The Stacks project

Remark 42.35.5. 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 $X = \coprod _{i \in I} X_ i$ and $Y = \coprod _{j \in J} Y_ j$ be the decomposition of $X$ and $Y$ into their connected components (the connected components are open as $X$ and $Y$ are locally Noetherian, see Topology, Lemma 5.9.6 and Properties, Lemma 28.5.5). Let $a(i) \in J$ be the index such that $f(X_ i) \subset Y_{a(i)}$. Then $A^ p(X \to Y) = \prod A^ p(X_ i \to Y_{a(i)})$ by Lemma 42.35.4. In this setting it is convenient to set

\[ A^*(X \to Y)^\wedge = \prod \nolimits _ i A^*(X_ i \to Y_{a(i)}) \]

This “completed” bivariant group is the subset

\[ A^*(X \to Y)^\wedge \quad \subset \quad \prod \nolimits _{p \geq 0} A^ p(X) \]

consisting of elements $c = (c_0, c_1, c_2, \ldots )$ such that for each connected component $X_ i$ the image of $c_ p$ in $A^ p(X_ i \to Y_{a(i)})$ is zero for almost all $p$. If $Y \to Z$ is a second morphism, then the composition $A^*(X \to Y) \times A^*(Y \to Z) \to A^*(X \to Z)$ extends to a composition $A^*(X \to Y)^\wedge \times A^*(Y \to Z)^\wedge \to A^*(X \to Z)^\wedge $ of completions. We sometimes call $A^*(X)^\wedge = A^*(X \to X)^\wedge $ the completed bivariant cohomology ring of $X$.

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