In the paper representations of general linear groups, there are two concepts: scheme-theoretical intersection and set-theoretical intersection. What are their differences and relations? Thank you very much.
Let me restrict the discussion to the affine case, from which the general case is easily extended.
An example Consider $X=Spec( k[S,T]) \subset \mathbb A^2_k$ , the affine plane over some field $k$.
Your article It seems very technical and I have certainly not read it. By very superficially browsing through it, I had the impression that the scheme theoretic intersection of equation (4.3) corresponds to the above description.
If $C_1, C_2$ are closed subset of a quasi-projective variety $X$ (according to Hartshorne's first chapter's definition), the set-theoretical intersection of $C_1$ and $C_2$ is the closed subset $C_1 \cap C_2$.
If $C_1, C_2$ are closed subschemes of a scheme $X$, the scheme-theoretical intersection of $C_1$ and $C_2$ is the closed subscheme $Y = C_1 \times_X C_2$. The underlying topological space of $Y$ is $C_1 \cap C_2$, but in this case also the structure sheaf is important.