This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
From dimension
An important example of closed subschemes is the vanishing scheme defined in §Closed Subschemes, ⁋Definition 7; the motivation for this is naturally the hypersurface \(f=0\) in Euclidean space \(\mathbb{R}^n\) defined by \(f^{-1}(0)\) for a function \(f: \mathbb{R}^n \rightarrow \mathbb{R}\).
Meanwhile, we are also interested more generally in the vanishing scheme \(Z(s_1,\ldots, s_k)\) defined by a (finite) family of global sections \(s_1,\ldots, s_k\in \Gamma(X, \mathscr{O}_X)\). Intuitively, this is obtained by first considering the vanishing scheme \(\iota_1:Z(s_1)\hookrightarrow X\) using the global section \(s_1\) on \(X\), then iterating the process of finding the vanishing scheme of \(s_2\vert_{Z(s_1)}\) on \(Z(s_1)\) through the global section
\[s_2\vert_{Z(s_1)}=\iota^\sharp(X)(s_2)\in(\iota_1)_\ast \mathscr{O}_{Z(s_1)}(X)=\Gamma(Z(s_1), \mathscr{O}_{Z(s_1)})\]; of course, for this to work, this 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 each \(U_i\), there exists a suitable \(s_i\in \Gamma(U_i, \mathscr{O}_X)\) 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 covering.
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 \(s_i\in A_i=\Gamma(U_i, \mathscr{O}_X)\) 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 generated (locally) 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.
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