The Stacks project

Lemma 42.14.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $U \subset X$ be an open subscheme, and denote $i : Y = X \setminus U \to X$ as a reduced closed subscheme of $X$. For every $k \in \mathbf{Z}$ the sequence

\[ \xymatrix{ Z_ k(Y) \ar[r]^{i_*} & Z_ k(X) \ar[r]^{j^*} & Z_ k(U) \ar[r] & 0 } \]

is an exact complex of abelian groups.

Proof. First assume $X$ is quasi-compact. Then $Z_ k(X)$ is a free $\mathbf{Z}$-module with basis given by the elements $[Z]$ where $Z \subset X$ is integral closed of $\delta $-dimension $k$. Such a basis element maps either to the basis element $[Z \cap U]$ or to zero if $Z \subset Y$. Hence the lemma is clear in this case. The general case is similar and the proof is omitted. $\square$

Comments (4)

Comment #177 by Adeel on

Above the second arrow in the diagram, it should be j^* instead of lower star.

Comment #2297 by Daniel on

Why cannot there be to different cycles and in both mapping to ??

Comment #2323 by on

@Daniel: If and are integral closed subschemes of which both meet and if , then we have . One argument to see this is if is the generic point, then nonempty implies , hence .

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