Complete Intersections

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

From dimension

An important example of a closed subscheme is the vanishing scheme defined in §Closed Subschemes, ⁋Definition 7; the motivation is naturally the hypersurface \(f=0\) in Euclidean space \(\mathbb{R}^n\), defined as \(f^{-1}(0)\) for a function \(f: \mathbb{R}^n \rightarrow \mathbb{R}\).

More generally, we are also interested in the vanishing scheme \(Z(s_1,\ldots, s_k)\) defined by a (finite) family of global sections \(s_1,\ldots, s_k\in \Gamma(X, \mathcal{O}_X)\). Intuitively, this is obtained by first forming the vanishing scheme \(\iota_1:Z(s_1)\hookrightarrow X\) using the global section \(s_1\) on \(X\), then iterating the process of taking the vanishing scheme of \(s_2\vert_{Z(s_1)}\) on \(Z(s_1)\) via the global section

\[s_2\vert_{Z(s_1)}=\iota^\sharp(X)(s_2)\in(\iota_1)_\ast \mathcal{O}_{Z(s_1)}(X)=\Gamma(Z(s_1), \mathcal{O}_{Z(s_1)})\]

; of course, for this to be well-defined, the process must yield the same scheme regardless of the order of \(s_1, \ldots, s_k\).

Locally principal embedding

Definition 1 A closed embedding \(\iota: Z \hookrightarrow X\) is said to be locally principal if there exists an open cover \(\{U_i\}\) of \(X\) such that, for each of the closed embeddings

\[\iota\vert^{U_i}: \iota^{-1}(U_i) \rightarrow U_i\]

obtained by restricting the codomain of \(\iota\) to \(U_i\), there exists a suitable \(s_i\in \Gamma(U_i, \mathcal{O}_X)\) such that the two closed embeddings \(\iota\vert^{U_i}\) and \(Z(s_i)\hookrightarrow U_i\) are isomorphic.

Now if \(\iota: Z\hookrightarrow X\) is locally principal, then by covering each of the \(U_i\) in the definition with affine open sets and restricting the \(s_i\) to these, we may assume that \(\{U_i\}\) is an affine open cover.

Definition 2 A closed embedding \(\iota: Z \hookrightarrow X\) is called an effective Cartier divisor if there exists an affine open cover \(\{U_i=\Spec A_i\}\) of \(X\) such that, for each of the closed embeddings

\[\iota\vert^{U_i}:\iota^{-1}(U_i) \rightarrow U_i\]

there exists a suitable non-zerodivisor \(s_i\in A_i=\Gamma(U_i, \mathcal{O}_X)\) such that the two closed embeddings \(\iota^{U_i}\) and \(Z(s_i)\hookrightarrow U_i\) are isomorphic.

By definition, a locally principal embedding is roughly one whose ideal sheaf is (locally) generated by a single element, i.e., a principal ideal; an effective Cartier divisor is one for which this single element can be taken to be a non-zerodivisor over a suitable affine cover. Thus every effective Cartier divisor is locally principal, but the converse does not hold.