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Integrality over ideal semifiltrations
arXiv.org
13 Jul 2019
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.
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Details
- Title
- Integrality over ideal semifiltrations
- Creators
- Darij Grinberg
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862242004721