The Stacks project

Remark 42.33.8. There is a more general type of bivariant class that doesn't seem to be considered in the literature. Namely, suppose we are given a diagram

\[ X \longrightarrow Z \longleftarrow Y \]

of schemes locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Let $p \in \mathbf{Z}$. Then we can consider a rule $c$ which assigns to every $Z' \to Z$ locally of finite type maps

\[ c \cap - : \mathop{\mathrm{CH}}\nolimits _ k(Y') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - p}(X') \]

for all $k \in \mathbf{Z}$ where $X' = X \times _ Z Z'$ and $Y' = Z' \times _ Z Y$ compatible with

  1. proper pushforward if given $Z'' \to Z'$ proper,

  2. flat pullback if given $Z'' \to Z'$ flat of fixed relative dimension, and

  3. gysin maps if given $D' \subset Z'$ as in Definition 42.29.1.

We omit the detailed formulations. Suppose we denote the collection of all such operations $A^ p(X \to Z \leftarrow Y)$. A simple example of the utility of this concept is when we have a proper morphism $f : X_2 \to X_1$. Then $f_*$ isn't a bivariant operation in the sense of Definition 42.33.1 but it is in the above generalized sense, namely, $f_* \in A^0(X_1 \to X_1 \leftarrow X_2)$.

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