Lemma 42.9.1. Let (S, \delta ) be as in Situation 42.7.1. Let X be locally of finite type over S. Let Z \subset X be a closed subscheme.
Let Z' \subset Z be an irreducible component and let \xi \in Z' be its generic point. Then
If \dim _\delta (Z) \leq k and \xi \in Z with \delta (\xi ) = k, then \xi is a generic point of an irreducible component of Z.
Proof. Let Z' \subset Z, \xi \in Z' be as in (1). Then \dim (\mathcal{O}_{Z, \xi }) = 0 (for example by Properties, Lemma 28.10.3). Hence \mathcal{O}_{Z, \xi } is Noetherian local ring of dimension zero, and hence has finite length over itself (see Algebra, Proposition 10.60.7). Hence, it also has finite length over \mathcal{O}_{X, \xi }, see Algebra, Lemma 10.52.5.
Assume \xi \in Z and \delta (\xi ) = k. Consider the closure Z' = \overline{\{ \xi \} }. It is an irreducible closed subscheme with \dim _\delta (Z') = k by definition. Since \dim _\delta (Z) = k it must be an irreducible component of Z. Hence we see (2) holds. \square
Definition 42.9.2. Let (S, \delta ) be as in Situation 42.7.1. Let X be locally of finite type over S. Let Z \subset X be a closed subscheme.
For any irreducible component Z' \subset Z with generic point \xi the integer m_{Z', Z} = \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{Z, \xi } (Lemma 42.9.1) is called the multiplicity of Z' in Z.
Assume \dim _\delta (Z) \leq k. The k-cycle associated to Z is
where the sum is over the irreducible components of Z of \delta -dimension k. (This is a k-cycle by Divisors, Lemma 31.26.1.)
It is important to note that we only define [Z]_ k if the \delta -dimension of Z does not exceed k. In other words, by convention, if we write [Z]_ k then this implies that \dim _\delta (Z) \leq k.
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