45.3 Correspondences
Let $k$ be a field. For schemes $X$ and $Y$ over $k$ we denote $X \times Y$ the product of $X$ and $Y$ in the category of schemes over $k$. In this section we construct the graded category over $\mathbf{Q}$ whose objects are smooth projective schemes over $k$ and whose morphisms are correspondences.
Let $X$ and $Y$ be smooth projective schemes over $k$. Let $X = \coprod X_ d$ be the decomposition of $X$ into the open and closed subschemes which are equidimensional with $\dim (X_ d) = d$. We define the $\mathbf{Q}$-vector space of correspondences of degree $r$ from $X$ to $Y$ by the formula:
\[ \text{Corr}^ r(X, Y) = \bigoplus \nolimits _ d \mathop{\mathrm{CH}}\nolimits ^{d + r}(X_ d \times Y) \otimes \mathbf{Q} \subset \mathop{\mathrm{CH}}\nolimits ^*(X \times Y) \otimes \mathbf{Q} \]
Given $c \in \text{Corr}^ r(X, Y)$ and $\beta \in \mathop{\mathrm{CH}}\nolimits _ k(Y) \otimes \mathbf{Q}$ we can define the pullback of $\beta $ by $c$ using the formula
\[ c^*(\beta ) = \text{pr}_{1, *}(c \cdot \text{pr}_2^*\beta ) \quad \text{in}\quad \mathop{\mathrm{CH}}\nolimits _{k - r}(X) \otimes \mathbf{Q} \]
This makes sense because $\text{pr}_2$ is flat of relative dimension $d$ on $X_ d \times Y$, hence $\text{pr}_2^*\beta $ is a cycle of dimension $d + k$ on $X_ d \times Y$, hence $c \cdot \text{pr}_2^*\alpha $ is a cycle of dimension $k - r$ on $X_ d \times Y$ whose pushforward by the proper morphism $\text{pr}_1$ is a cycle of the same dimension. Similarly, switching to grading by codimension, given $\alpha \in \mathop{\mathrm{CH}}\nolimits ^ i(X) \otimes \mathbf{Q}$ we can define the pushforward of $\alpha $ by $c$ using the formula
\[ c_*(\alpha ) = \text{pr}_{2, *}(c \cdot \text{pr}_1^*\alpha ) \quad \text{in}\quad \mathop{\mathrm{CH}}\nolimits ^{i + r}(Y) \otimes \mathbf{Q} \]
This makes sense because $\text{pr}_1^*\alpha $ is a cycle of codimension $i$ on $X \times Y$, hence $c \cdot \text{pr}_1^*\alpha $ is a cycle of codimension $i + d + r$ on $X_ d \times Y$, which pushes forward to a cycle of codimension $i + r$ on $Y$.
Given a three smooth projective schemes $X, Y, Z$ over $k$ we define a composition of correspondences
\[ \text{Corr}^ s(Y, Z) \times \text{Corr}^ r(X, Y) \longrightarrow \text{Corr}^{r + s}(X, Z) \]
by the rule
\[ (c', c) \longmapsto c' \circ c = \text{pr}_{13, *}(\text{pr}_{12}^*c \cdot \text{pr}_{23}^*c') \]
where $\text{pr}_{12} : X \times Y \times Z \to X \times Y$ is the projection and similarly for $\text{pr}_{13}$ and $\text{pr}_{23}$.
Lemma 45.3.1. We have the following for correspondences:
composition of correspondences is $\mathbf{Q}$-bilinear and associative,
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,
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,
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).
$\square$
Example 45.3.2. Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Denote $\Gamma _ f \subset X \times Y$ the graph of $f$. More precisely, $\Gamma _ f$ is the image of the closed immersion
\[ (f, \text{id}_ Y) : Y \longrightarrow X \times Y \]
Let $X = \coprod X_ d$ be the decomposition of $X$ into its open and closed parts $X_ d$ which are equidimensional of dimension $d$. Then $\Gamma _ f \cap (X_ d \times Y)$ has pure codimension $d$. Hence $[\Gamma _ f] \in \mathop{\mathrm{CH}}\nolimits ^*(X \times Y) \otimes \mathbf{Q}$ is contained in $\text{Corr}^0(X \times Y)$, i.e., $[\Gamma _ f]$ is a correspondence of degree $0$ from $X$ to $Y$.
Lemma 45.3.3. Smooth projective schemes over $k$ with correspondences and composition of correspondences as defined above form a graded category over $\mathbf{Q}$ (Differential Graded Algebra, Definition 22.25.1).
Proof.
Everything is clear from the construction and Lemma 45.3.1 except for the existence of identity morphisms. Given a smooth projective scheme $X$ consider the class $[\Delta ]$ of the diagonal $\Delta \subset X \times X$ in $\text{Corr}^0(X, X)$. We note that $\Delta $ is equal to the graph of the identity $\text{id}_ X : X \to X$ which is a fact we will use below.
To prove that $[\Delta ]$ can serve as an identity we have to show that $[\Delta ] \circ c = c$ and $c' \circ [\Delta ] = c'$ for any correspondences $c \in \text{Corr}^ r(Y, X)$ and $c' \in \text{Corr}^ s(X, Y)$. For the second case we have to show that
\[ c' = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta ] \cdot \text{pr}_{23}^*c') \]
where $\text{pr}_{12} : X \times X \times Y \to X \times X$ is the projection and simlarly for $\text{pr}_{13}$ and $\text{pr}_{23}$. We may write $c' = \sum a_ i [Z_ i]$ for some integral closed subschemes $Z_ i \subset X \times Y$ and rational numers $a_ i$. Thus it clearly suffices to show that
\[ [Z] = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta ] \cdot \text{pr}_{23}^*[Z]) \]
in the chow group of $X \times Y$ for any integral closed subscheme $Z$ of $X \times Y$. After replacing $X$ and $Y$ by the irreducible component containing the image of $Z$ under the two projections we may assume $X$ and $Y$ are integral as well. Then we have to show
\[ [Z] = \text{pr}_{13, *}([\Delta \times Y] \cdot [X \times Z]) \]
Denote $Z' \subset X \times X \times Y$ the image of $Z$ by the morphism $(\Delta , 1) : X \times Y \to X \times X \times Y$. Then $Z'$ is a closed subscheme of $X \times X \times Y$ isomorphic to $Z$ and $Z' = \Delta \times Y \cap X \times Z$ scheme theoretically. By Chow Homology, Lemma 42.62.51 we conclude that
\[ [Z'] = [\Delta \times Y] \cdot [X \times Z] \]
Since $Z'$ maps isomorphically to $Z$ by $\text{pr}_{13}$ also we conclude. The verification that $[\Delta ] \circ c = c$ is similar and we omit it.
$\square$
Lemma 45.3.4. There is a contravariant functor from the category of smooth projective schemes over $k$ to the category of correspondences which is the identity on objects and sends $f : Y \to X$ to the element $[\Gamma _ f] \in \text{Corr}^0(X, Y)$.
Proof.
In the proof of Lemma 45.3.3 we have seen that this construction sends identities to identities. To finish the proof we have to show if $g : Z \to Y$ is another morphism of smooth projective schemes over $k$, then we have $[\Gamma _ g] \circ [\Gamma _ f] = [\Gamma _{f \circ g}]$ in $\text{Corr}^0(X, Z)$. Arguing as in the proof of Lemma 45.3.3 we see that it suffices to show
\[ [\Gamma _{f \circ g}] = \text{pr}_{13, *}([\Gamma _ f \times Z] \cdot [X \times \Gamma _ g]) \]
in $\mathop{\mathrm{CH}}\nolimits ^*(X \times Z)$ when $X$, $Y$, $Z$ are integral. Denote $Z' \subset X \times Y \times Z$ the image of the closed immersion $(f \circ g, g, 1) : Z \to X \times Y \times Z$. Then $Z' = \Gamma _ f \times Z \cap X \times \Gamma _ g$ scheme theoretically and we conclude using Chow Homology, Lemma 42.62.5 that
\[ [Z'] = [\Gamma _ f \times Z] \cdot [X \times \Gamma _ g] \]
Since it is clear that $\text{pr}_{13, *}([Z']) = [\Gamma _{f \circ g}]$ the proof is complete.
$\square$
Lemma 45.3.6. Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Let $[\Gamma _ f] \in \text{Corr}^0(X, Y)$ be as in Example 45.3.2. Then
pushforward of cycles by the correspondence $[\Gamma _ f]$ agrees with the gysin map $f^! : \mathop{\mathrm{CH}}\nolimits ^*(X) \to \mathop{\mathrm{CH}}\nolimits ^*(Y)$,
pullback of cycles by the correspondence $[\Gamma _ f]$ agrees with the pushforward map $f_* : \mathop{\mathrm{CH}}\nolimits _*(Y) \to \mathop{\mathrm{CH}}\nolimits _*(X)$,
if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$, then
pushforward of cycles by the correspondence $[\Gamma _ f^ t]$ of Remark 45.3.5 corresponds to pushforward of cycles by $f$, and
pullback of cycles by the correspondence $[\Gamma _ f^ t]$ of Remark 45.3.5 corresponds to the gysin map $f^!$.
Proof.
Proof of (1). Recall that $[\Gamma _ f]_*(\alpha ) = \text{pr}_{2, *}([\Gamma _ f] \cdot \text{pr}_1^*\alpha )$. We have
\[ [\Gamma _ f] \cdot \text{pr}_1^*\alpha = (f, 1)_*((f, 1)^! \text{pr}_1^*\alpha ) = (f, 1)_*((f, 1)^! \text{pr}_1^!\alpha ) = (f, 1)_*(f^!\alpha ) \]
The first equality by Chow Homology, Lemma 42.62.6. The second by Chow Homology, Lemma 42.59.5. The third because $\text{pr}_1 \circ (f, 1) = f$ and Chow Homology, Lemma 42.59.6. Then we coclude because $\text{pr}_{2, *} \circ (f, 1)_* = 1_*$ by Chow Homology, Lemma 42.12.2.
Proof of (2). Recall that $[\Gamma _ f]_*(\beta ) = \text{pr}_{1, *}([\Gamma _ f] \cdot \text{pr}_2^*\beta )$. Arguing exactly as above we have
\[ [\Gamma _ f] \cdot \text{pr}_2^*\beta = (f, 1)_*\beta \]
Thus the result follows as before.
Proof of (3). Proved in exactly the same manner as above.
$\square$
Example 45.3.7. Let $X = \mathbf{P}^1_ k$. Then we have
\[ \text{Corr}^0(X, X) = \mathop{\mathrm{CH}}\nolimits ^1(X \times X) \otimes \mathbf{Q} = \mathop{\mathrm{CH}}\nolimits _1(X \times X) \otimes \mathbf{Q} \]
Choose a $k$-rational point $x \in X$ and consider the cycles $c_0 = [x \times X]$ and $c_2 = [X \times x]$. A computation shows that $1 = [\Delta ] = c_0 + c_2$ in $\text{Corr}^0(X, X)$ and that we have the following rules for composition $c_0 \circ c_0 = c_0$, $c_0 \circ c_2 = 0$, $c_2 \circ c_0 = 0$, and $c_2 \circ c_2 = c_2$. In other words, $c_0$ and $c_2$ are orthogonal idempotents in the algebra $\text{Corr}^0(X, X)$ and in fact we get
\[ \text{Corr}^0(X, X) = \mathbf{Q} \times \mathbf{Q} \]
as a $\mathbf{Q}$-algebra.
The category of correspondences is a symmetric monoidal category. Given smooth projective schemes $X$ and $Y$ over $k$, we define $X \otimes Y = X \times Y$. Given four smooth projective schemes $X, X', Y, Y'$ over $k$ we define a tensor product
\[ \otimes : \text{Corr}^ r(X, Y) \times \text{Corr}^{r'}(X', Y') \longrightarrow \text{Corr}^{r + r'}(X \times X', Y \times Y') \]
by the rule
\[ (c, c') \longmapsto c \otimes c' = \text{pr}_{13}^*c \cdot \text{pr}_{24}^*c' \]
where $\text{pr}_{13} : X \times X' \times Y \times Y' \to X \times Y$ and $\text{pr}_{24} : X \times X' \times Y \times Y' \to X' \times Y'$ are the projections. As associativity constraint
\[ X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z \]
we use the usual associativity constraint on products of schemes. The commutativity constraint will be given by the isomorphism $X \times Y \to Y \times X$ switching the factors.
Lemma 45.3.8. The tensor product of correspondences defined above turns the category of correspondences into a symmetric monoidal category with unit $\mathop{\mathrm{Spec}}(k)$.
Proof.
Omitted.
$\square$
Lemma 45.3.9. Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Assume $X$ and $Y$ equidimensional of dimensions $d$ and $e$. Denote $a = [\Gamma _ f] \in \text{Corr}^0(X, Y)$ and $a^ t = [\Gamma _ f^ t] \in \text{Corr}^{d - e}(Y, X)$. Set $\eta _ X = [\Gamma _{X \to X \times X}] \in \text{Corr}^0(X \times X, X)$, $\eta _ Y = [\Gamma _{Y \to Y \times Y}] \in \text{Corr}^0(Y \times Y, Y)$, $[X] \in \text{Corr}^{-d}(X, \mathop{\mathrm{Spec}}(k))$, and $[Y] \in \text{Corr}^{-e}(Y, \mathop{\mathrm{Spec}}(k))$. The diagram
\[ \xymatrix{ X \otimes Y \ar[r]_{a \otimes \text{id}} \ar[d]_{\text{id} \otimes a^ t} & Y \otimes Y \ar[r]_{\eta _ Y} & Y \ar[d]^{[Y]} \\ X \otimes X \ar[r]^{\eta _ X} & X \ar[r]^{[X]} & \mathop{\mathrm{Spec}}(k) } \]
is commutative in the category of correspondences.
Proof.
Recall that $\text{Corr}^ r(W, \mathop{\mathrm{Spec}}(k)) = \mathop{\mathrm{CH}}\nolimits _{-r}(W)$ for any smooth projective scheme $W$ over $k$ and given $c \in \text{Corr}^ s(W', W)$ the composition with $c$ agrees with pullback by $c$ as a map $\mathop{\mathrm{CH}}\nolimits _{-r}(W) \to \mathop{\mathrm{CH}}\nolimits _{-r - s}(W')$ (Lemma 45.3.1). Finally, we have Lemma 45.3.6 which tells us how to convert this into usual pushforward and pullback of cycles. We have
\[ (a \otimes \text{id})^* \eta _ Y^* [Y] = (a \otimes \text{id})^* [\Delta _ Y] = (f \times \text{id})_*\Delta _ Y = [\Gamma _ f] \]
and the other way around we get
\[ (\text{id} \otimes a^ t)^* \eta _ X^* [X] = (\text{id} \otimes a^ t)^* [\Delta _ X] = (\text{id} \times f)^![\Delta _ X] = [\Gamma _ f] \]
The last equality follows from Chow Homology, Lemma 42.59.8. In other words, going either way around the diagram we obtain the element of $\text{Corr}^ d(X \times Y, \mathop{\mathrm{Spec}}(k))$ corresponding to the cycle $\Gamma _ f \subset X \times Y$.
$\square$
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