Recently, I am reading the book written by Mumford, Algebraic Geometry II.
In the book, it mentioned a definition of local complete intersection.
For a closed subscheme $Y$ of noetherian scheme $X$, denote $\mathscr{I}_Y$ the corresponding sheaf of ideals on $\mathscr{O}_X$. Then we say that $Y$ is a local complete intersection if there exists a open cover of $Y$, such that $\mathscr{I}_Y$ is generated by a regular sequence of length $r$ on each open set, where r is the codimension.
Is the definition equivalent to $\mathscr{I}_{x,Y}$ is generated by regular sequences of length $r$ in $\mathscr{O}_{x,X}$ for every point $x\in Y$?
Suffices to show that given a point $x\in Y$, there is a neighborhood of $x$ in $X$ where the ideal sheaf is generated by a regular sequence of length $r$. Since this is true at $x$, we can extend these functions to a neighborhood of $x$ and further assume these generate the ideal sheaf of $Y$ in this neighborhood. Set of points in this open set where these $r$ functions form a regular sequence is an open set containing $x$, if you like, by using Koszul complex.