Understanding the concept of rational sections is crucial when studying Cartier divisors, especially within the context of integral schemes and invertible sheaves. This article aims to clarify the definition of a rational section of a line bundle, explore its various interpretations, and discuss their equivalence.
Let $X$ be an integral scheme with generic point $eta$. Let $mathscr{L}$ be an invertible sheaf on $X$. We will examine the precise definition of a rational section in this setting, addressing common interpretations and their relationships.
There are several ways to define a rational section of $mathscr{L}$:
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Section over a Dense Open Set: A rational section is a section $s$ of $mathscr{L}$ over a dense open subset $U$ of $X$. Since $X$ is integral, any non-empty open set is dense. This means $s in Gamma(U, mathscr{L})$ for some open $U subseteq X$.
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Equivalence Class of Sections: A rational section is an equivalence class of sections of $mathscr{L}$ defined on open sets, where two sections $s in Gamma(U, mathscr{L})$ and $t in Gamma(V, mathscr{L})$ are equivalent if they agree on an open subset $W subseteq U cap V$, i.e., $s|_W = t|_W$. This equivalence relation captures the idea that rational sections are determined by their behavior on sufficiently large open sets.
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Global Section of a Modified Sheaf: A rational section is a global section $s in Gamma(X, mathscr{L} otimes_{mathcal{O}_X} mathcal{K})$, where $mathcal{K}$ is the constant sheaf of rational functions on $X$. This definition involves tensoring the invertible sheaf $mathscr{L}$ with the sheaf of rational functions, providing a way to extend sections to the entire scheme $X$ in a “rational” sense.
It is crucial to demonstrate the equivalence of these definitions to solidify our understanding.
Let’s consider the relationship between definition 2 and the stalk at the generic point. Given an equivalence class as described in definition 2, we can associate it with an element of the stalk $mathscr{L}eta$ at the generic point $eta$. Specifically, if $s in Gamma(U, mathscr{L})$ represents an equivalence class, then the germ $seta$ is an element of $mathscr{L}eta$. Conversely, any element in $mathscr{L}eta$ can be represented by a section over some open set containing $eta$. This suggests that definition 2 essentially identifies a rational section with an element of the stalk $mathscr{L}eta$. Therefore, yes, considering definition 2 as an element of the stalk $mathscr{L}eta$ at the generic point is a correct interpretation.
An illustration of an invertible sheaf, demonstrating how sections are defined over open sets.
To show the equivalence between definitions 2 and 3, consider the natural map
$$Gamma(X, mathscr{L} otimes_{mathcal{O}X} mathcal{K}) to (mathscr{L} otimes{mathcal{O}X} mathcal{K})eta cong mathscr{L}eta otimes{mathcal{O}{X,eta}} mathcal{K}eta cong mathscr{L}eta otimes{mathcal{O}{X,eta}} K(X),$$
where $K(X)$ is the function field of $X$. An element of $Gamma(X, mathscr{L} otimes{mathcal{O}X} mathcal{K})$ determines, in particular, an element of $mathscr{L}eta otimes{mathcal{O}{X,eta}} K(X)$, which can be viewed as an element of $mathscr{L}_eta$.
Conversely, given a section $s in Gamma(U, mathscr{L})$, where $U$ is an open subset of $X$, we can consider the element $s otimes 1$ in $Gamma(U, mathscr{L}) otimes_{mathcal{O}X(U)} mathcal{K}(U)$. The rational section $s$ extends to a global section of $mathscr{L} otimes{mathcal{O}_X} mathcal{K}$.
The universal property for sheafification might seem applicable, but the issue arises if the line bundle $mathscr{L}$ does not have global sections. However, tensoring with $mathcal{K}$ allows us to circumvent this issue by considering rational functions, which always exist.
In summary, the three definitions of a rational section are equivalent. Understanding this equivalence allows for flexibility in working with rational sections in different contexts.
Rational sections are foundational for understanding Cartier divisors. With a clear understanding of rational sections, we can now delve deeper into the intricacies of Cartier divisors and their properties. Rational sections provide the necessary framework to explore further concepts in algebraic geometry.
