I am trying to learn about scheme theoretic algebraic geometry, because I actually want to study the basics of interseciton theory. I stumbled across the term "local ring $\mathcal{O}_{V,X}$ of a scheme $X$ along a closed subscheme/subvariety/primedivisor $V$" many times but I just can't find a definition of this term in my textbooks on schemes. (I have looked at Bosch, Eisenbud & Harris, FOAG by Vakil,...)
I am sure they explain it but I just don't see it. Could anybody help me out here?
By this question, or by Stacks Project Lemma 01IS, every irreducible closed subspace of a scheme has a unique generic point, i.e. the topological space underlying a scheme is sober. I think by the local ring $\mathcal O_{V, X}$ of a scheme $X$ along a sub-variety $V$ is meant the local ring $\mathcal O_{X, x}$ of $X$ with respect to the unique generic point $x$ of $V$.
Hope this helps.