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.
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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.