is a $n+2$-term exact sequence which gives a relation in $K_0$ between any $\mathcal{O}(k+1),\dots,\mathcal{O}(k+n+2)$ by applying a shift and taking the coherent sheaf on $\mathbb{P}^n$ associated to a graded module. Therefore we see that $H := [\mathcal{O}(-1)]$ and $H^2,\dots,H^n$ and $H^0 = [\mathcal{O}]$ together form a generating set. There are no more relations between these generators.

The proof of the non-trivial case is given by Grothendieck and Berthelot in SGA6 Exposé VI. Such a bundle formula also holds for higher algebraic K-Theory (due to Quillen):

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, $K_\bullet(\mathbb{P}(E))$ is a $K_0(X)$-module which is module-isomorphic to $K_\bullet(X)[H]/(H^{n+1})$, with $H$ in degree $1$.

Chow Motive

If we're talking about bundles over a smooth projective (or smooth complete) base, we can talk about the Chow motive of the total space in relation to the base.

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, $h(\mathbb{P}(E)) = \bigoplus_{s=0}^n h(X)(s)[2s]$ ($=h(X)[H]/(H^{n+1})$, with $H = \mathbb{Z}(1)[2]$).

Manin's Identity Principle implies that a morphism of Chow motives $M\to N$ is an isomorphism iff the morphism of associated functors $\omega_M \to \omega_{N}$ is an isomorphism, where
$\omega_M : \mathcal{P}(k) \ni Y \mapsto M_\sim(h(Y),M(\ast)) := \bigoplus_{r \in \mathbb{Z}} M_\sim(h(Y),M(r))$

We guess the motive of a projective bundle $\mathbb{P}(E)$ of rank $n$ over a base $X$ to be $M := \bigoplus_{s=0}^n h(X)(s)[2s]$, then from the projective bundle formula for Chow groups we see that $\omega_M(\mathbb{P}(E)) \simeq \omega_{\mathbb{P}(E)}(\mathbb{P}(E))$, so the identity morphism on the right hand side yields a morphism $h(\mathbb{P}(E)) \to M$ which induces an isomorphism on the functors $\omega$, thus is an isomorphism.

Voevodsky Motive

If we look at an arbitrary smooth base (no properness assumption) we need Voevodsky's triangulated motives. I assume that the base is a $k$-scheme for $k$ a perfect field.

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, the canonical morphism $\bigoplus_{s=0}^n \mathbb{Z}_{tr}(X)(s)[2s] \to \mathbb{Z}_{tr}(\mathbb{P}(E))$ is an isomorphism.

One can take a local trivialization of $E$ over a cover $\mathcal{U}$ of $X$, which allows (by Mayer-Vietoris triangles) to compute the motive of $\mathbb{P}(E)$ in terms of $\mathbb{P}(E)|_U$ for $U \in \mathcal{U}$. Thus one can look at the trivial case, where $\mathbb{P}(E) \simeq \mathbb{P}^n_X$ and the formula holds.

From this projective bundle formula one can deduce a Gysin triangle:
Gysin/localization triangle:
Let $X$ be a smooth scheme, $Z$ a smooth closed subscheme of codimension $c$. Then
$C_\ast \mathbb{Z}_{tr}(X \setminus Z) \to C_\ast \mathbb{Z}_{tr}(X) \to C_\ast \mathbb{Z}_{tr}(Z)(c)[2c] \to [1]$
is a distinguished triangle.

This can be proved by looking at the situation étale-locally, where $X= Y \times \mathbb{A}^c$ and $Z = Y \times 0$. Then the morphism $C_\ast \mathbb{Z}_{tr}(X) \to C_\ast \mathbb{Z}_{tr}(Z)(c)[2c]$ is just $C_\ast(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n\setminus 0)) \to \mathbb{Z}(n)[2n]$ from the projective bundle formula, tensored with $C_\ast\mathbb{Z}_{tr}(Y)$.

 

More general bundle formulas

One could ask what happens for a general $G$-bundle $E \to X$, let's say for $G$ a (split semi-simple) reductive linear algebraic group, if we take $E/P \to X$ for any subgroup $P$ of $G$. Maybe in the case of a parabolic $P$ we can describe the motive of $E/P$ in terms of $X$. At least in the case of a certain maximal parabolic of $G=SL_{n+1}$ this yields $G/P = \mathbb{P}^n$, so the analogy should be clear.

Using localization techniques becomes more interesting, since a $G$-bundle which is étale-locally trivial needn't be Zariski- (or even Nisnevich-) locally trivial (though this is the case for $G=GL_n$ and $G=SL_n$). Of course, one can restrict attention to Zariski-locally trivial bundles first.

Köck has further developed the techniques used to prove the projective bundle formula for Chow motives, and computed the motive of $G/P$ itself.
Theorem:
Let $G$ be a split reductive group over a field $k$ and $P$ a parabolic subgroup, with $Y:=G/P$ the quotient homogeneous space. Denote by $Y_w := BwB/P$ the Bruhat cells in $Y = \bigcup_{w \in W} Y_w$. Then $h(Y) \simeq \bigoplus_{w \in W} \mathbb{Z}(-dim(Y_w))$.

Habibi and Rad have recently proved for a connected reductive group $G$ over a characteristics $0$ field $k$ that the motive of a Zariski-locally trivial $G$-bundle over an irreducible base with mixed Tate motive is itself a mixed Tate motive.

Is there more known about $G/P$-bundles? I would love to see a formula that takes the type of the parabolic (i.e. the corresponding subset of roots) and spits out a motivic decomposition. I have the impression that one should be able to prove this along the lines of the projective bundle formula for Voevodsky motives. Locally, one has just $G/P$ and there Köck described the motive. Alternatively, one could use the Gysin sequence.

 

Example

To compute the Voevodsky motive of $\mathbb{P}^n$ from the bundle formula above would be cheating, since I did the proof by reduction to the motive of $\mathbb{P}^n$ (as it is done in the book of Mazza-Voevodsky-Weibel). I wrote about the Motive of $\mathbb{P}^n$ without a bundle formula before.

Flag varieties

One can see $\mathbb{P}^n$ as a special kind of partial flag variety which parametrizes flags of the form $0 = V_0 \subset V_1 \subset V_n = V$ with $V_1$ one-dimensional and $V$ an $n$-dimensional vector space (affine space).

Every partial flag variety $X$ parametrizing flags of the form $0 = V_0 \subset V_{i_1} \subset \cdots \subset V_{i_k} \subset V_n = V$ with $0 < i_1 < \cdots < i_k < n$ can be written as homogeneous space $X = GL_n/P(i_1,\dots,i_k)$, where $P(i_1,\dots,i_k)$ denotes a standard parabolic in $GL_n$. The $1$-flags in a vector space $V$ (of dimension $n+1$) form a variety $Fl_1$ which maps to a point, which we can consider as the space of $0$-flags $Fl_0$. This map $Fl_1 \to Fl_0$ is obviously just $\mathbb{P}^{n} \to \mathbb{P}^0$, hence we can compute the motive of $Fl_1$. Now $Fl_{1,2}$, the space of $1,2$-flags, fibers over $Fl_1$, since over each $1$-flag there is a projective space of $2$-flags containing this $1$-flag. This means $Fl_{1,2} \to Fl_1$ is a projective bundle, and we can compute the motive. And so on. Explicitly, this gives us $h(Fl_{1,2}) = \bigoplus_{s_2=0}^n \left(\bigoplus_{s_1=0}^n \mathbb{Z}(-s_1)[-2s_1] \right)(-s_2)[-2s_2]$ $= \bigoplus_{s_1,s_2=0}^n \mathbb{Z}(-s_1-s_2)[-2s_1-2s_2]$ and in general we have $h(Fl_{1,\dots,n}) = \bigoplus_{s_1,\dots,s_n=0}^n \mathbb{Z}(-(\sum s_i))[-2\sum(s_i)]$ reflecting the structure of the Bruhat cells of $Fl_{1,\dots,n}(V) \simeq GL(V)/B$ ($B$ a Borel subgroup). It would be nice if one could use a projective bundle formula to compute the motive of any homogeneous space $G/P$ for $P$ a parabolic, but if one tries to do that, the bundles one encounters are no longer projective. Do you know of other neat applications of the projective bundle formula?