Line bundles, ideal class group of an extension and Picard group

TL;DR

The paper generalizes the classical ideal class group to arbitrary ring extensions by introducing the ideal class group $\mathfrak{C}(A,B)$ as invertible $A$-submodules of $B$ modulo principal invertible ideals, and relates it to the Picard groups via a natural exact sequence. A key contribution is showing $\mathfrak{C}(A,B)\cong \ker\bigl(\operatorname{Pic}(A)\to\operatorname{Pic}(B)\bigr)$, framing Picard groups as special cases of ideal class groups, and proving that every ring has a faithfully flat extension with trivial Picard group using torsor algebras $A(L)=\bigoplus_{n\in\mathbb{Z}}L^{\otimes n}$ and a direct-limit construction. The authors also develop deep structural results for towers of extensions, show that $\operatorname{Pic}$ commutes with direct limits and products, and establish several vanishing and reduction theorems (e.g., $\operatorname{Pic}$ trivial for certain extensions, $\mathfrak{C}(A)\cong \operatorname{Pic}(A)$ under reductions, and $\mathfrak{C}(A)\to \mathfrak{C}(A_{\mathrm{red}})$ injectivity). Collectively, these results unify classical class-group and line-bundle theories and provide powerful algebraic tools for studying invertible ideals across extensions with broad implications for commutative algebra and algebraic geometry.

Abstract

For any extension of commutative rings $A\subseteq B$ we first naturally define a group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the basic properties of this group. Next, using ideas from algebraic geometry, we prove that every commutative ring has a (faithfully flat) ring extension whose Picard group is trivial. These results have several interesting applications. In particular, ...

Theorems & Definitions (83)

• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof
• Corollary 3.5
• proof