Suppose $X$ is a noetherian scheme. A cycle on $X$ is an element $(n_x)_{x\in X}$ of the direct sum $\mathbb Z^{(X)}$, and its support is given by the union of all closed irreducible subsets $\overline{\{x\}}$ for which $n_x\neq 0$.
Both in [EGA IV_4, 21.6] and [Liu, Algebraic Geometry and Arithmetic Curves, 7.2.1] such a cycle is called purely of codimension $p$ (in $X$) if its support is purely of codimension $p$ (i.e., all irreducible components are of codimension $p$).
It seems to me that this gives an undesirable definition. E.g., in $\mathbb A_k^2$, with coordinate functions $x$ and $x$, the cycle $[(x,y)] +[(x)]$ would be purely of codimension 1.
Would you agree, or am I missing something?