Let, $X$ be a nonsingular closed subscheme in $\mathbb{P}^{n}$(Over an algebraically closed field $\mathbb K$ of characteristic $0$) and $Y$ be a locally complete intersection of $X$.
We know that the length of a closed subscheme $Y$ is defined as dimension of the $\mathbb K$ vector space $\mathcal O_Y(Y)$.
At this point my question is :Is it true that if in the above situation $Y$ is a zero dimensional scheme,then length($Y$) = $|Y|$?
(Here, $|Y|: =$cardinality of $Y$).
I know that if a subscheme is a finite collection of $d$ distinct point in projective space then it has degree $d$,but I don't see how to extend it in case of length.
Any help from anyone is welcome.