The Stacks project

Similar to [Definition 17.1, F]

Definition 42.32.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. A bivariant class $c$ of degree $p$ for $f$ is given by a rule which assigns to every locally of finite type morphism $Y' \to Y$ and every $k$ a map

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

where $X' = Y' \times _ Y X$, satisfying the following conditions

  1. if $Y'' \to Y'$ is a proper, then $c \cap (Y'' \to Y')_*\alpha '' = (X'' \to X')_*(c \cap \alpha '')$ for all $\alpha ''$ on $Y''$ where $X'' = Y'' \times _ Y X$,

  2. if $Y'' \to Y'$ is flat locally of finite type of fixed relative dimension, then $c \cap (Y'' \to Y')^*\alpha ' = (X'' \to X')^*(c \cap \alpha ')$ for all $\alpha '$ on $Y'$, and

  3. if $(\mathcal{L}', s', i' : D' \to Y')$ is as in Definition 42.28.1 with pullback $(\mathcal{N}', t', j' : E' \to X')$ to $X'$, then we have $c \cap (i')^*\alpha ' = (j')^*(c \cap \alpha ')$ for all $\alpha '$ on $Y'$.

The collection of all bivariant classes of degree $p$ for $f$ is denoted $A^ p(X \to Y)$.


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