The Stacks project

62.10 Proper and equidimensional relative cycles

Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. We say a relative $r$-cycle $\alpha $ on $X/S$ is a proper and equidimensional relative cycle if $\alpha $ is both equidimensional (Definition 62.7.1) and proper (Definition 62.9.1). The group of all proper, equidimensional relative $r$-cycles on $X/S$ is denoted $c_{equi}(X/S, r)$.

Similarly we say a relative $r$-cycle $\alpha $ on $X/S$ is a proper and effective relative cycle if $\alpha $ is both effective (Definition 62.8.1) and proper (Definition 62.9.1). The monoid of all proper, effective relative $r$-cycles on $X/S$ is denoted $c^{eff}(X/S, r)$. Observe that these are equidimensional by Lemma 62.8.7.

Thus we have the following diagram of inclusion maps

\[ \xymatrix{ c^{eff}(X/S, r) \ar[r] \ar[d] & c_{equi}(X/S, r) \ar[r] \ar[d] & c(X/S, r) \ar[d] \\ z^{eff}(X/S, r) \ar[r] & z_{equi}(X/S, r) \ar[r] & z(X/S, r) } \]

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