The Serre Construction in Codimension Two
نویسنده
چکیده
Let X be a nonsingular algebraic variety. Suppose Z ⊆ X is a closed subscheme of X, with ideal sheaf IZ . When Z has codimension one in X, everything is as nice as it could be: IZ is a locally free sheaf, in fact a line bundle, and Z can locally be defined by a single equation. But starting in codimension two, all these pleasant things are usually false. To begin with, not every closed subscheme Z of codimension r ≥ 2 can be defined locally by r equations. When this is possible, in other words, if the ideal sheaf IZ is locally generated by r elements, the subscheme Z is called a local complete intersection in X (see [1, Definition on p. 185] for details). Since X is nonsingular, any such Z is automatically Cohen-Macaulay by [1, II.8.23], and as such has several useful properties. Now let us suppose that Z ⊆ X is such a local complete intersection of codimension two. We first look at the local situation near points x ∈ Z. If we let (A,m) be the local ring of the point x on X, the stalk of the ideal sheaf, I = IZ,x, can be generated by two elements, say f, g ∈ m. Because Z is Cohen-Macaulay, f and g form a regular sequence (see [1, II.8.21A]), and so the Koszul complex
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