Powers of ghost ideals

TL;DR

This paper develops a theory of ordinal powers of ghost ideals in exact categories, introduces the generalized generating hypothesis, and explores its implications for module theory and ring properties.

Contribution

It introduces the concept of ordinal powers of ghost ideals and establishes the generalized generating hypothesis for these ideals in various categorical contexts.

Findings

Every inductive power of a ghost ideal is an object-special preenveloping ideal under mild conditions.

The generalized generating hypothesis holds for ghost ideals in locally presentable Grothendieck categories.

If pure projective modules are extension-closed, then all left FP-projective modules are pure projective.

Abstract

A theory of ordinal powers of the ideal $\mathfrak{g}_{\mathcal{S}}$ of $\mathcal{S}$-ghost morphisms is developed by introducing for every ordinal $\lambda$, the $\lambda$-th inductive power $\mathcal{J}^{(\lambda)}$ of an ideal $\mathcal{J}.$ The Generalized $\lambda$-Generating Hypothesis ($\lambda$-GGH) for an ideal $\mathcal J$ of an exact category $\mathcal{A}$ is the proposition that the $\lambda$-th inductive power ${\mathcal{J}}^{(\lambda)}$ is an object ideal. It is shown that under mild conditions every inductive power of a ghost ideal is an object-special preenveloping ideal. When $\lambda$ is infinite, the proof is based on an ideal version of Eklof's Lemma. When $\lambda$ is an infinite regular cardinal, the Generalized $\lambda$-Generating Hypothesis is established for the ghost ideal $\mathfrak{g}_{\mathcal{S}}$ for the case when $\mathcal A$ a locally $\lambda$-presentable Grothendieck category and $\mathcal{S}$ is a set of $\lambda$-presentable objects in $\mathcal A$ such that $^\perp (\mathcal{S}^\perp)$ contains a generating set for $\mathcal A.$ As a consequence of $\lambda$-GGH for the ghost ideal $\mathfrak{g}_{R\mbox{-}\mathrm{mod}}$ in the category of modules $R\mbox{-}\mathrm{Mod}$ over a ring, it is shown that if the class of pure projective left $R$-modules is closed under extensions, then every left FP-projective module is pure projective. A restricted version $n$-GGH($\mathfrak{g}(\mathbf{C}(R))$) for the ghost ideal in $\mathbf{C}(R))$ is also considered and it is shown that $n$-GGH($\mathfrak{g}(\mathbf{C}(R))$) holds for $R$ if and only if the $n$-th power of the ghost ideal in the derived category $\mathbf{D}(R)$ is zero if and only if the global dimension of $R$ is less than $n.$ If $R$ is coherent, then the Generating Hypothesis holds for $R$ if and only if $R$ is von Neumann regular.