Subschemes of projective varieties I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective variety any zero-locus  $X$ of homogeneous polynomials in the complex projective space and, in particular, they consider Chow varieties $\mathcal{C}_{p,d}(X)$ only for varieties of this type.
At page 8 they give the following definition-lemma.

Then they say:

From this one see that $Z\in \mathcal{C}_{m+t,d}(T\times X)$ and this leads to my question:
Question: Can one see a subscheme of a projective variety as an algebraic  cycle?
 A: If $X$ is a Noetherian scheme, there is a map
\begin{align*}
\{\text{subschemes of X}\} &\to \operatorname{CH}_*(X)\\
Z &\mapsto [Z],
\end{align*}
defined by
$$[Z] = \sum_{i=1}^r m_i [Z_i],$$
where the $Z_i \subseteq Z$ are the (reduced) irreducible components of $Z$, and
$$m_i = \ell_{\mathcal O_{Z,\eta_i}}(\mathcal O_{Z,\eta_i})$$
is the length of the scheme $Z$ at the generic point $\eta_i$ of $Z_i$.
Example. If $Z$ is reduced, then $m_i = 1$ for all $i$. The converse is not true; for example the subscheme
$$Z = \operatorname{Spec} k[x,y]/(x^2,xy) \subseteq \mathbb A^2_k$$
is not reduced, but $[Z] = [V(x)]$ is the $y$-axis with multiplicity one. This example also shows that the map $Z \mapsto [Z]$ is not injective. 
It is injective on reduced subschemes when viewed as a map to $Z_*(X)$: any reduced subscheme $Z \subseteq X$ is uniquely determined by the collection $\{Z_i\}$ of components of $Z$. But on Chow groups this is again false; for example any two lines in $\mathbf P^2$ have the same cycle class.
Further reading.
See also Fulton's Intersection theory [Ful], §1.5.

References.
[Ful]  Fulton, William, Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 2. Berlin etc.: Springer-Verlag. XI, 470 p. DM 118 (1984). ZBL0541.14005.
