Integrality over ideal semifiltrations

Abstract

We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for integrality over a ring, the transitivity of integrality, and the theorem that sums and products of integral elements are again integral. Then, we define the notion of integrality over an ideal semifiltration (a sequence $\left( I_0,I_1,I_2,\ldots\right)$ of ideals satisfying $I_0 =A$ and $I_a I_b \subseteq I_{a+b}$ for all $a,b\in\mathbb{N}$), which generalizes both integrality over a ring and integrality over an ideal (as considered, e.g., in Swanson/Huneke, "Integral Closure of Ideals, Rings, and Modules"). We prove a criterion that reduces this general notion to integrality over a ring using a variant of the Rees algebra. Using this criterion, we study this notion further and obtain transitivity and closedness under sums and products for it as well. Finally, we prove the curious fact that if $u$, $x$ and $y$ are three elements of a (commutative) $A$-algebra (for $A$ a ring) such that $u$ is both integral over $A\left[ x\right]$ and integral over $A\left[ y\right]$, then $u$ is integral over $A\left[ xy\right]$. We generalize this to integrality over ideal semifiltrations, too.