The Stacks project

42.43 The splitting principle

In our setting it is not so easy to say what the splitting principle exactly says/is. Here is a possible formulation.

Lemma 42.43.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}_ i$ be a finite collection of locally free $\mathcal{O}_ X$-modules of rank $r_ i$. There exists a projective flat morphism $\pi : P \to X$ of relative dimension $d$ such that

  1. for any morphism $f : Y \to X$ the map $\pi _ Y^* : \mathop{\mathrm{CH}}\nolimits _*(Y) \to \mathop{\mathrm{CH}}\nolimits _{* + d}(Y \times _ X P)$ is injective, and

  2. each $\pi ^*\mathcal{E}_ i$ has a filtration whose successive quotients $\mathcal{L}_{i, 1}, \ldots , \mathcal{L}_{i, r_ i}$ are invertible ${\mathcal O}_ P$-modules.

Moreover, when (1) holds the restriction map $A^*(X) \to A^*(P)$ (Remark 42.34.2) is injective.

Proof. We may assume $r_ i \geq 1$ for all $i$. We will prove the lemma by induction on $\sum (r_ i - 1)$. If this integer is $0$, then $\mathcal{E}_ i$ is invertible for all $i$ and we conclude by taking $\pi = \text{id}_ X$. If not, then we can pick an $i$ such that $r_ i > 1$ and consider the morphism $\pi _ i : P_ i = \mathbf{P}(\mathcal{E}_ i) \to X$. We have a short exact sequence

\[ 0 \to \mathcal{F} \to \pi _ i^*\mathcal{E}_ i \to \mathcal{O}_{P_ i}(1) \to 0 \]

of finite locally free $\mathcal{O}_{P_ i}$-modules of ranks $r_ i - 1$, $r_ i$, and $1$. Observe that $\pi _ i^*$ is injective on chow groups after any base change by the projective bundle formula (Lemma 42.36.2). By the induction hypothesis applied to the finite locally free $\mathcal{O}_{P_ i}$-modules $\mathcal{F}$ and $\pi _{i'}^*\mathcal{E}_{i'}$ for $i' \not= i$, we find a morphism $\pi : P \to P_ i$ with properties stated as in the lemma. Then the composition $\pi _ i \circ \pi : P \to X$ does the job. Some details omitted. $\square$

Remark 42.43.2. The proof of Lemma 42.43.1 shows that the morphism $\pi : P \to X$ has the following additional properties:

  1. $\pi $ is a finite composition of projective space bundles associated to locally free modules of finite constant rank, and

  2. for every $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we have $\alpha = \pi _*(\xi _1 \cap \ldots \cap \xi _ d \cap \pi ^*\alpha )$ where $\xi _ i$ is the first Chern class of some invertible $\mathcal{O}_ P$-module.

The second observation follows from the first and Lemma 42.36.1. We will add more observations here as needed.

Let $(S, \delta )$, $X$, and $\mathcal{E}_ i$ be as in Lemma 42.43.1. The splitting principle refers to the practice of symbolically writing

\[ c(\mathcal{E}_ i) = \prod (1 + x_{i, j}) \]

The symbols $x_{i, 1}, \ldots , x_{i, r_ i}$ are called the Chern roots of $\mathcal{E}_ i$. In other words, the $p$th Chern class of $\mathcal{E}_ i$ is the $p$th elementary symmetric function in the Chern roots. The usefulness of the splitting principle comes from the assertion that in order to prove a polynomial relation among Chern classes of the $\mathcal{E}_ i$ it is enough to prove the corresponding relation among the Chern roots.

Namely, let $\pi : P \to X$ be as in Lemma 42.43.1. Recall that there is a canonical $\mathbf{Z}$-algebra map $\pi ^* : A^*(X) \to A^*(P)$, see Remark 42.34.2. The injectivity of $\pi _ Y^*$ on Chow groups for every $Y$ over $X$, implies that the map $\pi ^* : A^*(X) \to A^*(P)$ is injective (details omitted). We have

\[ \pi ^*c(\mathcal{E}_ i) = \prod (1 + c_1(\mathcal{L}_{i, j})) \]

by Lemma 42.40.4. Thus we may think of the Chern roots $x_{i, j}$ as the elements $c_1(\mathcal{L}_{i, j}) \in A^*(P)$ and the displayed equation as taking place in $A^*(P)$ after applying the injective map $\pi ^* : A^*(X) \to A^*(P)$ to the left hand side of the equation.

To see how this works, it is best to give some examples.

Lemma 42.43.3. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee $. Then

\[ c_ i(\mathcal{E}^\vee ) = (-1)^ i c_ i(\mathcal{E}) \]

in $A^ i(X)$.

Proof. Choose a morphism $\pi : P \to X$ as in Lemma 42.43.1. By the injectivity of $\pi ^*$ (after any base change) it suffices to prove the relation between the Chern classes of $\mathcal{E}$ and $\mathcal{E}^\vee $ after pulling back to $P$. Thus we may assume there exist invertible $\mathcal{O}_ X$-modules ${\mathcal L}_ i$, $i = 1, \ldots , r$ and a filtration

\[ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ r = \mathcal{E} \]

such that $\mathcal{E}_ i/\mathcal{E}_{i - 1} \cong \mathcal{L}_ i$. Then we obtain the dual filtration

\[ 0 = \mathcal{E}_ r^\perp \subset \mathcal{E}_1^\perp \subset \mathcal{E}_2^\perp \subset \ldots \subset \mathcal{E}_0^\perp = \mathcal{E}^\vee \]

such that $\mathcal{E}_{i - 1}^\perp /\mathcal{E}_ i^\perp \cong \mathcal{L}_ i^{\otimes -1}$. Set $x_ i = c_1(\mathcal{L}_ i)$. Then $c_1(\mathcal{L}_ i^{\otimes -1}) = - x_ i$ by Lemma 42.25.2. By Lemma 42.40.4 we have

\[ c(\mathcal{E}) = \prod \nolimits _{i = 1}^ r (1 + x_ i) \quad \text{and}\quad c(\mathcal{E}^\vee ) = \prod \nolimits _{i = 1}^ r (1 - x_ i) \]

in $A^*(X)$. The result follows from a formal computation which we omit. $\square$

Lemma 42.43.4. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ and $\mathcal{F}$ be a finite locally free $\mathcal{O}_ X$-modules of ranks $r$ and $s$. Then we have

\[ c_1(\mathcal{E} \otimes \mathcal{F}) = r c_1(\mathcal{F}) + s c_1(\mathcal{E}) \]
\[ c_2(\mathcal{E} \otimes \mathcal{F}) = r c_2(\mathcal{F}) + s c_2(\mathcal{E}) + {r \choose 2} c_1(\mathcal{F})^2 + (rs - 1) c_1(\mathcal{F})c_1(\mathcal{E}) + {s \choose 2} c_1(\mathcal{E})^2 \]

and so on in $A^*(X)$.

Proof. Arguing exactly as in the proof of Lemma 42.43.3 we may assume we have invertible $\mathcal{O}_ X$-modules ${\mathcal L}_ i$, $i = 1, \ldots , r$ ${\mathcal N}_ i$, $i = 1, \ldots , s$ filtrations

\[ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ r = \mathcal{E} \quad \text{and}\quad 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F}_ s = \mathcal{F} \]

such that $\mathcal{E}_ i/\mathcal{E}_{i - 1} \cong \mathcal{L}_ i$ and such that $\mathcal{F}_ j/\mathcal{F}_{j - 1} \cong \mathcal{N}_ j$. Ordering pairs $(i, j)$ lexicographically we obtain a filtration

\[ 0 \subset \ldots \subset \mathcal{E}_ i \otimes \mathcal{F}_ j + \mathcal{E}_{i - 1} \otimes \mathcal{F} \subset \ldots \subset \mathcal{E} \otimes \mathcal{F} \]

with successive quotients

\[ \mathcal{L}_1 \otimes \mathcal{N}_1, \mathcal{L}_1 \otimes \mathcal{N}_2, \ldots , \mathcal{L}_1 \otimes \mathcal{N}_ s, \mathcal{L}_2 \otimes \mathcal{N}_1, \ldots , \mathcal{L}_ r \otimes \mathcal{N}_ s \]

By Lemma 42.40.4 we have

\[ c(\mathcal{E}) = \prod (1 + x_ i), \quad c(\mathcal{F}) = \prod (1 + y_ j), \quad \text{and}\quad c(\mathcal{E} \otimes \mathcal{F}) = \prod (1 + x_ i + y_ j), \]

in $A^*(X)$. The result follows from a formal computation which we omit. $\square$

Remark 42.43.5. The equalities proven above remain true even when we work with finite locally free $\mathcal{O}_ X$-modules whose rank is allowed to be nonconstant. In fact, we can work with polynomials in the rank and the Chern classes as follows. Consider the graded polynomial ring $\mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$ where $r$ has degree $0$ and $c_ i$ has degree $i$. Let

\[ P \in \mathbf{Z}[r, c_1, c_2, c_3, \ldots ] \]

be a homogeneous polynomial of degree $p$. Then for any finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ on $X$ we can consider

\[ P(\mathcal{E}) = P(r(\mathcal{E}), c_1(\mathcal{E}), c_2(\mathcal{E}), c_3(\mathcal{E}), \ldots ) \in A^ p(X) \]

see Remark 42.38.10 for notation and conventions. To prove relations among these polynomials (for multiple finite locally free modules) we can work locally on $X$ and use the splitting principle as above. For example, we claim that

\[ c_2(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{E})) = P(\mathcal{E}) \]

where $P = 2rc_2 - (r - 1)c_1^2$. Namely, since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{E}) = \mathcal{E} \otimes \mathcal{E}^\vee $ this follows easily from Lemmas 42.43.3 and 42.43.4 above by decomposing $X$ into parts where the rank of $\mathcal{E}$ is constant as in Remark 42.38.10.

Example 42.43.6. For every $p \geq 1$ there is a unique homogeneous polynomial $P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ]$ of degree $p$ such that, for any $n \geq p$ we have

\[ P_ p(s_1, s_2, \ldots , s_ p) = \sum x_ i^ p \]

in $\mathbf{Z}[x_1, \ldots , x_ n]$ where $s_1, \ldots , s_ p$ are the elementary symmetric polynomials in $x_1, \ldots , x_ n$, so

\[ s_ i = \sum \nolimits _{1 \leq j_1 < \ldots < j_ i \leq n} x_{j_1}x_{j_2} \ldots x_{j_ i} \]

The existence of $P_ p$ comes from the well known fact that the elementary symmetric functions generate the ring of all symmetric functions over the integers. Another way to characterize $P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ]$ is that we have

\[ \log (1 + c_1 + c_2 + c_3 + \ldots ) = \sum \nolimits _{p \geq 1} (-1)^{p - 1}\frac{P_ p}{p} \]

as formal power series. This is clear by writing $1 + c_1 + c_2 + \ldots = \prod (1 + x_ i)$ and applying the power series for the logarithm function. Expanding the left hand side we get

\begin{align*} & (c_1 + c_2 + \ldots ) - (1/2)(c_1 + c_2 + \ldots )^2 + (1/3)(c_1 + c_2 + \ldots )^3 - \ldots \\ & = c_1 + (c_2 - (1/2)c_1^2) + (c_3 - c_1c_2 + (1/3)c_1^3) + \ldots \end{align*}

In this way we find that

\begin{align*} P_1 & = c_1, \\ P_2 & = c_1^2 - 2c_2, \\ P_3 & = c_1^3 - 3c_1c_2 + 3c_3, \\ P_4 & = c_1^4 - 4c_1^2c_2 + 4c_1c_3 + 2c_2^2 - 4c_4, \end{align*}

and so on. Since the Chern classes of a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ are the elementary symmetric polynomials in the Chern roots $x_ i$, we see that

\[ P_ p(\mathcal{E}) = \sum x_ i^ p \]

For convenience we set $P_0 = r$ in $\mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$ so that $P_0(\mathcal{E}) = r(\mathcal{E})$ as a bivariant class (as in Remarks 42.38.10 and 42.43.5).

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