The Stacks project

Lemma 45.3.1. We have the following for correspondences:

  1. composition of correspondences is $\mathbf{Q}$-bilinear and associative,

  2. there is a canonical isomorphism

    \[ \mathop{\mathrm{CH}}\nolimits _{-r}(X) \otimes \mathbf{Q} = \text{Corr}^ r(X, \mathop{\mathrm{Spec}}(k)) \]

    such that pullback by correspondences corresponds to composition,

  3. there is a canonical isomorphism

    \[ \mathop{\mathrm{CH}}\nolimits ^ r(X) \otimes \mathbf{Q} = \text{Corr}^ r(\mathop{\mathrm{Spec}}(k), X) \]

    such that pushforward by correspondences corresponds to composition,

  4. composition of correspondences is compatible with pushforward and pullback of cycles.

Proof. Bilinearity follows immediately from the linearity of pushforward and pullback and the bilinearity of the intersection product. To prove associativity, say we have $X, Y, Z, W$ and $c \in \text{Corr}(X, Y)$, $c' \in \text{Corr}(Y, Z)$, and $c'' \in \text{Corr}(Z, W)$. Then we have

\begin{align*} c'' \circ (c' \circ c) & = \text{pr}^{134}_{14, *}( \text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{12}c \cdot \text{pr}^{123, *}_{23}c') \cdot \text{pr}^{134, *}_{34}c'') \\ & = \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}(\text{pr}^{123, *}_{12}c \cdot \text{pr}^{123, *}_{23}c') \cdot \text{pr}^{134, *}_{34}c'') \\ & = \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} (\text{pr}^{1234, *}_{12}c \cdot \text{pr}^{1234, *}_{23}c') \cdot \text{pr}^{134, *}_{34}c'') \\ & = \text{pr}^{134}_{14, *} \text{pr}^{1234}_{134, *} ((\text{pr}^{1234, *}_{12}c \cdot \text{pr}^{1234, *}_{23}c') \cdot \text{pr}^{1234, *}_{34}c'') \\ & = \text{pr}^{1234}_{14, *}( (\text{pr}^{1234, *}_{12}c \cdot \text{pr}^{1234, *}_{23}c') \cdot \text{pr}^{1234, *}_{34}c”) \end{align*}

Here we use the notation

\[ p^{1234}_{134} : X \times Y \times Z \times W \to X \times Z \times W \quad \text{and}\quad p^{134}_{14} : X \times Z \times W \to X \times W \]

the projections and similarly for other indices. The first equality is the definition of the composition. The second equality holds because $\text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *} = \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}$ by Chow Homology, Lemma 42.15.1. The third equality holds because intersection product commutes with the gysin map for $p^{1234}_{123}$ (which is given by flat pullback), see Chow Homology, Lemma 42.62.3. The fourth equality follows from the projection formula for $p^{1234}_{134}$, see Chow Homology, Lemma 42.62.4. The fourth equality is that proper pushforward is compatible with composition, see Chow Homology, Lemma 42.12.2. Since intersection product is associative by Chow Homology, Lemma 42.62.1 this concludes the proof of associativity of composition of correspondences.

We omit the proofs of (2) and (3) as these are essentially proved by carefully bookkeeping where various cycles live and in what (co)dimension.

The statement on pushforward and pullback of cycles means that $(c' \circ c)^*(\alpha ) = c^*((c')^*(\alpha ))$ and $(c' \circ c)_*(\alpha ) = (c')_*(c_*(\alpha ))$. This follows on combining (1), (2), and (3). $\square$

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